-- MySQL dump 10.19  Distrib 10.3.39-MariaDB, for debian-linux-gnu (x86_64)
--
-- Host: localhost    Database: jsarkar
-- ------------------------------------------------------
-- Server version	10.3.39-MariaDB-0ubuntu0.20.04.2

/*!40101 SET @OLD_CHARACTER_SET_CLIENT=@@CHARACTER_SET_CLIENT */;
/*!40101 SET @OLD_CHARACTER_SET_RESULTS=@@CHARACTER_SET_RESULTS */;
/*!40101 SET @OLD_COLLATION_CONNECTION=@@COLLATION_CONNECTION */;
/*!40101 SET NAMES utf8mb4 */;
/*!40103 SET @OLD_TIME_ZONE=@@TIME_ZONE */;
/*!40103 SET TIME_ZONE='+00:00' */;
/*!40014 SET @OLD_UNIQUE_CHECKS=@@UNIQUE_CHECKS, UNIQUE_CHECKS=0 */;
/*!40014 SET @OLD_FOREIGN_KEY_CHECKS=@@FOREIGN_KEY_CHECKS, FOREIGN_KEY_CHECKS=0 */;
/*!40101 SET @OLD_SQL_MODE=@@SQL_MODE, SQL_MODE='NO_AUTO_VALUE_ON_ZERO' */;
/*!40111 SET @OLD_SQL_NOTES=@@SQL_NOTES, SQL_NOTES=0 */;

--
-- Table structure for table `js_answers`
--

DROP TABLE IF EXISTS `js_answers`;
/*!40101 SET @saved_cs_client     = @@character_set_client */;
/*!40101 SET character_set_client = utf8 */;
CREATE TABLE `js_answers` (
  `id` int(11) NOT NULL AUTO_INCREMENT,
  `chapter_id` int(11) NOT NULL,
  `question_id` int(11) NOT NULL,
  `image_link` varchar(500) NOT NULL,
  `date_of_upload` varchar(100) NOT NULL,
  PRIMARY KEY (`id`)
) ENGINE=InnoDB DEFAULT CHARSET=utf8 COLLATE=utf8_unicode_ci;
/*!40101 SET character_set_client = @saved_cs_client */;

--
-- Dumping data for table `js_answers`
--

LOCK TABLES `js_answers` WRITE;
/*!40000 ALTER TABLE `js_answers` DISABLE KEYS */;
/*!40000 ALTER TABLE `js_answers` ENABLE KEYS */;
UNLOCK TABLES;

--
-- Table structure for table `js_chapters`
--

DROP TABLE IF EXISTS `js_chapters`;
/*!40101 SET @saved_cs_client     = @@character_set_client */;
/*!40101 SET character_set_client = utf8 */;
CREATE TABLE `js_chapters` (
  `id` int(11) NOT NULL AUTO_INCREMENT,
  `cname` varchar(500) NOT NULL,
  `cslug` varchar(500) NOT NULL,
  `tutorial_update_dt` varchar(200) NOT NULL DEFAULT '',
  `tutorial_link_pdf` varchar(1000) NOT NULL DEFAULT '',
  `publish` int(11) NOT NULL DEFAULT 0,
  `last_refresh_dt` varchar(30) NOT NULL DEFAULT '',
  `last_upload_dt` varchar(30) NOT NULL DEFAULT '',
  `ques_filename` varchar(200) NOT NULL DEFAULT '',
  `total_ques` int(11) NOT NULL DEFAULT 0,
  `assesment_ques` int(11) NOT NULL DEFAULT 0,
  `languages` varchar(20) NOT NULL DEFAULT '',
  `ques_filename_bn` varchar(200) NOT NULL DEFAULT '',
  `ques_filename_hi` varchar(200) NOT NULL DEFAULT '',
  PRIMARY KEY (`id`),
  KEY `cname` (`cname`)
) ENGINE=InnoDB AUTO_INCREMENT=21 DEFAULT CHARSET=utf8 COLLATE=utf8_unicode_ci;
/*!40101 SET character_set_client = @saved_cs_client */;

--
-- Dumping data for table `js_chapters`
--

LOCK TABLES `js_chapters` WRITE;
/*!40000 ALTER TABLE `js_chapters` DISABLE KEYS */;
INSERT INTO `js_chapters` VALUES (1,'Geometric Progression','geometric-progression','','Trigonometric-Formulas-.pdf',0,'','','',0,0,'1','',''),(2,'Derivative (2nd order)','2nd-order-derivative','','',1,'07/04/2024 18:24','10/04/2020 18:30','2nd-order-derivative-problems.pdf',74,0,'1','',''),(3,'Derivative (1st order)','1st-order-derivative','2020-05-26','1st-order-derivative-study-material.pdf',1,'20/08/2023 12:27','10/04/2020 17:38','1st-order-derivative-problems.pdf',148,0,'1','',''),(4,'Integration (Indefinite)','indefinite-integration','','',1,'05/10/2020 20:49','05/10/2020 20:49','indefinite-integration-problems.pdf',374,0,'1','',''),(5,'Set Theory','set-theory','2019-06-21','set-theory.pdf',1,'13/12/2021 21:53','15/10/2020 17:07','set-theory-problems.pdf',53,0,'1,2','',''),(6,'Probability','probability','','',1,'31/07/2023 17:40','','',142,0,'1,2','',''),(7,'Matrix','matrix','2020-05-30','matrix-study-material.pdf',1,'02/06/2020 17:48','04/05/2020 21:55','matrix-problems.pdf',130,0,'1','',''),(8,'Determinant','determinant','2021-12-11','determinant-study-material.pdf',1,'04/05/2020 22:09','04/05/2020 22:09','determinant-problems.pdf',139,0,'1','',''),(9,'Differential Equation','differential-equation','2020-08-24','differential-equation-study-material.pdf',1,'06/09/2020 15:11','21/08/2020 22:54','differential-equation-problems.pdf',193,0,'1','',''),(10,'Relation','relation','2020-07-26','relation-study-material.pdf',1,'21/12/2020 20:02','','',19,0,'1,2','',''),(11,'Function or Mapping','mapping','2021-12-05','mapping-study-material.pdf',1,'26/09/2020 14:18','','',19,0,'1','',''),(12,'Complex Number','complex-number','2021-02-12','complex-number.pdf',1,'11/02/2021 21:42','11/02/2021 21:42','complex-number-problems.pdf',84,0,'1','',''),(13,'Variation','variation','','',1,'20/06/2023 22:27','','',17,0,'1,2','',''),(14,'Vector','vector','2022-02-14','vector-study-material.pdf',0,'','','',0,0,'1','',''),(15,'Simple Interest','simple-interest','','',1,'13/06/2023 12:05','','',34,0,'1,2','',''),(16,'General Solution of Trigonometrical Equations','general-solution-trigonometrical-equations','','',1,'','','',2,0,'1','',''),(17,'Limit','limit','','',1,'02/07/2023 20:11','','',15,0,'1','',''),(18,'Straight Line (2D)','straight-line-2d','','',1,'15/09/2023 12:18','','',6,0,'1,2','',''),(19,'Properties of Triangle','properties-of-triangle','','',1,'21/01/2024 18:10','','',3,0,'1,2','',''),(20,'Quadratic Equation (One Variable)','quadratic-equation-one-variable','','',1,'13/02/2024 23:30','','',53,0,'1,2','','');
/*!40000 ALTER TABLE `js_chapters` ENABLE KEYS */;
UNLOCK TABLES;

--
-- Table structure for table `js_languages`
--

DROP TABLE IF EXISTS `js_languages`;
/*!40101 SET @saved_cs_client     = @@character_set_client */;
/*!40101 SET character_set_client = utf8 */;
CREATE TABLE `js_languages` (
  `id` int(11) NOT NULL AUTO_INCREMENT,
  `language_full` varchar(20) NOT NULL DEFAULT '',
  `language_short` varchar(10) NOT NULL DEFAULT '',
  PRIMARY KEY (`id`)
) ENGINE=InnoDB AUTO_INCREMENT=4 DEFAULT CHARSET=utf8 COLLATE=utf8_general_ci;
/*!40101 SET character_set_client = @saved_cs_client */;

--
-- Dumping data for table `js_languages`
--

LOCK TABLES `js_languages` WRITE;
/*!40000 ALTER TABLE `js_languages` DISABLE KEYS */;
INSERT INTO `js_languages` VALUES (1,'English','en'),(2,'Bengali','bn'),(3,'Hindi','hi');
/*!40000 ALTER TABLE `js_languages` ENABLE KEYS */;
UNLOCK TABLES;

--
-- Table structure for table `js_maths`
--

DROP TABLE IF EXISTS `js_maths`;
/*!40101 SET @saved_cs_client     = @@character_set_client */;
/*!40101 SET character_set_client = utf8 */;
CREATE TABLE `js_maths` (
  `id` int(11) NOT NULL AUTO_INCREMENT,
  `chapter_id` int(11) NOT NULL,
  `question_sl` decimal(10,3) NOT NULL,
  `question` longtext NOT NULL,
  `timestamp` int(11) NOT NULL DEFAULT 0,
  `answer` longtext DEFAULT NULL,
  `answer_img_nos` varchar(500) NOT NULL DEFAULT '',
  `answer_2` longtext NOT NULL,
  `answer_3` longtext NOT NULL,
  `answer_4` longtext NOT NULL,
  `self_assesment` int(11) NOT NULL DEFAULT 0,
  `question_bn` longtext NOT NULL,
  `question_hi` longtext NOT NULL,
  PRIMARY KEY (`id`)
) ENGINE=InnoDB AUTO_INCREMENT=1573 DEFAULT CHARSET=utf8 COLLATE=utf8_unicode_ci;
/*!40101 SET character_set_client = @saved_cs_client */;

--
-- Dumping data for table `js_maths`
--

LOCK TABLES `js_maths` WRITE;
/*!40000 ALTER TABLE `js_maths` DISABLE KEYS */;
INSERT INTO `js_maths` VALUES (1,1,1.000,'Is `320` be any term of G.P. `5,-10, 20,...` ?',0,'Yes','1','','','',0,'',''),(2,1,3.000,'If `x,2x+2,3x+3` be the first three terms of a G.P., then find its `4^{th}` term.',0,'`- \\frac{27}{2}`','2','','','',0,'',''),(3,1,32.000,'Put three G.M. (Geometric Means) between `-1` and `-16`',0,'`-2,-4,-8` or `2,-4,8`','3','','','',0,'',''),(4,1,8.000,'In a G.P. `n^{th}` term is `128`, common ratio `2` and the sum of first `n` terms is `255`. Find the value of `n`.',0,'','','','','',0,'',''),(5,1,9.000,'In a G.P. 1st, 2nd & 8th terms are `a^{-4}, a^m, a^52` respectively. Find the value of `m`. ',0,'','','','','',0,'',''),(6,1,16.000,'In a G.P., sum of first six terms is `9` times the sum of first three terms. Find its common ratio.',0,'','','','','',0,'',''),(7,1,44.000,'If `a, b, c` be any three different positive real numbers which are in G.P., then show that `a+c \\gt 2b`',0,'N.A.','7','','','',0,'',''),(8,1,31.000,'If `5, x, y, z, 405` are in G.P., then find the value of `x , y, z`',0,'','','','','',0,'',''),(9,1,33.000,'If the A.M. (Arithmetic Mean) & G.M. (Geometric Mean) of two positive real numbers are `15` & `12` respectively, then find the numbers.',0,'`6`, `24`','9','','','',0,'',''),(10,1,46.000,'If `a, b, c, d` are in G.P., then show that `a+b, b+c, c+d` are also in G.P.',0,'N.A.','10','','','',0,'',''),(11,1,45.000,'Can any three sides of a right angled triangle are in G.P. ?',0,'No','11','','','',0,'',''),(12,1,19.000,'Let `S_n` denotes the sum of first `n` terms of a G.P.. If `S_{2n} = 4S_n` then show that `S_{3n} : S_n = 13 : 1`',0,'N.A.','12','','','',0,'',''),(13,1,4.000,'If in a G.P., 3rd term is the square of 1st term and 2nd term is `8`. Find the 6th term.',0,'','','','','',0,'',''),(14,1,12.000,'If in a G.P. ratio of 3rd term & 7th term is `1:16` and 1st term be `7`. Find the 8th term.',0,'','','','','',0,'',''),(15,1,10.000,'If in a G.P., 5th, 8th & 11th terms are `p, q, s` respectively, then show that `q^2 = ps`.',0,'','','','','',0,'',''),(16,1,47.000,'In a G.P. `p^{th}, q^{th} & r^{th}` terms are `a, b, c` respectively. Then show that `a^{q-r} \\cdot b^{r-p} \\cdot c^{p-q} = 1`',0,'N.A.','16','','','',0,'',''),(17,1,34.000,'If `A` be the A.M. and `a, b` be the two G.M.s of two real numbers, then show that `A = \\frac{1}{2}(\\frac{a^2}{b}+\\frac{b^2}{a})`',0,'N.A.','17','','','',0,'',''),(18,1,30.000,'If in a G.P., `(p+q)^{th}` term be `m` and `(p-q)^{th}` term be `n`, then find `p^{th}` & `q^{th}` terms.',0,'`\\pm \\sqrt{mn}`, `m (\\frac{n}{m}) ^ \\frac{p}{2q}`','18','','','',0,'',''),(19,1,14.000,'If in a G.P., first term & last term be `a` & `l` respectively. If `n` be the number of terms then show that `n = 1 + \\frac{\\log l - \\log a}{\\log r}` where `r` be the common ratio and `a, l, r \\gt 0`',0,'N.A.','19','','','',0,'',''),(20,1,36.000,'If `A` & `G` be the A.M. & G.M. of two distinct real numbers, then show that `A \\gt G \\gt \\frac{G^2}{A}`, where `G \\gt 0`',0,'N.A.','20','','','',0,'',''),(21,1,27.000,'If in a G.P., 1st term, `n^{th}` term & the product of first `n` terms be `a, b, P` respectively, then show that `P^2 = (ab)^n` ',0,'N.A.','21','','','',0,'',''),(22,1,28.000,'If in a G.P., sum of first `5` terms be `S`, their product be `P` and sum of their reciprocals is `R`, then show that `P^2 = (\\frac{S}{R})^5`',0,'','','','','',0,'',''),(23,1,37.000,'If `G` be the G.M. & `p, q` be two A.M.s of two numbers, then show that `G^2 = (2p-q)(2q-p)`',0,'N.A.','23','','','',0,'',''),(24,1,38.000,'If the ratio of A.M and G.M. of two numbers `a, b` is `m : n`, then show that `a:b = (m + \\sqrt{m^2 - n^2}) : (m - \\sqrt{m^2 - n^2})`',0,'N.A.','24','','','',0,'',''),(25,1,48.000,'If `m^{th}, n^{th} & p^{th}` terms of both an A.P. & G.P. are `x, y, z` respectively, then show that `x^{y-z} \\cdot y^{z-x} \\cdot z^{x-y} = 1`',0,'N.A.','25','','','',0,'',''),(26,1,49.000,'If in a G.P., sum of consecutive three terms is `70`. If its 1st term & 3rd term are multiplied by `4` and its middle term is multiplied by `5`, then the products thus obtained, are in A.P. Find the three consecutive terms.',0,'`10,20,40` or `40,20,10`','26','','','',0,'',''),(27,1,50.000,'If `a, b, c` be three different numbers which are in G.P. and `\\frac{1}{a-b}, \\frac{1}{c-a}, \\frac{1}{b-c}` are in A.P., then show that `a+4b+c = 0`',0,'','','','','',0,'',''),(28,1,51.000,'If `a, b, c` be three unequal numbers which are in A.P. and `(b-a), (c-b), a` are in G.P., then find `a:b:c`',0,'','','','','',0,'',''),(29,1,52.000,'Find the sum upto `n` terms: `1+11+111+...`',0,'','','','','',0,'',''),(30,1,53.000,'Find the sum upto `n` terms: `0.6+0.66+0.666+...`',0,'`\\frac{2}{3} {n - \\frac{1}{9} (1 - \\frac{1}{10^n})}`','30','','','',0,'',''),(31,1,22.000,'If in a G.P., sum of first `n, 2n, 3n` number of terms are `S_1, S_2, S_3` respectively, then show that `S_1(S_3 - S_2) = (S_2 - S_1)^2`',0,'','','','','',0,'',''),(32,1,24.000,'Find the least value of `n` such that `3+6+12+24+... `(upto `n` terms) should greater that `3000`.',0,'','','','','',0,'',''),(33,1,2.000,'Is `4096` be a term of G.P. `2, 4, 8, 16, ...` ? ',0,'','','','','',0,'',''),(34,1,35.000,'If both of A.M. & G.M. of two numbers `x, y` is `2`, then find `x, y`. ',0,'','','','','',0,'',''),(35,1,20.000,'If in a G.P., `S_n = P`, `S_{2n} = 3P`, then show that `S_{3n} = 7P`',0,'','','','','',0,'',''),(36,1,62.000,'If in a G.P., `p^{th}` term is `q` and `q^{th}` term is `p`, then show that its `(2p-q)^{th}` term is `\\frac{q^2}{p}`',0,'N.A.','36','','','',0,'',''),(37,1,13.000,'If in a G.P., 1st term is `4` and `t_3 : t_7 = 1:81`, then find its 10th term.',0,'','','','','',0,'',''),(38,1,63.000,'If in a G.P., `p^{th}` term & `q^{th}` term be `p` & `q` respectively, then show that its `(p+q)^{th}` term is `(\\frac{p^p}{q^q})^\\frac{1}{(p-q)}`',0,'N.A.','38','','','',0,'',''),(39,1,11.000,'If in a G.P., `t_n, t_{2n}` & `t_{3n}` are `a, b, c` respectively, then show that `b^2 = ac`',0,'','','','','',0,'',''),(40,1,39.000,'If A.M. & G.M. of two positive real numbers are `A` & `G` respectively, then show that the numbers are `A \\pm \\sqrt{A^2 - G^2}`',0,'','','','','',0,'',''),(41,1,40.000,'If the A.M. is double of the G.M. of two numbers `a` & `b`, then show that `a:b = (2+\\sqrt{3}) : (2-\\sqrt{3})`',0,'','','','','',0,'',''),(42,1,42.000,'If `a,b,c` are in G.P. and `x,y` is the A.M. of `a,b` & `b,c` respectively, then show that `\\frac{1}{x} + \\frac{1}{y} = \\frac{2}{b}` and `\\frac{a}{x} + \\frac{c}{y} = 2`',0,'','','','','',0,'',''),(43,1,43.000,'If the A.M. of `m,n` and the G.M. of `a,b` are both `\\frac{ma+nb}{m+n}`, then show that `m = \\frac{2b\\sqrt{a}}{\\sqrt{a}+\\sqrt{b}}` and `n = \\frac{2a\\sqrt{b}}{\\sqrt{a}+\\sqrt{b}}`',0,'','','','','',0,'',''),(44,1,64.000,'If `p^{th}, q^{th} & r^{th}` terms of an A.P., are in G.P., then show that the common ratio of G.P. is `\\frac{r-q}{q-p}`',0,'N.A.','44','','','',0,'',''),(45,1,65.000,'If `a,b,c` are in A.P. & `x,y,z` are in G.P., then show that `x^{b-c} \\cdot y^{c-a} \\cdot z^{a-b} = 1`',0,'N.A.','45','','','',0,'',''),(46,1,55.000,'Find sum upto `50` terms: `7+7.7+7.77+7.777+...`',0,'`\\frac{7}{9} {500 - \\frac{10}{9} (1 - \\frac{1}{10^50})}`','46','','','',0,'',''),(47,1,57.000,'Find sum upto `n` terms: `1+3+7+15+31+...`',0,'`2^(n+1) - (n+2)`','47','','','',0,'',''),(48,1,56.000,'Find sum upto `n` terms: `(1)+(1+3)+(1+3+3^2)+...`',0,'','','','','',0,'',''),(49,1,54.000,'Find sum upto `n` terms: `0.6+0.06+0.006+0.0006+...`',0,'`\\frac{2}{3} (1 - \\frac{1}{10^n})`','49','','','',0,'',''),(50,1,58.000,'Find the sum upto `n` terms: `1+4+13+40+121+...` ',0,'`\\frac{3}{4}(3^n - 1) - \\frac{n}{2}`','50','','','',0,'',''),(51,1,60.000,'Find the sum upto `n` terms: `2+3 \\cdot 3 + 4 \\cdot 3^2 + 5 \\cdot 3^3 + ...`',0,'`\\frac{(2n+1)3^n - 1}{4}`','51','','','',0,'',''),(52,1,15.000,'If in a G.P., 4th term & 7th term is `24` & `192` respectively. Find the sum of its first `10` terms.',0,'','','','','',0,'',''),(53,1,17.000,'The sum of three consecutive terms of an A.P. is `15`. If `1,4,19` are added to these terms, then they are in G.P.. Find the terms.',0,'','','','','',0,'',''),(54,1,21.000,'If in a G.P., sum of first `n,2n,3n` number of terms are `S_1, S_2, S_3` respectively, then show that `S_1^2 + S_2^2 = S_1(S_2 + S_3)`.',0,'','','','','',0,'',''),(55,1,18.000,'If in a sequence, the sum of first `n` terms is `\\frac{4^n - 3^n}{3^n}`, then find its `n^{th}` term and show that the sequence is a G.P. ',0,'','','','','',0,'',''),(56,1,23.000,'Find the least value of `n` such that the sum of the geometric series `1+3+3^2+...` (upto `n` terms) is greater than `7000`.',0,'','','','','',0,'',''),(57,1,25.000,'If `S_n = 1 + \\frac{1}{2} + \\frac{1}{2^2} + ... + \\frac{1}{2^{n-1}}`, then find the least value of `n` such that `2 - S_n \\lt \\frac{1}{100}`',0,'','','','','',0,'',''),(58,1,29.000,'If in a G.P., sum of consecutive `n` terms be `S`, their products be `P` and sum of their reciprocals are `R`, then prove that `P^2 = (\\frac{S}{R})^n`',0,'N.A.','58','','','',0,'',''),(59,1,26.000,'If in a G.P., first term is `a`, common ratio be `r` and the sum of the first `n` terms is `S_n`, then show that `S_1 + S_2 + S_3 + ... + S_n = \\frac{ar(r^n-1)}{(r-1)^2} - \\frac{na}{r-1}`',0,'N.A.','59.001','','','',0,'',''),(60,1,61.000,'Show that `1 + (1+x) + (1+x+x^2) + (1+x+x^2+x^3)+ ...` (upto `n` terms) ` = \\frac{n}{1-x} - \\frac{x(1-x^n)}{(1-x)^2}`',0,'N.A.','60.001','','','',0,'',''),(61,1,5.000,'If the sum & product of three consecutive numbers, which are in G.P., is `35` & `1000` respectively. Find the three numbers.',0,'','','','','',0,'',''),(62,1,7.000,'If in a G.P., sum of three consecutive numbers is `21` and sum of their square is `189`. Find the three terms.',0,'','','','','',0,'',''),(63,1,6.000,'Divide `21` is three parts such that the parts are in G.P. and their product is `64`.',0,'','','','','',0,'',''),(64,1,66.000,'If `a,b,c` are in A.P. and `x,y,z` are in G.P., then prove that `(\\frac{y}{z})^a \\cdot (\\frac{z}{x})^b \\cdot (\\frac{x}{y})^c = 1`',0,'N.A.','64','','','',0,'',''),(65,1,67.000,'If `a,b,c` are in A.P. and `a,b,(c+1)` are the three consecutive terms in G.P., then show that `a = (a-b)^2` ',0,'N.A.','65','','','',0,'',''),(66,3,68.000,'If `xy = a(y + \\sqrt{y^2 - x^2})` then show that `x^3 \\frac{dy}{dx} = y^2(y + \\sqrt{y^2 - x^2})`',1559796983,'N.A.','66','','','',0,'',''),(67,1,59.000,'Find the sum upto `n` terms: `3+7+14+27+52+...`',1559916257,'`3(2^n-1) + \\frac{n(n-1)}{2}`','67','','','',0,'',''),(68,1,41.000,'Show that the sum of `n` G.M.s between `a` & `b` is `(\\frac{ab^\\frac{1}{n+1} - ba^\\frac{1}{n+1}}{a^\\frac{1}{n+1} - b^\\frac{1}{n+1}})`',1559923008,'N.A.','68.001','','','',0,'',''),(69,4,24.000,'`\\int \\ \\frac{\\cos^2x - \\sin^2x}{\\sqrt{1+\\cos 4x}} dx`',1560072822,'`\\frac{x}{\\sqrt{2}} + c`','69','','','',0,'',''),(70,4,21.000,'`\\int \\ \\frac{\\cos 5x + \\cos 4x}{1-2 \\cos 3x} \\ dx`',1560077464,'`-( \\frac{\\sin 2x}{2} + \\sin x ) + c`','70','','','',0,'',''),(71,4,25.000,'`\\int \\ \\sec^2 x \\cos^2 2x \\ dx`',1560078406,'`\\tan x + \\sin 2x - 2x + c`','71','','','',0,'',''),(72,4,16.000,'`\\int \\ \\frac{1+x+x^2}{x^2(1+x)} \\ dx`',1560082755,'`-\\frac{1}{x} + \\log |1+x| + c`','72','','','',0,'',''),(73,4,17.000,'`\\int \\ \\frac{1+2x^2}{x^2 (1+x^2)} \\ dx`',1560082935,'`-\\frac{1}{x} + \\tan^{-1}x + c`','73','','','',0,'',''),(74,4,18.000,'`\\int \\ \\frac{dx}{\\sqrt{ax+b} + \\sqrt{ax-b}}`',1560083925,'`\\frac{1}{3ab} {(ax+b)^{\\frac{3}{2}} - (ax-b)^{\\frac{3}{2}}} + c`','74','','','',0,'',''),(75,4,19.000,'`\\int \\ \\frac{\\cos 2x + \\sin x}{1 + 2 \\sin x} \\ dx`',1560161584,'`x + cosx + c`','','','','',0,'',''),(76,4,20.000,'`\\int \\ \\frac{\\cos x - \\cos 2x}{1 - \\cos x} \\ dx`',1560161706,'`x + 2sinx + c`','','','','',0,'',''),(77,4,26.000,'`\\int \\ \\frac{\\cos 2x - \\cos 2\\alpha}{\\cos x - \\cos \\alpha} dx`',1560161946,'`2 (sinx + x\\cos\\alpha) + c`','','','','',0,'',''),(78,4,27.000,'`\\int \\ \\sin^4 x \\ dx`',1560161995,'`1/8 (3x + \\frac{sin4x}{4} - 2sin2x) + c`','78','','','',0,'',''),(79,4,28.000,'`\\int \\ \\cos^4 x \\ dx`',1560162021,'`1/8 (3x + \\frac{sin4x}{4} + 2sin2x) + c`','','','','',0,'',''),(80,4,29.000,'`\\int \\ \\tan^2 x \\ dx`',1560162040,'`tanx - x + c`','','','','',0,'',''),(81,4,22.000,'`\\int \\ \\frac{\\cos 7x + \\cos 5x}{1 - 2 \\cos 4x} \\ dx`',1560162188,'`-sinx - \\frac{sin3x}{3} + c`','','','','',0,'',''),(82,4,23.000,'`\\int \\ \\frac{\\cos 8x - \\cos 7x}{1 + 2 \\cos 5x} \\ dx`',1560162328,'`\\frac{sin 2x}{2} - \\frac{sin 3x}{3} + c`','','','','',0,'',''),(83,4,30.000,'`\\int \\ \\frac{dx}{\\sin^2 x \\cos^2 x} \\ dx`',1560162789,'`tanx - cotx + c`','83','','','',0,'',''),(84,4,31.000,'`\\int \\ \\frac{dx}{\\sin^2 x \\cos x} \\ dx`',1560162891,'`log|secx + tanx| - cscx + c`','','','','',0,'',''),(85,4,32.000,'`\\int \\ \\frac{\\sec x}{\\sec x + \\tan x} \\ dx`',1560162952,'`tanx - secx + c`','','','','',0,'',''),(86,4,33.000,'`\\int \\ \\frac{\\sin^8 x - \\cos^8 x}{1 - 2 \\sin^2 x \\cos^2 x} \\ dx`',1560163089,'`-\\frac{sin2x}{2} + c`','86','','','',0,'',''),(87,4,34.000,'`\\int \\ \\frac{\\sin^6x - \\cos^6x}{1 - 2 \\sin^2x \\cos^2x} \\ dx`',1560188749,'`- \\frac{1}{4\\sqrt{2}}\\ log| \\frac{\\sqrt{2}+sin2x}{\\sqrt{2} - sin2x} | - 1/4 sin2x + c`','87','','','',0,'',''),(88,4,35.000,'`\\int \\ \\frac{\\sin^6x + \\cos^6x}{\\sin^2x \\cos^2x} \\ dx`',1560189491,'`tanx - cotx - 3x + c`','88','','','',0,'',''),(89,4,36.000,'`\\int \\ \\frac{\\sin^6x - \\cos^6x}{\\sin^2x \\cos^2x} \\ dx`',1560189597,'`2 csc2x + 1/2 sin2x + c`','89','','','',0,'',''),(90,4,37.000,'`\\int \\ \\frac{dx}{\\sin^4x + \\cos^4x}`',1560189741,'`1/\\sqrt{2} tan^{-1}(\\frac{tanx-cotx}{\\sqrt{2}})+c`','90','','','',0,'',''),(91,4,38.000,'`\\int \\ \\frac{\\cos 2x}{\\sin^4x + \\cos^4x} \\ dx`',1560189777,'`\\frac{1}{2\\sqrt{2}} log|\\frac{\\sqrt{2}+sin2x}{\\sqrt{2}-sin2x}| + c`','91','','','',0,'',''),(92,4,39.000,'`\\int \\ \\frac{\\sin 2x}{\\sqrt{1 + \\cos 2x}} \\ dx`',1560189867,NULL,'','','','',0,'',''),(93,4,40.000,'`\\int \\ \\frac{\\sin 4x}{\\sqrt{1 + \\cos 4x}} \\ dx`',1560189895,NULL,'','','','',0,'',''),(94,4,41.000,'`\\int \\ \\frac{\\sin x}{\\sqrt{1 + \\sin x}} \\ dx`',1560189924,'`2(sin\\frac{x}{2}-cos\\frac{x}{2}) - 1/\\sqrt{2} log |tan(x/2+\\pi/8)| + c`','94','','','',0,'',''),(95,4,42.000,'`\\int \\ \\frac{\\sin x}{\\sqrt{1 - \\sin x}} \\ dx`',1560189988,NULL,'','','','',0,'',''),(96,4,43.000,'`\\int \\ \\frac{1 + \\cos x}{1 - \\cos x} \\ dx`',1560190030,NULL,'','','','',0,'',''),(97,4,44.000,'`\\int \\ \\frac{1 - \\sin x}{1 + \\sin x} \\ dx`',1560190071,'`2tanx - x - 2secx+ c`','','','','',0,'',''),(98,4,45.000,'`\\int \\ \\frac{dx}{1 + \\sin x}`',1560190190,NULL,'','','','',0,'',''),(99,4,46.000,'`\\int \\ \\frac{dx}{1 - \\sin x}`',1560190217,NULL,'','','','',0,'',''),(100,4,47.000,'`\\int \\ \\frac{dx}{4 + \\sin x}`',1560190290,'`2/\\sqrt{15} tan^{-1}{4/\\sqrt{15} (tan \\frac{x}{2} + 1/4)} + c`','100','','','',0,'',''),(101,4,48.000,'`\\int \\ \\frac{dx}{1 + \\cos 2x}`',1560190322,NULL,'','','','',0,'',''),(102,4,49.000,'`\\int \\ \\frac{dx}{1 + \\cos 2x}`',1560190351,'`1/2 tanx + c`','','','','',0,'',''),(103,4,50.000,'`\\int \\ \\frac{dx}{2 + \\cos x}`',1560190382,'`2/\\sqrt{3} tan^{-1}(1/\\sqrt{3} tan\\frac{x}{2})+c`','103','','','',0,'',''),(104,4,52.000,'`\\int \\ \\frac{dx}{1 + \\cot x}`',1560190468,'`x/2 - 1/2 log |sinx + cosx| + c`','104','','','',0,'',''),(105,4,51.000,'`\\int \\ \\frac{dx}{1 + \\tan x}`',1560190509,'`x/2 + 1/2 log |sinx + cosx| + c`','','','','',0,'',''),(106,4,53.000,'`\\int \\ \\frac{dx}{a + b \\tan x}`',1560239112,'`\\frac{1}{a^2+b^2} [ax + b log |a cosx + b sinx | ] + c`','106,106.001','','','',0,'',''),(107,4,54.000,'`\\int \\ \\frac{dx}{1 + \\text{cosec} \\ x}`',1560239689,'`secx - tanx + x +c`','','','','',0,'',''),(108,4,55.000,'`\\int \\ \\sqrt{1 + \\sin \\frac{x}{2}} \\ dx`',1560239737,'`4(\\sin\\frac{x}{4} - \\cos\\frac{x}{4}) + c`','','','','',0,'',''),(109,4,56.000,'`\\int \\ \\sqrt{1 + \\text{cosec} \\ x} \\ dx`',1560239784,'`sin^{-1}(2\\sinx-1)+c`','109','','','',0,'',''),(110,4,57.000,'`\\int \\ \\sqrt{1 + \\sec x} \\ dx`',1560239819,'`sin^{-1}(1-2\\cosx)+c`','110','','','',0,'',''),(111,4,58.000,'`\\int \\ \\frac{dx}{1 + \\sin x - \\cos x}`',1560239868,'`log|\\frac{\\tan\\frac{x}{2}}{\\tan\\frac{x}{2}+1}|+c`','111','','','',0,'',''),(112,4,59.000,'`\\int \\ \\frac{dx}{2 + \\sin x + \\cos x}`',1560239915,NULL,'','','','',0,'',''),(113,4,60.000,'`\\int \\ \\frac{dx}{3 + 2 \\sin x + \\cos x}`',1560239968,NULL,'','','','',0,'',''),(114,4,61.000,'`\\int \\ \\frac{dx}{a \\sin x + b \\cos x}`',1560240017,NULL,'','','','',0,'',''),(115,4,62.000,'`\\int \\ \\frac{dx}{\\cos x + \\sqrt{3} \\sin x}`',1560240070,NULL,'','','','',0,'',''),(116,4,63.000,'`\\int \\ \\frac{dx}{3 \\sin x + 4 \\cos x}`',1560240119,NULL,'','','','',0,'',''),(117,4,64.000,'`\\int \\ \\frac{dx}{3 \\sin x - 4 \\cos x}`',1560240183,NULL,'','','','',0,'',''),(118,4,65.000,'`\\int \\ \\frac{2 + 3 \\cos x}{\\sin x + 2 \\cos x + 3} \\ dx`',1560240428,NULL,'','','','',0,'',''),(119,4,66.000,'`\\int \\ \\frac{dx}{(3 \\sin x + 4 \\cos x)^2}`',1560240586,NULL,'','','','',0,'',''),(120,4,67.000,'`\\int \\ \\frac{2 \\sin x + 3 \\cos x}{2 \\cos x + 3 \\sin x} \\ dx`',1560240685,NULL,'','','','',0,'',''),(121,4,68.000,'`\\int \\ \\frac{4 \\sin x + 5 \\cos x}{4 \\cos x + 5 \\sin x} \\ dx`',1560240790,NULL,'','','','',0,'',''),(122,4,69.000,'`\\int \\ \\frac{\\cos x + 2 \\sin x}{3 \\cos x + 4 \\sin x} \\ dx`',1560240863,NULL,'','','','',0,'',''),(123,4,70.000,'`\\int \\ \\frac{1 + \\cos 4x}{\\cot x - \\tan x} \\ dx`',1560240948,NULL,'','','','',0,'',''),(124,4,71.000,'`\\int \\ \\frac{d\\theta}{\\sin (\\theta - \\frac{\\pi}{2}) \\sin (\\theta - \\frac{\\pi}{4})}`',1560241092,NULL,'','','','',0,'',''),(125,4,72.000,'`\\int \\ \\frac{x^2 + \\sin^2 x}{1 + x^2} \\ \\sec^2 x \\ dx`',1560241191,NULL,'','','','',0,'',''),(126,4,73.000,'`\\int \\ \\frac{x^2 + \\sin^2 x}{1 + x^2} \\ dx`',1560241245,NULL,'','','','',0,'',''),(127,4,74.000,'`\\int \\ \\frac{\\cos \\theta}{(1 - \\sin \\theta)(2 - \\sin \\theta)} \\ d\\theta`',1560241321,NULL,'','','','',0,'',''),(128,4,75.000,'`\\int \\ \\frac{\\cos \\theta}{(1 + \\sin \\theta)(2 + \\sin \\theta)} \\ d\\theta`',1560241448,NULL,'','','','',0,'',''),(129,4,76.000,'`\\int \\ \\frac{1 + \\tan^2 x}{1 + \\cot^2 x} \\ dx`',1560241511,NULL,'','','','',0,'',''),(130,4,77.000,'`\\int \\ \\frac{\\tan x}{a + b \\tan^2 x} \\ dx`',1560241575,NULL,'','','','',0,'',''),(131,4,78.000,'`\\int \\ \\frac{\\tan x}{\\sqrt{a + b \\tan^2 x}} \\ dx \\ \\ \\ (b \\gt a)`   ',1560241611,'`-\\frac{1}{\\sqrt{b-a}} \\sin^{-1}{\\frac{(\\sqrt{b} - \\sqrt{a}) \\cos x}{\\sqrt{b}}} + c`','131','','','',0,'',''),(132,4,79.000,'`\\int \\ \\frac{\\tan x \\sec^2x}{a^2 + b^2 \\tan^2 x} \\ dx`',1560241742,NULL,'','','','',0,'',''),(133,4,80.000,'`\\int \\ \\frac{dx}{\\sec^2 x + \\tan^2 x} \\ dx`',1560241811,NULL,'','','','',0,'',''),(134,4,81.000,'`\\int \\ \\frac{\\sec^2 x}{\\sqrt{5 - \\sec^2 x}} \\ dx`',1560241910,NULL,'','','','',0,'',''),(135,4,82.000,'`\\int \\ \\frac{dx}{2 \\sin^2 x + \\cos^2 x}`',1560241984,NULL,'','','','',0,'',''),(136,4,83.000,'`\\int \\ \\sqrt{\\frac{\\sin x - \\sin^3 x}{1 - \\sin^3 x}} \\ dx`',1560242072,NULL,'','','','',0,'',''),(137,4,84.000,'`\\int \\ \\frac{\\sin x + \\sin^2 x + \\sin^3 x}{1 + \\sin x} \\ dx`',1560242145,NULL,'','','','',0,'',''),(138,4,86.000,'`\\int \\ \\frac{2 \\sin 2\\phi - \\cos \\phi}{6 - \\cos^2 \\phi - 4 \\sin \\phi} \\ d\\phi`',1560242265,NULL,'','','','',0,'',''),(139,4,87.000,'`\\int \\ \\tan x \\tan 2x \\tan 3x \\ dx`',1560242372,NULL,'','','','',0,'',''),(140,4,88.000,'`\\int \\ \\sin^{-1} (\\frac{2 \\tan \\sqrt{x}}{1 + \\tan^2 \\sqrt{x}}) \\ dx`',1560242504,NULL,'','','','',0,'',''),(141,4,89.000,'`\\int \\ \\sin^{-1} (\\frac{2x}{1 + x^2}) \\ dx`',1560242687,NULL,'','','','',0,'',''),(142,4,90.000,'`\\int \\ \\tan^{-1} (\\sec x + \\tan x) \\ dx`',1560242756,NULL,'','','','',0,'',''),(143,4,91.000,'`\\int \\ \\tan^{-1} (\\sec x - \\tan x) \\ dx`',1560242842,NULL,'','','','',0,'',''),(144,4,92.000,'`\\int \\ \\tan^{-1} (\\frac{\\sin x}{1 - \\cos x}) \\ dx`',1560243215,NULL,'','','','',0,'',''),(145,4,93.000,'`\\int \\ \\tan^{-1} (\\frac{1 + \\cos x}{\\sin x}) \\ dx`',1560243270,NULL,'','','','',0,'',''),(146,4,94.000,'`\\int \\ \\tan^{-1} \\sqrt{\\frac{1 - \\sin x}{1 + \\sin x}} \\ dx`',1560243346,NULL,'','','','',0,'',''),(147,4,95.000,'`\\int \\ \\frac{x^n}{x-1} \\ dx \\ \\ \\ (n \\in \\mathbb{N})`',1560243659,NULL,'','','','',0,'',''),(148,4,96.000,'`\\int \\ \\frac{\\sqrt{x}}{\\sqrt{x} + 2} \\ dx`',1560243800,NULL,'','','','',0,'',''),(149,4,97.000,'`\\int \\ \\frac{x^2 + 1}{(x+1)^2} \\ dx`',1560243845,NULL,'','','','',0,'',''),(150,4,98.000,'`\\int \\ \\frac{x^2}{(x^2 + a^2)(x^2 + b^2)} \\ dx`',1560244017,NULL,'','','','',0,'',''),(151,4,99.000,'`\\int \\ \\frac{2^{5x} + 2^{3x}}{2^x + 2^{-x}} \\ dx`',1560244104,NULL,'','','','',0,'',''),(152,4,100.000,'`\\int \\ (e^{a \\log_e x} + e^{x \\log_e a}) \\ dx`',1560244285,NULL,'','','','',0,'',''),(153,4,101.000,'`\\int \\ \\frac{(x+1)e^x}{\\cos^2 (xe^x)} \\ dx`',1560244355,NULL,'','','','',0,'',''),(154,4,102.000,'`\\int \\ \\frac{(x+1)}{x(1 + xe^x)} \\ dx`',1560244410,NULL,'','','','',0,'',''),(155,4,103.000,'`\\int \\ \\frac{e^{5x} + e^{3x}}{e^x + e^{-x}} \\ dx`',1560244498,NULL,'','','','',0,'',''),(156,4,104.000,'`\\int \\ \\frac{e^{6x} + e^{4x}}{e^x + e^{-x}} \\ dx`',1560244542,NULL,'','','','',0,'',''),(157,4,105.000,'`\\int \\ e^{x^3} x^5 \\ dx`',1560244589,NULL,'','','','',0,'',''),(158,4,106.000,'`\\int \\ \\frac{dx}{e^x (e^x + 1)^2}`',1560244641,NULL,'','','','',0,'',''),(159,4,107.000,'`\\int \\ \\sqrt{e^{2x} + ae^{x}} \\ dx`',1560244687,NULL,'','','','',0,'',''),(160,4,108.000,'`\\int \\ \\frac{dx}{e^x + e^{-x}}`',1560244786,NULL,'','','','',0,'',''),(161,4,109.000,'`\\int \\ \\frac{dx}{1 - e^x}`',1560244820,NULL,'','','','',0,'',''),(162,4,110.000,'`\\int \\ \\frac{dx}{1 + e^x}`',1560244858,NULL,'','','','',0,'',''),(163,4,111.000,'`\\int \\ \\frac{e^{2x}}{e^x + 1} \\ dx`',1560244909,NULL,'','','','',0,'',''),(164,4,112.000,'`\\int \\ \\frac{e^x}{1 - e^{2x}} \\ dx`',1560244957,NULL,'','','','',0,'',''),(165,4,113.000,'`\\int \\ \\frac{e^x - 1}{e^x + 1} \\ dx`',1560245012,NULL,'','','','',0,'',''),(166,4,114.000,'`\\int \\ \\frac{dx}{\\sqrt{e^x - 1}}`',1560245071,NULL,'','','','',0,'',''),(167,4,115.000,'`\\int \\ e^x (\\frac{1}{x} - \\frac{1}{x^2}) \\ dx`',1560245136,NULL,'','','','',0,'',''),(168,4,116.000,'`\\int \\ \\frac{e^x}{x} (1 + x \\log x) \\ dx`',1560245207,NULL,'','','','',0,'',''),(169,4,117.000,'`\\int \\ e^x [\\frac{1}{(1-x)} + \\frac{1}{(1-x)^2}] \\ dx`',1560245305,NULL,'','','','',0,'',''),(170,4,118.000,'`\\int \\ \\frac{x e^x}{(x+1)^2} \\ dx`',1560269090,NULL,'','','','',0,'',''),(171,4,119.000,'`\\int \\ \\frac{e^x(1+x^2)}{(1+x)^2} \\ dx`',1560269529,NULL,'','','','',0,'',''),(172,4,120.000,'`\\int \\ e^x (\\frac{1-x}{1+x^2})^2 \\ dx`',1560269689,NULL,'','','','',0,'',''),(173,4,121.000,'`\\int \\ e^x (\\frac{x+2}{x+4})^2 \\ dx`',1560269897,NULL,'','','','',0,'',''),(174,4,122.000,'`\\int \\ e^x \\frac{x-1}{(x+1)^3} \\ dx`',1560269963,NULL,'','','','',0,'',''),(175,4,123.000,'`\\int \\ e^x \\frac{(x-4)}{(x-2)^3} \\ dx`',1560270007,NULL,'','','','',0,'',''),(176,4,124.000,'`\\int \\ e^x (\\cot x + \\log \\sin x) \\ dx`',1560270614,NULL,'','','','',0,'',''),(177,4,125.000,'`\\int \\ e^x (\\tan x - \\log \\cos x) \\ dx`',1560270695,NULL,'','','','',0,'',''),(178,4,126.000,'`\\int \\ e^x \\frac{(x^3 + x + 1)}{(1 + x^2)^{\\frac{3}{2}}} \\ dx`',1560270810,NULL,'','','','',0,'',''),(179,4,127.000,'`\\int \\ e^x (\\frac{1 - \\sin x}{1 - \\cos x}) \\ dx`',1560270887,NULL,'','','','',0,'',''),(180,4,128.000,'`\\int \\ e^x (\\frac{1 + \\sin x}{1 + \\cos x}) \\ dx`',1560270916,NULL,'','','','',0,'',''),(181,4,129.000,'`\\int \\ e^x (\\frac{2 + \\sin 2x}{1 + \\cos 2x}) \\ dx`',1560271026,NULL,'','','','',0,'',''),(182,4,130.000,'`\\int \\ e^x (\\frac{2 - \\sin 2x}{1 - \\cos 2x}) \\ dx`',1560271080,NULL,'','','','',0,'',''),(183,4,131.000,'`\\int \\ e^{x + \\frac{1}{x}} (1 + x - \\frac{1}{x}) \\ dx`',1560271180,NULL,'','','','',0,'',''),(184,4,132.000,'`\\int \\ e^{x + \\frac{1}{x}} (1 + x - x^2) \\ dx`',1560271216,NULL,'','','','',0,'',''),(185,4,133.000,'`\\int \\ \\frac{d(\\log x)}{x+1}`',1560271260,NULL,'','','','',0,'',''),(186,4,134.000,'`\\int \\ (x^4 + x^2 +1) \\ d(x^2)`',1560271306,NULL,'','','','',0,'',''),(187,4,135.000,'`\\int \\ x \\sqrt{x^2 + 1} \\ dx`',1560271512,NULL,'','','','',0,'',''),(188,4,136.000,'`\\int \\ \\frac{x^3}{x^2 + 4} \\ dx`',1560271647,NULL,'','','','',0,'',''),(189,4,137.000,'`\\int \\ \\frac{x^2 + 1}{(x^2 - 1)^2} \\ dx`',1560271731,NULL,'','','','',0,'',''),(190,4,138.000,'`\\int \\ \\frac{x^2 - 1}{(x^2 + 1)^2} \\ dx`',1560271817,NULL,'','','','',0,'',''),(191,4,139.000,'`\\int \\ \\frac{dx}{x^3 \\sqrt{x^2 - 1}}`',1560272114,NULL,'','','','',0,'',''),(192,4,140.000,'`\\int \\ \\frac{dx}{x \\sqrt{x^4 - 1}}`',1560272951,NULL,'','','','',0,'',''),(193,4,141.000,'`\\int \\ \\frac{dx}{x \\sqrt{x^6 - 1}}`',1560348546,NULL,'','','','',0,'',''),(194,4,142.000,'`\\int \\ \\frac{dx}{x \\sqrt{x^3 + 1}}`',1560348600,NULL,'','','','',0,'',''),(195,4,143.000,'`\\int \\ \\frac{dx}{x \\sqrt{x^4 + 1}}`',1560348649,NULL,'','','','',0,'',''),(196,4,144.000,'`\\int \\ \\frac{x^2 - 1}{x \\sqrt{x^4 + 1}} \\ dx`',1560348680,NULL,'','','','',0,'',''),(197,4,145.000,'`\\int \\ \\frac{x \\ dx}{\\sqrt{x} + x\\sqrt{x}}`',1560348735,NULL,'','','','',0,'',''),(198,4,146.000,'`\\int \\ \\frac{dx}{\\sqrt{x} + x\\sqrt{x}}`',1560348752,NULL,'','','','',0,'',''),(199,4,147.000,'`\\int \\ \\frac{dx}{x + \\sqrt{x}}`',1560348784,NULL,'','','','',0,'',''),(200,4,148.000,'`\\int \\ \\frac{dx}{x^4 - 1}`',1560348830,NULL,'','','','',0,'',''),(201,4,149.000,'`\\int \\ \\frac{dx}{x^4 + 1}`',1560348850,NULL,'','','','',0,'',''),(202,4,150.000,'`\\int \\ \\frac{x^4}{x^2 + 1} \\ dx`',1560348885,NULL,'','','','',0,'',''),(203,4,151.000,'`\\int \\ \\frac{dx}{x^2 \\sqrt{x+1}}`',1560348985,NULL,'','','','',0,'',''),(204,4,152.000,'`\\int \\ \\frac{dx}{x^2 (x^4 + 1)^{\\frac{3}{4}}}`',1560349837,NULL,'','','','',0,'',''),(205,4,153.000,'`\\int \\ \\frac{dx}{x(a+bx^n)^2}`',1560349901,NULL,'','','','',0,'',''),(206,4,154.000,'`\\int \\ \\frac{x^2 - 1}{x^4 + 1} \\ dx`',1560349965,NULL,'','','','',0,'',''),(207,4,155.000,'`\\int \\ \\frac{x^4+1}{x^6+1} \\ dx`',1560350057,NULL,'','','','',0,'',''),(208,4,156.000,'`\\int \\ \\frac{dx}{(1+x^2)\\sqrt{1+x^2}}`',1560350790,NULL,'','','','',0,'',''),(209,4,157.000,'`\\int \\ \\frac{dx}{(1+x^2)\\sqrt{1-x^2}}`',1560350810,NULL,'','','','',0,'',''),(210,4,158.000,'`\\int \\ \\frac{x^2 - 1}{(x^2 + 1)\\sqrt{1+x^4}} \\ dx`',1560776600,NULL,'','','','',0,'',''),(211,4,159.000,'`\\int \\ \\frac{dx}{\\sqrt{1+x^2} \\sqrt{\\log_e (x+\\sqrt{1+x^2})}}`',1560776673,NULL,'','','','',0,'',''),(212,4,160.000,'`\\int \\ \\frac{\\sqrt{x^2+1}[\\log (x^2+1) - 2 \\log x]}{x^4} \\ dx`',1560776757,NULL,'','','','',0,'',''),(213,4,161.000,'$$ \\int \\frac{x+\\sqrt[3]{x^2} + 2\\sqrt[6]{x}}{x(1+\\sqrt[3]{x})} \\ dx $$',1560777187,NULL,'','','','',0,'',''),(214,4,162.000,'`\\int \\ \\frac{x(1-x^2)}{x^4+1} \\ dx`',1560777280,NULL,'','','','',0,'',''),(215,4,163.000,'`\\int \\ \\frac{\\sqrt{x^2+1}}{x^4} \\ dx`',1560777352,NULL,'','','','',0,'',''),(216,4,164.000,'$$ \\int \\frac{\\sqrt[4]{(x^4 - x)}}{x^5} \\ dx $$',1560777751,NULL,'','','','',0,'',''),(217,4,165.000,'`\\int \\ \\sqrt{x+\\sqrt{x^2+1}} \\ dx`',1560777809,NULL,'','','','',0,'',''),(218,4,166.000,'`\\int \\ (x+1)(2x-1)^{\\frac{3}{2}} \\ dx`',1560777884,NULL,'','','','',0,'',''),(219,4,167.000,'`\\int \\ \\sqrt{\\frac{1+x}{1-x}} \\ dx`',1560777921,NULL,'','','','',0,'',''),(220,4,168.000,'`\\int \\ \\sqrt{\\frac{2x+1}{3x+2}} \\ dx`',1560777989,NULL,'','','','',0,'',''),(221,4,169.000,'`int \\ \\frac{x+2}{x^2 - 3x + 2} \\ dx`',1560778034,NULL,'','','','',0,'',''),(222,4,170.000,'`\\int \\ \\frac{1-x^2}{x(1-2x)} \\ dx`',1560778274,NULL,'','','','',0,'',''),(223,4,171.000,'`\\int \\ \\frac{(x+1)}{\\sqrt{1-2x-x^2}}`',1560778407,NULL,'','','','',0,'',''),(224,4,172.000,'`\\int \\ \\frac{2x+3}{\\sqrt{x^2+4x+5}} \\ dx`',1560778463,NULL,'','','','',0,'',''),(225,4,173.000,'`\\int \\ \\frac{2x+1}{\\sqrt{x^2+4x+3}} \\ dx`',1560778542,NULL,'','','','',0,'',''),(226,4,174.000,'`\\int \\ \\frac{(x^2-1)}{x\\sqrt{x^4+3x^2+1}} \\ dx`',1560778606,NULL,'','','','',0,'',''),(227,4,175.000,'`\\int \\ \\frac{dx}{\\sqrt{4x-3x^2+5}}`',1560782613,NULL,'','','','',0,'',''),(228,4,176.000,'`\\int \\ \\frac{dx}{\\sqrt{3-5x-2x^2}}`',1560782647,NULL,'','','','',0,'',''),(229,4,177.000,'`\\int \\ \\frac{dx}{\\sqrt{1-x-x^2}}`',1560782866,NULL,'','','','',0,'',''),(230,4,178.000,'`\\int \\ \\frac{dx}{\\sqrt{5-4x-2x^2}}`',1560782942,NULL,'','','','',0,'',''),(231,4,179.000,'`\\int \\ \\frac{dx}{\\sqrt{x^2+6x+5}}`',1560782971,NULL,'','','','',0,'',''),(232,4,180.000,'`\\int \\ \\frac{dx}{\\sqrt{(x-2)(x-3)}}`',1560783005,NULL,'','','','',0,'',''),(233,4,181.000,'`\\int \\ \\frac{x^2}{\\sqrt{1-x^6}} \\ dx`',1560783094,NULL,'','','','',0,'',''),(234,4,182.000,'`\\int \\ \\frac{dx}{x\\sqrt{2ax-x^2}}`',1560794276,NULL,'','','','',0,'',''),(235,4,183.000,'`\\int \\ \\frac{x^2}{(x-1)^3(x+1)} \\ dx`',1560794324,NULL,'','','','',0,'',''),(236,4,184.000,'$$ \\int \\frac{\\sqrt{x}}{1+\\sqrt[4]{x^3}} \\ dx $$',1560794500,'','','','','',0,'',''),(237,4,185.000,'`\\int \\ \\sqrt{\\frac{a+x}{a-x}} \\ dx`',1560794596,NULL,'','','','',0,'',''),(238,4,186.000,'`\\int \\ \\sqrt{\\frac{x}{a-x}} \\ dx`',1560794733,NULL,'','','','',0,'',''),(239,4,187.000,'`\\int \\ \\sqrt{\\frac{a-x}{x}} \\ dx`',1560794756,NULL,'','','','',0,'',''),(240,4,188.000,'`\\int \\ x \\sqrt{\\frac{4-x}{4+x}} \\ dx`',1560794805,NULL,'','','','',0,'',''),(241,4,189.000,'`\\int \\ x \\sqrt{\\frac{a-x}{a+x}} \\ dx`',1560794822,NULL,'','','','',0,'',''),(242,4,190.000,'`\\int \\ \\sqrt{4x^2-9} \\ dx`',1560794850,NULL,'','','','',0,'',''),(243,4,191.000,'`\\int \\ \\frac{\\sin \\sqrt{x}}{\\sqrt{x}} \\ dx`',1560794896,NULL,'','','','',0,'',''),(244,4,192.000,'`\\int \\ \\frac{\\cos \\sqrt{\\theta}}{\\sqrt{\\theta} \\sin^2 \\theta} \\ d\\theta`',1560794974,NULL,'','','','',0,'',''),(245,4,193.000,'`\\int \\ \\frac{\\tan^4\\sqrt{x} \\sec^2\\sqrt{x}}{\\sqrt{x}} \\ dx`',1560795046,NULL,'','','','',0,'',''),(246,4,194.000,'$$ \\int \\frac{1-\\sin x}{\\sqrt[3]{x+\\cos x}} \\ dx$$',1560795131,NULL,'','','','',0,'',''),(247,4,195.000,'`\\int \\ \\frac{dx}{a^2 \\cos^2x + b^2 \\sin^2x}`',1560795220,NULL,'','','','',0,'',''),(248,4,196.000,'`\\int \\ \\frac{dx}{1+3\\sin^2x+5\\cos^2x}`',1560795271,NULL,'','','','',0,'',''),(249,4,197.000,'`\\int \\ \\frac{\\sin 2x}{a^2 \\cos^2x + b^2 \\sin^2x} \\ dx`',1560795337,NULL,'','','','',0,'',''),(250,4,198.000,'`\\int \\ \\frac{\\sin2x}{(a^2\\cos^2x+b^2\\sin^2x)^2} \\ dx`',1560795410,NULL,'','','','',0,'',''),(251,4,199.000,'`\\int \\ \\frac{dx}{1-\\sin^4x}`',1560795441,NULL,'','','','',0,'',''),(252,4,200.000,'`\\int \\ \\frac{dx}{\\sin(x-\\alpha)\\cos(x-\\beta)}`',1560795494,NULL,'','','','',0,'',''),(253,4,201.000,'`\\int \\ \\frac{dx}{\\sin(x-\\alpha)\\sin(x-\\beta)}`',1560795521,NULL,'','','','',0,'',''),(254,4,202.000,'`\\int \\ \\frac{dx}{\\cos(x-\\alpha)\\cos(x-\\beta)}`',1560795545,NULL,'','','','',0,'',''),(255,4,203.000,'`\\int \\ \\frac{\\sinx}{\\sin(x+\\alpha)} \\ dx`',1560795598,NULL,'','','','',0,'',''),(256,4,204.000,'`\\int \\ \\frac{\\sinx}{\\sin(x-\\alpha)} \\ dx`',1560795623,NULL,'','','','',0,'',''),(257,4,205.000,'`\\int \\ \\frac{\\sinx+\\cosx}{\\sin(x-\\alpha)} \\ dx`',1560795653,NULL,'','','','',0,'',''),(258,4,206.000,'`\\int \\ \\frac{\\sinx-\\cosx}{\\sqrt{\\sin2x}}\\ dx`',1560795709,NULL,'','','','',0,'',''),(259,4,207.000,'`\\int \\ \\frac{\\sinx+\\cosx}{\\sqrt{\\sin2x}}\\ dx`',1560795729,NULL,'','','','',0,'',''),(260,4,208.000,'`\\int \\ \\frac{1-\\tanx}{1+\\tanx}\\ dx`',1560795768,NULL,'','','','',0,'',''),(261,4,209.000,'`\\int \\ \\frac{\\tanx-\\tan\\alpha}{\\tanx+\\tan\\alpha}\\ dx`',1560795831,NULL,'','','','',0,'',''),(262,4,210.000,'`\\int \\ \\frac{dx}{\\sec2x-\\tan2x}`',1560795873,NULL,'','','','',0,'',''),(263,4,211.000,'`\\int \\ \\frac{dx}{\\sinx+\\secx}`',1560795921,NULL,'','','','',0,'',''),(264,4,212.000,'`\\int \\ \\sin4x \\ e^{\\tan^2x}\\ dx`',1560795992,NULL,'','','','',0,'',''),(265,4,213.000,'`\\int \\ \\frac{\\sin2x}{\\sin5x\\sin3x}\\ dx`',1560796091,NULL,'','','','',0,'',''),(266,4,214.000,'`\\int \\ (\\tanx - x)\\ \\tan^2x \\ dx`',1560796148,NULL,'','','','',0,'',''),(267,4,215.000,'`\\int \\ (x+\\cotx)\\ \\cot^2x \\ dx`',1560796184,NULL,'','','','',0,'',''),(268,4,216.000,'`\\int \\ \\secx \\log(\\secx+\\tanx) \\ dx`',1560796290,NULL,'','','','',0,'',''),(269,4,217.000,'`\\int \\ \\frac{\\cosx+x\\sinx}{x(x+\\cosx)} \\ dx`',1560796348,NULL,'','','','',0,'',''),(270,4,218.000,'`\\int \\ \\frac{\\sinx-x\\cosx}{x(x-\\sinx)} \\ dx`',1560796393,NULL,'','','','',0,'',''),(271,4,219.000,'`\\int \\ \\frac{\\sinx-x\\cosx}{x(x+\\sinx)} \\ dx`',1560796412,NULL,'','','','',0,'',''),(272,4,220.000,'`int \\ \\frac{2\\sinx}{5+3\\cosx} \\ dx`',1560796474,NULL,'','','','',0,'',''),(273,4,221.000,'`int \\ \\frac{2\\sinx}{5+3\\cosx} \\ dx`',1560796486,NULL,'','','','',0,'',''),(274,4,222.000,'`\\int \\ \\frac{dx}{5+4\\cosx}`',1560796523,NULL,'','','','',0,'',''),(275,4,223.000,'`\\int \\ \\frac{dx}{5+4\\sinx}`',1560796541,NULL,'','','','',0,'',''),(276,4,224.000,'`\\int \\ \\frac{\\cos^4x}{\\sin^2x} \\ dx`',1560796612,NULL,'','','','',0,'',''),(277,4,225.000,'`\\int \\ \\frac{\\sqrt{\\tanx}}{\\cos^4x}\\ dx`',1560796672,NULL,'','','','',0,'',''),(278,4,226.000,'`\\int \\ \\frac{\\sec^4x}{\\sqrt{\\tanx}}\\ dx`',1560796718,NULL,'','','','',0,'',''),(279,4,227.000,'`\\int \\ \\frac{\\sqrt{\\tanx}}{\\sin2x}\\ dx`',1560796762,NULL,'','','','',0,'',''),(280,4,228.000,'`\\int \\ \\sin^3x\\ \\cos^4x \\ dx`',1560796839,NULL,'','','','',0,'',''),(281,4,229.000,'`\\int \\ \\cos^3x\\ \\sin^{\\frac{3}{2}}x \\ dx`',1560796892,NULL,'','','','',0,'',''),(282,4,230.000,'`\\int \\ \\sec^{\\frac{8}{9}}x\\ \\text{cosec}^{\\frac{10}{9}}x \\ dx`',1560797003,NULL,'','','','',0,'',''),(283,4,231.000,'`\\int \\ \\frac{\\sin^5x}{\\cos^2x} \\ dx`',1560862185,NULL,'','','','',0,'',''),(284,4,232.000,'`\\int \\ \\frac{\\sin^2x}{\\cos^4x} \\ dx`',1560862207,NULL,'','','','',0,'',''),(285,4,233.000,'`\\int \\ \\sqrt{\\frac{\\sinx}{\\cos^5x}} \\ dx`',1560862244,NULL,'','','','',0,'',''),(286,4,234.000,'`\\int \\ \\frac{dx}{\\sqrt{\\sinx\\cos^7x}}`',1560862303,NULL,'','','','',0,'',''),(287,4,235.000,'`\\int \\ \\tan^3x\\ \\sec^4x \\ dx`',1560862347,NULL,'','','','',0,'',''),(288,4,236.000,'`\\int \\ \\frac{1+x\\cosx}{x(1-x^2e^{2\\sinx})} \\ dx`',1560862411,NULL,'','','','',0,'',''),(289,4,237.000,'`\\int \\ \\sqrt{\\tanx} \\ dx`',1560862440,NULL,'','','','',0,'',''),(290,4,238.000,'`\\int \\ \\sqrt{\\cotx} \\ dx`',1560862457,NULL,'','','','',0,'',''),(291,4,239.000,'`\\int \\ (\\sqrt{\\tanx}+\\sqrt{\\cotx}) \\ dx`',1560862498,NULL,'','','','',0,'',''),(292,4,240.000,'`\\int \\ (\\sqrt{\\tanx}-\\sqrt{\\cotx}) \\ dx`',1560862510,NULL,'','','','',0,'',''),(293,4,241.000,'`\\int \\ \\frac{\\sqrt{\\cos2x}}{\\sinx} \\ dx`',1560862559,NULL,'','','','',0,'',''),(294,4,242.000,'`\\int \\ \\frac{\\cosx}{\\sqrt{\\cos2x}} \\ dx`',1560862598,NULL,'','','','',0,'',''),(295,4,243.000,'`\\int \\ \\frac{\\sin(\\tan^{-1}x)}{1+x^2} \\ dx`',1560862669,NULL,'','','','',0,'',''),(296,4,244.000,'`\\int \\ \\sin(2\\tan^{-1}\\sqrt{\\frac{3-x}{3+x}}) \\ dx`',1560862752,NULL,'','','','',0,'',''),(297,4,245.000,'`\\int \\ \\cos(2\\cot^{-1}\\sqrt{\\frac{1-x}{1+x}}) \\ dx`',1560862790,NULL,'','','','',0,'',''),(298,4,246.000,'`\\int \\ \\frac{dx}{(1+x^2)\\sqrt{\\tan^{-1}x+6}}`',1560862861,NULL,'','','','',0,'',''),(299,4,247.000,'$$\\int \\ [f(x){g}\'(x)-{f}\'(x)g(x)]\\ dx$$',1560863144,NULL,'','','','',0,'',''),(300,4,248.000,'`\\int \\ xe^x \\ dx`',1560863178,NULL,'','','','',0,'',''),(301,4,249.000,'`\\int \\ \\logx \\ dx`',1560863208,NULL,'','','','',0,'',''),(302,4,250.000,'`\\int \\ \\log_10x \\ dx`',1560863227,NULL,'','','','',0,'',''),(303,4,251.000,'`\\int \\ \\log(x+\\sqrt{x^2+a^2}) \\ dx`',1560863300,NULL,'','','','',0,'',''),(304,4,252.000,'`\\int \\ \\frac{\\log(x+\\sqrt{x^2+a^2})}{\\sqrt{x^2+a^2}} \\ dx`',1560863348,NULL,'','','','',0,'',''),(305,4,253.000,'`\\int \\ (\\logx)^2 \\ dx`',1560863389,NULL,'','','','',0,'',''),(306,4,254.000,'`\\int \\ (\\log\\sqrt{x})^2 \\ dx`',1560863416,NULL,'','','','',0,'',''),(307,4,255.000,'`\\int \\ (x+1)^2 \\logx \\ dx`',1560863801,NULL,'','','','',0,'',''),(308,4,256.000,'`\\int \\ {\\log(\\logx)+(\\logx)^{-2}} \\ dx`',1560863901,NULL,'','','','',0,'',''),(309,4,257.000,'`\\int \\ x^4 (\\logx)^2 \\ dx`',1560863990,NULL,'','','','',0,'',''),(310,4,258.000,'`\\int \\ \\frac{\\log(1-x)}{x^2} \\ dx`',1560864117,NULL,'','','','',0,'',''),(311,4,259.000,'`\\int \\ \\log(1+x)^{1+x} \\ dx`',1560864173,NULL,'','','','',0,'',''),(312,4,260.000,'`\\int \\ \\sin\\sqrt{x} \\ dx`',1560864200,NULL,'','','','',0,'',''),(313,4,261.000,'`\\int \\ \\cos\\sqrt{x} \\ dx`',1560864233,NULL,'','','','',0,'',''),(314,4,262.000,'`\\int \\ \\sinx \\log(\\secx+\\tanx) \\ dx`',1560864349,NULL,'','','','',0,'',''),(315,4,263.000,'`\\int \\ xe^x\\sinx \\ dx`',1560864380,NULL,'','','','',0,'',''),(316,4,264.000,'`\\int \\ xe^x\\cosx \\ dx`',1560864404,NULL,'','','','',0,'',''),(317,4,265.000,'`\\int \\ x\\tanx\\sec^2x \\ dx`',1560864464,NULL,'','','','',0,'',''),(318,4,266.000,'`\\int \\ x\\sin^{-1}x \\ dx`',1560864497,NULL,'','','','',0,'',''),(319,4,267.000,'`\\int \\ x^2\\sin^{-1}3x \\ dx`',1560864521,NULL,'','','','',0,'',''),(320,4,268.000,'`\\int \\ x^3\\sinx \\ dx`',1560864610,NULL,'','','','',0,'',''),(321,4,269.000,'`\\int \\ x^2\\cosx \\ dx`',1560864664,NULL,'','','','',0,'',''),(322,4,270.000,'`\\int \\ 3^x\\cos^3x \\ dx`',1560864700,NULL,'','','','',0,'',''),(323,4,271.000,'`\\int \\ \\sec^3x \\ dx`',1560864723,NULL,'','','','',0,'',''),(324,4,272.000,'`\\int \\ \\sec^5x \\ dx`',1560864746,NULL,'','','','',0,'',''),(325,4,273.000,'`\\int \\ \\text{cosec}^3x \\ dx`',1560864780,NULL,'','','','',0,'',''),(326,4,274.000,'`\\int \\ \\text{cosec}^4x \\ dx`',1560864795,NULL,'','','','',0,'',''),(327,4,275.000,'`\\int \\ \\frac{x}{1+\\sinx} \\ dx`',1560864828,NULL,'','','','',0,'',''),(328,4,276.000,'`\\int \\ \\frac{x+\\sinx}{1+\\cosx} \\ dx`',1560864860,NULL,'','','','',0,'',''),(329,4,277.000,'`\\int \\ \\frac{x+\\cosx}{1-\\sinx} \\ dx`',1560864884,NULL,'','','','',0,'',''),(330,4,278.000,'`\\int \\ \\frac{x^2}{(x\\sinx+\\cosx)^2} \\ dx`',1560864957,'`-\\frac{x}{\\cosx(x\\sinx+\\cosx)}+\\tanx+c`','330','','','',0,'',''),(331,4,279.000,'`\\int \\ \\frac{x^2}{(x\\cosx-\\sinx)^2} \\ dx`',1560864999,NULL,'','','','',0,'',''),(332,4,280.000,'`\\int \\ \\frac{x}{(1-\\cosx)^2} \\ dx`',1560865042,NULL,'','','','',0,'',''),(333,4,281.000,'`\\int \\ \\sqrt{\\frac{\\sin(x-\\alpha)}{\\sin(x+\\alpha)}} \\ dx`',1560865103,NULL,'','','','',0,'',''),(334,4,282.000,'`\\int \\ \\frac{1+\\sinx}{\\sinx(1+\\cosx)} \\ dx`',1560865167,NULL,'','','','',0,'',''),(335,4,283.000,'`\\int \\ \\frac{\\cosx}{\\sin^2x}(1-3\\cos^3x) \\ dx`',1560865248,NULL,'','','','',0,'',''),(336,4,284.000,'`\\int \\ \\frac{dx}{\\sqrt{\\sin^3x\\cos(x+\\alpha)}}`',1560865324,NULL,'','','','',0,'',''),(337,4,285.000,'`\\int \\ \\frac{dx}{\\sqrt{\\sin^3x\\cos(x-\\alpha)}}`',1560865445,NULL,'','','','',0,'',''),(338,4,286.000,'`\\int \\ \\frac{dx}{\\sqrt{\\sin^3x\\sin(x+\\alpha)}}`',1560865469,NULL,'','','','',0,'',''),(339,4,287.000,'`\\int \\ \\frac{dx}{1+\\cos\\alpha\\cosx}` where  `(0\\lt\\alpha\\lt\\frac{\\pi}{2})`',1560865628,NULL,'','','','',0,'',''),(340,4,288.000,'`\\int \\ \\frac{dx}{1-\\cos\\alpha\\cosx}` where  `(0\\lt\\alpha\\lt\\frac{\\pi}{2})`',1560865663,NULL,'','','','',0,'',''),(341,4,289.000,'`\\int \\ \\frac{dx}{\\cosx+\\cos\\alpha}`',1560865707,NULL,'','','','',0,'',''),(342,4,290.000,'`\\int \\ \\frac{\\sqrt{1+\\cos\\theta}}{(1-\\cos\\theta)^{\\frac{5}{2}}} \\ d\\theta`',1560865789,NULL,'','','','',0,'',''),(343,4,291.000,'`\\int \\ \\frac{\\sin^{-1}x}{x^2} \\ dx`',1560865842,NULL,'','','','',0,'',''),(344,4,292.000,'`\\int \\ \\frac{x\\sin^{-1}x}{\\sqrt{1-x^2}} \\ dx`',1560865902,NULL,'','','','',0,'',''),(345,4,293.000,'`\\int \\ (\\sin^{-1}x)^2 \\ dx`',1560865938,NULL,'','','','',0,'',''),(346,4,294.000,'`\\int \\ (\\sin^{-1}x)^3 \\ dx`',1560865951,NULL,'','','','',0,'',''),(347,4,295.000,'`\\int \\ \\frac{\\sin^{-1}x}{(1-x^2)^{\\frac{3}{2}}} \\ dx`',1560866019,NULL,'','','','',0,'',''),(348,4,296.000,'`\\int \\ \\frac{\\tan^{-1}x}{(1+x^2)^{\\frac{3}{2}}} \\ dx`',1560866053,NULL,'','','','',0,'',''),(349,4,297.000,'`\\int \\ \\frac{x^2\\sin^{-1}x}{(1-x^2)^{\\frac{5}{2}}} \\ dx`',1560866084,NULL,'','','','',0,'',''),(350,4,298.000,'`\\int \\ x\\sin^{-1}(\\frac{1}{2}\\sqrt{\\frac{2a-x}{a}}) \\ dx`',1560866170,NULL,'','','','',0,'',''),(351,4,299.000,'`\\int \\ \\sin^{-1}\\sqrt{\\frac{x}{a+x}} \\ dx`',1560866238,NULL,'','','','',0,'',''),(352,4,300.000,'`\\int \\ \\sin^{-1}\\sqrt{\\frac{a}{a+x}} \\ dx`',1560866255,NULL,'','','','',0,'',''),(353,4,301.000,'`\\int \\ \\cos^{-1}x \\ dx`',1560866286,NULL,'','','','',0,'',''),(354,4,302.000,'`\\int \\ \\cos^{-1}(\\frac{1}{x}) \\ dx`',1560866316,NULL,'','','','',0,'',''),(355,4,303.000,'`\\int \\ \\cos^{-1}\\sqrt{x} \\ dx`',1560866426,NULL,'','','','',0,'',''),(356,4,304.000,'`\\int \\ \\frac{\\sin^{-1}\\sqrt{x}-\\cos^{-1}\\sqrt{x}}{\\sin^{-1}\\sqrt{x}+\\cos^{-1}\\sqrt{x}} \\ dx` where `(0\\lex\\le1)`',1560866564,NULL,'','','','',0,'',''),(357,4,305.000,'`\\int \\ (\\cos^{-1}x)^3 \\ dx`',1560866683,NULL,'','','','',0,'',''),(358,4,306.000,'`\\int \\ \\cos^{-1}\\sqrt{\\frac{x}{a+x}} \\ dx`',1560866954,NULL,'','','','',0,'',''),(359,4,307.000,'`\\int \\ \\sec^{-1}x \\ dx`',1560867024,NULL,'','','','',0,'',''),(360,4,308.000,'`\\int \\ \\tan^{-1}\\sqrt{x} \\ dx`',1560867054,NULL,'','','','',0,'',''),(361,4,309.000,'Find `\\int \\ \\tan^{-1}x \\ dx`; hence find `\\int \\ \\cot^{-1}x \\ dx`',1560867220,NULL,'','','','',0,'',''),(362,4,310.000,'`\\int \\ x\\tan^{-1}x \\ dx`',1560867257,NULL,'','','','',0,'',''),(363,4,311.000,'`\\int \\ \\tan^{-1}\\sqrt{\\frac{1-x}{1+x}} \\ dx`',1560867297,NULL,'','','','',0,'',''),(364,4,312.000,'`\\int \\ \\cot^{-1}(1-x+x^2) \\ dx`',1560867346,NULL,'','','','',0,'',''),(365,4,313.000,'`\\int \\ [\\cot^{-1}(1-x+x^2)+\\tan^{-1}(x-1)] \\ dx`',1560867402,NULL,'','','','',0,'',''),(366,4,314.000,'`\\int \\ \\frac{e^{\\tan^{-1}x}}{(1+x^2)^2} \\ dx`',1560867470,NULL,'','','','',0,'',''),(367,4,315.000,'`\\int \\ \\e^{2\\tan^{-1}x}\\ \\frac{(1+x)^2}{1+x^2} \\ dx`',1560867548,NULL,'','','','',0,'',''),(368,4,316.000,'`\\int \\ \\e^{\\tan^{-1}x}\\ (\\frac{1+x+x^2}{1+x^2}) \\ dx`',1560867610,NULL,'','','','',0,'',''),(369,4,317.000,'`\\int \\ \\frac{e^{m\\tan^{-1}x}}{(1+x^2)^{\\frac{3}{2}}}\\ dx`',1560867997,NULL,'','','','',0,'',''),(370,4,318.000,'`\\int \\ \\sqrt{(x-\\alpha)(\\beta-x)} \\ dx` where `(\\alpha\\ltx\\lt\\beta)`',1560868097,NULL,'','','','',0,'',''),(371,4,319.000,'`\\int \\ \\frac{dx}{\\sqrt{(x-\\alpha)(\\beta-x)}}` where `(\\alpha\\ltx\\lt\\beta)`',1560868205,NULL,'','','','',0,'',''),(372,4,320.000,'`\\int \\ \\frac{dx}{\\sqrt{(x-\\alpha)(x-\\beta)}}` where `(\\alpha,\\beta\\gt0)`',1560868345,NULL,'','','','',0,'',''),(373,4,321.000,'`\\int \\ \\frac{(x+3)\\sqrt{x+1}}{\\sqrt{x-1}}\\ dx`',1560870837,NULL,'','','','',0,'',''),(374,4,322.000,'`\\int \\ \\frac{(x+1)\\sqrt{x+2}}{\\sqrt{x-2}}\\ dx`',1560871209,NULL,'','','','',0,'',''),(375,4,323.000,'`\\int \\ \\sqrt{\\frac{1-\\sqrt{x}}{1+\\sqrt{x}}}\\ dx`',1560871333,NULL,'','','','',0,'',''),(376,4,324.000,'`\\int \\ (3x-2)\\sqrt{x^2+x+1}\\ dx`',1560871734,NULL,'','','','',0,'',''),(377,4,325.000,'`\\int \\ \\frac{x^2+x+1}{\\sqrt{x^2+2x+3}}\\ dx`',1560871945,NULL,'','','','',0,'',''),(378,4,326.000,'`\\int \\ \\frac{\\sqrt{x^2+1}}{x^4}\\ dx`',1560872019,NULL,'','','','',0,'',''),(379,4,327.000,'`\\int \\ \\frac{\\log(1+x)}{(1+x)^2}\\ dx`',1560872167,NULL,'','','','',0,'',''),(380,4,328.000,'`\\int \\ \\frac{\\log(1+x^2)}{(1+x)^2}\\ dx`',1560872216,NULL,'','','','',0,'',''),(381,4,329.000,'`\\int \\ \\frac{\\logx}{(1+\\logx)^2}\\ dx`',1560872574,NULL,'','','','',0,'',''),(382,4,330.000,'`\\int \\ \\frac{\\cos(\\logx)}{x^3}\\ dx`',1560872692,NULL,'','','','',0,'',''),(383,4,331.000,'`\\int \\ \\sin(\\logx)\\ dx`',1560873007,NULL,'','','','',0,'',''),(384,4,332.000,'`\\int \\ \\cos2x \\log(\\frac{\\cosx+\\sinx}{\\cosx-\\sinx})\\ dx`',1560873122,NULL,'','','','',0,'',''),(385,4,333.000,'`\\int \\ \\cos2x \\log(1+\\tanx)\\ dx`',1560873168,NULL,'','','','',0,'',''),(386,4,334.000,'`\\int \\ \\log(\\sqrt{1+x}+\\sqrt{1-x})\\ dx`',1560873231,NULL,'','','','',0,'',''),(387,4,335.000,'If `I_n = \\int \\ \\tan^nx \\ dx`, then show that `I_{n-1}=\\frac{\\tan^{n-1}x}{n-1}-I_{n-2}` & then find `\\int \\ \\tan^5x\\ dx`',1560873588,NULL,'','','','',0,'',''),(388,4,336.000,'If `I_n = \\int \\ \\cos^nx \\ dx`, then show that `I_{n}=\\frac{\\cos^{n-1}x\\sinx}{n}+\\frac{n-1}{n}I_{n-2}` & then find `\\int \\ \\cos^7x\\ dx`',1560873899,NULL,'','','','',0,'',''),(389,4,337.000,'If `I_{m,n} = \\int \\ \\cos^mx\\ \\cosnx\\ dx`, then show that `(m+n)I_{m,n}=\\cos^mx\\ \\sinnx+mI_{m-1,n-1}`',1560874229,NULL,'','','','',0,'',''),(390,4,338.000,'`\\int \\ \\frac{x^2}{x^4+3x^2+1}\\ dx`',1560874484,NULL,'','','','',0,'',''),(391,4,339.000,'`\\int \\ \\frac{x^2}{2x^4-7x^2-4}\\ dx`',1560874579,NULL,'','','','',0,'',''),(392,4,340.000,'`\\int \\ \\frac{x^2}{x^4+x^2-2}\\ dx`',1560874683,NULL,'','','','',0,'',''),(393,4,341.000,'`\\int \\ \\frac{x^2-1}{x^4+3x^2+1}\\ dx`',1560874721,NULL,'','','','',0,'',''),(394,4,342.000,'`\\int \\ \\frac{x^2+1}{x^4+x^2+1}\\ dx`',1560874958,NULL,'','','','',0,'',''),(395,4,343.000,'`\\int \\ \\frac{x^3}{x^2-4x+3}\\ dx`',1560875150,NULL,'','','','',0,'',''),(396,4,344.000,'`\\int \\ \\frac{x}{(1+x)(1+x^2)}\\ dx`',1560875325,NULL,'','','','',0,'',''),(397,4,345.000,'`\\int \\ \\frac{dx}{(1+x)\\sqrt{1+2x-x^2}}`',1560917790,NULL,'','','','',0,'',''),(398,4,346.000,'`\\int \\ \\frac{dx}{(1+x)\\sqrt{1+x-x^2}}`',1560917818,NULL,'','','','',0,'',''),(399,4,347.000,'`\\int \\ \\frac{dx}{(1+x)\\sqrt{x^2+x-1}}`',1560917847,NULL,'','','','',0,'',''),(400,4,348.000,'`\\int \\ \\frac{dx}{(1+x^2)\\sqrt{x^2+x-1}}`',1560917878,NULL,'','','','',0,'',''),(401,4,349.000,'`\\int \\ \\frac{dx}{\\sqrt{\\frac{2}{3}x^3-x^2+\\frac{1}{3}}}`',1560917939,NULL,'','','','',0,'',''),(402,4,350.000,'`\\int \\ \\frac{dx}{(1+x)\\sqrt{1-x^2}}`',1560918040,NULL,'','','','',0,'',''),(403,4,351.000,'`\\int \\ \\frac{dx}{(1-x)\\sqrt{1-x^2}}`',1560918067,NULL,'','','','',0,'',''),(404,4,352.000,'`\\int \\ \\frac{\\cosx}{\\sqrt{5\\sin^2x-12\\sinx+4}}\\ dx`',1560918127,NULL,'','','','',0,'',''),(405,4,353.000,'`\\int \\ \\frac{x^2}{x^6-5x^3+6}\\ dx`',1560918247,NULL,'','','','',0,'',''),(406,4,354.000,'`\\int \\ \\frac{dx}{x(x^4-1)}`',1560918290,NULL,'','','','',0,'',''),(407,4,355.000,'`\\int \\ \\frac{dx}{x(x^n+1)}`',1560918315,NULL,'','','','',0,'',''),(408,4,356.000,'`\\int \\ \\frac{dx}{x\\sqrt{x^n+1}}`',1560918351,NULL,'','','','',0,'',''),(409,4,357.000,'`\\int \\ \\frac{dx}{x^2+x+1}`',1560918387,NULL,'','','','',0,'',''),(410,4,358.000,'`\\int \\ \\frac{dx}{x^4+x^2+1}`',1560918414,NULL,'','','','',0,'',''),(411,4,359.000,'`\\int \\ \\frac{dx}{x^3+x^2+x+1}`',1560918440,NULL,'','','','',0,'',''),(412,4,360.000,'`\\int \\ \\frac{x}{x^4-x^2+1}\\ dx`',1560918538,NULL,'','','','',0,'',''),(413,4,361.000,'`\\int \\ \\frac{dx}{(2x+1)\\sqrt{4x+3}}`',1560918582,NULL,'','','','',0,'',''),(414,4,362.000,'`\\int \\ \\frac{dx}{(2x+3)\\sqrt{x+5}}`',1560918615,NULL,'','','','',0,'',''),(415,4,363.000,'`\\int \\ \\frac{2^x}{\\sqrt{4^x-2^{x+2}+5}}\\ dx`',1560918695,NULL,'','','','',0,'',''),(416,4,364.000,'`\\int \\ \\frac{3x+5}{x^3-x^2-x+1}\\ dx`',1560918751,NULL,'','','','',0,'',''),(417,4,365.000,'`\\int \\ \\frac{3x-5}{x^2-2x+10}\\ dx`',1560918795,NULL,'','','','',0,'',''),(418,4,366.000,'`\\int \\ \\frac{3x+1}{2x^2+x+1}\\ dx`',1560918844,NULL,'','','','',0,'',''),(419,4,367.000,'`\\int \\ \\frac{x^2-3}{x^3-7x+6}\\ dx`',1560918888,NULL,'','','','',0,'',''),(420,4,368.000,'If `\\int \\ \\frac{x^3-3}{x^3-2x^2-x+2}\\ dx=\\frac{A}{1494}\\log_e(c|\\frac{(x-2)(x-1)^3}{(x+1)}|)` where `c` is constant of integration. Find the value of `A`.',1560919090,NULL,'','','','',0,'',''),(421,4,369.000,'`\\int \\ \\frac{x^3}{x^4+x^2-12}\\ dx`',1560919130,NULL,'','','','',0,'',''),(422,4,370.000,'`\\int \\ \\frac{x^3+3x+2}{(x^2+1)^2(x+1)}\\ dx`',1560919172,NULL,'','','','',0,'',''),(423,4,371.000,'`\\int \\ \\frac{5-2x}{5+4x-x^2}\\ dx`',1560919307,NULL,'','','','',0,'',''),(424,4,372.000,'`\\int \\ \\frac{3x^2-1}{x^2-3x+2}\\ dx`',1560919377,NULL,'','','','',0,'',''),(425,4,373.000,'`\\int \\ \\frac{x^2+x-1}{x^3+x^2-6x}\\ dx`',1560919426,NULL,'','','','',0,'',''),(426,4,374.000,'`\\int \\ \\frac{x^2}{(x-1)(x-2)(x-3)}\\ dx`',1560923492,NULL,'','','','',0,'',''),(427,4,1.000,'If `f(x)=\\frac{1-x}{1+x}` then find `\\int \\ f{f(\\frac{1}{x})}\\ dx` where `x\\ne0`',1560923628,'`log|x| + c`','','','','',0,'',''),(428,4,2.000,'If ${f}\'(x)=x$, `f(1)=-3` then find `f(x)`.',1560925142,'`f(x) = 1/2 (x^2 - 7)`','','','','',0,'',''),(429,4,15.000,'If $f\'(x)=x+\\frac{1}{x^2}$, `f(3)=4`, then find `f(x)`.',1560925730,'`f(x) = \\frac{x^2}{2} - 1/x - 1/6`','','','','',0,'',''),(430,4,14.000,'If $f\'\'(x)=\\cos^2x+3$, $f\'(0)=0=f(0)$, then find `f(x)`.',1560926026,'`f(x) = \\frac{7x^2}{4} - \\frac{cos2x}{8} + 1/8`','','','','',0,'',''),(431,4,13.000,'If slope of any curve at the point `(x,y)` be `e^x+\\frac{1}{1+x^2}` and the curve passing through the point `(0,1)`, then find the equation of the curve.',1560926179,'`y = e^x + tan^{-1}x`','','','','',0,'',''),(432,4,12.000,'If slope of any curve at the point `(x,y)` be `-y` and the curve passing through the point `(2,1)`, then find the equation of the curve.',1560926249,'`log|y| = 2-x`','','','','',0,'',''),(433,4,11.000,'If slope of any curve at the point `(x,y)` be `-\\frac{x}{y}` and the curve passing through the point `(1,1)`, then find the equation of the curve.',1560926304,'`x^2 + y^2 = 2`','','','','',0,'',''),(434,4,10.000,'If slope of any curve at the point `(x,y)` be `x^2-2` and the curve passing through the point `(3,8)`, then find the equation of the curve.',1560926351,'`y = \\frac{x^3}{3} -2x + 5`','','','','',0,'',''),(435,4,9.000,'If slope of any curve at the point `(x,y)` be `2\\sin^2px+2^{3x}+\\frac{1}{2x}`, then find the equation of the curve.',1560926440,'`y = x - \\frac{sin2px}{2p} + \\frac{2^{3x}}{3log2} + 1/2 log|x| + c`','435','','','',0,'',''),(436,4,8.000,'If `\\frac{dy}{dx}=3\\sqrt{x}-\\frac{1}{\\sqrt{x}}`, then express `y` as a function of `x`, where it is given that `y=12` when `x=4`',1560926582,'`y = 2x^{3/2} - 2\\sqrt{x}`','','','','',0,'',''),(437,4,7.000,'If `\\frac{dy}{dx}=\\frac{x^3+1}{x^2(x+1)}`, then express `y` as a function of `x`.',1560926674,'`y = x - log|x| - 1/x + c`','','','','',0,'',''),(438,4,6.000,'Find the value of `\\int \\ e^{3\\logx}\\ dx`',1560926731,'`\\frac{x^4}{4}+c`','','','','',0,'',''),(439,4,5.000,'Is the value of `\\frac{d}{dx}[\\int \\ f(x)\\ dx]` and `\\int \\ \\frac{d}{dx}[f(x)]\\ dx` are equal? Justify.',1560926861,'No','','','','',0,'',''),(440,4,4.000,'If `\\int \\ f(x)\\ dx=\\phi(x)` then which one is true? \r\n<br />(a)`f(x)=\\phi(x)`, (b)$\\phi\'(x)=f(x)$, (c)$f\'(x)=\\phi(x)$, (d)`f(x)+\\phi(x)=`cosntant.',1560927186,'(b)','','','','',0,'',''),(441,4,3.000,'If `\\int \\ f(x)\\ dx=\\phi(x)`, then find `\\int \\ x^5f(x^3)\\ dx`',1560927253,'`1/3 x^3 \\phi(x^3) - \\int x^2 \\phi(x^3) dx + c`','','','','',0,'',''),(442,4,85.000,'`\\int \\ frac{\\cos^3x+\\cos^5x}{\\sin^2x+\\sin^4x}\\ dx`',1561010982,NULL,'','','','',0,'',''),(443,5,8.000,'If `A\\cupB=A\\capB`, then which one is true?\r\n<br /><br />\r\n(a) `A\\subseteqB`   (b) `B\\subseteqA`   (c) `A=B`   (d) None of these',1561206920,'(c) `A=B`','443','','','',0,'',''),(444,5,9.000,'The set `A\\cap(A\\cupB)^c` equals to - \r\n<br /><br />\r\n(a) `A`   (b) `A^c`   (c) `B^c`   (d) `\\phi`',1561207655,'(d) `\\phi`','444','','','',0,'',''),(445,5,11.000,'If `A-B=\\phi`, then which one is true?\r\n<br /><br />\r\n(a) `A\\neB`   (b) `A\\subsetB`   (c) `B\\subsetA`   (d) `A\\capB=\\phi`',1561207812,'(b) `A\\subsetB`','445','','','',0,'',''),(446,5,14.000,'If `A\\capB=B`, then which one is true?\r\n<br /><br />\r\n(a) `A\\subseteqB`   (b) `B\\subseteqA`   (c) `A=B`   (d) `A=\\phi`',1561207936,'(b) `B\\subseteqA`','446','','','',0,'',''),(447,5,17.000,'The set `(A\\cupB^c)^c` equals to -\r\n<br /><br />\r\n(a) `\\phi`   (b) `B`   (c) `A-B`   (d) `B-A`',1561208279,'(d) `B-A`','447','','','',0,'',''),(448,5,19.000,'If `A\\subsetB`, then `A^c\\capB^c` equals to - \r\n<br /><br />\r\n(a) `A^c`   (b) `B^c`   (c) `\\phi`   (d) None of these',1561208669,'(b) `B^c`','448','','','',0,'',''),(449,5,20.000,'The set `(A\\capB)\\cap(A^c\\cupB^c)` equals to - \r\n<br /><br />\r\n(a) `A^c`   (b) `B^c`   (c) `\\phi`   (d) None of these',1561208810,'(c) `\\phi` ','449','','','',0,'',''),(450,5,15.000,'If `A\\capB=A` then `A\\cupB` equals to - \r\n<br /><br />\r\n(a) `A`   (b) `B`   (c) `\\phi`   (d) None of these',1561208961,'(b) `B`','450','','','',0,'',''),(451,5,21.000,'For two non-empty `A` and `B`, if `A-B=A`, then which of the following is always true ?-<br />\r\n(a) `A \\subset B`, (b) `B \\subset A`, (c) `A \\cap B = \\phi`, (d) `B-A=\\phi`',1561209098,'(c) `A \\cap B = \\phi`','451','','','',0,'দুটি সেট `A` ও `B` এর জন্য `A-B=A` হলে, প্রদত্ত কোনটি সর্বদা সত্য ?-<br />\r\n(a) `A subset B`, (b) `B subset A`, (c) `A cap B = phi`, (d) `B-A=phi`',''),(452,5,12.000,'If `A\\capB^c=\\phi`, then which is true?\r\n<br /><br />\r\n(a) `A\\subsetB`   (b) `B\\subsetA`   (c) None of these',1561209242,'(a) `A\\subsetB`','452','','','',0,'',''),(453,5,24.000,'If `A={{1},{2},{1,2}}` then which of the following is true?\r\n<br /><br />\r\n(a) `1\\inA`   (b) `{1}\\inA`   (c) `{1,2}\\subsetA`   (d) `\\phi\\inA`',1561209558,'(b) `{1}\\inA`','453','','','',0,'',''),(454,5,29.000,'For any set, which is true?\r\n<br /><br />\r\n(a) `{1}\\in{1,2,3}`   (b) `1\\notin{1,2,3}`   (c) `1\\subset{1,2,3}`   (d) `{1}\\subset{1,2,3}`',1561209809,'(d) `{1}\\subset{1,2,3}`','454','','','',0,'',''),(455,5,34.000,'If `a\\mathbb{N}={ax | x\\in\\mathbb{N}}`, then find `3\\mathbb{N} \\cap 7\\mathbb{N}`',1561210146,NULL,'','','','',0,'',''),(456,5,35.000,'If `B\\subseteqA`, then prove that `B-A=\\phi`',1561210619,'N.A.','456','','','',0,'',''),(457,5,36.000,'Find the H.C.F. of `15,40,105` and L.C.M of `12,15,20` by Set Theory.',1561210772,NULL,'','','','',0,'',''),(458,5,38.000,'If `U` be the universal set and `A,B` are two subsets of `U` where `U={1,3,5,7,9,12,15}`, `A={1,5,9,15}`, `B={3,7,9,12,15}`, then find the following sets.\r\n<br />\r\n(a) `A^c`,   (b) `A\\cupB`,   (c) `A\\capB`,   (d) `A-B`,   (e) `B-A`,   (f) `A^c\\cupB^c`,   (g) `A^c\\capB^c`,   (h) `(A\\capB)\\cup(A-B)`',1561211304,'(a) `A^c = {3, 7,12}`<br />\r\n(b) `A \\cup B = {1, 3, 5, 7, 9, 12, 15}`<br />\r\n(c) `A \\cap B = {9, 15}`<br />\r\n(d) `A - B = {1, 5}`<br />\r\n(e) `B - A = {3, 7, 12}`<br />\r\n(f) `A^c \\cup B^c = {1, 3, 5, 7, 12}`<br />\r\n(g) `A^c \\cap B^c = \\phi`<br />\r\n(h) `(A \\cap B) \\cup (A-B) = {1, 5, 9, 15}`','','','','',0,'',''),(459,5,39.000,'If `A={1,2,3,4}`, `B={3,4,5}`, `C={1,4,5}` then verify the following relations.\r\n<br />\r\n(a) `A-(B\\cupC)=(A-B)\\cap(A-C)`    (b) `A\\cup(B-C)=(A\\cupB)-(A\\cupC)`',1561211719,'(a) Valid, (b) Invalid','','','','',0,'',''),(460,5,40.000,'If `A={1,3}`, `B={3,5}`, `C={5,10}` then verify the following relations.\r\n<br />\r\n(a) `A\\xx(B\\cupC)=(A\\xxB)\\cup(A\\xxC)`,   (b) `A\\xx(B\\capC)=(A\\xxB)\\cap(A\\xxC)`',1561212119,'(a) Valid, (b) Valid','','','','',0,'',''),(461,5,41.000,'For any two set `A` & `B`, prove that -<br /> \r\n<br />(a) `A\\cap(B-A)=\\phi`\r\n<br />(b) `A\\cup(B-A)=A\\cupB`\r\n<br />(c) If `A\\cupB=A\\capB` then prove that `A=B`\r\n<br />(d) If `A\\capB^c=\\phi` and `A^c\\capB=\\phi` then show that `A=B`\r\n<br />(e) `A-B=A-(A\\capB)`',1561212833,'N.A.','461.001,461.002,443,461.003,461.004','','','',0,'',''),(462,5,45.000,'For any three sets `A,B,C`, prove that - <br />\r\n<br />(a) `A\\cap(B-C)=(A\\capB)-(A\\capC)`\r\n<br />(b) `(A\\cupB)-C=(A-C)\\cup(B-C)`\r\n<br />(c) `(A\\capB)-C=(A-C)\\cap(B-C)`\r\n<br />(d) `A-(B\\cupC)=(A-B)\\cap(A-C)`\r\n<br />(e) `A-(B\\capC)=(A-B)\\cup(A-C)`\r\n<br />(f) `A-(B-C)=(A-B)\\cup(A\\capC)`',1561215516,'N.A.','462_a,462_b,462_c,462_d,462_e','','','',0,'',''),(463,5,47.000,'For any three sets `A,B,C`, if `A\\capC=B\\capC=\\phi` and `A\\cupC=B\\cupC`, then prove that `A=B`',1561215639,'N.A,','463','','','',0,'',''),(464,5,48.000,'For any three sets `A,B,C`, if `A\\capC=B\\capC=\\phi` and `A\\capC^c=B\\capC^c`, then prove that `A=B`',1561215698,'N.A.','464','','','',0,'',''),(465,5,49.000,'For any three sets `A,B,C`, if `A\\capB=A\\capC` and `A\\cupB=A\\cupC`, then prove that `B=C`',1561215801,'N.A.','465','','','',0,'',''),(466,5,42.000,'For any two sets `A` & `B`, prove that - <br />\r\n<br />(a) `A\\subseteq A\\cupB` and `B\\subseteq A\\cupB`\r\n<br />(b) `A\\capB \\subseteq A` and `A\\capB \\subseteq B`',1561216795,'N.A.','466.001,466.002','','','',0,'',''),(467,5,43.000,'For any two sets `A` & `B`, if `A\\subseteqB`, then show that `A\\cupB=B` and `A\\capB=A`\r\n',1561217735,'N.A.','467','','','',0,'',''),(468,5,44.000,'For any three sets `A,B,C` -<br />\r\n<br />(a) if `A\\subseteqC` and `B\\subseteqC`, then show that  `A\\cupB\\subseteqC`\r\n<br />(b) if `C\\subseteqA` and `C\\subseteqB`, then show that `C\\subseteq A\\capB`',1561218340,'N.A.','468.001,468.002','','','',0,'',''),(469,3,61.000,'If `x=t\\log\\t` and `y=\\frac{log\\t}{t}` then find the value of `(frac{dy}{dx})_{t=1}`',1586262883,'','','','','',0,'',''),(470,3,69.000,'If `tany=\\frac{tanx+secx-1}{tanx-secx+1}` then show that `\\frac{dy}{dx}=1/2`',1586263206,'','','','','',0,'',''),(471,2,27.000,'If `y=e^x(sinx+cosx)` then prove that `\\frac{d^2y}{dx^2}-2\\frac{dy}{dx}+2y=0`',1586264495,'N.A.','471','','','',0,'',''),(472,3,55.000,'Find the derivative w.r.t. `x` of following,  `y=log_sinx secx + 10^{x^2}`',1586264776,NULL,'','','','',0,'',''),(473,2,55.000,'If `y=sin(2sin^{-1}x)` then show that `(1-x^2)\\frac{d^2y}{dx^2}=x\\frac{dy}{dx}-4y`',1586264946,NULL,'','','','',0,'',''),(474,3,24.000,'If `y=log_10x`, then find the value of `\\frac{dy}{dx}`',1586265153,NULL,'','','','',0,'',''),(475,3,73.000,'If `f(x) = (\\frac{1+x}{2+x})^{3+2x}` then find `f\\\'(0)`',1586265370,NULL,'','','','',0,'',''),(476,3,30.000,'If `f(x)=log(tan\\frac{x}{2})` then find `f\\\'(x)`',1586265583,'`csc x`','476','','','',0,'',''),(477,3,75.000,'If `x=y\\log(xy)` then prove that `\\frac{dy}{dx}=\\frac{y(x-y)}{x(x+y)}`',1586265691,NULL,'','','','',0,'',''),(478,3,54.000,'Find the derivative of $\\frac{5x}{\\sqrt[3]{1-x^2}}+sin^2(2x-3)$ w.r.t. `x`',1586267959,'`\\frac{5(3-x^2)}{(1-x^2)^{4/3}}+2sin(4x-6)`','478','','','',0,'',''),(479,3,74.000,'If `f(x)=(\\frac{a+x}{b+x})^{a+b+2x}` then prove that `f\\\'(0)=[2log(\\frac{a}{b})+\\frac{b^2-a^2}{ab}](\\frac{a}{b})^{a+b}`',1586268357,'N.A.','479','','','',0,'',''),(480,3,64.000,'If `x=a\\sec^2\\theta` and `y=a\\tan^3\\theta` then find the value of `\\frac{dy}{dx}` at `\\theta=\\frac{\\pi}{4}`',1586268677,NULL,'','','','',0,'',''),(481,2,17.000,'If `x=a\\cos\\2t`, `y=a\\sin\\2t` then find the value of `\\frac{d^2y}{dx^2}` in form of `t`',1586268834,NULL,'','','','',0,'',''),(482,2,28.000,'If `y=x^3\\log\\frac{1}{x}` then prove that `x\\frac{d^2y}{dx^2}-2\\frac{dy}{dx}+3x^2=0`',1586268991,NULL,'','','','',0,'',''),(483,3,117.000,'If `x=cos^{-1}(8t^4-8t^2+1)` and `y=sin^{-1}(3t-4t^3)` where `(0\\lt t \\lt \\frac{1}{2})` then find `\\frac{dy}{dx}`',1586270986,NULL,'','','','',0,'',''),(484,2,70.000,'If `x=\\frac{1}{z}` and `y=f(x)` then show that `\\frac{d^2f}{dx^2}=2z^3\\frac{dy}{dz}+z^4\\frac{d^2y}{dz^2}`',1586271280,'N.A.','484','','','',0,'',''),(485,2,66.000,'If `y = a\\cos(logx)+b\\sin(logx)`, where `a` & `b` are constants, then prove that `x^2\\frac{d^2y}{dx^2}+x\\frac{dy}{dx}+y=0`',1586271905,NULL,'','','','',0,'',''),(486,3,78.000,'If `y=\\sqrt{sinx+\\sqrt{sinx+\\sqrt{sinx+\\cdots\\ \\infty}}}` then show that `\\frac{dy}{dx}=\\frac{cosx}{2y-1}`',1586272581,NULL,'','','','',0,'',''),(487,2,23.000,'If `x=a(\\theta-\\sin\\theta)`, `y=a(1-\\cos\\theta)` where `a` is constant. then prove that `2y\\frac{d^2y}{dx^2}+\\csc^2\\frac{\\theta}{2}=0` [`csc` is abbreviation of cosec]',1586273089,NULL,'','','','',0,'',''),(488,2,18.000,'If `x=sint`, `y=sin\\kt` then show that `(1-x^2)\\frac{d^2y}{dx^2}-x\\frac{dy}{dx}+k^2y=0` where `k` is a constant',1586273361,'N.A.','488','','','',0,'',''),(489,3,81.000,'If `x\\sqrt{1+y}+y\\sqrt{1+x}=0` then prove that `\\frac{dy}{dx}=-\\frac{1}{(1+x)^2}`',1586273495,NULL,'','','','',0,'',''),(490,3,90.000,'If `cosy=xcos(a+y)`, then prove that `\\frac{dy}{dx}=\\frac{cos^2(a+y)}{sina}` where `a\\ne0` is a constant',1586273760,NULL,'','','','',0,'',''),(491,3,2.000,'Find the value of `\\frac{d}{dx}(x^x)`',1586273900,'`x^x(1+logx)`','491','','','',0,'',''),(492,3,91.000,'If `siny=xsin(a+y)`, then show that `\\frac{dy}{dx}=\\frac{sin^2(a+y)}{sina}=\\frac{sina}{1-2x\\cosa+x^2}` where `a\\nen\\pi`, `n` is an integer.',1586274132,'N.A.','492,492.001','','','',0,'',''),(493,2,61.000,'If `y=(tan^{-1}x)^2` then show that `(1+x^2)^2y_2+2x(1+x^2)y_1-2=0`',1586274321,'N.A.','493','','','',0,'',''),(494,2,57.000,'If `y=cos(2sin^{-1}x)` then prove that `(1-x^2)\\frac{d^2y}{dx^2}-x\\frac{dy}{dx}+4y=0`',1586274865,'N.A.','494','','','',0,'',''),(495,3,63.000,'If `x=a\\cos^3\\theta`, `y=a\\sin^3\\theta` then find `\\frac{dy}{dx}`',1586275164,NULL,'','','','',0,'',''),(496,3,59.000,'If `y=\\frac{x}{\\sqrt{1+x^2}}`, then prove that `x^3\\frac{dy}{dx}=y^3`',1586275337,'N.A.','496','','','',0,'',''),(497,2,62.000,'If `y=(cot^{-1}x)^2`, then prove that `(x^2+1)^2\\frac{d^2y}{dx^2}+2x(x^2+1)\\frac{dy}{dx}-2=0`',1586276491,NULL,'','','','',0,'',''),(498,3,67.000,'If `tany=\\frac{2t}{1-t^2}`, `sinx=\\frac{2t}{1+t^2}` then find the value of `\\frac{dy}{dx}`',1586276699,NULL,'','','','',0,'',''),(499,3,65.000,'If `x=a(\\theta+sin\\theta)`, `y=a(1+cos\\theta)` then show that `(\\frac{dy}{dx})_{\\theta=\\frac{\\pi}{2}}=-1`',1586276890,NULL,'','','','',0,'',''),(500,3,36.000,'If `log\\frac{x^2-y^2}{x^2+y^2}=a` (where `a` constant) then find the value of `\\frac{dy}{dx}`',1586277191,NULL,'','','','',0,'',''),(501,3,57.000,'If `siny+e^{-x\\cosy}=e` then the value of `\\frac{dy}{dx}` at `(1,\\pi)`',1586277512,NULL,'','','','',0,'',''),(502,3,79.000,'If `x=e^{y+e^{y+e^{y+\\cdots\\ \\infty}}}` where `x\\gt0` then find `\\frac{dy}{dx}`',1586278231,NULL,'','','','',0,'',''),(503,3,98.000,'If `(sinx)^y=x+y` then prove that `\\frac{dy}{dx}=\\frac{1-(x+y)y\\cotx}{(x+y)\\ln(sinx)-1}`',1586278824,NULL,'','','','',0,'',''),(504,2,56.000,'If `y=sin(m\\sin^{-1}x)`, then show that `(1-x^2)y_2-xy_1+m^2y=0`',1586279131,NULL,'','','','',0,'',''),(505,3,42.000,'If `x^2y^3=(x+y)^5` then find the value of `\\frac{dy}{dx}`',1586279239,'`y/x`','505','','','',0,'',''),(506,3,28.000,'If `y=log|x|` then find the value of `\\frac{dy}{dx}`',1586279822,'`1/x`','506','','','',0,'',''),(507,3,22.000,'If `x^y=e^{x-y}`, then show that `\\frac{dy}{dx}=\\frac{logx}{(logex)^2}`',1586280064,'N.A.','507','','','',0,'',''),(508,3,27.000,'If `y=10^{10^x}` then find the value of `\\frac{dy}{dx}`',1586280429,'`10^{10^x}10^x(log_e10)^2`','508','','','',0,'',''),(509,2,36.000,'If `ax^2+2hxy+by^2=1`, then show that `\\frac{d^2y}{dx^2}=\\frac{h^2-ab}{(hx+by)^3}`',1586280652,NULL,'','','','',0,'',''),(510,3,146.000,'Find the derivative of `tan^{-1}\\frac{\\sqrt{1+x^2}-1}{x}` with respect to `tan^{-1}\\frac{2x\\sqrt{1-x^2}}{1-2x^2}` at `x=0`',1586280944,NULL,'','','','',0,'',''),(511,2,58.000,'If `y=cos(m\\sin^{-1}x)` then show that `(1-x^2)y_2-xy_1+m^2y=0`',1586281072,'N.A.','511','','','',0,'',''),(512,2,7.000,'If `\\frac{dx}{dy}=u` and `\\frac{d^2x}{dy^2}=v`, then show that `\\frac{d^2y}{dx^2}=-\\frac{v}{u^3}`',1586281663,'N.A.','512','','','',0,'',''),(513,3,14.000,'If `f(x)=(x+1)^x+x^{x+1}`, then find the value of `f\'(1)`',1586323992,'`3 + 2 log 2`','513','','','',0,'',''),(514,2,32.000,'If `y^{\\frac{1}{3}}+y^{-\\frac{1}{3}}=2x`, then prove that `(x^2-1)\\frac{d^2y}{dx^2}+x\\frac{dy}{dx}=9y`',1586324282,'N.A.','514','','','',0,'',''),(515,3,95.000,'If `y=(sinx)^{cosx}+(cosx)^{sinx}` then find the value of `frac{dy}{dx}`',1586324431,NULL,'','','','',0,'',''),(516,3,46.000,'If `x^y=y^x` then find the value of `\\frac{dy}{dx}`',1586324590,'`\\frac{y(x\\logy-y)}{x(y\\logx-x)}`','516','','','',0,'',''),(517,3,101.000,'If `y=1+\\frac{a}{x-a}+\\frac{bx}{(x-a)(x-b)}+\\frac{cx^2}{(x-a)(x-b)(x-c)}` then show that `\\frac{dy}{dx}=\\frac{y}{x}[\\frac{a}{a-x}+\\frac{b}{b-x}+\\frac{c}{c-x}]` where `x\\nea,b,c`',1586325060,'N.A.','517,517.001','','','',0,'',''),(518,2,20.000,'If `x=e^tsint` and `y=e^tcost` then show that `(x+y)^2\\frac{d^2y}{dx^2}=2(x\\frac{dy}{dx}-y)`',1586325284,'N.A.','518','','','',0,'',''),(519,3,71.000,'If `y=\\frac{x\\sinx+cosx}{x\\cosx-sinx}`, then show that `\\frac{dy}{dx}=\\frac{x^2}{(x\\cosx-sinx)^2}`',1586325583,NULL,'','','','',0,'',''),(520,2,21.000,'If `x=a(\\theta+sin\\theta)`, `y=a(1-cos\\theta)` then the value of `\\frac{d^2y}{dx^2}` at `\\theta=\\frac{\\pi}{4}`',1586325890,'`\\frac{2(3-2\\sqrt{2})}{a}`','520','','','',0,'',''),(521,3,85.000,'If `\\sqrt{1-x^4}+\\sqrt{1-y^4}=k(x^2-y^2)` where `k` constant, then show that `\\frac{dy}{dx}=\\frac{x\\sqrt{1-y^4}}{y\\sqrt{1-x^4}}`',1586326307,NULL,'','','','',0,'',''),(522,3,70.000,'If `y=\\frac{1+sinx-cosx}{1+sinx+cosx}` then show that `\\frac{dy}{dx}=\\frac{1}{1+cosx}`',1586326656,NULL,'','','','',0,'',''),(523,2,12.000,'If `F(x)=f(x)\\cdot\\phi(x)` and `f\'(x)\\cdot\\phi\\\'(x)=a` where `a` constant, then prove that `\\frac{F\'\'}{F}=\\frac{f\'\'}{f}+\\frac{\\phi\\\'\'}{\\phi}+\\frac{2a}{f\\phi}`',1586327131,NULL,'','','','',0,'',''),(524,2,10.000,'If `y=\\frac{log\\x}{x}` then find the value of `(\\frac{d^2y}{dx^2})_{x=1}`',1586327176,'','','','','',0,'',''),(525,3,43.000,'If `x^m\\y^n=(x+y)^{m+n}`, then prove that `\\frac{dy}{dx}=\\frac{y}{x}`',1586327576,NULL,'','','','',0,'',''),(526,3,116.000,'If `y=cos^{-1}(8x^4-8x^2+1)` then show that `\\frac{dy}{dx}+\\frac{4}{\\sqrt{1-x^2}}=0`',1586327948,NULL,'','','','',0,'',''),(527,3,84.000,'If `y\\sqrt{1-x^2}+x\\sqrt{1-y^2}=1`, then show that `\\frac{dy}{dx}=-\\sqrt{\\frac{1-y^2}{1-x^2}}`',1586328455,NULL,'','','','',0,'',''),(528,2,37.000,'If `x\\sqrt{1+y}+y\\sqrt{1+x}=0`, then show that `(1+x)\\frac{d^2y}{dx^2}+2\\frac{dy}{dx}=0` and `\\frac{d^2y}{dx^2}=\\frac{2}{(1+x)^3}`',1586328999,'N.A.','528','','','',0,'',''),(529,3,137.000,'Find `\\frac{dy}{dx}` where `x=csc^{-1}(\\frac{1+t^2}{2t})`, `y=tan^{-1}(\\frac{3t-t^3}{1-3t^2})` where `csc` is abbreviation of cosec',1586329193,NULL,'','','','',0,'',''),(530,3,44.000,'If `x^2+y^2=t-\\frac{1}{t}` and `x^4+y^4=t^2+\\frac{1}{t^2}`, then show that `x^3y\\frac{dy}{dx}=1`',1586330763,NULL,'','','','',0,'',''),(531,2,30.000,'If `x+y=e^{x-y}` then show that `\\frac{d^2y}{dx^2}=\\frac{4(x+y)}{(x+y+1)^3}`',1586331841,'N.A.','531','','','',0,'',''),(532,3,119.000,'If `y=2\\tan^{-1}\\sqrt{\\frac{x-a}{b-x}}` where `a\\ltx\\ltb`, then show that `(\\frac{dy}{dx})^2+\\frac{1}{(x-a)(x-b)}=0`',1586332215,NULL,'','','','',0,'',''),(533,3,82.000,'If `\\sqrt{1-x^2}+\\sqrt{1-y^2}=a(x-y)`, then show that `\\frac{dy}{dx}=\\frac{\\sqrt{1-y^2}}{\\sqrt{1-x^2}}`',1586332627,'N.A.','533','','','',0,'',''),(534,2,29.000,'If `y=x\\log(\\frac{x}{a+bx})` then show that `x^3\\frac{d^2y}{dx^2}=(y-x\\frac{dy}{dx})^2`',1586332828,'N.A.','534','','','',0,'',''),(535,3,76.000,'If `y=\\sqrt{x+\\sqrt{x+\\sqrt{x+\\cdots\\ \\infty}}}`, then find the value of `\\frac{dy}{dx}`',1586333009,NULL,'','','','',0,'',''),(536,3,96.000,'If `y=(tanx)^{cotx}+(cotx)^{tanx}` then find the value of `\\frac{dy}{dx}`',1586334804,'`(tanx)^{cotx}[csc^2x(1-log\\tanx)]+(cotx)^{tanx}[sec^2x(log\\cotx-1)]`','536,536.001','','','',0,'',''),(537,3,47.000,'If `x^y\\cdot\\y^x=1` then find the value of `\\frac{dy}{dx}`',1586334984,'`\\-frac{y^2(1-logx)}{x^2(1-logy)}`','537','','','',0,'',''),(538,3,12.000,'If `y=e^{sin^{-1}x}` and `z=e^{-cos^{-1}x}` then find the value of `\\frac{dy}{dz}`',1586335160,'`e^{\\frac{\\pi}{2}}`','538','','','',0,'',''),(539,3,139.000,'Show that the derivative of `sec^{-1}(\\frac{1}{2x^2-1})` w.r.t. `\\sqrt{1-x^2}` at `x=\\frac{1}{2}` is 4',1586335432,NULL,'','','','',0,'',''),(540,2,40.000,'If `y=(x+\\sqrt{x^2-1})^m` then show that `(1-x^2)y_2-xy_1+m^2y=0`',1586335736,'N.A.','540','','','',0,'',''),(541,3,13.000,'If `x=\\sqrt{a^{sin^{-1}t}}`, `y=\\sqrt{a^{cos^{-1}t}}` then show that `\\frac{dy}{dx}=-\\frac{y}{x}`',1586335977,'N.A.','541','','','',0,'',''),(542,2,68.000,'If `y=tan(x+y)` then show that `\\frac{d^2y}{dx^2}=-\\frac{2(1+y^2)}{y^5}`',1586336371,'N.A.','542','','','',0,'',''),(543,3,124.000,'If `y=tan^{-1}\\frac{4x}{1+5x^2}+tan^{-1}\\frac{2+3x}{3-2x}` then show that `\\frac{dy}{dx}=\\frac{5}{1+25x^2}`',1586336577,NULL,'','','','',0,'',''),(544,2,6.000,'If `y=tan^{-1}(secx+tanx)` then find the value of `\\frac{d^2y}{dx^2}` at `x=\\frac{\\pi}{4}`',1586336781,'`0`','544','','','',0,'',''),(545,3,143.000,'Find the derivative of `tan^{-1}(\\frac{t}{\\sqrt{1-t^2}})` with respect to `sec^{-1}(\\frac{1}{2t^2-1})`',1586337053,NULL,'','','','',0,'',''),(546,2,1.000,'If `x^2+y^2=25` then find the value of `\\frac{d^2y}{dx^2}` at `x=0`',1586337388,'`\\pm1/5`','546','','','',0,'',''),(547,2,51.000,'If `y=e^{m\\sin^{-1}x}` where `(-1\\le\\x\\le1)`, then show that `(1-x^2)\\frac{d^2y}{dx^2}-x\\frac{dy}{dx}=m^2y`',1586338300,'','','','','',0,'',''),(548,3,134.000,'Find the derivative of `x^{sin^{-1}x}` w.r.t. `sin^{-1}x`',1586338577,NULL,'','','','',0,'',''),(549,3,131.000,'if `y=cot^{-1}(\\frac{\\sqrt{1+sinx}+\\sqrt{1-sinx}}{\\sqrt{1+sinx}-\\sqrt{1-sinx}})` then show that `\\frac{dy}{dx}=\\pm\\frac{1}{2}`',1586338866,NULL,'','','','',0,'',''),(550,3,93.000,'If `y=(tanx)^{sinx}` then find the value of `\\frac{dy}{dx}` at `x=\\frac{\\pi}{4}`',1586339147,NULL,'','','','',0,'',''),(551,3,132.000,'If `y=\\frac{2}{\\sqrt{a^2-b^2}}tan^{-1}[\\sqrt{\\frac{a-b}{a+b}}\\tan\\frac{x}{2}]` then show that `\\frac{dy}{dx}=\\frac{1}{a+b\\cosx}`',1586340412,'N.A.','551','','','',0,'',''),(552,2,31.000,'If `x^2+y^2=a^2` then show that `\\frac{(1+y_1^2)^\\frac{3}{2}}{y_2}=-a` where `a` is constant and `y_1`, `y_2` are first & second order derivative',1586341002,'N.A.','552','','','',0,'',''),(553,2,19.000,'If `x=tant` and `y=tanpt`, then show that `(1+x^2)\\frac{d^2y}{dx^2}+2(x-py)\\frac{dy}{dx}=0`',1586341315,NULL,'','','','',0,'',''),(554,3,140.000,'Find the derivative of `sin^{-1}\\frac{2x}{1+x^2}` w.r.t. `tan^{-1}\\frac{2x}{1-x^2}` where `x\\gt1`',1586344445,'','','','','',0,'',''),(555,3,40.000,'If `y=(x+\\sqrt{x^2+a^2})^n`, then show that `\\frac{dy}{dx}=\\frac{ny}{\\sqrt{x^2+a^2}}`',1586344693,NULL,'','','','',0,'',''),(556,3,94.000,'If `y=x^{cosx}+(cosx)^{sinx}` then find the value of `\\frac{dy}{dx}`',1586345182,NULL,'','','','',0,'',''),(557,3,126.000,'If `y=tan^{-1}[\\frac{\\sqrt{1+t^2}+\\sqrt{1-t^2}}{\\sqrt{1+t^2}-\\sqrt{1-t^2}}]` then find the value of `\\frac{dy}{dt}`',1586345476,'','','','','',0,'',''),(558,3,3.000,'Find the value of `frac{d}{dx}5^{f(x)}`',1586345661,NULL,'','','','',0,'',''),(559,2,24.000,'If `x=a(cos\\theta+\\theta\\sin\\theta)`, `y=a(sin\\theta-\\theta\\cos\\theta)` then find the value of `\\frac{d^2y}{dx^2}` at `\\theta=\\frac{\\pi}{4}`',1586345930,NULL,'','','','',0,'',''),(560,3,114.000,'If `sin^{-1}(\\frac{x^2-y^2}{x^2+y^2})=k` (where `k` is constant), then show that `\\frac{dy}{dx}=\\frac{y}{x}`',1586346274,'`y/x`','560','','','',0,'',''),(561,3,51.000,'If `y\\cdot\\e^y=x` then show that `\\frac{dy}{dx}=\\frac{y}{x(1+y)}`',1586347332,'N.A.','561','','','',0,'',''),(562,2,22.000,'If `x=2\\cos\\theta-cos2\\theta` and `y=2\\sin\\theta-sin2\\theta`, then find the value of `\\frac{d^2y}{dx^2}` at `\\theta=\\frac{\\pi}{2}`',1586347736,NULL,'','','','',0,'',''),(563,2,8.000,'If `pv^a=c` where `a`, `c` are constants, then show that `v^2\\frac{d^2p}{dv^2}=a(a+1)p`',1586348016,NULL,'','','','',0,'',''),(564,3,50.000,'If `y=5^x\\cdot\\x^5`, then show that `x\\frac{dy}{dx}=(5+x\\log_e5)y`',1586348560,NULL,'','','','',0,'',''),(565,2,49.000,'If `y=x\\sinx` then show that `x^2y_2-2xy_1+(2+x^2)y=0`',1586349173,NULL,'','','','',0,'',''),(566,3,97.000,'If `y={(tanx)^{tanx}}^{tanx}`, then find the value of `\\frac{dy}{dx}` at `x=\\frac{\\pi}{4}`',1586349403,'`2`','566,566.001','','','',0,'',''),(567,3,23.000,'If `y^x=e^{y-x}` then show that `\\frac{dy}{dx}=\\frac{(log\\ey)^2}{logy}`',1586349831,'N.A.','567.003,567.002','','','',0,'',''),(568,3,141.000,'Find the derivative of `sin^{-1}(2ax\\sqrt{1-a^2\\x^2})` w.r.t. `\\sqrt{1-a^2\\x^2}` where `-frac{1}{\\sqrt{2}}\\lt\\ax\\lt\\frac{1}{\\sqrt{2}}`',1586350180,NULL,'','','','',0,'',''),(569,3,31.000,'If `f(x)=log(cot\\frac{x}{2})` then find `f\\\'(x)`',1586350365,'`-\\text{cosec}\\ x`','569','','','',0,'',''),(570,2,43.000,'If `y=A(x+\\sqrt{x^2-1})^n+B(x-\\sqrt{x^2-1})^n` then prove that `(1-x^2)\\frac{d^2y}{dx^2}-x\\frac{dy}{dx}+n^2y=0`',1586350804,'N.A.','570','','','',0,'',''),(571,3,115.000,'If `y=sin^{-1}\\frac{a+b\\cosx}{b+a\\cosx}`, then prove that `\\frac{dy}{dx}=-\\frac{\\sqrt{b^2-a^2}}{b+a\\cosx}`',1586351258,NULL,'','','','',0,'',''),(572,3,144.000,'Find the derivative of `tan^{-1]\\frac{\\sqrt{1+x^2}-1}{x}` with respect to `tan^{-1}x`',1586351567,NULL,'','','','',0,'',''),(573,3,147.000,'Find the derivative of  with `tan^{-1}\\frac{2x\\sqrt{1-x^2}}{1-2x^2}` with respect to `tan^{-1}\\frac{\\sqrt{1+x^2}-1}{x}` where `x\\ne0`',1586351928,NULL,'','','','',0,'',''),(574,3,92.000,'If `f(x)=(\\frac{a+x}{b+x})^x+(sinx)^{tanx}` then find the value of `f\'(0)`',1586352450,NULL,'','','','',0,'',''),(575,2,67.000,'If `y=x^{n-1}logx` then show that `x^2\\frac{d^2y}{dx^2}+(3-2n)x\\frac{dy}{dx}+(n-1)^2y=0`',1586353723,'N.A.','575','','','',0,'',''),(576,3,89.000,'If `xy=cot(x+y)` then show that `\\frac{dy}{dx}=\\frac{1+y+x^2y^2}{1+x+x^2y^2}`',1586354164,NULL,'','','','',0,'',''),(577,2,60.000,'If `y=(cos^{-1}x)^2` then prove that `(1-x^2)y_2-xy_1-2=0`',1586354473,'N.A.','577,577.001','','','',0,'',''),(578,3,133.000,'If `cos^{-1}y=2tan^{-1}\\sqrt{\\frac{1+x}{1-x}}`, then show that `\\frac{dy}{dx}=-1`',1586357151,NULL,'','','','',0,'',''),(579,3,127.000,'If `y=tan^{-1}[\\frac{\\sqrt{1+x^2}-\\sqrt{1-x^2}}{\\sqrt{1+x^2}+\\sqrt{1-x^2}}]` then show that `\\frac{dy}{dx}=\\frac{x}{\\sqrt{1-x^4}}`',1586357632,'','','','','',0,'',''),(580,2,63.000,'If `y=(sin^{-1]x)^2+(cos^{-1}x)^2`, then show that `(1-x^2)\\frac{d^2y}{dx^2}-x\\frac{dy}{dx}=4`',1586358388,NULL,'','','','',0,'',''),(581,3,16.000,'If `f(x)=(sinx)^x+x^x` then find `f\'(x)`',1586359079,'`x^x(1+logx)+(sinx)^x(x\\cotx+log\\sinx)`','581','','','',0,'',''),(582,3,56.000,'If `y=log_{sinx}secx+10^{x^2}`, then find `\\frac{dy}{dx}`',1586359602,'`\\frac{tanx\\log(sinx)-cotx\\log(secx)}{{log(sinx)}^2}+2x10^{x^2}log_e10`','582.01','','','',0,'',''),(583,2,15.000,'If `x=t^2+2t`, `y=t^3-3t` then show that `\\frac{d^2y}{dx^2}=\\frac{3}{4(t+1)}`',1586359867,'','','','','',0,'',''),(584,3,72.000,'If `f(x)=\\frac{cos^2x}{1+sin^2x}`, then prove that `f(\\frac{\\pi}{4})-3f\'(\\frac{\\pi}{4})=3`',1586361062,NULL,'','','','',0,'',''),(585,3,60.000,'If `y=\\sqrt{1+\\sqrt{1+x^4}}`, then prove that `y(y^2-1)\\frac{dy}{dx}=x^3`',1586361378,NULL,'','','','',0,'',''),(586,3,145.000,'Differentiate `tan^{-1}\\frac{\\sqrt{1+x^2}-1}{x}` w.r.t. `cos^{-1}(2x\\sqrt{1-x^2})` at `x=0`',1586361650,NULL,'','','','',0,'',''),(587,2,59.000,'If `y=(sin^{-1}x)^2`, then find the value of `(1-x^2)\\frac{d^2y}{dx^2}-x\\frac{dy}{dx}+4`',1586361883,NULL,'','','','',0,'',''),(588,3,77.000,'If `y=x+\\frac{1}{x+\\frac{1}{x+\\frac{1}{x+ \\cdots \\ infty}}}`, prove that `\\frac{dy}{dx}=\\frac{y}{2y-x}`',1586365683,'N.A.','588','','','',0,'',''),(589,3,142.000,'Find `\\frac{dy}{dz}` where `y=tan^{-1}(\\frac{\\sqrt{1+x^2}-\\sqrt{1-x^2}}{\\sqrt{1+x^2}+\\sqrt{1-x^2}})` and `z=cos^{-1}x^2`',1586366098,NULL,'','','','',0,'',''),(590,3,45.000,'If `x^2+y^2=t+\\frac{1}{t}` and `x^4+y^4=t^2+\\frac{1}{t^2}`, then show that `\\frac{dy}{dx}=-\\frac{y}{x}=-frac{1}{x^3y}`',1586366858,NULL,'','','','',0,'',''),(591,2,50.000,'If `logy=sin^{-1}x` then show that `(1-x^2)y_2-xy_1-y=0`',1586367062,NULL,'','','','',0,'',''),(592,2,71.000,'If `y=f(x)` and `logx=z`, then show that `x^2\\frac{d^2y}{dx^2}=\\frac{d^2y}{dz^2}-\\frac{dy}{dz}`',1586367548,'N.A.','592','','','',0,'',''),(593,3,49.000,'If `x^y+y^x=2`, then show that `\\frac{dy}{dx}=-[\\frac{yx^{y-1}+y^x\\logy}{x^y\\logx+xy^{x-1}}]`',1586368019,NULL,'','','','',0,'',''),(594,2,41.000,'If `y=(x+\\sqrt{1+x^2})^m` then show that `(1+x^2)y_2+xy_1-m^2y=0`. Then find the value of `(y_2)_0`',1586368274,'`m^2`','594','','','',0,'',''),(595,2,33.000,'If `2x=y^{frac{1}{m}}+y^{-\\frac{1}{m}}`, then show that `(x^2-1)y_2+xy_1=m^2y`',1586412681,'N.A.','595','','','',0,'',''),(596,2,13.000,'If `x=f(t)` and `y=g(t)` then show that `\\frac{d^2y}{dx^2}=\\frac{x_1y_2-y_1x_2}{x_1^3}`',1586412867,NULL,'','','','',0,'',''),(597,3,138.000,'Find `\\frac{dy}{dx}`, where `x=sec^{-1}(\\frac{1+t^2}{1-t^2})` and `y=tan^{-1}(\\frac{3t-t^2}{1-3t^2})`',1586413188,NULL,'','','','',0,'',''),(598,2,16.000,'If `x=cost`, `y=logt` then show that `\\frac{d^2y}{dx^2}+(\\frac{dy}{dx})^2=0` at `t=\\frac{\\pi}{2}`',1586413484,NULL,'','','','',0,'',''),(599,3,125.000,'If `y=sin(cos^{-1}x)+\\frac{1}{2}sin^{-1}\\frac{2x}{1+x^2}`, then show that `\\frac{dy}{dx}+\\frac{1}{\\sqrt{1-x^2}}=\\frac{1}{1+x^2}`',1586413811,NULL,'','','','',0,'',''),(600,2,53.000,'If `y=sin^{-1}x` then find the value of `(1-x^2)y_2-xy_1`',1586414182,NULL,'','','','',0,'',''),(601,3,99.000,'If `x=sec\\theta-cos\\theta` and `y=sec^n\\theta-cos^n\\theta`, then show that `(x^2+4)(\\frac{dy}{dx})^2=n^2(y^2+4)`',1586416274,'N.A.','601','','','',0,'',''),(602,3,88.000,'If `xy=tan(x+y)` then find `\\frac{dy}{dx}`',1586416487,'`\\frac{y-1-x^2y^2}{1-x+x^2y^2}`','602','','','',0,'',''),(603,2,54.000,'If `y=\\frac{sin^{-1}x}{\\sqrt{1-x^2}}`, then show that `(1-x^2)\\frac{d^2y}{dx^2}-3x\\frac{dy}{dx}-y=0`',1586417118,NULL,'','','','',0,'',''),(604,3,121.000,'If `y=tan^{-1}\\frac{1}{1+x+x^2}+tan^{-1}\\frac{1}{3+3x+x^2}+cot^{-1](7+5x+x^2)` then show that `(\\frac{dy}{dx})_{x=0}=-\\frac{9}{16}`',1586417517,NULL,'','','','',0,'',''),(605,3,123.000,'If `y=tan^{-1}\\frac{3x}{1+4x^2}+tan^{-1}\\frac{2+5x}{5-2x}`, then show that `\\frac{dy}{dx}=\\frac{4}{1+16x^2}`',1586417646,NULL,'','','','',0,'',''),(606,3,118.000,'Find `\\frac{dy}{dx}` where `y=tan^{-1}(\\sqrt{1+x^2}+x)`',1586418477,'','','','','',0,'',''),(607,3,39.000,'If `y=\\frac{1}{1+x^{q-p}+x^{r-p}}+\\frac{1}{1+x^{p-q}+x^{r-q}}+\\frac{1}{1+x^{p-r}+x^{q-r}}` then show that `\\frac{dy}{dx}=0`',1586418482,'N.A.','607','','','',0,'',''),(608,2,48.000,'If `y=sin(sinx)`, then show that `\\frac{d^2y}{dx^2}+tanx\\frac{dy}{dx}+y\\cos^2x=0`',1586419601,NULL,'','','','',0,'',''),(609,3,6.000,'If `y=\\sqrt{sin\\sqrt{x}}` then find `\\frac{dy}{dx}`',1586420016,'`frac{1}{4\\sqrt{x}}\\frac{cos\\sqrt{x}}{\\sqrt{sin\\sqrt{x}}}`','609','','','',0,'',''),(610,3,21.000,'If `y=x^{e^x}` then find `\\frac{dy}{dx}`',1586420226,'`x^{e^x}\\cdot\\e^x(1/x+logx)`','610','','','',0,'',''),(611,3,80.000,'If `|x|\\lt1` and `y=1+x+x^2+\\ \\cdots\\ \\infty` then prove that `y^2(\\frac{dy}{dx}-y)=x`',1586420688,NULL,'','','','',0,'',''),(612,3,102.000,'If `f(x)=(1+x)(1+x^2)(1+x^4)(1+x^8)` then find the value of `f\'(1)`',1586421108,NULL,'','','','',0,'',''),(613,2,35.000,'If `y=\\frac{1}{1+x+x^2+x^3}` then find the value of `[\\frac{dy}{dx}]_{x=0}` and `[\\frac{d^2y}{dx^2}]_{x=0}`',1586421693,NULL,'','','','',0,'',''),(614,2,46.000,'If `p^2=a^2\\cos^2\\theta+b^2\\sin^2\\theta` then show that `p+\\frac{d^2p}{d\\theta^2}=\\frac{a^2b^2}{p^3}`',1586422315,NULL,'','','','',0,'',''),(615,2,38.000,'If `\\sqrt{y+x}+\\sqrt{y-x}=c` (`c` is a non-zero constant), then show that `\\frac{d^2y}{dx^2}=\\frac{2}{c^2}` ',1586422671,NULL,'','','','',0,'',''),(616,2,34.000,'If `(a+bx)e^{\\frac{x}{y}}=x`, then show that `x^3\\frac{d^2y}{dx^2}=(x\\frac{dy}{dx}-y)^2`',1586423038,NULL,'','','','',0,'',''),(617,2,45.000,'If `sinx+cosy=1` then show that `\\frac{d^2y}{dx^2}=-\\frac{sin^2x+cosy}{sin^3y}`',1586425428,'N.A.','617','','','',0,'',''),(618,3,66.000,'If `y=2cos2t(1-cos2t)` and `x=3sin2t(1+cos2t)`, then find the value of `\\frac{dy}{dx}`',1586426330,NULL,'','','','',0,'',''),(619,2,52.000,'If `y=e^{mcos^{-1}x}`, then show that `(1-x^2)y_2-xy_1-m^2y=0`',1586426497,NULL,'','','','',0,'',''),(620,3,48.000,'If `x^y\\cdot\\y^x=e^{xy}-3x`, then find the value of `\\frac{dy}{dx}`',1586427020,NULL,'','','','',0,'',''),(621,3,58.000,'If `y=x^{n-1}logx`, then find the value of `x\\frac{dy}{dx}+(1-n)y`',1586427453,NULL,'','','','',0,'',''),(622,3,130.000,'if `y=tan^{-1}(\\frac{\\sqrt{1+sinx}-\\sqrt{1-sinx}}{\\sqrt{1+sinx}+\\sqrt{1-sinx}})` where `0\\lt\\x\\lt\\frac{\\pi}{2}`, then find the value of `\\frac{dy}{dx}`',1586427728,NULL,'','','','',0,'',''),(623,2,9.000,'If `V=\\frac{A}{r}+B` (where `A`, `B` are constants) then show that `\\frac{d^2V}{dr^2}+\\frac{2}{r}\\frac{dV}{dr}=0`',1586428190,NULL,'','','','',0,'',''),(624,3,32.000,'If `f(x)=log_e\\tan(\\frac{\\pi}{4}+\\frac{x}{2})`, then show that `f\'(x)=secx`',1586431415,'N.A.','624','','','',0,'',''),(625,3,25.000,'Find the value of `\\frac{d}{dx}{log_7(log_7\\x)}`',1586431548,'`frac{log_7e}{x\\ logx}`','625.01','','','',0,'',''),(626,3,135.000,'Find the derivative of `log_10x` with respect to `tan^{-1}x` ',1586431769,NULL,'','','','',0,'',''),(627,3,26.000,'If `f(x)=log_ex`, then find the derivative of `f(log_ex)` with respect to `x`',1586432138,'`\\frac{1}{x\\log_ex}`','627','','','',0,'',''),(628,3,33.000,'If `f(x)=log_e(\\frac{1-x}{1+x})`, then find the value of `f\'(0)`',1586432770,NULL,'','','','',0,'',''),(629,3,136.000,'Find the derivative of `x^2` with respect to `logx`',1586433078,NULL,'','','','',0,'',''),(630,2,26.000,'If `x=a\\cot\\theta`, `y=\\frac{1}{x^2+a^2}`, then show that `\\frac{d^2y}{dx^2}=\\frac{2}{a^4}sin^3\\theta\\sin3\\theta`',1586433936,'N.A.','630','','','',0,'',''),(631,3,34.000,'If `y=log_5\\sqrt{\\frac{1-cos2x}{1+cos2x}}`, then find the value of `\\frac{dy}{dx}`',1586434290,'`\\frac{2log_e5}{sin2x}`','631','','','',0,'',''),(632,3,120.000,'If `y=tan^{-1}\\frac{5-x}{1+5x}`, then find the value of `\\frac{dy}{dx}`',1586436918,NULL,'','','','',0,'',''),(633,3,18.000,'If `f(x)=|1-2x|` then find the value of `f\'(2)`',1586437979,'2','633','','','',0,'',''),(634,3,1.000,'If `y=sinx°` then find `\\frac{dy}{dx}`',1586439606,'','','','','',0,'',''),(635,3,4.000,'If `f(x)` is an even function and `f\'(0)` exists, then find the value of `f\'(0)`',1586439898,'`0`','635','','','',0,'',''),(636,3,35.000,'If `y=log\\sqrt{tanx}` then find the value of `\\frac{dy}{dx}` at `x=\\frac{\\pi}{4}`',1586440113,'`1`','636','','','',0,'',''),(637,3,7.000,'Find the value of `\\frac{d}{dx}{cos(logx)}`',1586440274,'`-\\frac{\\sin(logx)}{x}`','637','','','',0,'',''),(638,3,20.000,'If `f(x)=|x|^3`, then find `f\'(0)`',1586440429,'`0`','638','','','',0,'',''),(639,2,14.000,'If `x\\sin\\theta+y\\cos\\theta=a` and `x\\cos\\theta-y\\sin\\theta=b`, then show that value of `\\frac{dx}{d\\theta}\\cdot\\frac{d^2y}{d\\theta^2}-\\frac{d^2x}{d\\theta^2}\\cdot\\frac{dy}{d\\theta}` is constant.',1586440747,'N.A.','639','','','',0,'',''),(640,3,9.000,'If `x=\\sqrt{t}+\\frac{1}{\\sqrt{t}}`, then find the value of `2t\\frac{dx}{dt}+x`',1586440957,'`2\\sqrt{t}`','640','','','',0,'',''),(641,3,103.000,'If `y=(1+x)(1+x^2)(1+x^4)\\cdots(1+x^{2n})`, then find the value of `\\frac{dy}{dx}` at `x=0`',1586441118,NULL,'','','','',0,'',''),(642,2,2.000,'If `x=acosnt-bsinnt`, then find the value of `\\frac{d^2x}{dt^2}`',1586441670,NULL,'','','','',0,'',''),(643,3,111.000,'If `y=tan^{-1]\\frac{cosx+sinx}{cosx-sinx}` then find the value of `\\frac{dy}{dx}`',1586443741,NULL,'','','','',0,'',''),(644,3,87.000,'If `xy=sin(x+y)` then show that `\\frac{dy}{dx}=\\frac{\\sqrt{1-x^2y^2}-y}{x-\\sqrt{1-x^2y^2}}`',1586444036,NULL,'','','','',0,'',''),(645,3,62.000,'If `x=t\\cos\\t` and `y=\\frac{log\\t}{t}` then find the value of `(frac{dy}{dx})_{t=1}`',1586444378,NULL,'','','','',0,'',''),(646,3,19.000,'If `f(x)=x\\|x|`, then show that `f\'(x)=2|x|`',1586444543,'N.A.','646','','','',0,'',''),(647,3,112.000,'If `y=tan^{-1}\\frac{a\\cosx-b\\sinx}{b\\cosx+a\\sinx}` where `-\\frac{\\pi}{2}\\lt\\x\\lt\\frac{\\pi}{2}` and `\\frac{a}{b}tanx\\gt-1`, then show that `\\frac{dy}{dx}=-1`',1586445124,NULL,'','','','',0,'',''),(648,3,15.000,'If `y=(logx)^x+x^{logx}`, then find `\\frac{dy}{dx}`',1586445439,NULL,'','','','',0,'',''),(649,3,38.000,'If `y=\\frac{1+x^2+x^4}{1+x+x^2}` and `\\frac{dy}{dx}=ax+b`, then find the value of `a`, `b`',1586445656,'`a=2, b=-1`','649','','','',0,'',''),(650,2,65.000,'If `y=tan^{-1}(\\frac{1-2logx}{1+2logx})+tan^{-1}(\\frac{3+2logx}{1-6logx})`, then show that `\\frac{d^2y}{dx^2}=0`',1586494202,'N.A.','650','','','',0,'',''),(651,3,11.000,'If `x^{\\frac{3}{2}}+y^{\\frac{3}{2}}=a^{\\frac{3}{2}}`, then show that `\\frac{dy}{dx}+\\sqrt{\\frac{x}{y}}=0`',1586494749,'N.A.','651','','','',0,'',''),(652,2,39.000,'If `y=\\sqrt{x+1}-\\sqrt{x-1}`, then show that `(x^2-1)\\frac{d^2y}{dx^2}+x\\frac{dy}{dx}=\\frac{1}{4}y`',1586495340,NULL,'','','','',0,'',''),(653,3,10.000,'If `\\sqrt{x}+\\sqrt{y}=4` then find the value of `\\frac{dy}{dx}` at `x=1`',1586496052,'`-3`','653','','','',0,'',''),(654,3,8.000,'If `u=e^{ax}\\sinbx` and `v=e^{ax}\\cosbx`, then show that `(\\frac{du}{dx})^2+(\\frac{dv}{dx})^2=(a^2+b^2)e^{2ax}`',1586496053,NULL,'','','','',0,'',''),(655,2,25.000,'If `x=a\\sin2\\theta(1+cos2\\theta)` and `y=a\\cos2\\theta(1-cos2\\theta)`, then prove that `1+(\\frac{dy}{dx})^2=sec^2\\theta`. Then show that `4acos3\\theta\\frac{d^2y}{dx^2}=[1+(\\frac{dy}{dx})^2]^{\\frac{3}{2}}`',1586496567,'N.A.','655','','','',0,'',''),(656,3,113.000,'If `y=cot^{-1}\\sqrt{\\frac{1-sinx}{1+sinx}}` where `(0\\lt\\x\\lt\\frac{\\pi}{2})`, then find `\\frac{dy}{dx}`',1586497202,NULL,'','','','',0,'',''),(657,2,44.000,'If `y=cos[m\\log_e(x+\\sqrt{x^2+1})]`, then show that `(x^2+1)\\frac{d^2y}{dx^2}+x\\frac{dy}{dx}+m^2y=0`',1586497587,'N.A.','657.01','','','',0,'',''),(658,2,3.000,'If `y=\\frac{1}{2}x\\frac{dy}{dx}` then show that `\\frac{d^2y}{dx^2}` is constant',1586497736,'N.A.','658','','','',0,'',''),(659,3,148.000,'Show that the derivative of `cot^{-1}\\frac{2\\sqrt{1+x^2}-5\\sqrt{1-x^2}}{5\\sqrt{1+x^2}+2\\sqrt{1-x^2}}` with respect to `cos^{-1]\\sqrt{1-x^4}` is `-\\frac{1}{2}`',1586498730,'','','','','',0,'',''),(660,2,64.000,'If `y=a\\sin^{-1}x+b\\cos^{-1}x`, then prove that `(1-x^2)y_2-xy_1=0`',1586499338,NULL,'','','','',0,'',''),(661,3,86.000,'If `\\sqrt{1-x^6}+\\sqrt{1-y^6}=a^3(x^3-y^3)` where `a` is constant, then show that `\\frac{dy}{dx}=\\frac{x^2}{y^2}\\sqrt{\\frac{1-y^6}{1-x^6}}`',1586499718,NULL,'','','','',0,'',''),(662,3,100.000,'If `x=csc\\theta-sin\\theta`, `y=csc^n\\theta-sin^n\\theta`, then show that `(x^2+4)(\\frac{dy}{dx})^2=n^2(y^2+4)`',1586500147,NULL,'','','','',0,'',''),(663,3,17.000,'If `y=\\frac{3x-2}{2x+1}` where `(x\\ne\\frac{1}{2})`, then justify the statement `\\frac{dy}{dx}=1\\div(\\frac{dx}{dy})`',1586500437,'N.A.','663','','','',0,'',''),(664,3,41.000,'If `10^x+10^y=10^{x+y}`, then find the value of `10^x\\frac{dy}{dx}+10^y`',1586500822,'`0`','664','','','',0,'',''),(665,3,108.000,'If `y=2sin^{-1}\\frac{x-2}{\\sqrt{6}}-\\sqrt{2+4x-x^2}`, then show that the value of `\\frac{dy}{dx}` at `x=2` is `\\frac{2}{\\sqrt{6}}`',1586501961,NULL,'','','','',0,'',''),(666,2,72.000,'If `y=e^u` and `u=f(x)`, then show that `\\frac{d^2y}{dx^2}=e^u[\\frac{d^2u}{dx^2}+(\\frac{du}{dx})^2]`',1586502525,'N.A.','666','','','',0,'',''),(667,3,107.000,'Find `\\frac{dy}{dx}` where `y=cos^{-1}(\\frac{2\\cosx-3\\sinx}{\\sqrt{13}})`',1586502826,NULL,'','','','',0,'',''),(668,2,4.000,'If `2y=x(1+\\frac{dy}{dx})`, then show that `\\frac{d^2y}{dx^2}` is constant.',1586503079,'N.A.','668','','','',0,'',''),(669,3,110.000,'If `y=tan^{-1](\\frac{cosx}{1+sinx})`, then find `\\frac{dy}{dx}`',1586503396,NULL,'','','','',0,'',''),(670,3,52.000,'If `y=e^{\\frac{y}{x}}`, then show that `\\frac{dy}{dx}=\\frac{y^2}{x(y-x)}`',1586503881,'N.A.','670','','','',0,'',''),(671,3,29.000,'If `f(x)=log\\sin(\\sqrt{x^2+1})`, then find `f\'(x)`.',1586504689,'`cot(\\sqrt{x^2+1}) \\cdot \\frac{x}{\\sqrt{x^2+1}}`','','','','',0,'',''),(672,2,5.000,'If `y=sin3x\\ \\cos4x` then find `\\frac{d^2y}{dx^2}`',1586504897,NULL,'','','','',0,'',''),(673,3,128.000,'If `y=cot^{-1}[\\frac{\\sqrt{1+x^2}-\\sqrt{1-x^2}}{\\sqrt{1+x^2}+\\sqrt{1-x^2}}]` then find the value of `\\frac{dy}{dx}`',1586505408,NULL,'','','','',0,'',''),(674,3,122.000,'If `y=tan^{-1]\\frac{1}{1+x+x^2}+tan^{-1]\\frac{1}{x^2+3x+3}+tan^{-1]\\frac{1}{x^2+5x+7}+\\ \\cdots` upto `n` terms, then find the value of `\\frac{dy}{dx}`',1586505881,NULL,'','','','',0,'',''),(675,3,104.000,'If `log(x^2+y^2)=2tan^{-1}(\\frac{y}{x})`, then show that `\\frac{dy}{dx}=\\frac{x+y}{x-y}`',1586506292,NULL,'','','','',0,'',''),(676,3,53.000,'If `f(x)=sin(logx)` and `y=f(\\frac{2x+3}{3-2x})`, then find `\\frac{dy}{dx}`.',1586506669,NULL,'','','','',0,'',''),(677,2,11.000,'If `y=e^{\\frac{1}{x}}`, then find `\\frac{d^2y}{dx^2}`',1586510296,NULL,'','','','',0,'',''),(678,3,105.000,'If `y=sin^{-1}(\\frac{5x+12\\sqrt{1-x^2}}{13})`, then find `\\frac{dy}{dx}`',1586510598,NULL,'','','','',0,'',''),(679,2,42.000,'If `y=[log(x+\\sqrt{x^2+1})]^2`, then show that `(1+x^2)\\frac{d^2y}{dx^2}+x\\frac{dy}{dx}=2`',1586512539,NULL,'','','','',0,'',''),(680,3,83.000,'If `\\sqrt{1+x^2}+\\sqrt{1+y^2}=k(x-y)` where `k` constant, then show that `\\frac{dy}{dx}=\\sqrt{\\frac{1+y^2}{1+x^2}}`',1586513497,NULL,'','','','',0,'',''),(681,2,69.000,'If `sin(x+y)=kx` where `k` is a non-zero constant, then show that `\\frac{d^2x}{dy^2}+x(\\frac{dx}{dy}+1)^3=0`',1586513743,'N.A.','681','','','',0,'',''),(682,2,74.000,'If `s=\\frac{1}{2}log(\\frac{1+t}{1-t})`, then prove that `\\frac{d^2s}{dt^2}=-(\\frac{ds}{dt})^3\\frac{d^2t}{ds^2}`',1586514094,'N.A.','682','','','',0,'',''),(683,3,37.000,'If `y\\sqrt{x^2+1}=log(\\sqrt{x^2+1]-x)`, then show that `(x^2+1)\\frac{dy}{dx}+xy+1=0`',1586514512,'N.A.','683','','','',0,'',''),(684,3,129.000,'If `y=cot^{-1}[\\frac{\\sqrt{1+x}-\\sqrt{1-x}}{\\sqrt{1+x}+\\sqrt{1-x}}]`, then show that value of `\\frac{dy}{dx}` at `x=\\frac{1}{2}` is `-\\frac{1}{\\sqrt{3}}`.',1586514838,NULL,'','','','',0,'',''),(685,3,106.000,'Find `\\frac{dy}{dx}` where `y=cos^{-1}(\\frac{x-1/x}{x+1/x})`',1586519866,NULL,'','','','',0,'',''),(686,3,109.000,'Find `\\frac{dy}{dx}` where `y=cos^{-1]2x+2cos^{-1}\\sqrt{1-4x^2}`',1586520085,NULL,'','','','',0,'',''),(687,3,5.000,'If `y=\\sqrt{2x}-\\sqrt{2/x}+frac{x+4}{4-x}`, then find `(\\frac{dy}{dx})_{x=2}`',1586520483,'`2\\3/4`','687.01','','','',0,'',''),(688,2,73.000,'If `x=e^t` and `\\frac{d^2y}{dt^2}+p^2y=0`, then show that `x^2\\frac{d^2y}{dx^2}+x\\frac{dy}{dx}+p^2y=0`',1586521109,'N.A.','688','','','',0,'',''),(689,7,70.000,'If $A= \\begin{bmatrix} 4 & 5 \\\\ 2 & 1  \\end{bmatrix}$ then show that `6A^{-1}+5I=A`',1586776468,'N.A.','689','','','',0,'',''),(690,7,36.000,'If $ 2 \\begin{bmatrix}  1 & 3 \\\\ 0 & x  \\end{bmatrix} + \\begin{bmatrix}  y & 0 \\\\ 1 & 2  \\end{bmatrix} = \\begin{bmatrix}  5 & 6 \\\\ 1 & 8  \\end{bmatrix}$ then find the value of `x`, `y`',1586778269,NULL,'','','','',0,'',''),(691,8,96.000,'Prove that  $ \\begin{vmatrix}  a^2+\\lambda & ab & ac \\\\ ab & b^2+\\lambda & bc \\\\ ac & bc & c^2+\\lambda \\end{vmatrix} = \\lambda^2(a^2+b^2+c^2+\\lambda) $',1586778817,NULL,'','','','',0,'',''),(692,8,19.000,'Solve for `x`. $ \\begin{vmatrix}  x+2 & 3 & 3 \\\\ 3 & x+4 & 5 \\\\ 3 & 5 & x+4 \\end{vmatrix} = 0 $',1586778936,'0, 1, -11','692','','','',0,'',''),(693,7,75.000,'If $ A=\\begin{bmatrix}  2 & 1 \\\\ 3 & 4  \\end{bmatrix} $ and $ B=\\begin{bmatrix}  1 & -2 \\\\ -1 & 1  \\end{bmatrix} $ then find the value of `(AB)^{-1}`',1586779108,NULL,'','','','',0,'',''),(694,7,91.000,'Show that matrix $A=\\begin{bmatrix}  1 & 2 & 2 \\\\ 2 & 1 & 2 \\\\ 2 & 2 & 1 \\end{bmatrix}$ satisfies the equation `A^2-4A-5I_3=O`. Hence find `A^{-1}`',1586779240,'','','','','',0,'',''),(695,7,24.000,'`A` is a square matrix such that `A^2=A`. Then find the value of `(I+A)^3-7A`',1586779599,'`I`','695','','','',0,'',''),(696,8,63.000,'Without expanding prove that $ \\begin{vmatrix}  1 & a & a^2 \\\\ 1 & b & b^2 \\\\ 1 & c & c^2 \\end{vmatrix} = \\begin{vmatrix}  1 & a & bc \\\\ 1 & b & ca \\\\ 1 & c & ab \\end{vmatrix} = (a-b)(b-c)(c-a)$',1586779858,'','','','','',0,'',''),(697,7,58.000,'If $ A = \\begin{pmatrix}  1  \\\\ 2 \\\\ 3  \\end{pmatrix} $ then find `A\\A^T`',1586780087,'','','','','',0,'',''),(698,7,43.000,'If $A=\\begin{pmatrix}  2 & -1 \\\\ 3 & 2  \\end{pmatrix}$ and $B=\\begin{pmatrix}  0 & 4 \\\\ -1 & 7  \\end{pmatrix}$ then find the matrix `3A^2-2B+I`',1586780371,NULL,'','','','',0,'',''),(699,7,114.000,'Solve by using inverse matrix method. `x+2y+z=7`, `x+3z=11`, `2x-y=1`',1586780516,NULL,'','','','',0,'',''),(700,8,115.000,'Prove that $ \\begin{vmatrix}  -1 & b & c \\\\ a & -1 & c \\\\ a & b & -1 \\end{vmatrix} = (a+1)(b+1)(c+1)(\\frac{a}{a+1}+\\frac{b}{b+1}+\\frac{c}{c+1}-1) $',1586780804,NULL,'','','','',0,'',''),(701,8,66.000,'Without expanding prove that $ \\begin{vmatrix} a & a^2 & bc \\\\ b & b^2 & ca \\\\ c & c^2 & ab \\end{vmatrix} = \\begin{vmatrix}  1 & a^2 & a^3 \\\\ 1 & b^2 & b^3 \\\\ 1 & c^2 & c^3 \\end{vmatrix}  = (a-b)(b-c)(c-a)(ab+bc+ca)$',1586781072,'','','','','',0,'',''),(702,7,6.000,'Show that `A-A^T` is a skew-symmetric matrix where $A=\\begin{bmatrix} 2 & 3 \\\\ 4 & 5 \\end{bmatrix}$',1586781375,'N.A.','702','','','',0,'',''),(703,7,115.000,'Solve by using inverse matrix method. `2x+3y+5z=5`, `x-2y+z=-4`, `3x-y-2z=3`',1586781870,NULL,'','','','',0,'',''),(704,8,106.000,'Prove (without expanding). $ \\begin{vmatrix} (b+c)^2 & a^2 & a^2 \\\\ b^2 & (c+a)^2 & b^2 \\\\ c^2 & c^2 & (a+b)^2 \\end{vmatrix} = 2abc\\ (a+b+c)^3 $',1586782200,'','','','','',0,'',''),(705,8,94.000,'If `x\\ne\\y\\nez` and $ \\begin{vmatrix} x & x^2 & 1+x^3 \\\\ y & y^2 & 1+y^3 \\\\ z & z^2 & 1+z^3 \\end{vmatrix} =0 $, then show that `1+xyz=0`',1586782513,NULL,'','','','',0,'',''),(706,8,2.000,'Let `A` be a square matrix of order `n\\times\\n`, and `k` is a scalar,  then `|kA|` is equal to -- (a) `k|A|`    (b) `nk|A|`   (c) `n^k|A|`   (d) `k^n|A|`',1586782823,'','','','','',0,'',''),(707,8,40.000,'Without expanding prove that $ \\begin{vmatrix} 9 & 9 & 12 \\\\ 1 & -3 & -4 \\\\ 1 & 9 & 12 \\end{vmatrix} = 0 $',1586783133,NULL,'','','','',0,'',''),(708,8,95.000,'Prove that $ \\begin{vmatrix} \\alpha & \\beta & \\gamma \\\\ \\alpha^2 & \\beta^2 & \\gamma^2 \\\\ \\beta+\\gamma & \\gamma+\\alpha & \\alpha+\\beta \\end{vmatrix} = (\\alpha-\\beta)(\\beta-\\gamma)(\\gamma-\\alpha)(\\alpha+\\beta+\\gamma)$',1586783585,NULL,'','','','',0,'',''),(709,8,136.000,'Solve by Cramer\'s rule: `1/x+1/y+1/z=1`, `2/x+5/y+3/z=0`, `1/x+2/y+4/z=0`',1586783808,NULL,'','','','',0,'',''),(710,8,135.000,'Solve by Cramer\'s rule: `x+y+z=1`, `ax+by+cz=k`, `a^2x+b^2y+c^2z=k^2`  where  [`a\\neb\\nec`]',1586784084,NULL,'','','','',0,'',''),(711,8,116.000,'Prove that $ \\begin{vmatrix} 1+x & 1 & 1 \\\\ 1 & 1+y & 1 \\\\ 1 & 1 & 1+z \\end{vmatrix} = xy+yz+zx+xyz = xyz\\ (1+\\frac{1}{x}+\\frac{1}{y}+\\frac{1}{z}) $',1586784363,'','','','','',0,'',''),(712,7,25.000,'If `A` is a symmetric matrix, then `A^n` (where `n` is positive integer) is _____ matrix.',1586784609,'symmetric','712','','','',0,'',''),(713,8,35.000,'Without expanding prove that $ \\begin{vmatrix} 5 & 2 & 3 \\\\ 7 & 3 & 4 \\\\ 9 & 4 & 5 \\end{vmatrix} = 0 $',1586784828,'N.A.','713','','','',0,'',''),(714,7,81.000,'If $ A = \\begin{pmatrix} 1 & 1 \\\\ 2 & 2 \\\\ 3 & 3 \\end{pmatrix} $ then find `A\\A^{-1}`',1586785196,NULL,'','','','',0,'',''),(715,7,76.000,'If $ A = \\begin{pmatrix} 2 & 1 \\\\ 3 & 4 \\end{pmatrix} $ and $ B = \\begin{pmatrix} 1 & -2 \\\\ -1 & 1 \\end{pmatrix} $ then show that `(AB)^{-1}=B^{-1}\\cdotA^{-1]`',1586785576,NULL,'','','','',0,'',''),(716,7,44.000,'If $ A = \\begin{pmatrix} 4 & 2 \\\\ -1 & 1 \\end{pmatrix} $ and $ I = \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} $ then prove that `(A-2I)(A-3I)=`$\\begin{pmatrix} 0 & 0 \\\\ 0 & 0 \\end{pmatrix}$',1586786087,NULL,'','','','',0,'',''),(717,7,116.000,'Solve by using inverse matrix method. `2/x-3/y+3/z=10`,  `1/x+1/y+1/z=10`,  `3/x-1/y+2/z=13`',1586786341,NULL,'','','','',0,'',''),(718,7,109.000,'If $ A = \\begin{bmatrix} 1 & -1 & 2 \\\\ 0 & 2 & -3 \\\\ 3 & -2 & 4 \\end{bmatrix}$ then find `A^{-1}` and also show that `A\\A^{-1} = A^{-1}A = I`',1586790700,'','','','','',0,'',''),(719,7,40.000,'If $ A = \\begin{bmatrix} 3 & -2 \\\\ 4 & -2 \\end{bmatrix}$ and `A^2=kA-2I`, then find the value of `k`.',1586790850,NULL,'','','','',0,'',''),(720,8,132.000,'Solve by Cramer\'s rule: `3x+y+z=2`, `2x-4y+3z=-1`, `4x+y-3z=-11`',1586791176,NULL,'','','','',0,'',''),(721,7,26.000,'The matrix $ \\begin{bmatrix} cos\\ x & sin\\ x \\\\ -sin\\ x & cos\\ x \\end{bmatrix} $ is - (a) symmetric & singular;  (b) unit & skew-symmetric;  (c) Unit & orthogonal;  (d) Non-singular & orthogonal matrix. ',1586791563,'(d) Non-singular & orthogonal matrix. ','','','','',0,'',''),(722,8,72.000,'Prove that $ \\begin{vmatrix} 10! & 11! & 12! \\\\ 11! & 12! & 13! \\\\ 12! & 13! & 14! \\end{vmatrix} = 2\\times10!\\times11!\\times12! $',1586791809,NULL,'','','','',0,'',''),(723,7,71.000,'If $ A^{-1} = \\begin{pmatrix} 3 & 1 \\\\ -1 & 2 \\end{pmatrix} $ then find the matrix `A`',1586791974,NULL,'','','','',0,'',''),(724,7,63.000,'Express the following matrix (`A`) as the sum of symmetric & skew-symmetric matrix. $ A = \\begin{bmatrix} -3 & 4 & 1 \\\\ 2 & 3 & 0 \\\\ 1 & 4 & 5 \\end{bmatrix}$',1586792282,'$ \\begin{bmatrix} -3 & 3 & 1 \\\\ 3 & 3 & 2 \\\\ 1 & 2 & 5 \\end{bmatrix} + \\begin{bmatrix} 0 & 1 & 0 \\\\ -1 & 0 & -2 \\\\ 0 & 2 & 0 \\end{bmatrix}$','724','','','',0,'',''),(725,7,111.000,'If $ A = \\begin{bmatrix} 1 & 2 & 3 \\\\ 1 & 3 & -1 \\\\ -1 & 1 & -7 \\end{bmatrix}$ find `A^{-1}`. Hence solve the following equations. `x+y-z=3`,  `2x+3y+z=10`,  `3x-y-7z=1`',1586792606,'','','','','',0,'',''),(726,8,102.000,'Prove that $ \\begin{vmatrix} a+b+2c & a & b \\\\ c & b+c+2a & b \\\\ c & a & c+a+2b \\end{vmatrix} = 2\\ (a+b+c)^3 $',1586792845,NULL,'','','','',0,'',''),(727,7,112.000,'From the matrix equation `AX=B` find the matrix `X`, given that $ A = \\begin{bmatrix} 1 & -1 & 0 \\\\ 0 & 1 & -1 \\\\ 1 & 1 & 1 \\end{bmatrix} $ and $ B = \\begin{bmatrix} 2 \\\\ 1 \\\\ 7 \\end{bmatrix}$',1586793146,NULL,'','','','',0,'',''),(728,7,117.000,'By using the matrix method, show that the following set of equations have an infinite number of solutions. `x+2y+3z=1`,  `3x+4y+5z=2`,  `5x+6y+7z=3`',1586793362,NULL,'','','','',0,'',''),(729,8,105.000,'Prove that $ \\begin{vmatrix} a+b+c & -c & -b \\\\ -c & a+b+c & -a \\\\ -b & -a & a+b+c \\end{vmatrix} = 2\\ (a+b)\\ (b+c)\\ (c+a) $',1586793573,NULL,'','','','',0,'',''),(730,8,58.000,'Find the value of  $ \\begin{vmatrix} 1 & \\omega^3 & \\omega^2 \\\\ \\omega^3 & 1 & \\omega \\\\ \\omega^2 & \\omega & 1 \\end{vmatrix} $ where `\\omega` is the imaginary cube root of unity.',1586793894,NULL,'','','','',0,'',''),(731,7,59.000,'If $ A = \\begin{pmatrix} 1 & 2 & 3 \\end{pmatrix} $ then find `A\\A^T`',1586794180,NULL,'','','','',0,'',''),(732,7,33.000,'If  $ A = \\begin{bmatrix} \\cos\\alpha & \\sin\\alpha \\\\ -\\sin\\alpha & \\cos\\alpha \\end{bmatrix} $ then prove that `A\\cdotA^T = I`. Hence find `A^{-1]`',1586794456,NULL,'','','','',0,'',''),(733,7,99.000,'Find the value of `p` such that $ \\begin{bmatrix} 1 & p & 1 \\end{bmatrix} \\times \\begin{bmatrix} 1 & 3 & 2 \\\\ 2 & 5 & 1 \\\\ 15 & 3 & 2 \\end{bmatrix} \\times \\begin{bmatrix} 1 \\\\ 2 \\\\ p \\end{bmatrix} = 0 $',1586794902,'-2, -14','','','','',0,'',''),(734,8,92.000,'Show that (without expanding) $ \\begin{vmatrix} 1 & x^2+yz & x^3 \\\\ 1 & y^2+zx & y^3 \\\\ 1 & z^2+xy & z^3 \\end{vmatrix} = -(x-y) (y-z) (z-x) (x^2+y^2+z^2)$',1586795455,NULL,'','','','',0,'',''),(735,7,53.000,'If  $ A = \\begin{pmatrix} i & - i \\\\ - i & i \\end{pmatrix} $ and $ B = \\begin{pmatrix} 1 & -1 \\\\ -1 & 1 \\end{pmatrix} $ then show that `A^8=128\\ B`',1586796002,NULL,'','','','',0,'',''),(736,7,95.000,'If  $ A = \\begin{bmatrix} 1 & -\\tan \\frac{\\theta}{2} \\\\ \\tan \\frac{\\theta}{2} & 1 \\end{bmatrix} $ and  $ B = \\begin{bmatrix} 1 & \\tan \\frac{\\theta}{2} \\\\ -\\tan \\frac{\\theta}{2} & 1 \\end{bmatrix} $ then prove that `AB^{-1} = `$ \\begin{bmatrix} \\cos \\theta & -\\sin \\theta \\\\ \\sin \\theta & \\cos \\theta \\end{bmatrix} $',1586796861,NULL,'','','','',0,'',''),(737,8,113.000,'Prove that $ \\begin{vmatrix} 1 & b+c & b^2+c^2 \\\\ 1 & c+a & c^2+a^2 \\\\ 1 & a+b & a^2+b^2 \\end{vmatrix} = (a-b) (b-c) (c-a) $',1586797052,NULL,'','','','',0,'',''),(738,8,122.000,'Prove that $ \\begin{vmatrix} b+c & a-b & a \\\\ c+a & b-c & b \\\\ a+b & c-a & c \\end{vmatrix} = 3abc - (a^3+b^3+c^3)$',1586797249,NULL,'','','','',0,'',''),(739,8,77.000,'Find the value (without expanding). $ \\begin{vmatrix} a^{-1} & a^2 & bc \\\\ b^{-1} & b^2 & ca \\\\ c^{-1} & c^2 & ab \\end{vmatrix} $',1586842706,NULL,'','','','',0,'',''),(740,8,137.000,'Solve by Cramer\'s rule: `1/x+2/y+1/z=1/2`,  `4/x+2/y-3/z=2/3`,  `3/x-4/y+4/z=1/3`',1586842911,NULL,'','','','',0,'',''),(741,7,120.000,'Find the inverse of the following matrix by using Elementary row operations. $ \\begin{bmatrix} 1 & 2 & 3 \\\\ 2 & 5 & 7 \\\\ -2 & -4 & -5 \\end{bmatrix} $',1586843215,NULL,'','','','',0,'',''),(742,7,102.000,'Show that the matrix $ P = \\frac{1}{3}\\begin{pmatrix} -1 & 2 & -2 \\\\ -2 & 1 & 2 \\\\ 2 & 2 & 1 \\end{pmatrix}$ is proper orthogonal, and hence find `P^{-1}`',1586843494,'`P^{-1} = P^T` $ = \\frac{1}{3}\\begin{pmatrix} -1 & -2 & 2 \\\\ 2 & 1 & 2 \\\\ -2 & 2 & 1 \\end{pmatrix}$','742','','','',0,'',''),(743,8,16.000,'If $ \\begin{vmatrix} -5 & 5 & 10 \\\\ 5 & -5 & x \\\\ 0 & 10 & 5 \\end{vmatrix} =0 $ then find `x`',1586843696,'`-10`','743','','','',0,'',''),(744,7,66.000,'If $ A = \\begin{pmatrix} 1 & 3 \\\\ 2 & 4 \\end{pmatrix}$ then show that `A^2-5A-2I_2=O` where $ I_2 = \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix}$ and $ O = \\begin{pmatrix} 0 & 0 \\\\ 0 & 0 \\end{pmatrix}$. Hence find `A^{-1}`',1586844149,'$ A^{-1} = \\frac{1}{2} \\begin{pmatrix} -4 & 3 \\\\ 2 & -1 \\end{pmatrix}$ ','','','','',0,'',''),(745,8,110.000,'Prove that $ \\begin{vmatrix} a & b-c & c+b \\\\ a+c & b & c-a \\\\ a-b & b+a & c \\end{vmatrix} = (a+b+c) (a^2+b^2+c^2)$',1586844526,NULL,'','','','',0,'',''),(746,8,25.000,'Solve :  $ \\begin{vmatrix} x+4 & 2x & 2x \\\\ 2x & x+4 & 2x \\\\ 2x & 2x & x+4 \\end{vmatrix} = 0 $',1586844844,NULL,'','','','',0,'',''),(747,8,28.000,'Solve : $ \\begin{vmatrix} x-2 & 2x-3 & 3x-4 \\\\ x-4 & 2x-9 & 3x-16 \\\\ x-8 & 2x-27 & 3x-64 \\end{vmatrix} = 0 $',1586846165,'4','747','','','',0,'',''),(748,7,10.000,'If `A` be a square matrix, then which of the following is false: (a) `(A^T)^T = A`,   (b) `A+A^T` is symmetric,   (c) `A-A^T` is skew-symmetric,   (d) `(AB)^T = B^T\\cdotA^T` ',1586846511,'All options are correct','','','','',0,'',''),(749,8,73.000,'Show that $ \\begin{vmatrix} a^2 & 2a & 1 \\\\ 1 & a^2 & 2a \\\\ 2a & 1 & a^2 \\end{vmatrix} $ is a perfect square quantity.',1586846891,NULL,'','','','',0,'',''),(750,8,133.000,'Solve by Cramer\'s rule: `x-2y+z=-1`,   `3x+y-2z=4`,   `y-z=1`',1586847070,NULL,'','','','',0,'',''),(751,7,51.000,'If `A`, `B`, `C` are 3 given matrices such that $ A = \\begin{pmatrix} 3 & 5 \\\\ 2 & a \\end{pmatrix}$, $ B = \\begin{pmatrix} 4 & b \\\\ 2 & 9 \\end{pmatrix}$ and $ C = \\begin{pmatrix} 22 & 14 \\\\ a & -1 \\end{pmatrix}$, find `a` and `b` such that `A\\cdot\\B = C^T`.',1586847500,NULL,'','','','',0,'',''),(752,8,103.000,'Prove that $ \\begin{vmatrix} 2a & a-b-c & 2a \\\\ 2b & 2b & b-c-a \\\\ c-a-b & 2c & 2c \\end{vmatrix} = (a+b+c)^3 $',1586847803,NULL,'','','','',0,'',''),(753,8,18.000,'Solve : $ \\begin{vmatrix} x+4 & 3 & 5 \\\\ 3 & x+4 & 5 \\\\ 3 & 5 & x+4 \\end{vmatrix} = 0 $',1586847981,'`x = \\pm1, -12`','','','','',0,'',''),(754,7,11.000,'If `A` & `B` be two symmetric matrices, then the matrix `AB` will be symmetric if - (a) `AB=O`   (b) `AB=BA`   (c) `|AB|=0`   (d) None of these.',1586851848,'(b) `AB=BA`','754','','','',0,'',''),(755,8,53.000,'Find the value (without expanding)  $ \\begin{vmatrix} 49 & 1 & 6 \\\\ 39 & 7 & 4 \\\\ 26 & 2 & 3 \\end{vmatrix} $',1586851994,NULL,'','','','',0,'',''),(756,7,124.000,'Prove that any square matrix can be expressed as the sum of a symmetric matrix and a skew-symmetric matrix.',1586852291,NULL,'','','','',0,'',''),(757,7,37.000,'If $ A = \\begin{bmatrix} 0 & 2 \\\\ 3 & -4 \\end{bmatrix}$ and $ kA = \\begin{bmatrix} 0 & 3a \\\\ 2b & 24 \\end{bmatrix}$ then find the values of `k, a, b`.',1586852862,NULL,'','','','',0,'',''),(758,7,54.000,'If $ A = \\begin{bmatrix} 1 & 1 \\\\ 1 & 1 \\end{bmatrix} $ and `n in mathbb{N}`, then show that `A^n=2^{n-1}A`',1586853175,'N.A.','758','','','',0,'',''),(759,8,130.000,'Show that the condition for which equations `x+ay+az=0`, `bx+y+bz=0`, `cx+cy+z=0` have non-zero solution is `\\frac{a}{1-a]+\\frac{b}{1-b}+\\frac{c}{1-c}=-1`',1586875281,'','','','','',0,'',''),(760,8,64.000,'Prove that $ \\begin{vmatrix} 1 & x & x^2-yz \\\\ 1 & y & y^2-zx \\\\ 1 & z & z^2-xy \\end{vmatrix} = 0 $',1586875486,NULL,'','','','',0,'',''),(761,8,78.000,'If $ \\Delta = \\begin{vmatrix} a-x & c & b \\\\ c & b-x & a \\\\ b & a & c-x \\end{vmatrix} =0 $ and `a+b+c=0`, then show that either `x=0` or `x=\\pm\\sqrt{3/2(a^2+b^2+c^2)}`',1586958963,'','','','','',0,'',''),(762,7,62.000,'If $ A = \\begin{bmatrix} 1 & 2 & 3 & 4 \\end{bmatrix} $ and $ B = \\begin{bmatrix} 1 \\\\ 2 \\\\ 3 \\\\ 4 \\end{bmatrix}  $ then find `AB` and `BA`',1586959222,NULL,'','','','',0,'',''),(763,7,89.000,'If $ A = \\begin{bmatrix} 1 & x & -2 \\\\ 2 & 2 & 4 \\\\ 0 & 0 & 2 \\end{bmatrix} $ and `A^2+2I_3=3A`, then find the value of `x`. Here `I_3` is the unit matrix of order 3.',1586959521,NULL,'','','','',0,'',''),(764,8,79.000,'Prove that $ \\begin{vmatrix} a & b & c \\\\ b & c & a \\\\ c & a & b \\end{vmatrix} = -(a+b+c)(a+b\\omega+c\\omega^2)(a+b\\omega^2+c\\omega) $ where `\\omega` is the imaginary cube root of unity.',1586959983,NULL,'','','','',0,'',''),(765,7,8.000,'If $ A = \\begin{bmatrix} -i & 0 \\\\ 0 & i \\end{bmatrix} $ then `A^TA` is equal to - (a) `A`   (b) `-A`   (c) `I`   (d) `-I`',1586962360,'(d) `-I`','765','','','',0,'',''),(766,7,77.000,'If $ A = \\begin{bmatrix} 1 & 2 \\\\ 3 & 5 \\end{bmatrix} $ and $ B = \\begin{bmatrix} 1 \\\\ 2 \\end{bmatrix} $ , then find `A^{-1}\\cdot\\B`',1586963897,NULL,'','','','',0,'',''),(767,7,18.000,'`A` and `B` are two matrices such that `A\\cdotB=O` (where `O` is zero matrix). Can we deduce that either `A` or `B` is a zero matrix? Illustrate by an example.',1587017647,'`A` or `B` may not be zero matrix','767','','','',0,'',''),(768,8,100.000,'Prove that $ \\begin{vmatrix} 1+a^2-b^2 & 2ab & -2b \\\\ 2ab & 1-a^2+b^2 & 2a \\\\ 2b & -2a & 1-a^2-b^2 \\end{vmatrix} = (1+a^2+b^2)^3 $',1587018397,NULL,'','','','',0,'',''),(769,7,69.000,'If $ A = \\begin{pmatrix} 2 & -3 \\\\ -4 & 7 \\end{pmatrix} $ and `2A^{-1}=kI-A` where `I` is unit matrix. Find the value of `k`.',1587019033,NULL,'','','','',0,'',''),(770,8,134.000,'Solve by Cramer\'s rule:  `2x-y+z=6`,   `x+2y+3z=3`,   `3x+y-z=4`',1587019344,NULL,'','','','',0,'',''),(771,7,49.000,'If $ \\begin{pmatrix} 2 & 1 \\\\ 3 & 4 \\end{pmatrix} \\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\begin{pmatrix} 1 \\\\ -1 \\end{pmatrix} $ then find the value of `x` and `y`.',1587019603,NULL,'','','','',0,'',''),(772,7,103.000,'Show that the matrix $ \\begin{pmatrix} 3 & 2 & 1 \\\\ 0 & 4 & 5 \\\\ 3 & 6 & 6 \\end{pmatrix} $ is non-singular.',1587019792,NULL,'','','','',0,'',''),(773,8,120.000,'Prove that $ \\begin{vmatrix} \\frac{a^2+b^2}{c} & c & c \\\\ a & \\frac{b^2+c^2}{a} & a \\\\ b & b & \\frac{c^2+a^2}{b} \\end{vmatrix} = 4abc $',1587020230,NULL,'','','','',0,'',''),(774,7,52.000,'Find `A^100` if $ A = \\begin{pmatrix} \\omega & 0 \\\\ 0 & \\omega \\end{pmatrix} $ , where `\\omega` is the imaginary cube root of 1.',1587020728,'`A`','','','','',0,'',''),(775,7,23.000,'If $ A = \\begin{bmatrix} 1 & 2 \\\\ 2 & 1 \\end{bmatrix} $ and `f(x) = x^2-2x-3`, then show that `f(A)=0`',1587021346,NULL,'','','','',0,'',''),(776,7,106.000,'If $ A =\\frac{1}{9} \\begin{bmatrix} -8 & 1 & 4 \\\\ 4 & 4 & 7 \\\\ 1 & -8 & 4 \\end{bmatrix} $ , then show that `A^{-1} = A^T`.',1587021558,NULL,'','','','',0,'',''),(777,7,29.000,'If $ A = \\begin{bmatrix} 0 & 6 \\\\ 0 & 0 \\end{bmatrix} $ and `f(x) = 1+x+x^2+ \\cdots +x^20`, then show the value of `f(A)` is -- \r\n(a) $ \\begin{bmatrix} 1 & 6 \\\\ 0 & 1 \\end{bmatrix} $   \r\n(b) $ \\begin{bmatrix} 1 & 6 \\\\ 0 & 0 \\end{bmatrix} $   \r\n(c) $ \\begin{bmatrix} 1 & 6 \\\\ 1 & 0 \\end{bmatrix} $   \r\n(d) $ \\begin{bmatrix} 1 & 0 \\\\ 6 & 1 \\end{bmatrix} $',1587021990,'(a)','777','','','',0,'',''),(778,7,65.000,'If $ A = \\begin{bmatrix} 1 & x \\\\ x^2 & 4y \\end{bmatrix} $,  $ B = \\begin{bmatrix} -3 & 1 \\\\ 1 & 0 \\end{bmatrix} $  and  $ adj(A) + B = \\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\end{bmatrix} $ then find the value of `x` and `y`.',1587022544,NULL,'','','','',0,'',''),(779,7,64.000,'Express the following matrix (`A`) as the sum of symmetric & skew-symmetric matrix. $ A = \\begin{bmatrix} 2 & 3 & 4 \\\\ 5 & 7 & 9 \\\\ -2 & 1 & 1 \\end{bmatrix}$',1587022817,NULL,'','','','',0,'',''),(780,8,129.000,'If `S_n = \\alpha^n+\\beta^n+\\gamma^n` then show that $ \\begin{vmatrix} S_0 & S_1 & S_2 \\\\ S_1 & S_2 & S_3 \\\\ S_2 & S_3 & S_4 \\end{vmatrix} = (\\alpha-\\beta)^2 (\\beta-\\gamma)^2 (\\gamma-\\alpha)^2 $',1587023171,NULL,'','','','',0,'',''),(781,7,72.000,'If $ A = \\begin{bmatrix} 4 & 1 \\\\ 7 & 2 \\end{bmatrix} $ then find the matrix `B` such that `AB = I` where `I` is the unit matrix of order 2.',1587023627,NULL,'','','','',0,'',''),(782,8,124.000,'If `a, b, c` are real numbers, then prove that $ \\begin{vmatrix} b+c & c+a & a+b \\\\ c+a & a+b & b+c \\\\ a+b & b+c & c+a \\end{vmatrix} = 2 \\begin{vmatrix} a & b & c \\\\ b & c & a \\\\ c & a & b \\end{vmatrix} = -2(a^3+b^3+c^3-3abc) $ and hence show that if the value of this determinant is zero, then either `a+b+c=0` or `a=b=c`.',1587024697,'','','','','',0,'',''),(783,8,55.000,'If `x=-9` is one root of the equation $ \\begin{vmatrix} x & 3 & 7 \\\\ 2 & x & 2 \\\\ 7 & 6 & x \\end{vmatrix} = 0 $, then find other two roots.',1587025574,NULL,'','','','',0,'',''),(784,7,101.000,'If $ A = \\begin{bmatrix} 1 & 0 & 2 \\\\ 0 & 2 & 1 \\\\ 2 & 0 & 3 \\end{bmatrix} $ and `f(x) = x^3-6x^2+7x+2`, then show that `A` is a root of `f(x)=0`',1587026138,'N.A.','784','','','',0,'',''),(785,8,131.000,'Show that the equations `(\\lambda+a)x+\\lambda\\y+\\lambda\\z=0`,   `\\lambda\\x+(\\lambda+b)y+\\lambda\\z=0` and  `\\lambda\\x+\\lambda\\y+(\\lambda+c)z=0` have non-zero solutions if `\\frac{1}{\\lambda}=-(1/a+1/b+1/c)` ',1587028800,NULL,'','','','',0,'',''),(786,7,129.000,'If `A` and `B` are two square matrices of same order and `A^{-1}`, `B^{-1}` exist. Then prove that inverse of `AB` also exists and `(AB)^{-1} = B^{-1]A^{-1}`.',1587029562,NULL,'','','','',0,'',''),(787,8,7.000,'If `A` is a skew-symmetric matrix of odd order, then value of `|A|` is -- (a) `1`  (b) `-1`  (c) `0`  (d) any real number.',1587029971,NULL,'','','','',0,'',''),(788,7,4.000,'If $ A = \\begin{pmatrix} 3x & x-1 \\\\ 2x+3 & x+2 \\end{pmatrix} $ is a symmetric matrix, then find the  value of `x`.',1587041081,'`-4`','788','','','',0,'',''),(789,7,19.000,'`A` is a matrix of order `2\\times\\m` and `B` is a matrix of order `3\\times\\n` . If `A\\cdot\\B` is defined and its order is `p\\times\\4`, then find the value of `m, n, p`.',1587041549,'m=3, n=4, p=2','','','','',0,'',''),(790,7,92.000,'If $ f(x) = \\begin{bmatrix} cos\\ x & - sin\\ x & 0 \\\\ sin\\ x & cos\\ x & 0 \\\\ 0 & 0 & 1 \\end{bmatrix}  $, then prove that `f(\\alpha)\\cdot\\f(\\beta) = f(\\alpha+\\beta)`.',1587042030,NULL,'','','','',0,'',''),(791,8,93.000,'If `c` is any constant, then prove that $ \\begin{vmatrix} x & x^2 & 1+cx^3 \\\\ y & y^2 & 1+cy^3 \\\\ z & z^2 & 1+cz^3 \\end{vmatrix} = (1+cxyz)(x-y)(y-z)(z-x) $',1587042468,NULL,'','','','',0,'',''),(792,7,1.000,'If `A = [a_{ij}]` is a matrix of order `2\\times\\2` where `a_{ij} = i+2j` then find the matrix `A`.',1587044384,'$ A= \\begin{bmatrix} 3 & 5 \\\\ 4 & 6  \\end{bmatrix} $','792','','','',0,'',''),(793,7,34.000,'Show that $ A = \\frac{1}{\\sqrt{2}} \\begin{bmatrix} 1 & 1 \\\\ -1 & 1 \\end{bmatrix} $ is a proper orthogonal matrix, and hence find `A^{-1}`.',1587044659,NULL,'','','','',0,'',''),(794,8,60.000,'Find the value of  $ \\begin{vmatrix} 1+2\\omega^{100}+\\omega^{200} & \\omega^2 & 1 \\\\ 1 & 1+\\omega^{100}+2\\omega^{200} & \\omega \\\\ \\omega & \\omega^2 & 2+\\omega^{100}+\\omega^{200} \\end{vmatrix} $ where `omega` is the imaginary cube root of unity.',1587045223,NULL,'','','','',0,'',''),(795,8,126.000,'Using properties of determinant, prove that $ \\begin{vmatrix} b+c & q+r & y+z \\\\ c+a & r+p & z+x \\\\ a+b & p+q & x+y \\end{vmatrix} = 2 \\begin{vmatrix} a & p & x \\\\ b & q & y \\\\ c & r & z \\end{vmatrix} $',1587121815,NULL,'','','','',0,'',''),(796,7,35.000,'If $ \\begin{pmatrix} x+y & 2 \\\\ 1 & 0 \\end{pmatrix} = \\begin{pmatrix} 2 & x-z \\\\ 2x-y & 0 \\end{pmatrix} $ then find the value of `z`.',1587122039,'`-1`','','','','',0,'',''),(797,8,13.000,'The determinant $ \\begin{vmatrix} a & b & a\\alpha+b \\\\ b & c & b\\alpha+c \\\\ a\\alpha+b & b\\alpha+c & 0 \\end{vmatrix} = 0 $ if `a, b, c` are in -- (a) A.P.   (b) G.P.   (c) H.P. (d) None of these.',1587122390,NULL,'','','','',0,'',''),(798,7,97.000,'If a matrix $ A = \\begin{pmatrix} 6 & 2 & -2 \\\\ -2 & 2 & 2 \\\\ 2 & 2 & 2 \\end{pmatrix} $, then show that `(A-2I)(A-4I) = 0` matrix and hence find `A^3`',1587123111,'$ \\begin{pmatrix} 120 & 56 & -56 \\\\ -56 & 8 & 56 \\\\ 56 & 56 & 8 \\end{pmatrix} $','798','','','',0,'',''),(799,8,97.000,'Show that $ \\begin{vmatrix} a^2 & bc & ac+c^2 \\\\ a^2+ab & b^2 & ac \\\\ ab & b^2+bc & c^2 \\end{vmatrix} = 4a^2b^2c^2 $',1587123662,NULL,'','','','',0,'',''),(800,7,126.000,'If `A` and `B` are two symmetric matrices of same order, then show that `(AB-BA)` is a skew-symmetric matrix.',1587123954,NULL,'','','','',0,'',''),(801,8,10.000,'`A` be a `3\\times\\3` matrix. If `A^{-1}` exists and `|A|=6`, then the value of `|Adj(A)|` is -- (a) 1   (b) 6   (c) 12   (d) 36',1587125561,NULL,'','','','',0,'',''),(802,8,17.000,'Solve: $ \\begin{vmatrix} x+a & b & c \\\\ b & x+c & a \\\\ c & a & x+b \\end{vmatrix} = 0 $',1587126435,'`x=-(a+b+c), \\pm\\frac{1}{\\sqrt{2}}\\sqrt{(a-b)^2+(b-c)^2+(c-a)^2}`','802','','','',0,'',''),(803,8,14.000,'Prove that $ \\begin{vmatrix} a & b & a\\alpha+b\\beta \\\\ b & c & b\\alpha+c\\beta \\\\ a\\alpha+b\\beta & b\\alpha+c\\beta & 0 \\end{vmatrix} = (b^2-ac) (a\\alpha^2+2b\\alpha\\beta+c\\beta^2) $',1587126924,NULL,'','','','',0,'',''),(804,8,6.000,'Find the value of $ \\begin{vmatrix} 0 & a & b \\\\ -a & 0 & c \\\\ -b & -c & 0 \\end{vmatrix} $',1587127406,NULL,'','','','',0,'',''),(805,7,94.000,'If `\\alpha - \\beta = (2n+1)\\frac{\\pi}{2}` where `n in \\mathbb{Z}` then show that product of $ \\begin{pmatrix} \\cos^2\\alpha & \\cos\\alpha\\sin\\alpha \\\\ \\cos\\alpha\\sin\\alpha & \\sin^2\\alpha \\end{pmatrix} $  and  $ \\begin{pmatrix} \\cos^2\\beta& \\cos\\beta\\sin\\beta \\\\ \\cos\\beta\\sin\\beta & \\sin^2\\beta \\end{pmatrix} $ is a zero matrix.',1587127926,'N.A.','805','','','',0,'',''),(806,8,20.000,'Solve: $ \\begin{vmatrix} 2-x & 2 & 3 \\\\ 2 & 5-x & 6 \\\\ 3 & 4 & 10-x \\end{vmatrix} = 0 $',1587128317,'`1, 8\\pm \\sqrt{37}`','806','','','',0,'',''),(807,8,44.000,'Show that the value of $ \\begin{vmatrix} x+1 & x+2 & x+4 \\\\ x+3 & x+5 & x+8 \\\\ x+5 & x+8 & x+12 \\end{vmatrix} $ , is not depend on `x`.',1587128851,NULL,'','','','',0,'',''),(808,8,83.000,'Prove without expanding. $ \\begin{vmatrix} \\cos(x-\\alpha) & \\cos(x+\\alpha) & cosx \\\\  \\sin(x+\\alpha) & \\sin(x-\\alpha) & sinx \\\\ \\cos\\alpha\\ tanx & \\cos\\alpha\\ cotx & \\csc\\ 2x \\end{vmatrix} = 0 $',1587134979,NULL,'','','','',0,'',''),(809,7,30.000,'Find a matrix `X` such that `2A+B+X = 0` where $ A = \\begin{bmatrix} -1 & 2 \\\\ 3 & 4 \\end{bmatrix} $ and $ B = \\begin{bmatrix} 3 & -2 \\\\ 1 & 5 \\end{bmatrix} $',1587135617,'$ \\begin{bmatrix} -1 & -2 \\\\ -7 & -13 \\end{bmatrix} $','','','','',0,'',''),(810,7,93.000,'If  $ A = \\begin{bmatrix} 0 & -\\tan \\frac{\\alpha}{2} \\\\ \\tan \\frac{\\alpha}{2} & 0 \\end{bmatrix} $ and `I` is the identity matrix of order 2, then show that `(I+A) = (I-A)`$ \\begin{bmatrix} \\cos\\alpha & -\\sin\\alpha \\\\ \\sin\\alpha & \\cos\\alpha \\end{bmatrix} $  (or)   `(I+A)(I-A)^{-1] = `$ \\begin{bmatrix} \\cos\\alpha & -\\sin\\alpha \\\\ \\sin\\alpha & \\cos\\alpha \\end{bmatrix} $ ',1587286268,'N.A.','810,810.001','','','',0,'',''),(811,8,85.000,'Prove that $ \\begin{vmatrix} 2\\cos\\theta & 1 & 0 \\\\ 1 & 2\\cos\\theta & 1 \\\\ 0 & 1 & 2\\cos\\theta \\end{vmatrix} = $`\\frac{\\sin4\\theta}{\\sin\\theta}` [$ \\theta \\ne n\\pi$]',1587287295,NULL,'','','','',0,'',''),(812,8,70.000,'Without expanding prove that: $ \\begin{vmatrix} 1 & bc & a(b+c) \\\\ 1 & ca & b(c+a) \\\\ 1 & ab & c(a+b) \\end{vmatrix} = 0 $',1587287665,NULL,'','','','',0,'',''),(813,7,110.000,'If $ A = \\begin{bmatrix} 2 & -3 & 5 \\\\ 3 & 2 & -4 \\\\ 1 & 1 & -2 \\end{bmatrix} $ then find `A^{-1}`. Hence solve the following equations. `2x-3y+5z=11`,   `3x+2y-4z=-5`,   `x+y-2z=-3`',1587288193,NULL,'','','','',0,'',''),(814,8,86.000,'If $ \\Delta = \\begin{vmatrix} 1 & \\sin\\theta & 1 \\\\ -\\sin\\theta & 1 & \\sin\\theta \\\\ 1 & -\\sin\\theta & 1 \\end{vmatrix} $ then show that `2 \\le \\Delta \\le 4`',1587288885,NULL,'','','','',0,'',''),(815,7,41.000,'If $ A = \\begin{pmatrix} 3 & 1 \\\\ 7 & 5 \\end{pmatrix} $ and `A^2=-xI+yA` (where `I` is the identity matrix). Then find the value of `x` and `y`.',1587289679,NULL,'','','','',0,'',''),(816,7,96.000,'If $ A = \\begin{pmatrix} 1 & \\tan\\ x \\\\ -\\tan\\ x & 1 \\end{pmatrix} $ then show that $ A^T \\cdot A^{-1} = \\begin{pmatrix} \\cos2x & -\\sin2x \\\\ \\sin2x & \\cos2x \\end{pmatrix} $',1587290049,NULL,'','','','',0,'',''),(817,8,41.000,'Without expanding, prove that: $ \\begin{vmatrix} 0 & 2016 & -2017 \\\\ -2016 & 0 & 2018 \\\\ 2017 & -2018 & 0 \\end{vmatrix} = 0 $',1587290447,'N.A.','817','','','',0,'',''),(818,8,5.000,'Value of the determinant $ \\begin{vmatrix} 2^{10} & 2^{11} & 2^{12} \\\\ 2^{11} & 2^{12} & 2^{13} \\\\ 2^{12} & 2^{13} & 2^{14} \\end{vmatrix} $ is -- (a) `2^10`   (b) `0`   (c) `1`   (d) None of these.',1587291808,'','','','','',0,'',''),(819,8,3.000,'Find co-factor of `-4` and `9` in determinant $ \\begin{vmatrix} -1 & -2 & 3 \\\\ -4 & -5 & -6 \\\\ -7 & 8 & 9 \\end{vmatrix} $',1587291813,NULL,'','','','',0,'',''),(820,7,31.000,'If $ A = \\begin{pmatrix} -1 & 5 \\\\ 3 & 2 \\end{pmatrix} $ and $ B = \\begin{pmatrix} 3 & -2 \\\\ 5 & 4 \\end{pmatrix} $ then show that `(A+B)^T = A^T + B^T` and `(AB)^T = B^T\\cdot A^T`',1587292537,NULL,'','','','',0,'',''),(821,8,91.000,'Prove that $ \\begin{vmatrix} y+z & z & y \\\\ z & z+x & x \\\\ y & x & y+x \\end{vmatrix} = 4xyz $',1587295162,NULL,'','','','',0,'',''),(822,8,117.000,'Without expanding prove that: $ \\begin{vmatrix} x & b & c \\\\ a & y & c \\\\ a & b & z \\end{vmatrix}  $`=(x-a)(y-b)(z-c)(\\frac{x}{x-a}+\\frac{y}{y-b}+\\frac{z}{z-c}-2)`',1587307377,NULL,'','','','',0,'',''),(823,7,90.000,'If $ P = \\begin{bmatrix} -1 & 3 & 5 \\\\ 1 & -3 & -5 \\\\ -1 & 3 & -5 \\end{bmatrix} $ then show that `P^2 = P` and then find a matrix `Q` such that `3P^2-2P+Q = I`.',1587310437,'$ \\begin{bmatrix} 2 & -3 & -5 \\\\ -1 & 4 & 5 \\\\ 1 & -3 & -4 \\end{bmatrix} $ ','823','','','',0,'',''),(824,8,127.000,'Justify the truth. $ \\begin{vmatrix} b+c & a-c & a-b \\\\ b-c & c+a & b-a \\\\ c-b & c-a & a+b \\end{vmatrix} = 8abc $',1587311622,NULL,'','','','',0,'',''),(825,7,100.000,'If `f(x) = x^2-5x+6` then find `f(A)` where $ A = \\begin{bmatrix} 2 & 0 & 1 \\\\ 0 & 1 & 3 \\\\ 1 & -1 & 0 \\end{bmatrix} $',1587312570,'$ \\begin{bmatrix} 1 & -1 & -3 \\\\ -1 & -1 & -10 \\\\ -5 & 4 & 4 \\end{bmatrix} $','825','','','',0,'',''),(826,7,121.000,'Find the inverse of the following matrix by using Elementary column operations.  $ \\begin{bmatrix} 1 & -1 & 2 \\\\ 0 & 2 & -3 \\\\ 3 & -2 & 4 \\end{bmatrix} $',1587313075,NULL,'','','','',0,'',''),(827,8,69.000,'Prove without expanding. $ \\begin{vmatrix} a & b & c \\\\ x & y & z \\\\ yz & zx & xy \\end{vmatrix} = \\begin{vmatrix} ax & by & cz \\\\ x^2 & y^2 & z^2 \\\\ 1 & 1 & 1 \\end{vmatrix} $',1587313893,NULL,'','','','',0,'',''),(828,7,22.000,'If $ A = \\begin{pmatrix} 2 & 4 \\\\ 3 & 2 \\end{pmatrix} $ and $ B = \\begin{pmatrix} 1 & 3 \\\\ -2 & 5 \\end{pmatrix} $ then show that `AB \\ne BA`',1587314451,NULL,'','','','',0,'',''),(829,7,60.000,'If $ P = \\begin{pmatrix} 1 & 2 & 1 \\\\ 1 & 3 & 1 \\end{pmatrix} $ and `Q = PP^T`, then find the matrix `Q`',1587315161,NULL,'','','','',0,'',''),(830,8,27.000,'Solve: $ \\begin{vmatrix} a+x & a-x & a-x \\\\ a-x & a+x & a-x \\\\ a-x & a-x & a+x \\end{vmatrix} = 0 $',1587315860,'0, 0, a','830','','','',0,'',''),(831,7,48.000,'If `\\alpha` and `\\beta` are two roots of the equation `x^2+x+1=0` then find the matrix `A` such that $ A = \\begin{bmatrix} 1 & \\beta \\\\ \\alpha & \\alpha \\end{bmatrix} \\times \\begin{bmatrix} \\alpha & \\beta \\\\ 1 & \\beta \\end{bmatrix} $',1587316180,NULL,'','','','',0,'',''),(832,8,56.000,'If $ \\begin{vmatrix} a & b & c \\\\ x & y & z \\\\ m & n & p \\end{vmatrix} = \\lambda $, then find the value of $ \\begin{vmatrix} 6a & 2b & 2c \\\\ 3x & y & z \\\\ 3m & n & p \\end{vmatrix} $ in terms of `\\lambda`',1587316536,NULL,'','','','',0,'',''),(833,8,26.000,'Solve for `x`: $ \\begin{vmatrix} x & c+x & b+x \\\\ c+x & x & a+x \\\\ b+x & a+x & x \\end{vmatrix} = 0 $ ',1587317236,'`\\frac{2abc}{a^2+b^2+c^2-2ab-2bc-2ca}`','833','','','',0,'',''),(834,8,65.000,'Without expanding prove that: $ \\begin{vmatrix} 1 & bc & b+c \\\\ 1 & ca & c+a \\\\ 1 & ab & a+b \\end{vmatrix} = \\begin{vmatrix} 1 & a & a^2 \\\\ 1 & b & b^2 \\\\ 1 & c & c^2 \\end{vmatrix} $',1587317514,'','','','','',0,'',''),(835,8,37.000,'Without expanding prove that: $ \\begin{vmatrix} 27 & 40 & 58 \\\\ 24 & 36 & 52 \\\\ 18 & 28 & 40 \\end{vmatrix} = 0 $',1587374201,NULL,'','','','',0,'',''),(836,8,128.000,'Prove that $ \\begin{vmatrix} (a+1)(a+2) & a+2 & 1 \\\\ (a+2)(a+3) & a+3 & 1 \\\\ (a+3)(a+4) & a+4 & 1 \\end{vmatrix} = -2 $',1587375148,NULL,'','','','',0,'',''),(837,8,34.000,'Without expanding, prove that: $ \\begin{vmatrix} \\frac{1}{a} & 1 & bc \\\\ \\frac{1}{b} & 1 & ca \\\\ \\frac{1}{c} & 1 & ab \\end{vmatrix} = 0 $',1587375557,NULL,'','','','',0,'',''),(838,7,39.000,'If $ A = \\begin{bmatrix} x & 1 \\\\ 1 & 0 \\end{bmatrix} $ and `A^2=I` then find the value of `x`.',1587379185,NULL,'','','','',0,'',''),(839,8,24.000,'Solve: $ \\begin{vmatrix} x & a & b \\\\ a & x & b \\\\ a & b & x \\end{vmatrix} = 0 $',1587379895,NULL,'','','','',0,'',''),(840,8,42.000,'Without expanding prove that: $ \\begin{vmatrix} a+1 & a+4 & a+2 \\\\ a+2 & a+5 & a+4 \\\\ a+3 & a+6 & a+6 \\end{vmatrix} = 0 $',1587380309,NULL,'','','','',0,'',''),(841,7,105.000,'If $ A = \\begin{bmatrix} 1 & 0 & 2 \\\\ 3 & 4 & -1 \\\\ 0 & 5 & 6 \\end{bmatrix} $ then show that `(A^T)^{-1} = (A^{-1})^T`.',1587380735,NULL,'','','','',0,'',''),(842,7,104.000,'If `|A|=2` and $ Adj(A) = \\begin{bmatrix} -2 & 3 & 1 \\\\ 6 & -8 & -2 \\\\ -4 & 7 & 2 \\end{bmatrix} $ then find the matrix `A`',1587381038,NULL,'','','','',0,'',''),(843,8,81.000,'Prove that: $ \\begin{vmatrix} a & a+b & a+b+c \\\\ 2a & 3a+2b & 4a+3b+2c \\\\ 3a & 6a+3b & 10a+6b+3c \\end{vmatrix} = a^3 $',1587381589,'','','','','',0,'',''),(844,8,104.000,'Prove that $ \\begin{vmatrix} -2a & a+b & a+c \\\\ b+a & -2b & b+c \\\\ c+a & c+b & -2c \\end{vmatrix} = 4(a+b)(b+c)(c+a) $',1587382309,NULL,'','','','',0,'',''),(845,8,52.000,'Without expanding, prove that $ \\begin{vmatrix} 3 & 4 & 5 \\\\ 4 & 3 & 7 \\\\ 5 & 7 & 5 \\end{vmatrix} $ is divisible by `23`.',1587382653,'','','','','',0,'',''),(846,8,67.000,'without expanding, prove that: $ \\begin{vmatrix} a^3 & a^2 & 1 \\\\ b^3 & b^2 & 1 \\\\ c^3 & c^2 & 1 \\end{vmatrix} = (ab+bc+ca) \\begin{vmatrix} a^2 & a & 1 \\\\ b^2 & b & 1 \\\\ c^2 & c & 1 \\end{vmatrix} $',1587382654,NULL,'','','','',0,'',''),(847,7,47.000,'If $ A = \\begin{bmatrix} \\alpha & 0 \\\\ 1 & 1 \\end{bmatrix} $ and $ B = \\begin{bmatrix} 1 & 0 \\\\ 3 & 1 \\end{bmatrix} $ and `A^2=B`, then find the value of `\\alpha` (if possible). Give reason for your answer.',1587383102,NULL,'','','','',0,'',''),(848,8,21.000,'Solve: $ \\begin{vmatrix} 11-x & -6 & 2 \\\\ -6 & 10-x & -4 \\\\ 2 & -4 & 6-x \\end{vmatrix} = 0 $',1587383668,'3, 6, 18','848','','','',0,'',''),(849,7,125.000,'If `A` is a square matrix, then show that the matrices `A\\cdotA^T` and `A^T\\cdotA` are symmetric.',1587384098,NULL,'','','','',0,'',''),(850,8,33.000,'Without expanding, show that: $ \\begin{vmatrix} a & b & c \\\\ x & y & z \\\\ p & q & r \\end{vmatrix} = \\begin{vmatrix} y & b & q \\\\ x & a & p \\\\ z & c & r \\end{vmatrix} $',1587384398,NULL,'','','','',0,'',''),(851,7,7.000,'If $ A = \\begin{pmatrix} 2 & -2 & -4 \\\\ -1 & 3 & 4 \\\\ 1 & -2 & -3 \\end{pmatrix} $ then show that `A+A^T` is a symmetric matrix.',1587384999,'N.A.','851','','','',0,'',''),(852,8,51.000,'Find the value of: $ \\begin{vmatrix} a-b & b-c & c-a \\\\ b-c & c-a & a-b \\\\ c-a & a-b & b-c \\end{vmatrix} $',1587386339,NULL,'','','','',0,'',''),(853,8,111.000,'Prove that $ \\begin{vmatrix} 3a & -a+b & -a+c \\\\ -b+a & 3b & -b+c \\\\ -c+a & -c+b & 3c \\end{vmatrix} = 3(a+b+c)(ab+bc+ca) $',1587386913,NULL,'','','','',0,'',''),(854,8,118.000,'If `a\\ne\\p`, `b\\ne\\q`, `c\\ne\\r` and $ \\begin{vmatrix} p & b & c \\\\ a & q & c \\\\ a & b & r \\end{vmatrix} = 0 $, then show that `\\frac{p}{p-a}+\\frac{q}{q-b}+\\frac{r}{r-c}=2`.',1587387492,'','','','','',0,'',''),(855,7,2.000,'If `A = [a_{ij}]` is a matrix of order `2\\times\\2` where `a_{ij} = \\frac{(i+j)^2}{2}`, then find the matrix `A`.',1587387850,'$ \\begin{pmatrix} 2 & 9/2 \\\\ 9/2 & 8 \\end{pmatrix} $','855','','','',0,'',''),(856,8,88.000,'Show that $ \\begin{vmatrix} \\sin^2A & sinA & \\cos^2A \\\\ \\sin^2B & sinB & \\cos^2B \\\\ \\sin^2C & sinC & \\cos^2C \\end{vmatrix} $`= (sinA-sinB)(sinB-sinC)(sinA-sinC)`',1587388622,'','','','','',0,'',''),(857,7,78.000,'If inverse of $ A = \\begin{pmatrix} k & 2 \\\\ 3 & 4 \\end{pmatrix} $ does not exist, then find the value of `k`.',1587389349,NULL,'','','','',0,'',''),(858,7,32.000,'If $ 2A - 3B = \\begin{bmatrix} 1 & 4 \\\\ 2 & 8 \\end{bmatrix} $ and $ 3A + 2B = \\begin{bmatrix} -1 & 2 \\\\ 0 & 4 \\end{bmatrix} $ , then find the matrix `A` and `B`.',1587391328,NULL,'','','','',0,'',''),(859,7,79.000,'If $ A = \\begin{bmatrix} 1 & 2 \\\\ 3 & -5 \\end{bmatrix} $ then show that `A(adj A) = |A|\\cdot\\I`',1587391671,NULL,'','','','',0,'',''),(860,7,108.000,'If $ A = \\begin{bmatrix} 1 & 3 & 3 \\\\ 1 & 4 & 3 \\\\ 1 & 3 & 4 \\end{bmatrix} $ then verify the relation `A\\cdot(adj\\ A) = |A|\\cdot I`. Hence find `A^{-1}`',1587392085,NULL,'','','','',0,'',''),(861,8,89.000,'If `A, B, C` are three angles of a triangle, then prove that $ \\begin{vmatrix} -1 & \\cos\\ C & \\cos\\ B \\\\ \\cos\\ C & -1 & \\cos\\ A \\\\ \\cos\\ B & \\cos\\ A & -1 \\end{vmatrix} = 0 $',1587392702,NULL,'','','','',0,'',''),(862,8,119.000,'If $ \\begin{vmatrix} p & q-y & r-z \\\\ p-x & q & r-z \\\\ p-x & q-y & r \\end{vmatrix} = 0 $ , then show that `p/x+q/y+r/z = 2`',1587403312,NULL,'','','','',0,'',''),(863,8,47.000,'Prove that $ \\begin{vmatrix} 1 & \\log_xy & \\log_xz \\\\ \\log_yx & 1 & \\log_yz \\\\ \\log_zx & \\log_zy & 1 \\end{vmatrix} = 0 $',1587405677,NULL,'','','','',0,'',''),(864,8,87.000,'Show that $ \\begin{vmatrix} 1 & 1 & 1 \\\\ \\sin\\alpha & \\sin\\beta & \\sin\\gamma \\\\ \\cos\\alpha & \\cos\\beta & \\cos\\gamma \\end{vmatrix} = $`-4sin\\frac{\\alpha-\\beta}{2}sin\\frac{\\beta-\\gamma}{2}sin\\frac{\\gamma-\\alpha}{2}`',1587477227,NULL,'','','','',0,'',''),(865,8,68.000,'Without expanding, prove that: $ \\begin{vmatrix} a^2 & b^2 & c^2 \\\\ (a+1)^2 & (b+1)^2 & (c+1)^2 \\\\ (a-1)^2 & (b-1)^2 & (c-1)^2 \\end{vmatrix} = 4 \\begin{vmatrix} a^2 & b^2 & c^2 \\\\ a & b & c \\\\ 1 & 1 & 1 \\end{vmatrix} $',1587477943,NULL,'','','','',0,'',''),(866,8,29.000,'Solve: $ \\begin{vmatrix} 1 & 3 & 9 \\\\ 1 & x & x^2 \\\\ 4 & 16 & 64 \\end{vmatrix} = 0 $',1587478628,NULL,'','','','',0,'',''),(867,8,9.000,'If $ A = \\begin{bmatrix} x & 0 & 0 \\\\ 0 & x & 0 \\\\ 0 & 0 & x \\end{bmatrix} $ then the value of `|A|\\cdot|adj(A)|` is -- (a) `x^9`   (b) `x^6`   (c) `x^3`   (d) `x`',1587485528,NULL,'','','','',0,'',''),(868,8,139.000,'If the coordinates of vertices of a triangle are `[m(m+1), m+1]`, `[(m+1)(m+2), m+2]` and `[(m+2)(m+3), m+3]`, then prove that the area of said triangle is not depend on `m`.',1587486635,NULL,'','','','',0,'',''),(869,7,98.000,'Find the value of $ \\begin{pmatrix} x & y & z \\end{pmatrix} \\times \\begin{pmatrix} a & h & g \\\\ h & b & f \\\\ g & f & c \\end{pmatrix} \\times \\begin{pmatrix} x \\\\ y \\\\ z \\end{pmatrix} $',1587486900,NULL,'','','','',0,'',''),(870,7,119.000,'Find the inverse of the following matrix by using elementary row transformations. $ \\begin{pmatrix} 3 & 0 & -1 \\\\ 2 & 3 & 0 \\\\ 0 & 4 & 1 \\end{pmatrix} $',1587487328,NULL,'','','','',0,'',''),(871,8,80.000,'Prove that `f(200)=0` where $ f(x) = \\begin{vmatrix} 1 & x & x+1 \\\\ 2x & x(x-1) & x(x+1) \\\\ 3x(x-1) & x(x-1)(x-2) & x(x+1)(x-1) \\end{vmatrix} $',1587487824,NULL,'','','','',0,'',''),(872,7,61.000,'If $ A = \\begin{pmatrix} -2 & 1 & 3 \\\\ 0 & 4 & -1 \\end{pmatrix} $ and $ B = \\begin{pmatrix} 2 & 1 \\\\ -3 & 0 \\\\ 4 & -5 \\end{pmatrix} $, then prove that `(AB)^T = B^TA^T`',1587488296,NULL,'','','','',0,'',''),(873,7,46.000,'If $ A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} $ and $ I = \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} $ , then prove that `A^2 - (a+d)A = (bc-ad)I`',1587489868,NULL,'','','','',0,'',''),(874,7,113.000,'Find the matrix `A` where $ \\begin{bmatrix} 2 & -1 \\\\ 1 & 0 \\\\ -3 & 4 \\end{bmatrix} A = \\begin{bmatrix} -1 & -8 & -10 \\\\ 1 & -2 & -5 \\\\ 9 & 22 & 15 \\end{bmatrix} $',1587490475,NULL,'','','','',0,'',''),(875,8,50.000,'Find the value of $ \\begin{vmatrix} 1+x & y & z \\\\ x & 1+y & z \\\\ x & y & 1+z \\end{vmatrix} $',1587491001,NULL,'','','','',0,'',''),(876,7,107.000,'If $ A = \\begin{bmatrix} 1 & -1 & 1 \\\\ 2 & -1 & 0 \\\\ 1 & 0 & 0 \\end{bmatrix} $ then show that `A^2 = A^{-1}`. Hence find `A^3`',1587491351,NULL,'','','','',0,'',''),(877,8,12.000,'If `A` is an orthogonal matrix of order 3, then value of `|A|` is -- (a) `0`   (b) `1`   (c) `-1`   (d) `\\pm1`',1587651477,NULL,'','','','',0,'',''),(878,7,87.000,'Show that the matrix $ A = \\begin{bmatrix} 2 & -2 & -4 \\\\ -1 & 3 & 4 \\\\ 1 & -2 & -3 \\end{bmatrix} $ is an idempotent matrix.',1587651737,'N.A.','878','','','',0,'',''),(879,8,11.000,'If `A = [a_{ij}]_{n\\timesn}` and `|A| = d \\ne0`, then `|\\adj\\ A|` is -- (a) `d`   (b) `d^n`   (c) `d^{n-1}`   (d) `d^{n+1}`',1587652053,NULL,'','','','',0,'',''),(880,8,109.000,'If `2s=a+b+c` then prove that $ \\begin{vmatrix} a^2 & (s-a)^2 & (s-a)^2 \\\\ (s-b)^2 & b^2 & (s-b)^2 \\\\ (s-c)^2 & (s-c)^2 & c^2 \\end{vmatrix} = 2s^3(s-a)(s-b)(s-c) $',1587703454,NULL,'','','','',0,'',''),(881,8,49.000,'If `p^{th}`, `q^{th}` and `r^{th}` terms of a geometric series are `a`, `b`, `c` respectively, then show that $ \\begin{vmatrix} \\log a & p & 1 \\\\ \\log b & q & 1 \\\\ \\log c & r & 1 \\end{vmatrix} = 0 $ where `a, b, c` are positive.',1587703914,NULL,'','','','',0,'',''),(882,8,46.000,'Prove that $ \\begin{vmatrix} 1 & 1 & 1 \\\\ C_1^m & C_1^{m+1} & C_1^{m+2} \\\\ C_2^m & C_2^{m+1} & C_2^{m+2} \\end{vmatrix} = 1 $',1587710805,NULL,'','','','',0,'',''),(883,8,36.000,'Without expanding prove that: $ \\begin{vmatrix} 1 & 3 & 5 \\\\ 2 & 6 & 10 \\\\ 31 & 11 & 38 \\end{vmatrix} =0 $',1587711951,NULL,'','','','',0,'',''),(884,7,57.000,'If $ A = \\begin{bmatrix} 1 & 0 \\\\ -1 & 1 \\end{bmatrix} $ then prove that `A^2-2A+I_2=0`. Hence find the matrix `A^50`',1587713427,'$ \\begin{bmatrix} 1 & 0 \\\\ -50 & 1 \\end{bmatrix} $','884','','','',0,'',''),(885,7,42.000,'If $ A = \\begin{bmatrix} 1 & -1 \\\\ 2 & -1 \\end{bmatrix} $, $ B = \\begin{bmatrix} a & 1 \\\\ b & -1 \\end{bmatrix} $ and `(A+B)^2 = A^2+B^2`, then find the value of `a` and `b`.',1587713685,'`a=1, b=4`','','','','',0,'',''),(886,7,83.000,'Find the matrices `A` and `B` such that, $ A+2B = \\begin{pmatrix} 1 & 2 & 0 \\\\ 6 & -3 & 3 \\\\ -5 & 3 & 1 \\end{pmatrix} $ and $ 2A - B = \\begin{pmatrix} 2 & -1 & 5 \\\\ 2 & -1 & 6 \\\\ 0 & 1 & 2 \\end{pmatrix} $',1587714227,NULL,'','','','',0,'',''),(887,7,9.000,'If `A` is a square matrix, then `A\\A^T + A^TA` is -- (a) unit matrix   (b) zero matrix   (c) symmetric matrix   (d) skew-symmetric matrix',1587714639,'(c) symmetric matrix','887','','','',0,'',''),(888,8,8.000,'If `A` is an invertible matrix of order `2`, then the value of `\\det(A^{-1})` is -- (a) `0`   (b) `1`   (c) `\\det A`   (d) `\\frac{1}{\\det A}`',1587714917,NULL,'','','','',0,'',''),(889,8,75.000,'Prove that $ \\begin{vmatrix} 2ab & a^2 & b^2 \\\\ a^2 & b^2 & 2ab \\\\ b^2 & 2ab & a^2 \\end{vmatrix} = -(a^3+b^3)^2 $',1587715620,NULL,'','','','',0,'',''),(890,8,99.000,'Prove that $ \\begin{vmatrix} -bc & b^2+bc & c^2+bc \\\\ a^2+ac & -ac & c^2+ac \\\\ a^2+ab & b^2+ab & -ab \\end{vmatrix} = (ab+bc+ca)^3 $',1587717059,NULL,'','','','',0,'',''),(891,8,32.000,'Without expanding, prove that: $ \\begin{vmatrix} a+x & y & z \\\\ x & a+y & z \\\\ x & y & a+z \\end{vmatrix} = a^2(a+x+y+z) $',1588050379,NULL,'','','','',0,'',''),(892,7,123.000,'`A` and `B` are two matrices such that `AB = BA`. Then show that `A^2+B^2 = A+B`',1588050819,NULL,'','','','',0,'',''),(893,7,50.000,'If $ A = \\begin{bmatrix} 3 & -1 \\\\ 1 & 2 \\end{bmatrix} $, $ B = \\begin{bmatrix} 3 \\\\ 1 \\end{bmatrix} $, $ C = \\begin{bmatrix} 1 \\\\ -2 \\end{bmatrix} $ and `AX=3B+2C`, then find the matrix `X`',1588051434,'$ X = \\begin{bmatrix} 3 \\\\ -2 \\end{bmatrix} $','893','','','',0,'',''),(894,7,45.000,'If $ A = \\begin{bmatrix} 4 & 2i \\\\ i & 1 \\end{bmatrix} $ then show that `(A-2I)(A-3I) = O` where `O` & `I` are zero & unit matrices. `i=\\sqrt{-1}`',1588051730,NULL,'','','','',0,'',''),(895,8,71.000,'without expanding, prove that: $ \\begin{vmatrix} 1 & ab & \\frac{1}{a}+\\frac{1}{b} \\\\ 1 & bc & \\frac{1}{b}+\\frac{1}{c} \\\\ 1 & ca & \\frac{1}{c}+\\frac{1}{a} \\end{vmatrix} = \\begin{vmatrix} b^2c^2 & bc & b+c \\\\ c^2a^2 & ca & c+a \\\\ a^2b^2 & ab & a+b \\end{vmatrix} = 0 $',1588052263,'N.A.','895.01','','','',0,'',''),(896,8,138.000,'If the points `(a_1, b_1)`, `(a_2, b_2)` and `(a_1+a_2, b_1+b_2)` are collinear, then show that `a_1b_2 = a_2b_1`',1588052491,NULL,'','','','',0,'',''),(897,7,73.000,'If $ A = \\begin{bmatrix} 2 & 5 \\\\ 1 & 3 \\end{bmatrix} $ and $ AB = \\begin{bmatrix} -13 & 8 \\\\ -8 & 5 \\end{bmatrix} $, then find the matrix `B`.',1588053978,NULL,'','','','',0,'',''),(898,8,15.000,'Without expanding, prove that: $ \\begin{vmatrix} a+bx & c+dx & p+qx \\\\ ax+b & cx+d & px+q \\\\ u & v & w \\end{vmatrix} = (1-x^2) \\begin{vmatrix} a & c & p \\\\ b & d & q \\\\ u & v & w \\end{vmatrix} $',1588054700,NULL,'','','','',0,'',''),(899,8,125.000,'Prove that $ \\begin{vmatrix} 2bc-a^2 & c^2 & b^2 \\\\ c^2 & 2ca-b^2 & a^2 \\\\ b^2 & a^2 & 2ab-c^2 \\end{vmatrix} = (a^3+b^3+c^3-3abc)^2$',1588055302,NULL,'','','','',0,'',''),(900,7,84.000,'Given $ A = \\begin{bmatrix} 1 & 0 & 0 \\\\ 0 & -1 & 0 \\\\ 0 & 0 & -1 \\end{bmatrix} $ and $ B = \\begin{bmatrix} 2 & x & x \\\\ x & 4 & 5 \\\\ x & 6 & 7 \\end{bmatrix} $. Then find the value of `x` (if possible) such that `AB = BA`',1588055764,'`x=0`','','','','',0,'',''),(901,7,118.000,'Given $ A = \\begin{bmatrix} 1 & -1 & 1 \\\\ 1 & -2 & -2 \\\\ 2 & 1 & 3 \\end{bmatrix} $ and $ B = \\begin{bmatrix} -4 & 4 & 4 \\\\ -7 & 1 & 3 \\\\ 5 & -3 & -1 \\end{bmatrix} $. Find `AB`. Hence from this result, solve the following set of equations. `x-y+z=4`,   `x-2y-2z=9`,   `2x+y+3z=1`',1588056088,'`x=3, y=-2, z=-1`','901,901.001','','','',0,'',''),(902,8,107.000,'Without expanding, prove that: $ \\begin{vmatrix} (a+b)^2 & ac & bc \\\\ ac & (b+c)^2 & ab \\\\ bc & ab & (c+a)^2 \\end{vmatrix} = 2abc(a+b+c)^3 $',1588061279,NULL,'','','','',0,'',''),(903,7,68.000,'If $ A = \\begin{pmatrix} 2x & 0 \\\\ x & x \\end{pmatrix} $ and $ A^{-1} = \\begin{pmatrix} 1 & 0 \\\\ -1 & 2 \\end{pmatrix} $, then find the value of `x`',1588061763,NULL,'','','','',0,'',''),(904,8,62.000,'If `a, b, c` are non-zero unequal real numbers and $ \\begin{vmatrix} bc & ca & ab \\\\ ca & ab & bc \\\\ ab & bc & ca \\end{vmatrix} = 0 $, then prove that `1/a+1/b+1/c=0`',1588062305,NULL,'','','','',0,'',''),(905,7,38.000,'If the matrices `A` & `B`, given by $ A = \\begin{bmatrix} x+y & y-z \\\\ 5-t & 7+x \\end{bmatrix} $, $ B = \\begin{bmatrix} t-x & z-t \\\\ z-y & x+z+t \\end{bmatrix} $, are equal, then find the values of `x, y, z, t`',1588062729,'`x=1, y=2, z=3, t=4`','905','','','',0,'',''),(906,8,30.000,'Solve. $ \\begin{vmatrix} 15-2x & 11-3x & 7-x \\\\ 11 & 17 & 14 \\\\ 10 & 16 & 13 \\end{vmatrix} = 0 $',1588063002,NULL,'','','','',0,'',''),(907,7,5.000,'If $ A = \\begin{bmatrix} 0 & x & 7 \\\\ -2 & z & 3 \\\\ y & w & 0 \\end{bmatrix} $ is a skew-symmetric matrix,then find the value of `x, y, z`',1588082030,'`x=2, y=-7, z=0`','907','','','',0,'',''),(908,7,12.000,'If `A` & `B` both are symmetric matrices, then the matrix `ABA` is -- (a) symmetric   (b) skew-symmetric   (c) diagonal   (d) none of these',1588082620,'(a) symmetric','908','','','',0,'',''),(909,7,3.000,'If `A = (a_{ij})_{n\\times\\n}` is a square matrix where `a_{ij} = i^2-j^2`, then matrix `A` is -- (a) zero matrix,   (b) unit matrix,   (c) symmetric matrix,   (d) skew-symmetric matrix',1588083123,'(d) skew-symmetric matrix','909','','','',0,'',''),(910,7,122.000,'For any square matrix `A`, wrong statement is -- (a) `(adj\\ A)^{-1} = adj\\ (A^{-1})`,   (b) `(A^T)^{-1} = (A^{-1})^T`,   (c) `(A^3)^{-1} = (A^{-1})^3`,   (d) none of these.',1588083386,NULL,'','','','',0,'',''),(911,7,14.000,'If the matrices `A`, `B` are such that `AB=A` and `BA=B`. Then `B^2` is -- (a) `B`,   (b) `A`,   (c) `I`,   (d) `O`',1588083600,'(a) `B`','911','','','',0,'',''),(912,8,84.000,'Prove that: $ \\begin{vmatrix} \\cos(x+y) & \\sin(x+y) & -\\cos(x+y) \\\\ \\sin(x-y) & \\cos(x-y) & \\sin(x-y) \\\\ \\sin2x & 0 & \\sin2y \\end{vmatrix} = \\sin2(x+y) $',1588092361,NULL,'','','','',0,'',''),(913,7,130.000,'Prove that inverse of any square matrix (if exists) is unique.',1588093733,NULL,'','','','',0,'',''),(914,8,112.000,'Prove that: $ \\begin{vmatrix} a & b-c & c+b \\\\ a+c & b & c-a \\\\ a-b & b+a & c \\end{vmatrix} = (a+b+c)(a^2+b^2+c^2) $',1588094752,NULL,'','','','',0,'',''),(915,8,38.000,'Without expanding, prove that: $ \\begin{vmatrix} 101 & 103 & 105 \\\\ 104 & 105 & 106 \\\\ 107 & 108 & 109 \\end{vmatrix} = 0 $',1588096487,NULL,'','','','',0,'',''),(916,7,86.000,'If $ A = \\begin{bmatrix} 1 & -2 & 2 \\\\ 0 & 1 & -1 \\\\ 0 & 0 & 1 \\end{bmatrix} $ and $ B = \\begin{bmatrix} 1 & 2 & 0 \\\\ 2 & 3 & -1 \\\\ 0 & -1 & -2 \\end{bmatrix} $, then show that the matrix `(A^TB)A` is a diagonal matrix.',1588098053,'N.A.','916','','','',0,'',''),(917,8,54.000,'Find the value (without expanding): $ \\begin{vmatrix} 41 & 1 & 5 \\\\ 79 & 7 & 9 \\\\ 29 & 5 & 3 \\end{vmatrix} $',1588098370,NULL,'','','','',0,'',''),(918,7,56.000,'If $ A = \\begin{bmatrix} 1 & 0 \\\\ \\frac{1}{2} & 1 \\end{bmatrix} $, then find the matrix `A^50`',1588102601,'$ \\begin{bmatrix} 1 & 0 \\\\ 25 & 1 \\end{bmatrix} $','918','','','',0,'',''),(919,7,28.000,'If the matrix $ \\begin{bmatrix} -x & x & 2 \\\\ 2 & x & -x \\\\ x & -2 & -x \\end{bmatrix} $ is non-singular, then value of `x` is -- (a) `-2\\le\\x\\le2`,   (b) Any real number except `\\pm2`,   (c) `x\\ge2`,   (d) `x\\le-2`',1588103058,'(b) Any real number except `\\pm2`','919','','','',0,'',''),(920,7,15.000,'If the matrix `A` is proper orthogonal, then value of `|A|` is -- (a) 0,  (b) 1,  (c) 2,   (d) 3',1588103442,'(b) 1','','','','',0,'',''),(921,7,13.000,'If a matrix `A` is both symmetric & skew-symmetric, then `A` should be -- (a) diagonal matrix,  (b) zero matrix,  (c) square matrix,   (d) none of these.',1588103773,'(b) zero matrix','921','','','',0,'',''),(922,7,20.000,'For any two square matrices `A` & `B`, when the matrix equation `A^2-B^2 = (A+B)(A-B)` holds true?',1588104378,'when `AB = BA`','922','','','',0,'',''),(923,7,16.000,'If `\\omega` is the imaginary cube root of 1 and $ A = \\begin{pmatrix} 1 & 0 \\\\ 0 & \\omega \\end{pmatrix} $, $ B = \\begin{pmatrix} \\omega & 0 \\\\ 0 & 1 \\end{pmatrix} $, then `(A+B)^47` is -- (a) `-I_2`,   (b) `\\omega\\I_2`,   (c) `-\\omega^2I_2`,   (d) `-\\omega\\I_2`',1588104951,'(d) `-\\omega\\I_2`','923','','','',0,'',''),(924,7,82.000,'If $ A = \\begin{bmatrix} k & 2 \\\\ 2 & k \\end{bmatrix} $ and `|A^3|=125`, then find value of `k`.',1588309855,NULL,'','','','',0,'',''),(925,8,45.000,'If `a, b, c` are in Arithmetic Progression (A.P.) , then show that $ \\begin{vmatrix} x+1 & x+2 & x+a \\\\ x+2 & x+3 & x+b \\\\ x+3 & x+4 & x+c \\end{vmatrix} = 0 $',1588311553,'','','','','',0,'',''),(926,8,61.000,'If `\\alpha`, `\\beta` are imaginary roots of `x^3-1=0`, then show that $ \\begin{vmatrix} \\lambda+1 & \\alpha & \\beta \\\\ \\alpha & \\lambda+\\beta & 1 \\\\ \\beta & 1 & \\lambda+\\alpha \\end{vmatrix} = \\lambda^3 $',1588312603,NULL,'','','','',0,'',''),(927,8,57.000,'If $ \\begin{vmatrix} x & z & y \\\\ a & c & b \\\\ 1 & 1 & 1 \\end{vmatrix} = 7 $, then show that $ \\begin{vmatrix} c-b & y-z & zb-yc \\\\ b-a & x-y & ay-xb \\\\ a-c & z-x & cx-az \\end{vmatrix}=49 $',1588319418,NULL,'','','','',0,'',''),(928,8,123.000,'Prove that:  $ \\begin{vmatrix} b+c & a+b & a \\\\ c+a & b+c & b \\\\ a+b & c+a & c \\end{vmatrix} = a^3+b^3+c^3-3abc $',1588319789,NULL,'','','','',0,'',''),(929,8,39.000,'Without expanding, prove that: $ \\begin{vmatrix} 441 & 442 & 443 \\\\ 445 & 446 & 447 \\\\ 448 & 449 & 450 \\end{vmatrix} = 0 $',1588320330,NULL,'','','','',0,'',''),(930,8,23.000,'If $ \\begin{vmatrix} x-1 & 1 & 1 \\\\ 1 & x+1 & 1 \\\\ -1 & 1 & x+1 \\end{vmatrix} =0 $, then show that values of `x` are `0, 1, -2`.',1588321517,NULL,'','','','',0,'',''),(931,7,55.000,'If $ A = \\begin{pmatrix} 3 & -4 \\\\ 1 & -1 \\end{pmatrix} $, then show that $ A^n = \\begin{pmatrix} 1+2n & -4n \\\\ n & 1-2n \\end{pmatrix} $ where `n \\in \\mathbb{N}` ',1588322739,'N.A.','931','','','',0,'',''),(932,8,108.000,'If `a^2+b^2+c^2=0` and $ \\begin{vmatrix} b^2+c^2 & ab & ac \\\\ ab & c^2+a^2 & bc \\\\ ca & bc & a^2+b^2 \\end{vmatrix} = ka^2b^2c^2 $, then find the value of `k`.',1588323904,NULL,'','','','',0,'',''),(933,7,74.000,'Find a matrix `A` such that $ A \\begin{bmatrix} 4 & -2 \\\\ 0 & 5 \\end{bmatrix} + \\begin{bmatrix} -1 & 3 \\\\ -9 & 6 \\end{bmatrix} = \\begin{bmatrix} 3 & 16 \\\\ 7 & 8 \\end{bmatrix} $',1588326566,NULL,'','','','',0,'',''),(934,8,90.000,'If `A, B, C` are three angles of a triangle, then prove that $ \\begin{vmatrix} \\sin\\ (A+B+C) & \\sin\\ (A+C) & \\cos\\ C \\\\ -\\sin\\ B & 0 & \\tan\\ A \\\\ \\cos\\ (A+B) & \\tan\\ (B+C) & 0 \\end{vmatrix} = 0 $',1588327212,NULL,'','','','',0,'',''),(935,7,21.000,'If $ A = \\begin{bmatrix} 1 & -1 \\\\ 2 & 3 \\end{bmatrix} $ and $ B = \\begin{bmatrix} 2 & 1 \\\\ 1 & 0 \\end{bmatrix} $, then show that `(A+B)^2 \\ne A^2+2AB+B^2`',1588505199,NULL,'','','','',0,'',''),(936,8,4.000,'If $ \\begin{vmatrix} 6i & -3i & 1 \\\\ 4 & 3i & -i \\\\ 20 & 3 & i \\end{vmatrix} = x+iy $, then find the values of `x` & `y` where `i=\\sqrt{-1}`',1588505563,NULL,'','','','',0,'',''),(937,8,1.000,'If $ A = \\begin{bmatrix} 11 & 20 \\\\ 31 & 41 \\end{bmatrix} $ and `|3A|=k|A|`, then find the value of `k`, where `|A|` is determinant of matrix `A`.',1588506067,NULL,'','','','',0,'',''),(938,7,67.000,'If $ A = \\begin{bmatrix} 3 & 2 \\\\ 1 & 1 \\end{bmatrix} $ and `a`, `b` are two real numbers such that `A^2+aA+bI=O` (where `I` & `O` are unit & zero matrix), then find `A^{-1}`',1588506752,'$ A^{-1} = \\begin{bmatrix} 7 & 2 \\\\ 1 & 5 \\end{bmatrix} $','','','','',0,'',''),(939,7,127.000,'If `A` is a skew symmetric matrix, then show that `A^2` is a symmetric matrix.',1588507569,NULL,'','','','',0,'',''),(940,7,88.000,'Show that the matrix $ \\begin{bmatrix} 1 & 1 & 3 \\\\ 5 & 2 & 6 \\\\ -2 & -1 & -3 \\end{bmatrix} $ is a Nilpotent matrix of index 3.',1588507979,'N.A.','940','','','',0,'',''),(941,8,74.000,'Prove that $ \\begin{vmatrix} 1 & a & a^2 \\\\ a^2 & 1 & a \\\\ a & a^2 & 1 \\end{vmatrix} = (a^3-1)^2 $',1588508683,NULL,'','','','',0,'',''),(942,8,59.000,'Find the value of $ \\begin{vmatrix} 1 & \\omega^n & \\omega^{2n} \\\\ \\omega^{2n} & 1 & \\omega^n \\\\ \\omega^n & \\omega^{2n} & 1 \\end{vmatrix} $',1588509253,NULL,'','','','',0,'',''),(943,7,27.000,'If $ A = \\begin{pmatrix} 2 & 0 & 0 \\\\ 0 & 2 & 0 \\\\ 0 & 0 & 2 \\end{pmatrix} $, then `A^5` is -- (a) `5A`,  (b) `10A`,  (c) `16A`,  (d) `32A`',1588509697,'(c) `16A`','943','','','',0,'',''),(944,8,82.000,'Prove that: $ \\begin{vmatrix} x^2+y^2+1 & x^2+2y^2+3 & x^2+3y^2+4 \\\\ y^2+2 & 2y^2+6 & 3y^2+8 \\\\ y^2+1 & 2y^2+3 & 3y^2+4 \\end{vmatrix} = x^2y^2 $',1588577225,NULL,'','','','',0,'',''),(945,8,121.000,'If `x^3=1`, then prove that $ \\begin{vmatrix} a & b & c \\\\ b & c & a \\\\ c & a & b \\end{vmatrix} = (a+bx+cx^2) \\begin{vmatrix} 1 & b & c \\\\ x^2 & c & a \\\\ x & a & b \\end{vmatrix} $',1588579534,NULL,'','','','',0,'',''),(946,7,128.000,'If `A` is a skew-symmetric matrix and `I+A` isa non-singular matrix, then show that `(I-A)(I+A)^{-1}` is an orthogonal matrix.',1588587626,NULL,'','','','',0,'',''),(947,8,22.000,'If $ A = \\begin{bmatrix} 1 & -1 & 0 \\\\ 1 & 2 & -1 \\\\ 3 & 2 & -2 \\end{bmatrix} $ then find the roots of `\\det\\ (A-xI_3)=0` where `I` is the unit matrix of order 3.',1588592655,NULL,'','','','',0,'',''),(948,7,80.000,'If $ A = \\begin{bmatrix} 2 & -1 \\\\ -4 & 3 \\end{bmatrix} $, then justify the following relation: `(A^2)^{-1} = (A^{-1})^2`',1588593132,NULL,'','','','',0,'',''),(949,8,31.000,'Without expanding prove that: $ \\begin{vmatrix} bc & a^2 & a^2 \\\\ b^2 & ca & b^2 \\\\ c^2 & c^2 & ab \\end{vmatrix} = \\begin{vmatrix} bc & ab & ca \\\\ ab & ca & bc \\\\ ca & bc & ab \\end{vmatrix} $',1588594301,'N.A.','949','','','',0,'',''),(950,8,48.000,'Without expanding prove that: $ \\begin{vmatrix} \\log_x\\ xyz & \\log_x\\ y & \\log_x\\ z \\\\ \\log_y\\ xyz & 1 & \\log_y\\ z \\\\ \\log_z\\ xyz & \\log_z\\ y & 1 \\end{vmatrix} = 0 $',1588595277,NULL,'','','','',0,'',''),(951,8,76.000,'Prove $ \\begin{vmatrix} a & b & 0 \\\\ 0 & a & b \\\\ b & 0 & a \\end{vmatrix} = a^3+b^3 $. Hence find the value of $ \\begin{vmatrix} 2ab & a^2 & b^2 \\\\ a^2 & b^2 & 2ab \\\\ b^2 & 2ab & a^2 \\end{vmatrix} $',1588598456,NULL,'','','','',0,'',''),(952,7,17.000,'If $ A = \\begin{pmatrix} 1 & 2 & x \\\\ 4 & -1 & 7 \\\\ 2 & 4 & -6 \\end{pmatrix} $ is a singular matrix, then value of `x` is -- (a) `0`,  (b) `1`,  (c) `3`,  (d)  `-3`',1588600398,'(d)  `-3`','952','','','',0,'',''),(953,7,85.000,'If $ A = \\begin{pmatrix} 0 & 2b & c \\\\ a & b & -c \\\\ a & -b & c \\end{pmatrix} $ and `A^TA=I`, then find the value of `a, b, c`',1588604089,'`a = \\pm \\frac{1}{\\sqrt{2}}, b = \\pm \\frac{1}{\\sqrt{6}}, c = \\pm \\frac{1}{\\sqrt{3}}`','','','','',0,'',''),(954,8,98.000,'Show that $ \\begin{vmatrix} -a^2 & ab & ac \\\\ ab & -b^2 & bc \\\\ ca & bc & -c^2 \\end{vmatrix} = 4a^2b^2c^2 $',1588604816,NULL,'','','','',0,'',''),(955,8,101.000,'Show that $ \\begin{vmatrix} -a(b^2+c^2-a^2) & 2b^3 & 2c^3 \\\\ 2a^3 & -b(c^2+a^2-b^2) & 2c^3 \\\\ 2a^3 & 2b^3 & -c(a^2+b^2-c^2) \\end{vmatrix} = abc(a^2+b^2+c^2)^3  $',1588606028,NULL,'','','','',0,'',''),(956,8,43.000,'Without expanding prove that: $ \\begin{vmatrix} a+b & 2a+b & 3a+b \\\\ 2a+b & 3a+b & 4a+b \\\\ 4a+b & 5a+b & 6a+b \\end{vmatrix} = 0 $',1588606614,NULL,'','','','',0,'',''),(957,8,114.000,'Prove that: $ \\begin{vmatrix} 1 & b+c & c^2 \\\\ 1 & c+a & a^2 \\\\ 1 & a+b & b^2 \\end{vmatrix} = (a-b)(b-c)(c-a) $',1588608693,NULL,'','','','',0,'',''),(958,9,125.000,'Solve. `xy \\frac{dy}{dx}-y^2=(x+y)^2e^{-y/x}`',1594174225,'`xe^{y/x}=(x+y)\\log|cx|`','958.001,958.002','','','',0,'',''),(959,9,176.000,'Let the rate of growth of a colony of bacteria be proportional to the square* root of the number of bacteria present at any time. When an experiment started, it was estimated that there were about 16 times of bacteria after 5 hours, then when it is 49 times? (*alt: cube)',1594174899,'','','','','',0,'',''),(960,9,3.000,'The order and degree of differential equation `\\frac{d^2y}{dx^2}+(\\frac{dy}{dx})^{1/3}+x^{1/4}=0` is -- (i) 2,3  (ii) 2,12  (iii) 2,6   (iv) 2,4',1594175170,'(i) 2,3','960','','','',0,'',''),(961,9,34.000,'Find the Integrating Factor of `(1+y^2)dx=(\\tan^{-1}y-x)dy`',1594175331,NULL,'','','','',0,'',''),(962,9,84.000,'Solve: `\\frac{dy}{dx}-y/x+\\csc\\frac{y}{x}=0` given `y=0` when `x=1`',1594175701,'`log|x|+1=\\cos\\frac{y}{x}`','962','','','',0,'',''),(963,9,157.000,'Show that the differential equation of all circles touching `y`-axis at origin is `2xy\\frac{dy}{dx}=y^2-x^2`',1594175817,NULL,'','','','',0,'',''),(964,9,135.000,'Solve: `1+y^2+(x-e^{-\\tan^{-1}y})\\frac{dy}{dx}=0`',1594176065,NULL,'','','','',0,'',''),(965,9,74.000,'Solve: `\\frac{dy}{dx}=\\frac{3x+2y}{2x-3y}` given `y(1)=0`',1594176255,NULL,'','','','',0,'',''),(966,9,7.000,'Order of differential equation `(\\frac{d^py}{dx^p})^2+(\\frac{d^qy}{dx^q})^s+y^{1/r}=0` (where `p\\gtq\\gtr\\gts`) is -- (i) 2 (ii) `s`  (iii) `2r`  (iv) `p`',1594176685,'(iv) `p`','','','','',0,'',''),(967,9,13.000,'Show that `v=A/r+B` satisfies the differential equation `\\frac{d^2v}{dr^2}+2/r\\cdot\\frac{dv}{dr}=0`',1594176848,'','','','','',0,'',''),(968,9,133.000,'Solve: `(\\tan^{-1}y-x)dy=(1+y^2)dx` given `y(0)=0`',1594177028,NULL,'','','','',0,'',''),(969,9,187.000,'If `\\gamma` be the increase in volume per degree centigrade of a cube of unit volume and `\\beta` be the increase in area per degree centigrade of each surface of the cube, then show that `2\\gamma=3\\beta`',1594177944,NULL,'','','','',0,'',''),(970,9,19.000,'Form the differential equation representing family of curves `y=A\\cos(x+B)` where `A` and `B` are parameters.',1594178216,'`\\frac{d^2y}{dx^2}+y=0`','','','','',0,'',''),(971,9,63.000,'Solve: `(x^2+y^2)dx-2xy\\ dy=0` given `y=0` when `x=1`',1594178370,'`y^2/x = x - 1`','','','','',0,'',''),(972,9,172.000,'A particle starts from the origin with a velocity 5 cm/s and moves in a straight line, its acceleration at time `t` seconds being `(3t^2-5t)` cm/`s^2`. Find velocity of the particle and its distance from the origin at the end of 4 seconds.',1594178834,'29 cm/sec, 30.67 cm (approx)','972','','','',0,'',''),(973,9,105.000,'Solve: `(1+x^2)\\frac{dy}{dx}+2xy=\\sqrt{x^2+4}`',1594883925,NULL,'','','','',0,'',''),(974,9,174.000,'A radio-active element satisfies the equation `\\frac{dv}{dt}=-kv` for its natural decay, where `v` is the volume of the element at time `t` and `k` is positive constant. If 40% of the substance disappear in 25 years, find the time it takes to disappear 60% of the substance.',1594884843,'44.8 years (approx)','974','','','',0,'',''),(975,9,65.000,'Solve: `(x^2y^2+xy)ydx+(x^2y^2-1)xdy=0`',1594885578,'`xy-\\log|y|=c`','975','','','',0,'',''),(976,9,188.000,'A right-circular conical water tank, with its vertex down and semi-vertical angle being `\\pi/6`, loses water out of a circular hole at its bottom at a rate of `\\pi\\ cm^3//sec`. Find the rate of change of radius of water level when it is 3 cm deep.',1594887172,NULL,'','','','',0,'',''),(977,9,5.000,'The order of the differential equation `(\\frac{d^4y}{dx^4})^3-\\frac{d^3y}{dx^3}=\\sqrt{1+\\frac{dy}{dx}}` is -- (i) 3  (ii) 4   (iii) 6   (iv) 7',1594887432,'(ii) 4','','','','',0,'',''),(978,9,38.000,'Find particular solution of `(x-y)(dx+dy)=dx-dy` where it is given that `y=-1` when `x=0`',1594887691,NULL,'','','','',0,'',''),(979,9,17.000,'Find the differential equation of all curves given by `y=Ae^{2x}+Be^{-x/2}` where `A` & `B` are non-zero parameters.',1594887874,'`2\\frac{d^2y}{dx^2}-3\\frac{dy}{dx}=2y`','','','','',0,'',''),(980,9,14.000,'Prove that `x=A\\cos\\sqrt{\\mu}t` is a solution of the differential equation `\\frac{d^2x}{dt^2}+\\mu\\x=0`',1594889741,'','','','','',0,'',''),(981,9,121.000,'Solve: `(1+x^2)\\frac{dy}{dx}+y=e^{\\tan^{-1}x}`',1594889844,NULL,'','','','',0,'',''),(982,9,91.000,'Solve: `\\frac{dy}{dx}=\\sin(x+y)+\\cos(x+y)`',1594890013,NULL,'','','','',0,'',''),(983,9,136.000,'Solve: `(xy^2-e^{1/x^3})dx-x^2ydy=0` when `y=0`, `x=1`',1594890290,NULL,'','','','',0,'',''),(984,9,112.000,'Solve: `\\frac{dy}{dx}+\\frac{y}{\\log_ex}=\\frac{2}{x}`',1594890339,NULL,'','','','',0,'',''),(985,9,127.000,'Solve: `ye^ydx=(y^3+2xe^y)dy` when `x=2`, `y=1`',1595309710,NULL,'','','','',0,'',''),(986,9,85.000,'Solve: `\\frac{dy}{dx}=\\frac{2x-y+2}{2y-4x+1}`',1595309948,NULL,'','','','',0,'',''),(987,9,64.000,'Solve: `x^2(y-1)dx+y^2(x-1)dy=0`',1595310108,NULL,'','','','',0,'',''),(988,9,104.000,'Solve: `x\\log\\x\\frac{dy}{dx}+y=2/x\\log\\x` where `x\\gt1`',1595310779,'','','','','',0,'',''),(989,9,137.000,'Solve: `y-x\\frac{dy}{dx}=2(1+x^2\\frac{dy}{dx})` given `y=1` when `x=1`',1595311734,NULL,'','','','',0,'',''),(990,9,77.000,'Solve: `(1+3e^{y/x})dy+3e^{y/x}(1-y/x)dx=0`',1595312229,'','','','','',0,'',''),(991,9,1.000,'The order & degree of the differential equation `[1+(\\frac{dy}{dx})^2]^{5/2}=\\frac{d^3y}{dx^3}` is -- (i) 2,3   (ii) 3,2   (iii) 3,5   (iv) 5,2',1595312776,'(ii) 3,2','','','','',0,'',''),(992,9,88.000,'Solve: `\\sin^{-1}(\\frac{dy}{dx})=x+y`',1595312944,NULL,'','','','',0,'',''),(993,9,55.000,'Solve: `xdx+ydy+\\frac{xdy-ydx}{x^2+y^2}=0` given `y=1` when `x=1`',1595313050,'`tan^{-1}(y/x) = - (\\frac{x^2+y^2}{2}) + \\frac{\\pi}{4} + 1`','','','','',0,'',''),(994,9,2.000,'The order & degree of the differential equation `(\\frac{d^3y}{dx^3})^{2/3}=a[1+(\\frac{dy}{dx})^6]^{3/2}` is -- (i) 3,4  (ii) 3,6  (iii) 3,2  (iv) 4,3',1595313298,'(i) 3,4','','','','',0,'',''),(995,9,75.000,'Solve: `\\frac{dy}{dx}=\\frac{y(2y-x)}{x(2y+x)}` given `y=1` when `x=1`',1595313652,NULL,'','','','',0,'',''),(996,9,83.000,'Solve: `\\frac{dy}{dx}=y/x{\\log\\(y/x)+1}`',1595314042,NULL,'','','','',0,'',''),(997,9,139.000,'Solve: `(x+\\tany)dy = sin2y\\ dx`',1595314298,NULL,'','','','',0,'',''),(998,9,106.000,'Solve: `(1+x^2)\\frac{dy}{dx}-2xy=(x^2+1)(x^2+2)`',1595314502,NULL,'','','','',0,'',''),(999,9,161.000,'The slope of a curve at point `(x,y)` is `\\frac{x^2+y^2}{2xy}` and the curve passing through the point `(1,0)`. Find the equation of curve.',1595314794,'`x^2-y^2=x`','999','','','',0,'',''),(1000,9,67.000,'Solve: `(3xy+y^2)dx+(x^2+xy)dy=0`',1595315139,NULL,'','','','',0,'',''),(1001,9,26.000,'If `\\frac{dy}{dx}+2x=e^{3x}` then find `y`',1595315406,NULL,'','','','',0,'',''),(1002,9,86.000,'Solve: `\\frac{dy}{dx}=(x+y)^2` given `y=1` when `x=0`',1595316124,NULL,'','','','',0,'',''),(1003,9,99.000,'Solve: `x\\frac{dy}{dx}-y=x^2`',1595316432,NULL,'','','','',0,'',''),(1004,9,27.000,'Find the values of `k` for which `y=k\\sinx` satisfies the differential equation `\\frac{d^2y}{dx^2}+2y=2`',1595604371,'`\\frac{2}{sinx}`','1004','','','',0,'',''),(1005,9,87.000,'Solve: `(x+y)^2\\frac{dy}{dx}=a^2`',1595604497,NULL,'','','','',0,'',''),(1006,9,134.000,'Solve: `(1+x^2)\\frac{dy}{dx}+y=\\tan^{-1}x`',1595605133,NULL,'','','','',0,'',''),(1007,9,129.000,'Solve: `x\\frac{dy}{dx}=y\\cdot(\\logy-\\logx+1)`',1595613910,NULL,'','','','',0,'',''),(1008,9,72.000,'Solve: `x(x-y)dy+y^2dx=0`',1595614337,NULL,'','','','',0,'',''),(1009,9,25.000,'Find general solution of `\\frac{dy}{dx}+y\\ g\'(x)=g(x)\\cdotg\'(x)`',1595643667,NULL,'','','','',0,'',''),(1010,9,41.000,'Solve: `\\frac{dy}{dx}=\\sqrt{4-y^2}`',1595644066,NULL,'','','','',0,'',''),(1011,9,130.000,'Solve: `\\frac{dy}{dx}-y\\tanx+y^2\\secx=0`',1595644454,NULL,'','','','',0,'',''),(1012,9,173.000,'If population growth rate is `5%`, in how many years, it becomes double?',1595644731,NULL,'','','','',0,'',''),(1013,9,8.000,'Find order & degree of differential equation `(\\frac{d^py}{dx^p})^2-(\\frac{d^qy}{dx^q})^{1/r}+(\\frac{d^sy}{dx^s})^t=0` where `p\\gt\\q\\gtr\\gts\\gtt`',1595646198,'Order = `p`, degree = `2r`','1013','','','',0,'',''),(1014,9,79.000,'Solve: `x\\frac{dy}{dx}=y+x\\tan\\frac{y}{x}` given `y=\\pi/2` when `x=1`',1595646666,NULL,'','','','',0,'',''),(1015,9,118.000,'Solve: `\\frac{dy}{dx}-3y\\cotx=\\sin2x`, given `y=2` when `x=\\pi/2`',1595647049,'``','','','','',0,'',''),(1016,9,128.000,'Solve: `(e^x+1)y\\ dy-(y^2+1)e^xdx=0`, given `y=0` when `x=0`',1595647651,NULL,'','','','',0,'',''),(1017,9,80.000,'Solve: `(x+y\\cos\\frac{y}{x})dx=x\\cos\\frac{y}{x}dy`',1595647988,NULL,'','','','',0,'',''),(1018,9,111.000,'Solve: `(x+1)\\frac{dy}{dx}-y=e^{3x}(x+1)^2`',1595653534,'`\\frac{y}{1+x} = \\frac{e^{3x}}{3} + c`','','','','',0,'',''),(1019,9,60.000,'Solve: `xdy-ydx=2\\sqrt{y^2-x^2}dx`',1595653743,'`- sin^{-1}(x/y) = 2x+c`','','','','',0,'',''),(1020,5,50.000,'Three daily newspapers X, Y, Z are published in a city. 65% of the people in the city read X magazine, 54% Y magazine and 45% Z magazine. 38% of people read X and Y; 32% of people read Y and Z; 28% read X and Z magazines and 12% do not read any of these three types of magazines. If the total population of the city is 1000000, determine how many people in the city read all three magazines.',1595749922,'220000','','','','',0,'',''),(1021,5,51.000,'In any exam, 70% of the candidates passed in Mathematics, 68% passed in Physics and 60% passed in both subjects. If 44 candidates fail in both the subjects, determine the total number of candidates.',1595750125,'200','','','','',0,'',''),(1022,5,52.000,'Out of 1000 students in a college, 540 play football, 465 play cricket and 370 play volleyball. Of the total students, 325 play football and cricket, 260 play football and volleyball, 235 play cricket and volleyball and 125 play each game. How many students (i) do not play any game (ii) play only one game and (iii) play exactly two games?',1595750369,'(i) 320, (ii) 110, (iii) 445 ','','','','',0,'',''),(1023,5,53.000,'There are several students in a group and everyone in the group can speak at least one of Bengali, Hindi and English. 65 students can speak Bengali, 54 can speak Hindi and 37 can speak English. 31 can speak both Bengali and Hindi, 17 can speak both Hindi and English and 18 can speak both Bengali and English. Determine the largest and smallest number of students in the group.',1595750621,'107, 90','1023','','','',0,'',''),(1024,9,61.000,'Solve: `x\\dy-y\\dx=\\sqrt{x^2+y^2}dx`',1597239498,'`tan^{-1}(y/x) = x+c`','','','','',0,'',''),(1025,9,78.000,'Solve: `(1+e^{x/y})dx+e^{x/y}(1-x/y)dy=0`',1597240092,'`x+ye^{\\frac{x}{y}}=c`','1025.001,1025.002','','','',0,'',''),(1026,9,12.000,'The differential equation of the family of lines passing through the origin is -- (i) `x\\frac{dy}{dx}=y`,  (ii) `x+\\frac{dy}{dx}=0`,  (iii) `\\frac{dy}{dx}=x`,  (iv) `x\\frac{dy}{dx}+y=0`',1597240563,NULL,'','','','',0,'',''),(1027,9,20.000,'Form the differential equation from `y=a\\secx+b\\tanx` where `a` & `b` are arbitrary constants.',1597240822,'`\\frac{d^2y}{dx^2}=y\\sec^2x+\\tanx\\frac{dy}{dx}`','1027','','','',0,'',''),(1028,9,81.000,'Solve: `x\\cos(y/x)(y\\dx+x\\dy)=y\\sin(y/x)(x\\dy-y\\dx)`',1597242015,'`xy\\cos\\frac{y}{x}=c`','1028.001,1028.002','','','',0,'',''),(1029,9,71.000,'Solve: `y\\dx+(x^2-xy)dy=0`',1597242442,NULL,'','','','',0,'',''),(1030,9,162.000,'Show that the equation of the curve, passing through the point `(1,0)` and satisfying the differential equation `(1+y^2)dx+xy\\dy=0` is `x^2-y^2=1`',1597242784,NULL,'','','','',0,'',''),(1031,9,140.000,'Solve: `\\frac{x+y\\frac{dy}{dx}}{x\\frac{dy}{dx}-y}=\\sqrt{\\frac{1-x^2-y^2}{x^2+y^2}}`',1597243228,'','','','','',0,'',''),(1032,9,184.000,'A spherical snowball melts at a rate proportional to its volume at that time. If half the ice melts in 30 minutes, then prove that the volume of the remaining ice is `1/8` part of original volume after 90 minutes of start of melting ice.',1597245876,NULL,'','','','',0,'',''),(1033,9,4.000,'Find order and degree of differential equation `2\\frac{d^2y}{dx^2}+3\\sqrt{1-(\\frac{dy}{dx})^3}-y=0`',1597249044,'Order = 2, degree = 2','1033','','','',0,'',''),(1034,9,131.000,'Solve: `dx+x\\dy=e^{-y}\\sec^2y\\ dy`',1597249535,NULL,'','','','',0,'',''),(1035,9,177.000,'Let the rate of growth of population in a city be proportional to number of citizen at any time. When an experiment started, it was estimated that there were about double of population after 30 years, then when it is three times? Given `\\log_e3=1.6\\log_e2`',1597249954,NULL,'','','','',0,'',''),(1036,9,141.000,'Solve: `(x+2y^3)\\frac{dy}{dx}=y`',1597250169,NULL,'','','','',0,'',''),(1037,9,94.000,'Solve: `e^\\{\\frac{dy}{dx}}=x+1` given `y(0)=3`',1597250387,NULL,'','','','',0,'',''),(1038,9,180.000,'The volume of a spherical balloon increases at a rate of 10 cc per second. Find the rate of change of its outer surface when its radius is 16 cm. ',1597250608,NULL,'','','','',0,'',''),(1039,9,96.000,'Solve: `\\log(\\frac{dy}{dx})=4x-2y-2` given `y(1)=1`',1597250964,NULL,'','','','',0,'',''),(1040,9,117.000,'Solve: `\\frac{dy}{dx}+y\\cotx=2\\cosx`',1597251349,'`y\\ sinx = - \\frac{cos2x}{2} + c`','','','','',0,'',''),(1041,9,93.000,'Solve: `x\\frac{dy}{dx}=y+\\sqrt{x^2+y^2}`',1597251826,NULL,'','','','',0,'',''),(1042,9,160.000,'The slope of a curve at a point `(x,y)` is `\\frac{3y+2x+4}{4x+6y+5}` and the curve passing through the point `(0,-1)`. Find the equation of the curve.',1597252222,'','','','','',0,'',''),(1043,9,170.000,'A circular ink drop increases at a rate of 2 square cm per second. Find the rate of change of its radius at time `2\\frac{6}{11}` sec.',1597252527,NULL,'','','','',0,'',''),(1044,9,6.000,'The order and degree of differential equation `\\sqrt{y+\\frac{d^2y}{dx^2}}=x+(\\frac{dy}{dx})^{3/2}` is -- (i) 2, 2   (ii) 2, 3   (iii) 2, 4   (iv) 2, 1',1597253041,'(i) 2, 2','1044','','','',0,'',''),(1045,9,123.000,'Solve: `(1+x)\\frac{dy}{dx}-\\tany=(1+x)^2e^x\\secy`',1597299757,NULL,'','','','',0,'',''),(1046,9,108.000,'Solve: `(x^2-1)\\frac{dy}{dx}+2xy=\\frac{2}{x^2-1}`',1597299969,NULL,'','','','',0,'',''),(1047,9,138.000,'Solve: `y-x\\frac{dy}{dx}=a(y^2+\\frac{dy}{dx})`',1597300187,NULL,'','','','',0,'',''),(1048,9,165.000,'A point is moving through a parabolic curve `y^2=4x`. At which point abscissa increases at twice the rate the ordinate increases?',1597300691,'','','','','',0,'',''),(1049,9,56.000,'Solve: `x^2(x\\ dx+y\\ dy)+2y(x\\ dy-y\\ dx)=0`',1597301471,NULL,'','','','',0,'',''),(1050,9,102.000,'Solve: `(1-x^2)\\frac{dy}{dx}-xy=x`; given `y=2` when `x=0`',1597301986,NULL,'','','','',0,'',''),(1051,9,132.000,'Solve: `(1+y^2)(1+\\logx)dx+x\\ dy=0`; given `y=1` when `x=1`',1597302533,NULL,'','','','',0,'',''),(1052,9,182.000,'The rate of change of radius of a sphere is `\\frac{1}{2\\pi}` with respect to time. Find the rate of change of area of its outer surface when radius is 5 cm.',1597302828,NULL,'','','','',0,'',''),(1053,9,103.000,'Solve: `(x+3)\\frac{dy}{dx}-3y=(x+3)^2`',1597303037,NULL,'','','','',0,'',''),(1054,9,186.000,'The volume of a cube changes in such a way that it remains a cube after that change. Show for for a cube (whose volume is 1 cube unit), rate of change of volume = `3/2 \\times` (rate of change of anyone surface area)',1597304209,NULL,'','','','',0,'',''),(1055,9,143.000,'Solve: `(x+2y^2)\\frac{dy}{dx}=y`, where `y\\gt0`',1597305798,NULL,'','','','',0,'',''),(1056,9,166.000,'A point moves through a curve `6y=x^3+2`. At which point ordinate increases at 8 times the rate of abscissa increases?',1597306379,NULL,'','','','',0,'',''),(1057,9,33.000,'Find the integrating factor of `x\\frac{dy}{dx}-y=x\\logx`',1597308362,NULL,'','','','',0,'',''),(1058,9,144.000,'Solve: `y^2+(x-1/y)\\frac{dy}{dx}=0`',1597309084,'','','','','',0,'',''),(1059,9,29.000,'Solve: `\\frac{dy}{dx}=e^{x+y}+x^2e^y` (alt: `\\frac{dy}{dx}=e^{x-y}+x^2e^{-y}`)',1597309671,'','','','','',0,'',''),(1060,9,70.000,'Solve: `(x^2+xy)dy=(x^2+y^2)dx`',1597309917,NULL,'','','','',0,'',''),(1061,9,46.000,'Solve: `(1+x^2)\\frac{dy}{dx}=x+\\tan^{-1}x`',1597310192,NULL,'','','','',0,'',''),(1062,9,58.000,'The general solution of differential equation `y^2(x\\ dy+y\\ dx)+(x\\ dy-y\\ dx)=0` is -- (i) `xy=x/y+c`,  (ii) `x+y=xy+c`,   (iii) `log|x+y|=cx`,  (iv) `log|x/y|=x/y+c`',1597311305,'(i) `xy=x/y+c`','','','','',0,'',''),(1063,9,154.000,'If the differential equation of all straight lines, whose distance from origin is10 unit, is `(y-x\\frac{dy}{dx})^2=A{1+(\\frac{dy}{dx})^2}`, then find value of `A`.',1597311765,NULL,'','','','',0,'',''),(1064,9,155.000,'Find the differential equation of all ellipses, who center is at origin. ',1597311966,NULL,'','','','',0,'',''),(1065,9,156.000,'Find the differential equation of all circles touching `x`-axis at origin.',1597312127,NULL,'','','','',0,'',''),(1066,9,73.000,'Solve: `\\frac{dy}{dx}+\\frac{y(x+y)}{x^2}=0`',1597312645,NULL,'','','','',0,'',''),(1067,9,44.000,'Solve: `\\frac{dy}{dx}=\\frac{\\cos^2y}{1+x^2}`; given that `y(0)=0`',1597312982,NULL,'','','','',0,'',''),(1068,9,89.000,'Solve: `\\cos^{-1}(\\frac{dy}{dx})=x+y`',1597313203,NULL,'','','','',0,'',''),(1069,9,49.000,'Solve: `\\cosy\\ dx+(1+e^{-x})\\siny\\ dy=0`; given `y=\\pi/4` when `x=0`',1597314440,NULL,'','','','',0,'',''),(1070,9,168.000,'If the area of a circle increases uniformly with respect to time, then show that the rate of increment of its circumference is inversely proportional to its radius. ',1597317456,'','','','','',0,'',''),(1071,9,50.000,'Solve: `e^x\\tany\\ dx+(1-e^x)\\sec^2y\\ dy=0`',1597332667,'`tany=c(1-e^x)`','1071','','','',0,'',''),(1072,9,90.000,'Solve: `\\tan^{-1}(\\frac{dy}{dx})=x+y`',1597332920,NULL,'','','','',0,'',''),(1073,9,95.000,'Solve: `\\log(\\frac{dy}{dx})=3x-5y`; given that `y=0` when `x=0`',1597333293,'`3e^{5y}=5e^{3x}-2`','1073','','','',0,'',''),(1074,9,142.000,'Solve: `(2x-10y^3)\\frac{dy}{dx}+y=0`',1597333878,NULL,'','','','',0,'',''),(1075,9,82.000,'Solve: `(y\\sin\\frac{y}{x}-x\\cos\\frac{y}{x})x\\ dy=(x\\cos\\frac{y}{x}+y\\sin\\frac{y}{x})y\\ dx`',1597335080,'`xy\\cos\\frac{y}{x}=c`','1028.001,1028.002','','','',0,'',''),(1076,9,124.000,'Solve: `x\\cosx\\frac{dy}{dx}+y(x\\sinx+\\cosx)=1`',1597336041,NULL,'','','','',0,'',''),(1077,9,30.000,'Solve: `\\frac{dy}{dx}=e^{4x-3y}`; given `y(0)=0`',1597336474,NULL,'','','','',0,'',''),(1078,9,145.000,'Solve: `(x+y+1)\\frac{dy}{dx}=1`; given that `y(0)=0`',1597336842,NULL,'','','','',0,'',''),(1079,9,21.000,'Find the differential equation of all curves satisfying equation `k(y+k)^2=x^3`, where `k` is a parameter.',1597337281,'`144xy^2(\\frac{dy}{dx})^10+648x^2y(\\frac{dy}{dx})^8+729x^3(\\frac{dy}{dx})^6=64x^3(\\frac{dy}{dx})^12`','1079','','','',0,'',''),(1080,9,66.000,'Solve: `(4x^2y-6)dx+x^3dy=0`',1597337966,'`xy + \\frac{3}{2x} = c`','','','','',0,'',''),(1081,9,23.000,'Find the differential equation of `x^2+y^2=2ax` where `a` is a parameter.',1597369404,NULL,'','','','',0,'',''),(1082,9,18.000,'Find the differential equation of `y=Ax+B/x` by eliminating two arbitrary constants `A` & `B`.',1597369969,'`x^2\\frac{d^2y}{dx^2}+x\\frac{dy}{dx}-y=0`','','','','',0,'',''),(1083,9,36.000,'The Integrating Factor of `(x+y+1)\\frac{dy}{dx}=1` is -- (i) `e^x`,   (ii) `e^{-x}`,   (iii) `e^y`,   (iv) `e^{-y}`',1597371042,NULL,'','','','',0,'',''),(1084,9,178.000,'Let `r` be the radius of any sphere. If `s` & `v` be the area of outer surface and volume of this sphere respectively, then show that `2\\frac{dv}{dt}=r\\frac{ds}{dt}`',1597573335,NULL,'','','','',0,'',''),(1085,9,92.000,'Solve: `\\log(x+y)^{x+y}(1+\\frac{dy}{dx})=1`',1597573613,NULL,'','','','',0,'',''),(1086,9,68.000,'Solve: `(xy^2+x)dx+(x^2y+y)dy=0` when `x=0`, `y=1`',1597824589,'`x^2y^2 + x^2 + y^2 = 1`','','','','',0,'',''),(1087,9,31.000,'Solve: `\\frac{dy}{dx}=1+e^{2x-y}` ; given `y(2)=2`',1597824772,'`e^{2x}-e^y=(e^2-1)e^x`','1087','','','',0,'',''),(1088,9,114.000,'Solve: `\\cos^2x\\frac{dy}{dx}+y=\\tanx` ;  where `0\\lex\\le\\frac{\\pi}{2}`',1597825178,'`y = tanx -1 +c e^{-tanx}`','','','','',0,'',''),(1089,9,115.000,'Solve: `\\frac{dy}{dx}+y\\secx=\\tanx` ; where `0\\le\\x\\le\\frac{\\pi}{2}`',1597825571,'`y\\ (secx+tanx) = secx + tanx - x + c`','','','','',0,'',''),(1090,9,28.000,'Solve: `(1+\\cosx)dy=(1-\\cosx)dx`',1597826759,'`y=2\\tan\\frac{x}{2}-x+c`','','','','',0,'',''),(1091,9,169.000,'If the circumference of a circle increases uniformly, then show that the rate of increment of its area is proportional to its radius. ',1597827123,NULL,'','','','',0,'',''),(1092,9,113.000,'Solve: `\\frac{dy}{dx}+\\frac{y}{x\\log_ex}=2/x`',1597827493,'`ylogx = (logx)^2 + c`','','','','',0,'',''),(1093,9,189.000,'A right-circular inverted conical water tank has height 18 inch and diameter of base 10 inch. Water is poured in this tank at a rate of 4 cubic inches per minute. Find the rate of increment of surface (alt, rate of increment of height) of water when depth of water is 12 inch?',1597828121,'','','','','',0,'',''),(1094,9,98.000,'Solve: `\\frac{dy}{dx}+2/xy=x^3`',1597828352,NULL,'','','','',0,'',''),(1095,9,122.000,'Solve: `\\frac{dy}{dx}=\\frac{x\\sqrt{x^2-1}+y}{\\sqrt{x^2-1}}` ; given `y=1` when `x=1`',1597829324,NULL,'','','','',0,'',''),(1096,9,9.000,'Find order & degree of differential equation `\\frac{d^4y}{dx^4}+\\sin(\\frac{d^3y}{dx^3})=0`',1597829469,'Order =4, degree = undefined','','','','',0,'',''),(1097,9,183.000,'A balloon which always remains spherical on inflation, is being inflated by pumping in 40 c.c. per minute. Find the rate of change of surface area when radius is 8 c.m. Find also the increase in radius in the next `1/2` minute.',1597830454,NULL,'','','','',0,'',''),(1098,9,148.000,'Solve: `\\sqrt{1+x^2+y^2+x^2y^2}+xy\\frac{dy}{dx}=0`',1597839019,NULL,'','','','',0,'',''),(1099,9,22.000,'Find the differential equation of `ax+by+c=0` by removing three arbitrary constants `a, b, c`.',1597839171,'`\\frac{d^2y}{dx^2}=0`','','','','',0,'',''),(1100,9,15.000,'Examine that `y=cx+2c^2` is a general solution of the differential equation `2(\\frac{dy}{dx})^2+x\\frac{dy}{dx}-y=0`.',1597839693,NULL,'','','','',0,'',''),(1101,9,16.000,'The differential equation representing the family of curves `y^2=2c(x+\\sqrt{c})` where `c\\gt0` is a parameter. Find the order and degree of this differential equation.',1597840337,'Order = 1, degree = 3','1101','','','',0,'',''),(1102,9,11.000,'Find order & degree of differential equation of `\\sqrt{\\sinx}(dx+dy)=\\sqrt{\\cosx}(dx-dy)`.',1597841058,'Order = 1, degree = 1','1102','','','',0,'',''),(1103,9,149.000,'Solve: `\\frac{dy}{dx}=\\frac{1}{x\\coty+\\secy}`',1597841342,NULL,'','','','',0,'',''),(1104,9,54.000,'Solve: `x\\ dy-y\\ dx=(x^2+y^2)dx`',1597841479,'`tan^{-1}(y/x) = x+c`','','','','',0,'',''),(1105,9,119.000,'Solve: `\\frac{dy}{dx}+y\\cotx=2x+x^2\\cotx` ; where `y(0)=0`',1597843150,'`y sinx = x^2 sinx`','','','','',0,'',''),(1106,9,179.000,'Spherical raindrops are evaporated. The rate of decrement of volume of raindrop is proportional to its surface area. Prove that the radius of spherical raindrop changes uniformly.',1597843521,NULL,'','','','',0,'',''),(1107,9,39.000,'Solve: `(x+y)(dx-dy)=dx+dy`',1597844090,NULL,'','','','',0,'',''),(1108,9,181.000,'The rate of increment of surface area of a spherical bubble is `2\\ cm^2`/`s`. Find the rate of increment of its volume when radius of the bubble is 6 cm.',1597845023,NULL,'','','','',0,'',''),(1109,9,100.000,'Solve: `x\\frac{dy}{dx}-y=\\logx` ; given `y(1)=0`',1597845225,NULL,'','','','',0,'',''),(1110,9,59.000,'General solution of `x^2(x\\ dy+y\\ dx)+(x\\ dy-y\\ dx)=0` is -- (i) `xy+y/x=c`,   (ii) `xy-y/x=c`,   (iii) `xy+x/y=c`,   (iv) `x/y-xy=c`',1597845612,'(i) `xy+y/x=c`','','','','',0,'',''),(1111,9,109.000,'Solve: `(1-x^2)\\frac{dy}{dx}-2xy=x - x^3`',1597845829,NULL,'','','','',0,'',''),(1112,9,24.000,'Show that the differential equation of `\\frac{x^2}{a^2+\\lambda}+\\frac{y^2}{b^2+\\lambda}=1` (`a, b` fixed constants & `\\lambda` arbitrary constant) is `(x+y\\frac{dy}{dx})(x\\frac{dy}{dx}-y)=(a^2-b^2)\\frac{dy}{dx}`',1597846895,NULL,'','','','',0,'',''),(1113,9,146.000,'Solve: `\\frac{dy}{dx}=\\frac{y^2+y+1}{x^2+x+1}`',1597847080,NULL,'','','','',0,'',''),(1114,9,163.000,'A particle is moving with velocity `u` through a straight line and its acceleration is equal to its displacement. At an instant, if displacement be `x` and velocity be `v`, then prove that `v^2=u^2+x^2`',1597847619,NULL,'','','','',0,'',''),(1115,9,35.000,'The integrating factor of `x(x-1)\\frac{dy}{dx}-y=x^2(x-1)^2` is -- (i) `\\frac{x-1}{x}`,   (ii) `\\frac{x}{x-1}`,   (iii) `\\frac{1-x}{x}`,   (iv) `\\frac{x}{1-x}`',1597851303,NULL,'','','','',0,'',''),(1116,9,10.000,'Find order & degree of differential equation `\\frac{d^3y}{dx^3}+x(\\frac{dy}{dx})^4=4\\ln(\\frac{d^4y}{dx^4})`.\r\n',1597851523,'Order =4, degree = undefined','','','','',0,'',''),(1117,9,164.000,'Find the differential equation of all parabolas, whose axes are along `y`-axis and vertices are at origin.',1597851982,NULL,'','','','',0,'',''),(1118,9,116.000,'Solve: `\\sinx\\frac{dy}{dx}+3y=\\cosx`',1597994287,NULL,'','','','',0,'',''),(1119,9,120.000,'Solve: `\\frac{dy}{dx}=-\\frac{x+y\\cosx}{1+\\sinx}`',1597994639,'`y\\ (1+sinx) +x^2/2 = c`','','','','',0,'',''),(1120,9,48.000,'Solve: `y\\ dx+\\sqrt{1-x^2}\\sin^{-1}x\\ dy=0`',1597994858,NULL,'','','','',0,'',''),(1121,9,97.000,'Solve: `\\frac{dy}{dx}-y=e^x` ; given `y(0)=1`',1597995653,NULL,'','','','',0,'',''),(1122,9,151.000,'Solve: `(xy\\cosxy+\\sinxy)dx+x^2\\cosxy\\ dy=0`',1597996073,NULL,'','','','',0,'',''),(1123,9,51.000,'Solve: `(e^x+1)y\\ dy-(y^2+1)e^xdx=0` ; given `y(0)=0`',1597996866,'','','','','',0,'',''),(1124,9,45.000,'Solve: `\\frac{dy}{dx}=\\frac{\\cos(\\logx)}{\\logy}`',1597997234,NULL,'','','','',0,'',''),(1125,9,57.000,'The general solution of differential equation `ydx+xdy=xy(dy-dx)` is -- (i) `\\ln(cxy)=x-y`,   (ii) `\\ln(xy)=cxy`,   (iii) `\\ln(cxy)=y-x`,   (iv) None of these.',1597997611,'(i) `\\ln(cxy)=x-y`','','','','',0,'',''),(1126,9,167.000,'If the rate of change of area of a circle is equal to rate of change of its diameter, then radius of the circle is -- (i) `\\frac{2}{\\pi}`,   (ii) `\\frac{1}{\\pi}`,   (iii) `\\frac{\\pi}{2}`,   (iv) `\\pi`',1597998112,NULL,'','','','',0,'',''),(1127,9,32.000,'The general solution of differential equation `\\frac{dy}{dx}+x^2=x^2e^3y` is -- (i) `1-e^3y=ce^{x^3}`,   (ii) `1-e^3y=ce^{-x^3}`,   (iii) `(1-e^{-3}y)^2=c^2e^{2x^3}`,   (iv) None of these.',1597999011,NULL,'','','','',0,'',''),(1128,9,62.000,'Solve: `(x-\\sqrt{xy})dy=y\\ dx`',1597999446,NULL,'','','','',0,'',''),(1129,9,171.000,'If the rate of increment of radius of a circle is `\\frac{1}{\\pi}`, then find the rate of change of its area when radius is 2 unit.',1598000072,NULL,'','','','',0,'',''),(1130,9,185.000,'The rate of change of volume of a cube is constant. Prove that the rate of change of total surface area of this cube is inversely proportional to its side. ',1598000474,NULL,'','','','',0,'',''),(1131,9,101.000,'Solve: `(1-x^2)\\frac{dy}{dx}-xy=1`',1598000663,NULL,'','','','',0,'',''),(1132,9,40.000,'Solve: `(x\\cosx)\\frac{dy}{dx}+(x\\sinx+\\cosx)y=1`',1598001006,NULL,'','','','',0,'',''),(1133,9,150.000,'Solve: `\\frac{dy}{dx}=\\frac{1}{x\\cosy+\\sin2y}`',1598001209,'','','','','',0,'',''),(1134,9,42.000,'Solve: `\\frac{dy}{dx}=1-x+y-xy`',1598002391,'`log|1+y|=x-x^2/2+c`','1134','','','',0,'',''),(1135,9,107.000,'Solve: `(1+x^2)\\frac{dy}{dx}+2xy=\\cotx`',1598003332,NULL,'','','','',0,'',''),(1136,9,152.000,'Solve: `\\frac{dy}{dx}+y/x\\logy=\\frac{y}{x^2}(logy)^2`',1598017671,'','','','','',0,'',''),(1137,9,69.000,'Solve: `(x^2-yx^2)dy+(y^2+x^2y^2)dx=0`',1598018162,'`log |y| +1/y + 1/x = x + c`','','','','',0,'',''),(1138,9,53.000,'Solve: `y\\ dx-x\\ dy=xy\\ dx`, when `x=1`, `y=1`',1598018466,'`log |x/y| = x - 1`','','','','',0,'',''),(1139,9,147.000,'Solve: `(1+y+x^2y)dx+(x+x^3)dy=0`',1598018707,NULL,'','','','',0,'',''),(1140,9,52.000,'Solve: `(1+e^{2x})dy+e^x(1+y^2)dx=0` ; given `y(0)=1`',1598018985,'`tan^{-1}y = -1/2 log(1+e^{2x})+\\frac{\\pi}{4} + 1/2 log2`','','','','',0,'',''),(1141,9,158.000,'Find the differential equation of all circles, lying at first co-ordinate and touch both axes.',1598019455,NULL,'','','','',0,'',''),(1142,9,175.000,'At any instant, the rate of disintegration of Uranium is proportional to its mass. If the amount of Uranium at time `t_1` and `t_2` is `m_1` and `m_2` respectively, then show that half life of uranium is `\\frac{(t_2-t_1)log2}{logm_1-logm_2}`',1598020265,NULL,'','','','',0,'',''),(1143,9,76.000,'Solve: `x^2y\\ dx-(x^3+y^3)dy=0`',1598023506,NULL,'','','','',0,'',''),(1144,9,47.000,'Solve: `y-x\\frac{dy}{dx}=y^2+\\frac{dy}{dx}`',1598023932,NULL,'','','','',0,'',''),(1145,9,159.000,'Find the differential equation of all circles whose centres are on `x`-axis and radius variable. ',1598024654,NULL,'','','','',0,'',''),(1146,9,37.000,'The integrating factor of `\\frac{dx}{dy}+x/y=\\frac{1}{y^2}`  is -- (i) `logy`,  (ii) `e^y`,   (iii) `y`,   (iv) `1/y`',1598024906,NULL,'','','','',0,'',''),(1147,9,43.000,'Solve: `x\\sqrt{1-y^2}dx+y\\sqrt{1-x^2}dy=0`',1598028262,NULL,'','','','',0,'',''),(1148,9,126.000,'Solve: `(xe^{y/x}+y)dx=xdy` ; given `y(1)=1`',1598028717,NULL,'','','','',0,'',''),(1149,9,110.000,'Solve: `(x-y^2)dx+2xy\\ dy=0`',1598029192,NULL,'','','','',0,'',''),(1150,9,153.000,'Solve: `x(\\frac{dy}{dx}+y)=1-y`',1598030655,NULL,'','','','',0,'',''),(1151,9,190.000,'Solve: `\\tanx\\frac{dy}{dx}=1+y^2`, given `y=1` when `x=\\frac{\\pi}{2}`',1599383936,'`\\tan^{-1}y-log|sinx|=\\frac{\\pi}{4}`','1151','','','',0,'',''),(1152,9,191.000,'Solve: `(e^y+1)cosx\\ dx+e^y\\sinx\\ dy=0`',1599383996,'`|sinx|(e^y+1)=|c|`','1152','','','',0,'',''),(1153,9,192.000,'Solve: `xy\\frac{dy}{dx}=\\frac{1+y^2}{1+x^2}(1+x+x^2)`',1599384066,'`log(1+y^2)=2\\tan^{-1}x+\\logx^2+c`','1153','','','',0,'',''),(1154,9,193.000,'Solve: `cosy\\log(secx+tanx)\\ dx=cosx\\log(secy+tany)\\ dy`',1599384721,'`[\\log(\\secx+\\tanx)]^2-[\\log(\\secy+\\tany)]^2=c`','1154','','','',0,'',''),(1155,10,18.000,'A relation `\\rho` is defined on the set of all natural numbers `\\mathbb{N}` by : `(x,y) \\in \\rho => (x-y)` is divisible by 5 `AA x,y \\in \\mathbb{N}`. Prove that `\\rho` is an equivalence relation on `\\mathbb{N}`',1600499048,'','','','','',0,'',''),(1156,10,5.000,'`\\mathbb{R}` is a set of all real numbers. If a relation `\\rho` over set `\\mathbb{R}` is defined such that `\\rho = {(a,b) : a-b \\lt 3, AA a,b \\in \\mathbb{R}}`, then the relation `\\rho` is -- (a) transitive,  (b) equivalence,  (c) reflexive,  (d) symmetric.',1600499058,'','','','','',0,'',''),(1157,11,19.000,'Let `\\mathbb{R}` be the set of all real numbers and for all `x \\in \\mathbb{R}`, the mapping `f : \\mathbb{R} -> \\mathbb{R}` is defined by `f(x)=ax+2`. If `(f \\circ f)=I_{\\mathbb{R}}`, then find the value of `a`',1600499805,'','','','','',0,'',''),(1158,11,3.000,'Let `\\mathbb{I}` be the set of all integers. A function `f : \\mathbb{I} -> \\mathbb{I}` is defined by `f(x)=3x^2-14x+10`. Then find `f^{-1}(2)`',1600500733,NULL,'','','','',0,'',''),(1159,11,7.000,'Let `A = \\mathbb{R} - {2}` and `B = \\mathbb{R} - {1}`. Show that the function `f : A -> B` defined by `f(x)=\\frac{x-3}{x-2}` is bijective, where `\\mathbb{R}` is the set of all real numbers.',1600501514,NULL,'','','','',0,'',''),(1160,10,1.000,'The total number of relations from set `A={a,b,c}` to set `B={d,e}` is -- (a) `2^6`,  (b) `2^8`,  (c) `2^4`,  (d) `2^15`',1600501876,NULL,'','','','',0,'',''),(1161,10,14.000,'Show that the relation `\\rho = {(x,y) : x\\ley}` defined over the set of all real numbers `\\mathbb{R}` is reflexive and transitive, but not symmetric.',1600502127,NULL,'','','','',0,'',''),(1162,11,1.000,'A function `f : \\mathbb{R} -> \\mathbb{R}` is defined by `f(x)=(x-1)(x-2)`. Then which of the following is true? (a) function is one-one but not onto,  (b) function is onto but not one-one,  (c) function is neither one-one nor onto,  (d) function is one-one and onto.',1600502905,NULL,'','','','',0,'',''),(1163,11,4.000,'Show that the modulus function `f : \\mathbb{R} \\rarr \\mathbb{R}`, given by `f(x) = |x|` is neither one-one nor onto, where $|x| = \\begin{cases} x, & \\text{when } x\\ge0 \\\\ -x, & \\text{when } x\\lt0 \\end{cases}$',1600518056,NULL,'','','','',0,'',''),(1164,11,12.000,'Let the function `f : \\mathbb{R} \\rarr \\mathbb{R}` be defined by `f(x)=x^3-6`, for all `x \\in \\mathbb{R}`. Show that `f` is bijective. Also find a formula that defines `f^{-1}(x)`',1600518288,NULL,'','','','',0,'',''),(1165,11,9.000,'A mapping `f : \\mathbb{R}-{1/2} \\rarr \\mathbb{R}-{1/2}` is defined by `f(x) = \\frac{x+2}{2x-1}`. Show that `f^{-1}` exists and find formula of `f^{-1}`. ',1600518592,NULL,'','','','',0,'',''),(1166,10,4.000,'Let `A = {8, 9, 10, 11}` and `B = {2, 3, 4, 5}` be two sets. A relation `\\rho` is defined from set `A` to set `B` by `x\\rhoy \\implies x` is divisible by `y`. Find domain of relation `\\rho`.',1600518958,NULL,'','','','',0,'',''),(1167,10,17.000,'A relation `\\rho` defined on plane `\\mathbb{R}^2` by : `\\rho = [ {(a,b),(c,d)} \\in \\mathbb{R}^2 \\times \\mathbb{R}^2 | a^2+b^2 = c^2+d^2 ]`. Show that the relation `\\rho` is an equivalence relation on `\\mathbb{R}^2`.',1600519409,NULL,'','','','',0,'',''),(1168,11,5.000,'Show that the mapping `f : \\mathbb{N} \\rarr \\mathbb{N}` defined by `f(x)=2^x` is injective, but not surjective mapping.',1600519779,NULL,'','','','',0,'',''),(1169,11,18.000,'If `f(x)=x^2`, `g(x)=\\tanx` and `h(x)=\\logx`, then value of  `(h \\circ (g \\circ f))(\\sqrt{\\frac{\\pi}{4}})` is -- (a) `0`,  (b) `1`,  (c) `\\frac{1}{\\pi}`,  (d) `1/2\\log\\frac{\\pi}{4}`',1600520087,NULL,'','','','',0,'',''),(1170,11,10.000,'Let `f : A \\rarr A` is a mapping (where `A = {x | -1 \\le x \\le1}`) defined by `f(x) = \\sin\\frac{\\pi x}{2}`. Show that the mapping `f` is a one-one, onto mapping. Also find a formula that defines `f^{-1}`',1601053755,NULL,'','','','',0,'',''),(1171,11,17.000,'Let `\\mathbb{R}` be a set of all real numbers and `f : \\mathbb{R} \\rarr \\mathbb{R}` is a mapping defined by `f(x) = 2x+1`. If `(g \\circ f)(x)=10x+10`, then find the mapping `g : \\mathbb{R} \\rarr \\mathbb{R}`',1601054285,NULL,'','','','',0,'',''),(1172,11,15.000,'Two mappings `f : \\mathbb{R} \\rarr \\mathbb{R}` and `g : \\mathbb{R} \\rarr \\mathbb{R}` are defined by `f(x) = x^2+1` and `g(x) = 3x-2` respectively. Then find the value of `(g \\circ f)(-1)`.',1601054736,NULL,'','','','',0,'',''),(1173,11,14.000,'Let `f : (-1,1) \\rarr \\mathbb{R}` be a mapping defined by `f(x) = \\frac{x}{1-|x|}`. Show that `f^{-1}` exists. Hence find `f^{-1}`.',1601055107,NULL,'','','','',0,'',''),(1174,11,6.000,'Let `f : \\mathbb{N} \\rarr \\mathbb{N}` be a mapping defend by $ f(n) = \\begin{cases} n+1, & \\text{when } n \\text{ is odd} \\\\ n-1, & \\text{when } n \\text{ is even} \\end{cases} $. Show that `f` is a bijective mapping.',1601055887,NULL,'','','','',0,'',''),(1175,11,11.000,'Let `f : \\mathbb{R} \\rarr \\mathbb{R}` be a mapping defined by `f(x) = x^3+x`. Prove that `f` is a bijective mapping.',1601056148,NULL,'','','','',0,'',''),(1176,10,3.000,'A relation `\\rho` defined on a set `A = {1, 2, 3}` by `\\rho = {(1,2), (2,3), (3,1)}`. Then the relation `\\rho` is -- (a) reflexive,  (b) symmetric,  (c) transitive,  (d) none of these.',1601056658,NULL,'','','','',0,'',''),(1177,11,13.000,'Let `f : \\mathbb{R} \\to \\mathbb{R}` be a mapping, defined by `f(x) = x + |x|`, where `x \\in \\mathbb{R}`. Show that `f` is neither one-one nor onto.',1601057098,NULL,'','','','',0,'',''),(1178,11,16.000,'A mapping `f : \\mathbb{R} \\to \\mathbb{R}` defined by $ f(x) = \\begin{cases} x, & \\text{when } x \\in \\mathbb{Q} \\\\ 1-x, & \\text{when } x \\notin \\mathbb{Q} \\end{cases}$, where `\\mathbb{Q}` is the set of rational number. Show that `f \\circ f = I_{\\mathbb{R}}`',1601105729,NULL,'','','','',0,'',''),(1179,10,15.000,'Show that the relation `\\rho = {(x,y) \\in \\mathbb{Z} \\times \\mathbb{Z} : 3x+4y \\text{ is divisible by } 7}` defined over the set of all integers `\\mathbb{Z}`, is an equivalence relation.',1601106152,NULL,'','','','',0,'',''),(1180,10,6.000,'`\\mathbb{R}` is a set of all real numbers. If a relation `\\rho` over set `\\mathbb{R}` is defined such that `\\rho = {(a,b) \\in \\mathbb{R} \\times \\mathbb{R} : ab \\gt 0}`. Then relation `\\rho` is -- (a) reflexive & symmetric, but not transitive,  (b) symmetric & transitive, but not reflexive,  (c) reflexive & transitive, but not symmetric,  (d) equivalence relation. ',1601106462,NULL,'','','','',0,'',''),(1181,11,8.000,'A mapping `f : \\mathbb{R}-{1/3} \\rarr \\mathbb{R}-{1/3}` is defined by `f(x) = \\frac{x+3}{3x-1}`. Then the value of `f^{-1}(4)` is -- (a) `{11/7}`,  (b) `{1/4}`,  (c) `{7/11}`,  (d) none of these. ',1601107544,'','','','','',0,'',''),(1182,10,12.000,'A relation `\\rho` defined on a set `A = {1, 2, 3}` by `\\rho = {(1,1), (2,3), (3,3)}`. Add minimum members to `\\rho` so that `\\rho` becomes an equivalence relation.',1601108355,NULL,'','','','',0,'',''),(1183,10,16.000,'Examine that the relation `\\rho = {(x,y) \\in \\mathbb{N} \\times \\mathbb{N} : x-y+\\sqrt{3} \\text{ is an irrational number}}` defined over the set of all natural numbers `\\mathbb{N}`, is an equivalence relation or not.',1601108906,'','','','','',0,'',''),(1184,10,2.000,'The total number of relations on set `A = {1, 2, 3, 4}` is -- (a) `2^4`,  (b) `2^8`,  (c) `2^12`,  (d) `2^16`',1601109467,NULL,'','','','',0,'',''),(1185,11,2.000,'Let `\\mathbb{Z}` be the set of all integers and a mapping `f : \\mathbb{Z} \\to \\mathbb{Z}` is defined by `f(x) = x^2`. Then `f^{-1}(-4)` is -- (a) `{2}`,  (b) `{-2}`,  (c) `{-2,2}`,  (d) `\\phi`',1601109947,NULL,'','','','',0,'',''),(1186,10,13.000,'Let `\\mathbb{Z}` be the set of all integers. A relation `\\rho` over set `\\mathbb{Z}` defined by `\\rho = {(x,y) : x+y \\text{ is an even number, where } x,y \\in \\mathbb{Z}}`. Show that `\\rho` is an equivalence relation.',1601110514,NULL,'','','','',0,'',''),(1187,5,46.000,'If `A = {1,2,3,4,5}` and `B \\cup C = {3,4, 6}`, then find `(A-B) \\cap (A-C)`.',1602761592,'`{1,2,5}`','','','','',0,'',''),(1188,5,37.000,'If `A = {x : -1 \\lt x \\le 5}` and `B = {x : -3 \\le x \\lt 4}`, then find `A \\cap B`',1602761768,NULL,'','','','',0,'',''),(1189,6,137.000,'There are 30 boys and 20 girls in a class. Half of the boys and half of the girls are of blue eyes. If one candidate is selected at random from the class, find the probability that it is either a boy or a blue eyed.',1603957849,'`4/5`','','','','',0,'একটি শ্রেণীতে 30 জন বালক এবং 20 জন বালিকা আছে। অর্ধেক বালক এবং অর্ধেক বালিকা নীল চক্ষুবিশিষ্ট। শ্রেণি থেকে উদ্দেশ্যহীনভাবে একজনকে নির্বাচন করা হলে, সে বালক অথবা নীল চক্ষু বিশিষ্ট হওয়ার সম্ভাবনা নির্ণয় করো।',''),(1190,12,6.000,'Which one is correct? i) `2+3i \\gt 1+4i`, ii) `3+3i \\gt 6+2i`,  iii) `5+9i \\gt 5+6i`,  iv) none of these.',1612792392,NULL,'','','','',0,'',''),(1191,12,38.000,'Find the value of --<br />\r\ni) `1+i+i^2+i^3+i^4`<br />\r\nii) `1+i^2+i^4+i^6+\\cdots + i^16`<br /> \r\niii) `i^n + i^{n+1} + i^{n+2} + i^{n+3}` where `n \\in \\mathbb{N}`<br />\r\niv) `\\frac{i+i^2+i^3+i^4}{1+i}`<br />\r\nv) `\\frac{i^2+i^4+i^6+i^7}{1+i^2+i^3}`<br />\r\nvi) `\\sum_{n=0}^225 i^n`<br />\r\nvii) `(1+i)^6(1+1/i)^6`<br />\r\nviii) `\\sqrt{-3 + \\sqrt{-3 + \\sqrt{-3 + \\cdots \\ \\infty}}}`',1612792462,'i) `1`<br />\r\nii) `1` <br />\r\niii) `0` <br />\r\niv) `0` <br />\r\nv) `1 - i` <br />\r\nvi) `1 + i` <br />\r\nvii) `64` <br />\r\nviii) `\\frac{1 \\pm i \\sqrt{11}}{2}` <br />','1191.01,1191.02,1191.03,1191.04','','','',0,'',''),(1192,12,77.000,'Find the value of --<br />\r\ni) `(1-\\omega) (1-\\omega^2) (1-\\omega^4)(1-\\omega^5)`<br />\r\nii) `(1-\\omega^2) (1-\\omega^4) (1-\\omega^8) (1-\\omega^10)`<br />\r\niii) `(3 + \\omega + 3\\omega^2)^4`<br />\r\niv) `1 + \\omega^28 + \\omega^29`<br />\r\nv) `\\omega^4 + \\omega^8 + \\omega^{-1}\\cdot\\omega^{-2}`<br />\r\nvi) `\\omega^{3n} + \\omega^{3n+1} + \\omega^{3n+2}` where `n \\in \\mathbb{N}`<br />\r\nvii) `(\\frac{-1+\\sqrt{-3}}{2})^19 + (\\frac{-1 - \\sqrt{-3}}{2})^19`<br />\r\nviii) `(x + y\\omega + z\\omega^2)^2 + (x\\omega + y\\omega^2 + z)^2 + (x\\omega^2 + y + z\\omega)^2`<br />\r\nix) `1\\cdot(2-\\omega)\\cdot(2-\\omega^2) + 2\\cdot(3-\\omega)\\cdot(3-\\omega^2)+ \\cdots + (n-1)\\cdot(n-\\omega)\\cdot(n-\\omega^2)`',1612863620,'','','','','',0,'',''),(1193,12,78.000,'Find the value of `\\alpha^4 + \\beta^4 + \\alpha^{-1} \\beta^{-1}` where `\\alpha`, `\\beta` are imaginary cube root of 1.',1612864297,NULL,'','','','',0,'',''),(1194,12,39.000,'i) If `a = \\frac{1+i}{\\sqrt{2}}` then show that `1+a^2+a^4+a^6=0`<br />\r\nii) If `x = 2-i\\sqrt{3}` then find the value of `2x^4 - 5x^3 - 3x^2 + 41x - 35` <br />\r\niii) If `x = 2 + 3i` then find the value of `x^3 - 4x^2 + 13x + 5`<br />\r\niv) If `x = -1 + i\\sqrt{2}` then find the value of `x^4 + 4x^3 + 6x^2 + 4x + 9` <br />',1612864620,'i) N.A. <br />\r\nii) `0` <br />\r\niii) `5` <br />\r\niv) `12` <br />','1194,1194.01,1194.02,1194.03','','','',0,'',''),(1195,12,4.000,'If `(x+iy)^5 = a+ib` then find the value of `(y+ix)^5`',1612873495,'`b+ia`','1195','','','',0,'',''),(1196,12,79.000,'Show that the value of `(a+\\omega+\\omega^2)(a + \\omega^2 + \\omega^4)(a + \\omega^4 + \\omega^8) \\cdots` upto `2n` number of factors is `(a-1)^{2n}`',1612873967,NULL,'','','','',0,'',''),(1197,12,80.000,'Show that `\\omega/9 [(1-\\omega)(1-\\omega^2)(1-\\omega^4)(1-\\omega^8) + 9(\\frac{c+a\\omega+b\\omega^2}{a\\omega^2+b+c\\omega})] = -1`',1612874272,NULL,'','','','',0,'',''),(1198,12,5.000,'Find conjugate of the following complex numbers  --<br />\r\ni) `-\\sqrt{5}+7i`<br />\r\nii) `\\frac{1}{1+i}`<br />\r\niii) `\\frac{2-i}{(1-3i)^2}`',1612874481,'iii) `\\frac{-1-2i}{10}`','1198.03','','','',0,'',''),(1199,12,1.000,'Covert the following complex number into `A + iB` form --<br />\r\ni) `(\\frac{1+i}{1-i})^3`<br />\r\nii) `(\\frac{1-i}{1+i})^100`<br />\r\niii) `\\frac{(1+i)^2}{3-i}`<br />\r\niv) `\\frac{\\sqrt{3}+i\\sqrt{2}}{2\\sqrt{3}+i\\sqrt{2}}`<br />\r\nv) `\\frac{1}{1-\\cos\\theta-i\\sin\\theta}`',1612874971,'i) `0+(-1)i` <br />\r\nii) `1+0.i` <br />\r\niv) `4/7 + i \\frac{\\sqrt{6}}{14}`<br />\r\nv) `1/2 + i/2 cot \\frac{\\theta}{2}`','1199.1,1199.02,1199.04,1199.03','','','',0,'',''),(1200,12,2.000,'If `\\frac{2+i}{2-3i} = A+iB` then find the value of `A^2+B^2`',1612875151,'`65/169`','1200','','','',0,'',''),(1201,12,47.000,'If `z = 3+2i` and `\\frac{2z-1}{z-2} = x+iy` (where `x, y` are real), then find the value of `x, y`.',1612875934,'`x=13/5`, `y= - 6/5`','1201','','','',0,'',''),(1202,12,3.000,'If `x+iy = \\frac{2}{3+cos\\theta+i\\sin\\theta}` then show that `2x^2+2y^2 = 3x-1`',1612876094,'N.A.','1202,1202.1','','','',0,'',''),(1203,12,12.000,'Find the modulus of following complex numbers --<br />\r\ni) `\\frac{1}{1-i}`<br />\r\nii) `\\frac{-i}{1+i}`<br />\r\niii) `\\frac{1-i}{\\sqrt{5}-i\\sqrt{3}}`<br />\r\niv) `(a-ib)^2`<br />\r\nv) `(3i-1)^2`',1612876328,NULL,'','','','',0,'',''),(1204,12,13.000,'Find argument or principal value of amplitude of following complex numbers --<br />\r\ni) `3i`<br />\r\nii) `-3-\\sqrt{3}i`<br />\r\niii) `\\frac{1}{1+i}`<br />\r\niv) `-2`',1612876475,NULL,'','','','',0,'',''),(1205,12,14.000,'Find modulus & amplitude of following complex numbers --<br />\r\ni) `1+i\\tan\\frac{3\\pi}{5}`<br />\r\nii) `1 + i\\tan\\frac{4\\pi}{7}`',1612876667,'i) Modulus = `- \\sec \\frac{3\\pi}{5}` and amplitude = `- \\frac{2\\pi}{5}` <br />\r\nii) Modulus = `- \\sec \\frac{4\\pi}{7}` and amplitude = `- \\frac{3\\pi}{7}` <br />','1205,1205.01','','','',0,'',''),(1206,12,30.000,'If `z = 1 + i\\tan\\alpha` (where `\\pi \\lt \\alpha \\lt \\frac{3\\pi}{2}`), then find `|z|`.',1612876814,NULL,'','','','',0,'',''),(1207,12,32.000,'If `z_1 = \\frac{1-i}{\\sqrt{2}}`, `z_2 = \\frac{1+i\\sqrt{3}}{2}`, then find the principal value of argument of `z_1z_2`',1612877053,'`15°`','1207','','','',0,'',''),(1208,12,31.000,'If `z_1 = 3i` and `z_2 = -1-i`, then find `arg(z_1/z_2)`',1612877137,'`-135°`','1208','','','',0,'',''),(1209,12,15.000,'Express the following complex numbers into modulus-amplitude form --<br />\r\ni) `\\sqrt{3}-i`<br />\r\nii) `\\frac{i}{1-i}`<br />\r\niii) `\\frac{\\sqrt{3}-i}{1-\\sqrt{3}i}`',1612877464,NULL,'','','','',0,'',''),(1210,12,29.000,'`z` is a complex number such that `|z|=4` and `arg(z) = \\frac{5\\pi}{6}`. Find the complex number `z`',1612877622,NULL,'','','','',0,'',''),(1211,12,9.000,'Find the least positive whole number of `n`, that satisfies the equation `(\\frac{1+i}{1-i})^n=1`',1612877850,NULL,'','','','',0,'',''),(1212,12,10.000,'If `(1+i)(2+i)(3+i)\\cdots(n+i)=a+ib`, then show that `2 \\cdot 5 \\cdot 10 \\ \\cdots (n^2+1) = a^2+b^2`',1612885571,NULL,'','','','',0,'',''),(1213,12,44.000,'If `y = \\sqrt{x^2+6x+8}` where `x \\gt 0`, then show that one of the values of `\\sqrt{1+iy} + \\sqrt{1-iy}` is `\\sqrt{2x+8}`',1612885739,NULL,'','','','',0,'',''),(1214,12,42.000,'Show that one of the values of `\\sqrt{i} + \\sqrt{-i}` is `\\sqrt{2}`',1612885821,NULL,'','','','',0,'',''),(1215,12,43.000,'Show that one of the values of `\\sqrt{1+i} - \\sqrt{1-i}` is `i\\sqrt{2(\\sqrt{2}-1)}`',1612885992,'N.A.','1215','','','',0,'',''),(1216,12,40.000,'Find square root of following complex number --<br />\r\ni) `i`<br />\r\nii) `-2i`<br />\r\niii) `\\frac{1+i}{1-i}`<br />\r\niv) `\\frac{-1+\\sqrt{-3}}{2}`<br />\r\nv) `7-24i`<br />\r\nvi) `y+\\sqrt{y^2-x^2}` where `y^2 \\lt x^2`<br />\r\nvii) `1+i\\sqrt{a^4-1}`<br />\r\nviii) `x-i\\sqrt{x^4+x^2+1}`<br />\r\nix) `1+i` <br />\r\nx) `1-i`',1612886379,'i) `\\pm \\frac{1}{\\sqrt{2}}(1+i)` <br />\r\nii) `\\pm (1-i)`<br />\r\niii) `\\pm \\frac{1}{\\sqrt{2}} (1+i)`<br />\r\niv) `\\pm 1/2 (1+i\\sqrt{3})` <br />\r\nv) `\\pm (4-3i)`<br />\r\nvi) `\\pm \\frac{1}{\\sqrt{2}} (\\sqrt{y+x}+\\sqrt{y-x})`<br />\r\nvii) `\\pm \\frac{1}{\\sqrt{2}} (\\sqrt{a^2+1}+i\\sqrt{a^2-1})`<br />\r\nviii) `\\pm \\frac{1}{\\sqrt{2}} (\\sqrt{x^2+x+1}-i\\sqrt{x^2-x+1})`<br />\r\nix) `\\pm (\\sqrt{\\frac{\\sqrt{2}+1}{2}} + i \\sqrt{\\frac{\\sqrt{2}-1}{2}})` <br />','1216.01,1216.02,1216.03,1216.04,1216.05,1216.06,1216.07,1216.08,1216.09','','','',0,'',''),(1217,12,45.000,'i) If `z = x+iy` where `x,y \\in \\mathbb{R}` and `i=\\sqrt{-1}` and `|z-2| = |2z-1|`, then prove that `x^2+y^2 = 1`<br />\r\nii) If `z = x+iy` where `x,y \\in \\mathbb{R}` and `i=\\sqrt{-1}` and `|z+6| = |2z+3|`, then prove that `x^2+y^2 = 9`',1612886617,'','','','','',0,'',''),(1218,12,46.000,'If `z = x+iy` (`x,y \\in \\mathbb{R}`) and `i=\\sqrt{-1}` and `|2z+1| = |z-2i|` then show that `3(x^2+y^2)+4(x+y)=3`',1612887165,'N.A.','1218','','','',0,'',''),(1219,12,48.000,'If `a, b, c, d, x, y` are real number and `(a+ib)(c+i``d) = (x+iy)`, then show that `(ac-bd)^2+(ad+bc)^2 = x^2+y^2`',1612887522,NULL,'','','','',0,'',''),(1220,12,52.000,'If `x+iy = \\sqrt{\\frac{a+ib}{c+i\\d}}`, then show that `(x^2+y^2)^2 = \\frac{a^2+b^2}{c^2+d^2}`',1612887813,'N.A.','1220','','','',0,'',''),(1221,12,53.000,'If `\\sqrt{x-iy} = a-ib` (where `x,y,a,b \\in \\mathbb{R}`) then show that `\\sqrt{x+iy} = a+ib`',1612889590,NULL,'','','','',0,'',''),(1222,12,54.000,'i) If `(a-ib)^{1/3} = p-iq` (where `a,b,p,q \\in \\mathbb{R}`) then show that `(a+ib)^{1/3}=p+iq`<br />\r\nii) If `(x-iy)^{1/3} = a-ib` (where `x,y,a,b \\in \\mathbb{R}`) then show that `x/a + y/b = 4(a^2-b^2)`',1612889958,NULL,'','','','',0,'',''),(1223,12,55.000,'If `x = cos\\theta + isin\\theta` (where `\\theta` real), then show that `x^2 + 1/x^2` is a real number.',1612890055,NULL,'','','','',0,'',''),(1224,12,57.000,'If `a = cos\\alpha + isin\\alpha` and `b = cos\\beta + isin\\beta` then show that `a/b + b/a = 2cos(\\alpha-\\beta)`',1612890208,NULL,'','','','',0,'',''),(1225,12,58.000,'If `x = cos\\theta + isin\\theta` (where `\\theta` real) and `1 + \\sqrt{1-a^2} = na`, then show that `a/{2n}(1+nx)(1+n/x) = 1+acos\\theta`',1612891547,'N.A.','1225','','','',0,'',''),(1226,12,56.000,'If `x - 1/x = 2isin\\theta`, then show that `x^4 - 1/x^4 = 2isin4\\theta`',1612922595,NULL,'','','','',0,'',''),(1227,12,11.000,'Find the condition to become purely real & purely imaginary of the expression `(a+ib)\\cdot(c+i\\d)`',1612922878,NULL,'','','','',0,'',''),(1228,12,49.000,'If `a^2+b^2=1` (where `a,b \\in \\mathbb{R}`), then show that a real value of `x` satisfies the equation `\\frac{1-ix}{1+ix}=a-ib`',1612923589,NULL,'','','','',0,'',''),(1229,12,50.000,'If a complex number `z` is such that `\\frac{z-1}{z+1}` is purely imaginary, then show that `|z| = 1`',1612923949,'N.A.','1229','','','',0,'',''),(1230,12,51.000,'If a complex number `z` is such that that `|z|=1`, then show that `\\frac{z-1}{z+1}` is purely imaginary.',1612924093,'N.A.','1230','','','',0,'',''),(1231,12,71.000,'If `z = x+iy` (where `x,y \\in \\mathbb{R}`) and `|\\frac{z-3}{z+3}|=2`, then find the locus of `z` in complex plane.',1612924682,NULL,'','','','',0,'',''),(1232,12,73.000,'If `z = x+iy` (`x,y \\in \\mathbb{R}`) and `\\frac{z+1}{z+i}` is purely imaginary, then show that locus of `z` is a circle with centre at `-1/2(1+i)` and radius `1/\\sqrt{2}`',1612924933,NULL,'','','','',0,'',''),(1233,12,72.000,'If `z = x+iy` and `\\frac{z-i}{z-1}=ia` (where `x,y,a` are real), then show that `(x-1/2)^2+(y-1/2)^2 = (1/\\sqrt{2})^2`',1612973423,NULL,'','','','',0,'',''),(1234,12,74.000,'If `z=x+iy` and `|z-2-i|=5`, then show that locus of `z` in complex plane is a circle. Also find its centre & radius.',1612973750,NULL,'','','','',0,'',''),(1235,12,65.000,'If `z = x+iy`, `w = \\frac{1-iz}{z-i}` and `|w|=1`, then show that `z` lies on real axis on the complex plane.',1612976033,NULL,'','','','',0,'',''),(1236,12,68.000,'If `z = x+iy` and `\\text{arg}(\\frac{z-1}{z+1})=\\pi/2`, then show that locus of `z` in complex plane is a circle.',1612976246,NULL,'','','','',0,'',''),(1237,12,69.000,'If `z = x+iy` and `\\text{arg}(\\frac{z-1}{z+1})=\\pi/4`, then show that locus of `z` in a complex plane is a circle.',1612976439,NULL,'','','','',0,'',''),(1238,12,70.000,'If `z_1 = 4+3i`, `z_2 = 7+4i` and `z` is another complex number such that `\\text{arg}(\\frac{z_1-z}{z-z_2}) = \\pi/4`; then show that `|z-6-2i| = \\sqrt{5}`',1613025837,NULL,'','','','',0,'',''),(1239,12,82.000,'Let `\\omega` be the imaginary cube root of 1. If `x=a+b`, `y=a\\omega + b\\omega^2`, `z = a\\omega^2+b\\omega`, then show that --<br />\r\ni) `xyz = a^3 + b^3`<br />\r\nii) `x^3 + y^3 + z^3 = 3(a^3 + b^3)`<br />\r\niii) `x^2 + y^2 + z^2 = 6ab`',1613026301,'','','','','',0,'',''),(1240,12,83.000,'Let `\\omega` be the imaginary cube root of 1. If `x = \\alpha + \\beta`, `y = \\alpha + \\beta\\omega`, `z = \\alpha + \\beta\\omega^2` , then show that `x^3 + y^3 + z^3 = 3(\\alpha^3 + \\beta^3)`',1613026540,'','','','','',0,'',''),(1241,12,84.000,'Let `\\omega` be the imaginary cube root of 1 and `a+b+c=0`. Then show that `(a+b\\omega+c\\omega^2)^3 + (a+b\\omega^2+c\\omega)^3 = 27abc`',1613026789,NULL,'','','','',0,'',''),(1242,12,81.000,'Find the roots of the equation `(x+5)^3 + 27=0`',1613027147,NULL,'','','','',0,'',''),(1243,12,76.000,'If `a,b,c` real number, `z` complex number and `a^2+b^2+c^2=1`, `b+ic = z(1+a)` then show that `\\frac{a+ib}{1+c} = \\frac{1+iz}{1-iz}`',1613027665,NULL,'','','','',0,'',''),(1244,12,75.000,'If `z = x+iy` then find the numerical value of area of circle determined by `z \\bar{z} + (3-4i)z + (3+4i)\\bar{z}=0`',1613027943,NULL,'','','','',0,'',''),(1245,12,16.000,'If `iz^2 - \\bar{z}=0`, then find `|z|`',1613028234,'0,1','1245,1245.01','','','',0,'',''),(1246,12,17.000,'If `8iz^3 + 12z^2 - 18z + 27i=0` then find `|z|`',1613030519,'`3/2`','1246','','','',0,'',''),(1247,12,8.000,'If `z_1` and `z_2` are two complex numbers, then show that--<br /> \r\ni) `|z_1 + z_2|^2 + |z_1 - z_2|^2 = 2[ |z_1|^2 + |z_2|^2 ]`<br />',1613031247,'','','','','',0,'',''),(1248,12,7.000,'For any complex number `z`, show that `|z| \\gt \\frac{|\\text{Re}(z)|+|\\text{Im}(z)|}{\\sqrt{2}}`',1613031580,'N.A.','1248','','','',0,'',''),(1249,12,28.000,'If `z_1, z_2, z_3` are three complex numbers such that `|z_1| = |z_2| = |z_3| = |1/z_1 + 1/z_2 + 1/z_3| = 1`, then find the value of `|z_1 + z_2 + z_3|`',1613032320,NULL,'','','','',0,'',''),(1250,12,18.000,'If `|z - 6/z| = 2` then find maximum & minimum value of `|z|`, where `|z|` is a complex number.',1613032618,'`1+\\sqrt{7}`, `0`','1250','','','',0,'',''),(1251,12,19.000,'If `z` is a complex number, then find the minimum value of `|z| + |z-1|`',1613032804,NULL,'','','','',0,'',''),(1252,12,20.000,'If `z` is a complex number and `|z + 2/z| = 2`, then find maximum value of `|z|`',1613032910,NULL,'','','','',0,'',''),(1253,12,21.000,'If `z` is a complex number and `|z-2| \\le 3`, then find maximum value of `|z|`',1613033007,NULL,'','','','',0,'',''),(1254,12,22.000,'If `z` is a complex number and `|z+4| le 3`, then find the maximum value of `|z+1|`',1613033204,NULL,'','','','',0,'',''),(1255,12,23.000,'If `z` is a complex number and `|z+2| \\le 2`, then find maximum & minimum value of `|z|`',1613033467,NULL,'','','','',0,'',''),(1256,12,24.000,'If `z` is a complex number and `|z+5| \\le 6`, then find maximum & minimum value of `|z+2|`',1613033591,NULL,'','','','',0,'',''),(1257,12,25.000,'If `z` is a complex number and `|z+2| + |z-2| \\le 6`, then find maximum value of `|z|`',1613033721,NULL,'','','','',0,'',''),(1258,12,27.000,'`z, z_0` are complex numbers and `|z-i| \\le 2`, `z_0 = 5+3i`; then find the maximum value of `|iz+z_0|`',1613049436,NULL,'','','','',0,'',''),(1259,12,26.000,'`z` is a complex number and `|z^2-9| = 8|z|`, then find maximum & minimum value of `|z|`',1613049617,NULL,'','','','',0,'',''),(1260,12,33.000,'If `z_1, z_2` are two complex numbers and `|z_1 + z_2| = |z_1| + |z_2|`, then prove that `\\text{arg}(z_1) = \\text{arg}(z_2)`',1613049823,NULL,'','','','',0,'',''),(1261,12,34.000,'If `z_1, z_2` are two complex numbers and `|z_1 + z_2| = |z_1 - z_2|`, then find the value of `\\text{amp}(z_1) - \\text{amp}(z_2)`',1613050051,NULL,'','','','',0,'',''),(1262,12,35.000,'If `z_1, z_2` are two complex numbers and `|z_1| = |z_2| = 1` and `\\text{arg}(z_1) + \\text{arg}(z_2) = 0`. Then show that `z_1 = 1/z_2`',1613050304,NULL,'','','','',0,'',''),(1263,12,36.000,'If `z_1, z_2` are two complex numbers and `|z_1| = |z_2|` and `\\text{arg}(z_1) ~ \\text{arg}(z_2) = \\pi`, then show that `z_1 + z_2 = 0`',1613050671,NULL,'','','','',0,'',''),(1264,12,37.000,'Prove that, `\\text{amp}(z) - \\text{amp}(-z) = \\pm\\pi` when `\\text{amp}(z)` be positive or negetive.',1613050879,NULL,'','','','',0,'',''),(1265,12,67.000,'If three points `z, -iz` and 1 are collinear, then show that `z` always lies on a circle.',1613052801,NULL,'','','','',0,'',''),(1266,12,59.000,'The three vertices of an equilateral triangle are represented by three complex numbers `z_1, z_2, z_3`. Then show that --<br />\r\ni) `\\frac{1}{z_1 - z_2} + \\frac{1}{z_2 - z_3} + \\frac{1}{z_3 - z_1} = 0`<br />\r\nii) `z_1^2 + z_2^2 + z_3^2 = z_1z_2 + z_2z_3 + z_3z_1`',1613055893,NULL,'','','','',0,'',''),(1267,12,60.000,'The three points in a complex plane are represented by `z_1, z_2, z_3` such that `|z_1| = |z_2| = |z_3|` and they form an equilateral triangle in complex plane. Prove that `z_1 + z_2 + z_3 = 0`',1613058245,NULL,'','','','',0,'',''),(1268,12,61.000,'In complex plane, three points are represented by three complex numbers `z_1, z_2, z_3` and `|z_1| = |z_2| = |z_3| = 1` and `z_1 + z_2 + z_3 = 0`. Then find the area of triangle formed by these three points.',1613058569,NULL,'','','','',0,'',''),(1269,12,62.000,'In a complex plane three points, represented by three complex numbers `z_1, z_2, z_3` and `\\frac{1}{z_1-z_2} + \\frac{1}{z_2-z_3} + \\frac{1}{z_3-z_1} = 0`. Show that the triangle formed by these three points are equilateral triangle.',1613059007,NULL,'','','','',0,'',''),(1270,12,63.000,'In a complex plane, the three points `z_1, z_2, z_3` are three vertices of an isosceles right angle triangle with right angle at `z_3`. Then show that `z_1^2 + 2z_2^2 + z_3^2 = 2z_2(z_1 + z_3)`',1613059425,NULL,'','','','',0,'',''),(1271,12,64.000,'`z_1, z_2` are two complex numbers and `z_1^2 + z_2^2 + 2z_1z_2cos\\theta=0`. Then show that the triangle formed by origin, `z_1` and `z_2` is an isosceles triangle. (where `\\theta \\in \\mathbb{R}`)',1613059743,NULL,'','','','',0,'',''),(1272,12,66.000,'Show that the three points `1+4i, 2+7i` and `3+10i` are collinear.',1613059833,NULL,'','','','',0,'',''),(1273,6,34.000,'If `P(A \\cap B) = 5/13`, then the value of `P(A^c \\cup B^c)` is --<br />\r\n(i) `4/13`,   (ii) `6/13`,   (iii) `7/13`,   (iv) `8/13`',1617850798,'iv) `8/13`','1273','','','',0,'',''),(1274,6,53.000,'If `A` & `B` are two independent events and `P(A) = 0.3` , `P(B) = 0.4` then find `P(A \\cup B)` & `P(B | A)`',1617851050,'`0.58`, `0.40`','','','','',0,'',''),(1275,6,99.000,'`A` speaks the truth 3 out of 4 times, and `B` 5 out of 6 times. What is the probability that they will contradict each other in stating the same fact?\r\n',1617851408,'`1/3`','','','','',0,'A, 4 বারের মধ্যে 3বার সত্যি কথা বলে এবং B, 6বারের মধ্যে 5বার সত্য কথা বলে । একই তথ্য বিবৃত করতে কত ক্ষেত্রে তারা পরস্পর বিরোধিতা করবে তার সম্ভাবনা নির্ণয় করো।',''),(1276,6,114.000,'There are two boxes. Box I contains 4 white, 3 red & 3 blue balls. Box II contains 5 white, 4 red & 3 blue balls. If one ball is drawn randomly from each box, then what is the probability  the both balls are of same color?',1617852234,'`41/120`','','','','',0,'',''),(1277,6,136.000,'A machine operates only when all of its three components function. The probabilities of the failures of the first , second and third component are 0.2, 0,3 and 0.5 respectively . What is the probability that the machine will fail ?\r\n',1617855663,'`0.72`','','','','',0,'',''),(1278,6,138.000,'An insurance company insured 2000 scooters and 3000 motor cycles. The probability of an accident involving a scooter is 0.01 and that of a motor cycle is 0.02. An insured vehicle met with an accident. Find the probability that the accidented vehicle was a motor cycle.\r\n',1617856404,'`3/4`','1278','','','',0,'',''),(1279,6,139.000,'In a factory there are three types of Machines `M_1, M_2, M_3` which produces 2000, 2500 and 4000 bolts daily respectively. `M_1, M_2, M_3` produces 3%, 4%  and 2.5% defective products respectively. A bolt is drawn at random from the product and is found to be defective. What is the probability of that it is manufactured by machine `M_2`?',1617857169,'`5/13`','1279','','','',0,'',''),(1280,6,56.000,'If `P(A^c \\cup B^c) = 5/6`, `P(A)  = 1/2` and `P(\\bar{B}) = 2/3`, then are `A` & `B` independent events?',1618978355,'Yes','','','','',0,'`P(A^c cup B^c) = 5/6`, `P(A)  = 1/2` এবং `P(\\bar{B}) = 2/3`, হলে `A` & `B` ঘটনা দুটি কি স্বাধীন?',''),(1281,6,78.000,'The events `A` & `B` are mutually independent and `P(A), P(B) \\gt 1/2`. `P(A \\cap B^c) = 3/25` & `P(A^c \\cap B) = 8/25`. Then find the value of `P(A)` & `P(B)`',1618978665,'`3/5, 4/5`','','','','',0,'',''),(1282,6,94.000,'The probability of passing of students in Physics is 70% and in Chemistry is 40%. What is the probability of passing only one subject?',1618978924,'`3/5`','','','','',0,'',''),(1283,6,40.000,'If `P(A) = 3/7`, `P(B) = 4/7` and `P(A \\cap B) = 2/9`, then the value of `P(A | B)` is --<br />\r\n(a) `7/18`, (b) `14/27`,  (c) `5/18`, (d) `4/9`',1618979250,'(a) `7/18`','1283','','','',0,'যদি `P(A) = 3/7`, `P(B) = 4/7` এবং `P(A cap B) = 2/9` হয়, তবে `P(A | B)` এর মান --<br />\r\n(a) `7/18`, (b) `14/27`,  (c) `5/18`, (d) `4/9`',''),(1284,6,111.000,'There are 5 while, 7 red & 3 black balls in a bag. Three balls are drawn from the bag one by one without replacement. What is the probability of no red balls?',1618979493,'`8/65`','','','','',0,'',''),(1285,6,49.000,'There are three children in a family. The probability of all the children are boys is --<br />\r\n(a) `1/3`, (b) `1/4`, (c) `1`, (d) `1/2`',1618980065,'(a) `1/3`','','','','',0,'',''),(1286,6,31.000,'If odds against an event is `2:3`, then the probability of that event is --<br />\r\n(a) `2/5`, (b) `3/5`, (c) `2/3`, (d) `1/5` ',1619016321,'(b) `3/5`','','','','',0,'',''),(1287,6,69.000,'If the events `A` & `B` are mutually independent, then prove that the events `A^c` & `B^c` are also mutually independent. <br />\r\nIf the events `A^c` & `B^c` are mutually independent, then prove that the events `A` & `B` are also mutually independent.',1619016528,'','','','','',0,'`A` ও `B` ঘটনা দুটি পরস্পর স্বাধীন হলে, প্রমাণ করো যে, `A^c` & `B^c` ঘটনা দুটিও পরস্পর স্বাধীন হবে। <br />\r\n`A^c` & `B^c` ঘটনা দুটি পরস্পর স্বাধীন হলে, প্রমাণ করো যে, `A` & `B` ঘটনা দুটিও পরস্পর স্বাধীন হবে। ',''),(1288,6,12.000,'An unbiased coin is tossed three times in succession, then the probability of getting just one head is --<br />\r\n(a) `1/2`, (b) `5/8`, (c) `3/4`,  (d) `3/8`',1619021945,'(d) `3/8`','','','','',0,'',''),(1289,6,107.000,'The odds against two events are `2:7` and `7:5` respectively. If the events are independent, then find the probability that at least one will occur. ',1619022464,'`47/54`','','','','',0,'',''),(1290,6,122.000,'There are two bags. Bag I contains 5 red & 4 black balls, bag II contains 3 red & 5 black balls. One bag is chosen at random and one ball is drawn from it. What is the probability that the ball is red?',1619023091,'`67/144`','','','','',0,'',''),(1291,6,127.000,'There are two bags. Bag I contains 4 red & 4 black balls, bag II contains 2 red & 6 black balls. One bag is chosen at random and a ball is drawn from it and it is found as a red ball. What is the probability that it is drawn from first bag?',1619023633,'`2/3`','1291','','','',0,'দুটি ব্যাগ আছে। প্রথম ব্যাগে 4 টি লাল ও 4 টি কালো বল আছে, দ্বিতীয় ব্যাগে 2 টি লাল ও 6 টি কালো বল আছে। যথেচ্ছভাবে একটি ব্যাগ নির্বাচন করা হল ও সেখান থেকে একটি বল তোলা হল। দেখা গেল বলটি লাল। বলটি প্রথম ব্যাগ থেকে তোলার সম্ভাবনা নির্ণয় করো। ',''),(1292,6,81.000,'If `\\bar{A}` and `\\bar{B}` are the events complementary to the events `A` & `B` respectively, then prove that `P(\\bar{A}\\ \\text{or}\\ \\bar{B}) = 1 - P(A) P(B | A)`',1619024301,NULL,'','','','',0,'',''),(1293,6,10.000,'If an unbiased die is throwing twice, then the probability of getting at least one `1` is --<br />\r\n(a) `25/36`,  (b) `1/3`,  (c) `11/36`,  (d) `1/36`',1619025032,'(c) `11/36`','','','','',0,'',''),(1294,6,67.000,'For any two events `A` & `B`, if `P(A) = 1/8`, `P(A | B) = 1/8` and `P(B | A) = 1/4`, then are these events mutually exclusive?',1619185659,'No','','','','',0,'',''),(1295,6,44.000,'`A` and `B` are two events such that `P(A) = 1/4`, `P(B) = 1/2` and `P(A \\cap B) = 1/8`, then `P(\\bar{A} \\cap \\bar{B})` is --<br />\r\n(a) `3/4`, (b) `3/8`, (c) `1/2`, (d) `1/3`',1619196833,'(b) `3/8`','','','','',0,'',''),(1296,6,92.000,'Find the probability of drawing a diamond card in each of the two consecutive draws from a well-shuffled pack of cards, if the card drawn is not replaced after the first draw.',1619197432,'`1/17`','','','','',0,'',''),(1297,6,142.000,'A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn and are found to be both diamonds. Find the probability of the lost card being a diamond.',1619198030,'`11/50`','1297','','','',0,'',''),(1298,6,37.000,'If the two events `A` & `B` are such that `2P(A) = P(B) = 5/13` and `P(A | B) = 2/3`, then find `P(A \\cup B)`',1619198337,'`25/78`','1298','','','',0,'যদি `A` & `B` দুটি ঘটনা এমন হয় যে `2P(A) = P(B) = 5/13` এবং `P(A | B) = 2/3`, তবে `P(A cup B)` এর মান নির্ণয় করো। ',''),(1299,6,123.000,'There are 3 identical urns. First urn contains 3 white and 2 red balls, second urn contains 7 white and 3 red balls and third urn contains 5 white and 3 red balls. Probability of choosing any urn is same. An urn is selected at random and a ball is drawn from it. What is the probability that the ball is white?',1619254266,'`77/120`','','','','',0,'',''),(1300,6,105.000,'One Number is chosen at random from the integers 1 to 50. What is the probability that it is divisible by 4 or 6?',1619255050,'`8/25`','','','','',0,'',''),(1301,6,93.000,'Four cards are drawn from a full pack of playing cards. Find the probability that there is at least one ace among them.',1619267448,NULL,'','','','',0,'',''),(1302,6,117.000,'An urn contains 4 white and 5 black balls, a second urn contains 5 white and 4 black balls. One ball is transferred from the first to the second urn, then a ball is drawn from the second urn. What the probability that it is white?',1619267761,'`49/90`','','','','',0,'একটি থলিতে 4 টি সাদা ও 5 টি কালো বাল আছে এবং দ্বিতীয় থলিতে 5 টি সাদা ও 4 টি কালো বল আছে। প্রথম থলি থেকে উদ্দেশ্যহীনভাবে একটি বল তুলে দ্বিতীয় থলিতে রাখা হল। এখন দ্বিতীয় থলি থেকে উদ্দেশ্যহীনভাবে একটি বল তোলা হলে, তোলা বলটি সাদা হওয়ার সম্ভাবনা নির্ণয় করো।',''),(1303,6,41.000,'If `P(A) = 1/4`, `P(B) = 1/3` & `P(A \\cap B) = 1/5`, then find `P(\\bar{B} | \\bar{A})`',1619689489,'`37/45`','1303','','','',0,'যদি `P(A) = 1/4`, `P(B) = 1/3` এবং `P(A cap B) = 1/5`, তবে `P(B^c|A^c)` এর মান নির্ণয় করো।',''),(1304,6,103.000,'Probability of solving a specific problem independently by three students are `1/2, 1/3` & `1/4`. If all are trying to solve the problem independently then find the probability that<br /> \r\ni) the problem is solved,\r\nii) exactly one of them solves the problem,\r\niii) exactly two of them solve the problem.',1619690306,'i) `1/2` , ii) `11/24`, iii) `6\\log_3e`','','','','',0,'',''),(1305,6,1.000,'Two unbiased dice are throwing. Find the probability of getting 1 as difference of two obtained numbers.',1619691659,'`1/4`','','','','',0,'',''),(1306,6,135.000,'Out of 20 games of chess played between two players A and B, A won 12, B won 4 and 4 ended in a tie. In a tournament of three games find the probability that (i) B wins at least one game, (ii) two games end in a tie.',1619693810,'(i) `61/125`, (ii) `12/125`','','','','',0,'',''),(1307,6,30.000,'Eight coins are thrown together. Find the probability that -<br />\r\ni) at least six heads obtained,<br />\r\nii) exactly three heads obtained,<br />',1619695086,'i) `37/256` ii) `7/32`','','','','',0,'',''),(1308,6,116.000,'In the boxes A, B, C there are 4 reds and 3 black balls, 5 red and 4 black balls, 4 red and 4 black balls respectively. From every box one ball is drawn. What is the probability that the among the three balls 2 balls are red and 1 ball is black.',1619695857,'','','','','',0,'',''),(1309,6,28.000,'`A, B, C` are three events which are mutually exclusive & exhaustive. If `P(B) = 3/2\\ P(A)`, `P(C) = 1/3\\ P(B)`, then value of `P(C)` is --<br />\r\n(a) `1/6`,  (b) `1/2`,  (c) `1/3`,  (d) `2/3`',1620119635,'(a) `1/6`','1309','','','',0,'A, B, C ঘটনা তিনটি পরস্পর পৃথক ও সম্পূর্ণ এবং `P(B) = 3/2 P(A)`, `P(C) = 1/3 P(B)`, হলে `P(C)` এর মান --<br />\r\n(a) `1/6`,  (b) `1/2`,  (c) `1/3`,  (d) `2/3`',''),(1310,6,8.000,'Two unbiased dice are thrown. The probability of getting total as 7 or 12 is --<br />\r\n(a) `2/9`,  (b) `5/9`,  (c) `5/36`, (d) `7/36`',1620120956,NULL,'','','','',0,'',''),(1311,6,90.000,'A couple has 2 children. Find the probability that both are boys, if it is known that --<br />(i) one of them is a boy,<br />(ii) the older child is a boy.',1620825077,'(i) `1/3`, (ii) `1/2`','','','','',0,'',''),(1312,6,113.000,'An urn contains 8 red & 5 white balls. 3 balls are drawn together for 2 times without replacement. What is the probability of 3 white balls in first drawn & 3 red balls in second drawn?',1620826270,'`7/420`','','','','',0,'একটি বাক্সে 8টি লাল বল ও 5টি সাদা বল আছে । বাক্সটি থেকে পরপর 2 বার বিনা প্রত্যর্পণে (without replacement) 3টি করে বলা তোলা হল । প্রথমবার 3টি বল সাদা হওয়ার এবং দ্বিতীয়বার 3টি বল লাল হওয়ার সম্ভাবনা নির্ণয় করো।',''),(1313,6,88.000,'The probability that a contractor will get a contract `A` is `3/5` and contract `B` is `1/3`; and the probability of getting both contracts is `1/8`. What is the probability that the contractor will get one contract between A or B?',1620827125,'`97/120`','','','','',0,'',''),(1314,6,9.000,'Two unbiased dice are thrown. What is the probability of getting odd number from first die and multiples of 3 from second die?',1620827538,'`1/6`','','','','',0,'',''),(1315,6,91.000,'A green die and a white die are thrown at the same time. What is the probability that the sum of the numbers appearing on the top of the dice is greater than 8 provided that 6 is appeared on the green die.',1621059149,'`2/3`','','','','',0,'',''),(1316,6,83.000,'`A, B` & `C` are three events such that `P(C) = 0.55`, `P(A | C) = 0.25`, `P(B | C) = 0.4` and `P(A \\cap B \\cap C) = 0.30` then find `P((A \\cup B) | C))`',1621059552,'0.1045','','','','',0,'',''),(1317,6,100.000,'`A` speaks the truth 8 times out of 10 times. A die is tossed and he reports that it was 5. What is the probability that It was actually 5?',1621059918,'`4/9`','','','','',0,'এক ব্যক্তি 10 বারের মধ্যে 8 বার সত্য বলে। একটি ছক্কা রোল করা হল এবং সে জানাল এটি 5 । এটি সঠিক 5 হওয়ার সম্ভাবনা নির্ণয় করো। ',''),(1318,6,76.000,'If `P(A | B) = 0.8`, `P(B | A) = 0.6` and `P(A^c \\cup B^c) = 0.7`, then find `P(A | B^c)` & `P(A\\cup B)`',1621060259,'0.32, `23/40`','','','','',0,'যদি `P(A | B) = 0.8`, `P(B | A) = 0.6` এবং `P(A^c cup B^c) = 0.7` হয়, তবে  `P(A | B^c)` & `P(A\\cup B)` এর মান নির্ণয় করো।',''),(1319,6,85.000,'Two events `A` & `B` are such that that `P(A^c) = 0.3`, `P(B) = 0.4` and `P(A \\cap B^c) = 0.5`. Then find `P(B | A \\cup B^c)`',1621060660,'`1/4`','','','','',0,'',''),(1320,6,101.000,'`A` speaks lie 3 times out of 4 times. A die is tossed and he reports that it was 6. What is the probability that It was actually 6?',1621061198,'`3/8`','','','','',0,'এক ব্যক্তি 4 বারে 3 বার সত্যি কথা বলে। তিনি একটি ছক্কা ছুড়লেন এবং বলেন যে ছয় পড়েছে। সত্যিই ছয় পড়েছে, তার সম্ভাবনা নির্ণয় করো। ',''),(1321,6,141.000,'In a class 5% of boys and 10% of girls have an I.Q of more than 150. In this class 60% of students are boys. If a student is selected at random and found to have an I.Q. of more than 150. Find the probability that the student is a boy.',1621061747,'`3/7`','','','','',0,'',''),(1322,6,133.000,'Two coins are tossed. What is the conditional probability that two heads result, given that there is at least one head ?',1621062212,'`1/3`','','','','',0,'',''),(1323,6,39.000,'If `P(A) = 2/3`, `P(B) = 1/2` and `P(A \\ cap B) = 1/6` then find `P(A \\cap B^c)`',1621065456,'`1/2`','1323','','','',0,'',''),(1324,6,106.000,'One Number is chosen at random from the integers 1 to 50. What is the probability that it is divisible by 4 or 5?',1621065603,'`2/5`','','','','',0,'',''),(1325,6,82.000,'Show that the probability of occurring exactly one events between `A` & `B` is `P(A) + P(B) - 2P(AB)` ',1621066545,NULL,'','','','',0,'',''),(1326,6,43.000,'If `P(A) = 7/13`, `P(B) = 9/13` and `P(A \\cap B) = 4/13` then find `P(\\bar{A} | B)`',1621067701,'`5/9`','1326','','','',0,'',''),(1327,6,11.000,'An unbiased die is thrown seven times. What is the probability of getting number `5` exactly 2 times?',1621068012,'`(7/12)(5/6)^5`','','','','',0,'',''),(1328,6,6.000,'An  unbiased die is throwing 3 times. If 5 obtained in first throwing, then the probability of getting total as 16 in three throwings are --<br />\r\n(a) `1/18`, (b) `1/9`, (c) `1/108`, (d) `5/16` ',1621080068,'(a) `1/18`','','','','',0,'',''),(1329,6,66.000,'Two events `A` & `B` are such that `P(A) = P(A | B) = 1/4`, `P(B | A) = 1/2`, then --<br />\r\n(a) `A` & `B` are independent, (b) `A` & `B` are not independent, (c) `P(A \\cup B) = 0`, (d) None of these',1621080430,'(a) `A` & `B` are independent,','','','','',0,'`A` ও `B` দুটি ঘটনা এমন যে, `P(A) = P(A | B) = 1/4` এবং `P(B | A) = 1/2` হলে -- <br />\r\n(a) `A` ও `B` ঘটনা দুটি স্বাধীন, (b) `A` ও `B` ঘটনা দুটি স্বাধীন নয়, (c) `P(A \\cup B) = 0`, (d) এদের কোনোটিই নয়। ',''),(1330,6,61.000,'If `A` & `B` are two events such that `P(A) = 1/4`, `P(A | B) = 1/2` and `P(B | A) = 2/3` then value of `P(B)` is --<br />\r\n(a) `1/2`, (b) `1/6`, (c) `2/3`, (d) `1/3`',1621083887,'(d) `1/3`','','','','',0,'',''),(1331,6,24.000,'Three integers are chosen at random from the first 20 integers. The probability that their product is even is --<br />\r\n(a) `2/19`, (b) `17/19`, (c) `15/19`, (d) none of these',1621084364,'(b) `17/19`','','','','',0,'',''),(1332,6,48.000,'`A` & `B` are two events such that `P(A \\cup B) = 4/5`, `P(\\bar{A} \\cup \\bar{B}) = 7/10` and `P(B) = 2/5`, then value of `P(A)` is --<br />\r\n(a) `9/10`, (b) `7/10`, (c) `3/5`, (d) `8/10`',1621085673,'(b) `7/10`','','','','',0,'',''),(1333,6,35.000,'If `P(A) = 1/4`, `P(B) = 1/2` and `P(A \\cup B) = 5/9`, then `P(A | B) = ` ?',1621233411,'`7/18`','1333','','','',0,'',''),(1334,6,121.000,'There are three bags each containing 5 red and 5 black balls. One ball is transferred blindly from the first to the second bag and then one ball is transferred blindly from the second to the third. A ball is now drawn from the third bag. Find the probability that the drawn ball is red.',1621233736,'`1/2`','','','','',0,'',''),(1335,6,26.000,'`A` & `B` are mutually exclusive events. If `P(A) = 1/2` & `P(A \\cup B) = 2/3`, then value of `P(B)` is --<br />\r\n(a) `1/4`, (b) `1/6`, (c) `1/3`, (d) `1/5`',1621234020,'(b) `1/6`','1335','','','',0,'',''),(1336,6,73.000,'If `A` & `B` are any two events of a random experiment, then prove that `P(\\bar{A} | B) = 1 - P(A | B)`',1621234300,NULL,'','','','',0,'',''),(1337,6,128.000,'There are three bags. Bag A contains 6 red & 4 white balls; bag B contains 2 red & 6 white balls; bag C contains 1 red & 5 white balls. One bag is chosen at random and a ball is drawn from it and it is found as a red ball. What is the probability that the ball is drawn from bag A?',1621234589,'`36/61`','1337.01','','','',0,'',''),(1338,6,19.000,'The probability that there will be 53 Sunday in a year of the 22nd century chosen at random is --<br />\r\n(a) `3/28`, (b) `2/28`, (c) `7/28`, (d) `5/28`',1621234922,'(d) `5/28`','','','','',0,'',''),(1339,6,32.000,'If the probability of an event is `5/8`, then odds against this event is --<br />\r\n(a) `5:13`, (b) `5:3`, (c) `3:5`, (d) `8:13`',1621235404,'(c) `3:5`','','','','',0,'একটি ঘটনা ঘটার সম্ভাবনা `5/8`, ঘটনাটির প্রতিকূলে সুযোগ হবে -- --<br />\r\n(a) `5:13`, (b) `5:3`, (c) `3:5`, (d) `8:13`',''),(1340,6,115.000,'There are three bags. Bag I contains 3 white & 1 black balls; bag II contains 2 white & 2 black balls; bag III contains 1 white & 3 black balls. One ball is drawn randomly from each bag. What is the probability of getting 2 white & 1 black balls?',1621238167,'`13/32`','','','','',0,'',''),(1341,6,57.000,'If `P(A) = 1/4`, `P(B) = 1/3` and `P(A - B) = 1/6`, then are `A` & `B` mutually independent?',1621239125,'Yes','','','','',0,'',''),(1342,6,46.000,'If `P(A) = 2/3`, `P(B) = 1/4` and `P(A | B) = 1/3`, then find the probability of occurring exactly one events of `A` & `B` ',1621241928,'`3/4`','','','','',0,'',''),(1343,6,125.000,'A bag contains 3 red & 7 black balls. Two balls are drawn randomly one by one without replacement. If second ball drawn is red color, then find the probability of first ball drawn is also red.',1621242514,'`1/15`','','','','',0,'',''),(1344,6,15.000,'Let `X` represents the difference between number of heads and number of tails obtained when a coin is tossed 6 times. What are possible values of `X`? ',1621242904,'`X = {0,2,4,6}`','','','','',0,'',''),(1345,6,140.000,'In a bulb factory, machines A, B and C manufacture 60%, 30% and 10% bulbs respectively. 1%, 2% and 3% of the bulbs produced respectively by machines A, B and C are found to be defective. A bulb is picked up at random from the total production and found to be defective. Find the probability that this bulb was produced by the machine A.',1621249397,'`2/5`','1345','','','',0,'',''),(1346,6,59.000,'If `P(A - B) = 1/3`, `P(A) = 1/2` and `P(B) = 1/3`, then find the probability of occurring exactly the event `B`.',1621251567,'`1/6`','','','','',0,'',''),(1347,6,62.000,'If `P(A) = 1/4`, `P(A | B) = P(B | A) = 1/2`, then find `P(A^c | B^c)`',1621252042,'`5/6`','','','','',0,'',''),(1348,6,51.000,'`A` & `B` are two mutually independent events. If `P(A) = 3/5` and `P(A \\cap B) = 4/9`, then value of `P(B)` is -<br />\r\n(a) `5/9`, (b) `8/9`, (c) `5/27`, (d) `20/27`',1621252849,'(d) `20/27`','','','','',0,'',''),(1349,6,36.000,'If `P(A) = 1/12`, `P(B) = 5/12` & `P(B | A) = 1/15`, then value of `P (A \\cup B)` is -<br />\r\n(a) `89/180`, (b) `91/180`, (c) `93/180`, (d) `87/180`',1621253285,'(a) `89/180`','1349','','','',0,'',''),(1350,6,52.000,'`A` & `B` are two mutually independent events. If `P(A) = 1/3` and `P(A \\cap B) = 1/6`, then value of `P(A^c \\cap B^c)` is -<br />\r\n(a) `1/6`, (b) `2/3`, (c) `1/3`, (d) `2/9`',1621253750,'(c) `1/3`','','','','',0,'`A` ও `B` হলো দুটি পরস্পর স্বাধীন ঘটনা। যদি `P(A) = 1/3` এবং `P(A cap B) = 1/6` হয়, তবে `P(A^c cap B^c)` এর মান -<br />\r\n(a) `1/6`, (b) `2/3`, (c) `1/3`, (d) `2/9`',''),(1351,6,2.000,'If two dice are rolled, then the probability of getting one number greater than 4 is -<br />\r\n(a) `4/9`, (b) `7/9`, (c) `1/9`, (d) `5/6`',1621254254,'(c) `1/9`','','','','',0,'',''),(1352,6,3.000,'A die is thrown twice. Let `A` be the event of getting multiple of 3 in 1st throwing and `B` be the event of getting total as less than 6 in two throwing. Then `P(A | B)` is - <br />\r\n(a) `5/19`, (b) `5/21`, (c) `5/23`, (d) `5/17`',1621254781,'(b) `5/21`','','','','',0,'',''),(1353,6,42.000,'If `P(A) = 0.5`, `P(B) = 0.4`, `P(A \\cap B) = 0.3`, then find the value of `P(A^c | B^c)`',1621256080,'`2/3`','','','','',0,'',''),(1354,6,84.000,'`A, B, C` are three events such hat `P(B) = 3/4`, `P(A^c \\cap B \\cap C^c) = 1/3` and `P(A \\cap B \\cap C^c) = 1/3`, then the value of `P(B \\cap C)` is -<br />\r\n(a) `1/12`, (b) `5/12`, (c) `1/4`, (d) `23/36`',1621581843,'(a) `1/12`','','','','',0,'',''),(1355,6,45.000,'If `P(A) = 0.3`, `P(B) = 0.4` and `P(A \\cup B) = 0.6`, then find `P(\\bar{A} | B)` & `P(\\bar{B} | A)`',1621583041,'`2/3, 4/7`','','','','',0,'কোনো যদৃচ্ছ পরীক্ষায় A ও B দুটি ঘটনা হলে `P(\\bar{A} | B)` এবং `P(\\bar{B} | A)` এর মান নির্ণয় করো যেখানে  `P(A) = 0.3`, `P(B) = 0.4` এবং `P(A \\cup B) = 0.6`।',''),(1356,6,55.000,'`A` & `B` are two mutually independent events. If `P(\\bar{A} \\cap B) = 2/15` and `P(A \\cap \\bar{B}) = 1/6`, then find `P(B)`',1621583481,'`1/6` or `4/5`','','','','',0,'',''),(1357,6,119.000,'There are two urns. First urn contains 2 red & 3 white balls. Second urn contains 3 red & 5 white balls. A ball is transferred randomly from first urn to second urn and then a ball is drawn randomly from second urn. If the ball drawn is red, then find the probability of transferring a white ball from first urn to second urn.',1621830053,'`9/17`','','','','',0,'',''),(1358,6,21.000,'Find the probability that the 13th day of a randomly chosen month is a Friday.',1621831194,'`1/84`','','','','',0,'',''),(1359,6,126.000,'A bag contains 4 white & 6 black balls. Two balls are drawn randomly one by one without replacement. If first ball drawn is white color, then find the probability of second ball drawn is also white.',1621831584,'`2/15`','','','','',0,'',''),(1360,6,120.000,'There are two urns. First urn contains 3 black & 4 white balls. Second urn contains 4 black & 5 white balls. A ball is transferred randomly from first urn to second urn and then a ball is drawn randomly from second urn. If the ball drawn is black, then find the probability of transferring a white ball from first urn to second urn.',1621832619,'`16/31`','','','','',0,'দুটি পাত্র আছে। প্রথম পাত্রে 3টি কালো এবং 4টি সাদা বল রয়েছে। দ্বিতীয় পাত্রে 4টি কালো এবং 5টি সাদা বল রয়েছে। উদ্দেশ্যহীনভাবে একটি বল প্রথম পাত্র থেকে দ্বিতীয় পাত্রে স্থানান্তরিত করা হল এবং তারপরে একটি বল দ্বিতীয় পাত্র থেকে উদ্দেশ্যহীনভাবে তোলা হয়। যদি তোলা বলটি কালো হয়, তাহলে প্রথম পাত্র থেকে দ্বিতীয় পাত্রে স্থানান্তরিত হওয়া বলটি সাদা হওয়ার সম্ভাবনা নির্ণয় করো।',''),(1361,6,17.000,'A student makes random guesses for 8 true/false questions. What is the probability of getting 5 correct answer?',1621835474,'`93/2^8`','','','','',0,'',''),(1362,6,71.000,'For any two events `A` & `B` (where `P(B) \\ne 0`), show that `P(A | \\bar{B}) = P(A)`',1621836961,NULL,'','','','',0,'',''),(1363,6,63.000,'If `P(A | B) = 0.75`, `P(B | A) = 0.6` and `P(A) = 0.4`, then find `P(\\bar{A} | \\bar{B})`',1621840461,'`13/17`','','','','',0,'',''),(1364,6,64.000,'If `P(A | B) = 0.8`. `P(B | A) = 0.6` & `P(A^c \\cup B^c) = 0.7`; then find `P(A \\cup B)`',1621841018,'`23/40`','','','','',0,'',''),(1365,6,72.000,'If `P(A) = a` and `P(B) = b`, then show that -<br />(i) `P(A | B) \\le a/b`<br />(ii) `P(A | B) \\gt (a+b-1)/b`',1621841427,'','','','','',0,'',''),(1366,6,79.000,'If `A_1, A_2, \\cdots, A_n` are mutually independent events and `P(A_i) = 1 - q_i` (`i = 1, 2, \\cdots, n`), then prove that `P(A_1 \\cup A_2 \\cup \\cdots \\cup A_n) = 1 - q_1 q_2 \\cdots q_n`',1621841752,'','','','','',0,'যদি `A_1, A_2, cdots, A_n` ঘটনাসমূহ পরস্পর স্বাধীন এবং `P(A_i) = 1 - q_i` (`i = 1, 2, cdots, n`) হয়, তবে প্রমাণ করো যে `P(A_1 cup A_2 cup cdots cup A_n) = 1 - q_1 q_2 cdots q_n`',''),(1367,6,29.000,'A coin is tossed 10 time. The probability of getting 6 heads - <br />\r\n(a) `C_5^10 1/2^10`, (b) `C_3^10 1/2^10`, (c) `C_4^10 1/2^10`, (d) `C_8^10 1/2^10`, ',1621842224,'(c) `C_4^10 1/2^10`','','','','',0,'',''),(1368,6,38.000,'If `P(A) = 1/5`, `P(B) = 2/3` & `P(A \\cap B) = 4/15`, then verify that `A` & `B` are exhaustive events.',1622256599,'Not mutually exhaustive','1368.01','','','',0,'',''),(1369,6,65.000,'If `P(A) = 1/3`, `P(A \\cap B^c) = 1/4`; then value of `P(B | A)` is -<br />\r\n(a) `1/4`, (b) `1/3`, (c) `1/12`, (d) `3/4`',1622266320,NULL,'','','','',0,'',''),(1370,6,20.000,'What is the probability that in a family of 5 members, exactly two members have birthday on Sunday?',1622271564,'`\\frac{(10 \\times 6^3)}{7^5}`','','','','',0,'5 সদস্য বিশিষ্ট একটি পরিবারের যে-কোনো 2 সদস্যের জন্মদিন রবিবার হওয়ার সম্ভাবনা নির্ণয় করো।  ',''),(1371,6,75.000,'If `P(A | B) = P(A)`, then prove that `P(A^c | B) = P(A^c)`',1622871911,NULL,'','','','',0,'',''),(1372,6,108.000,'The odds against a man who is 45 years old living till he is 70 are 7:5 and the odds against his wife who is now 36 living till she is 61 are 5:3 Find the probability that<br />\r\n(i) The couple will be alive 25 years hence,<br />\r\n(ii) At least one of them will be alive 25 years hence.',1622872775,'(i) `5/32`, (ii) `61/96`','','','','',0,'',''),(1373,6,33.000,'7 white balls and 3 black balls are randomly placed in a row. The probability that no two black balls are placed adjacently is - <br />\r\n(a) `1/2`, (b) `7/15`, (c) `2/15`, (d) `1/3`',1623383885,'(b) `7/15`','','','','',0,'',''),(1374,6,22.000,'The probability that a leap year will have 53 Fridays or 53 Saturdays is -<br />\r\n(a) `1/7` (b) `2/7`, (c) `3/7`, (d) `4/7`',1623385464,'(c) `3/7`','','','','',0,'',''),(1375,6,74.000,'If `A` & ``B are mutually exclusive and `P(B) \\ne 0` then `P(A | \\bar{B})` is - <br />\r\n(a) `\\frac{1}{P(B)}`  (b) `\\frac{1}{1 - P(B)}` (c) `\\frac{P(A)}{P(B)}` (d) `\\frac{P(A)}{1 - P(B)}`',1623386659,'(d) `\\frac{P(A)}{1 - P(B)}`','','','','',0,'',''),(1376,6,68.000,'`A` & `B` are two events such that `P(\\bar{B}) = 1/2`, `P(A \\cup B) = 5/6` and `P(A \\cap B) = 1/3`, then the events `A` & `B` are - <br />\r\n(a) independent (b) dependent (c) mutually exclusive (d) none of these',1623387249,'(a) independent','','','','',0,'যদি `A` ও `B` দুটি ঘটনা এবং `P(\\bar{B}) = 1/2`, `P(A \\cup B) = 5/6` and `P(A \\cap B) = 1/3` হয়, তবে `A` ও `B` ঘটনা দুটি - <br />\r\n(a) স্বাধীন (b) অধীন (c) পরস্পর পৃথক (d) কোনোটিই নয়।',''),(1377,6,14.000,'An unbiased coin is tossed three times. Find `P(E | F)` where `E`: Head in third toss, & `F`:  Head in first two toss.',1623392039,'`1/2`','','','','',0,'',''),(1378,6,58.000,'If `A` & `B` are mutually independent events and `P(A \\cup B^c) = 0.8` and `P(A) = 0.3`. Find `P(B)`',1623393101,'`2/7`','','','','',0,'',''),(1379,6,16.000,'What is the probability of two numbers randomly selected from the first 15 natural numbers both being odd, if their sum is even?',1623394338,'`4/7`','','','','',0,'',''),(1380,6,7.000,'Three unbiased dice are thrown at a time. What is the probability of getting total as at least 6 from three obtained numbers?',1623394897,'`209/216`','','','','',0,'',''),(1381,6,60.000,'Two events `A` & `B` are such that `P(A) = 1/3`, `P(B) = 3/4`. Find the probability of occurring only event `B` provided that `A` is not occurred.',1623395236,'Between `[5/8, 1]`','','','','',0,'',''),(1382,6,95.000,'The probability that `A` hits a target is `1/3` and the probability that `B` hits it, is `2/5`. What is the probability that the target will be hit, if each one of `A` and `B` shoots at the target?',1623396300,'`3/5`','','','','',0,'',''),(1383,6,97.000,'A can hit a target in 3 out of 4 shots and B can hit the target in 4 out of 5 shots. What is the probability of hitting the target ?',1623396691,'`19/20`','','','','',0,'',''),(1384,6,13.000,'A coin is tossed 4 times, if head occurs on first three tosses, then the probability of getting head on fourth toss is - <br />\r\n(a) `1/2` (b) `1/3` (c) `1/4` (d) `3/4`',1624013251,'(a) `1/2`','','','','',0,'',''),(1385,6,50.000,'`A` & `B` are two mutually independent events. If `P(A) = 0.2`, `P(A \\cup B) = 0.8` then value of `P(A \\cap B)` is - <br />\r\n(a) `0.12` (b) `0.15` (c) `0.3` (d) `0.25`',1624015233,'(b) `0.15`','','','','',0,'',''),(1386,6,124.000,'There are 3 white & 2 red balls in a bag. One ball is drawn. Then another ball is drawn without replacing the first ball. Then the probability of 2nd ball is red is -<br />\r\n(a) `8/25`, (b) `2/5` (c) `3/5`, (d) `21/25`',1624016047,'(b) `2/5`','','','','',0,'',''),(1387,6,23.000,'In a non-leap year the probability of getting 53 Sundays or 53 Tuesdays or 53 Thursdays is -<br />\r\n(a) `1/2` (b) `2/7` (c) `3/7` (d) `4/7`',1624016576,'(c) `3/7`','','','','',0,'',''),(1388,6,70.000,'If `A` & `B` are mutually independent events, then prove that `P(A | B^c) = P(A)`',1624016805,NULL,'','','','',0,'',''),(1389,6,86.000,'If `P(A) = 0.8`, `P(A | B) = 0.8` and `P(A \\cap B) = 0.5`, then find `P(A \\cap B | \\bar{B})`',1624018467,'`0`','','','','',0,'',''),(1390,6,18.000,'What is the probability of getting 5 Sundays in the month of January?',1624019151,'`3/7`','','','','',0,'',''),(1391,6,112.000,'There are 5 white, 7 red & 4 black balls in a bag. If 4 balls are drawn one by one (i) with replacement (ii) without replacement, then find the probability of no white balls. ',1624019421,'(i) `(11/16)^4`, (ii) `33/182`','','','','',0,'',''),(1392,6,89.000,'12 cards, numbered 1 to 12, are placed in a box, mixed up thoroughly and then a card is drawn at random from the box. If it is known that the number on the drawn card is more than 3, find the probability that it is an even number.',1624019948,'`5/9`','','','','',0,'',''),(1393,6,129.000,'There are three bags. Bag A contains 2 white, 1 black & 3 red balls; bag B contains 3 white, 2 black & 4 red balls; bag C contains 4 white, 3 black & 2 red balls. One bag is chosen randomly and 2 balls are drawn from it. If the balls drawn are 1 black & 1 red, then find the probability that they are drawn from bag B.',1624021857,'`20/53`','1393','','','',0,'',''),(1394,6,96.000,'The probabilities for A, B, and C hitting a target are `1/3`, `1/4` & `1/5` respectively. If they try together, find the probability of exactly one shot hitting the target.',1624022203,'`13/30`','','','','',0,'',''),(1395,6,5.000,'Two unbiased dice are thrown and the sum of the numbers obtained is 8. Find the conditional probability of getting at least one 5.',1624022849,'`2/5`','','','','',0,'',''),(1396,6,4.000,'Two unbiased dice are thrown. What is the probability of getting sum as 8 or more of 2 obtained numbers?',1624023058,'`15/36`','','','','',0,'দুটি পক্ষপাতশূন্য ছক্কা একবার রোল করা হল। দুটি ছক্কার দুটি তলে নির্দেশিত অঙ্ক দুটির সমষ্টি 8 (আট) বা তার বেশী হওয়ার সম্ভাবনা কত? ',''),(1397,6,98.000,'A can hit a target 3 times in 6 shots, B: 2 times in 6 shots and C: 4 times in 4 shots. They fix a volley. What is the probability that at least 2 shots hit? ',1624023405,'`2/3`','','','','',0,'',''),(1398,13,17.000,'The price of a precious gem is directly varies to square of its weight. This gem is broken into four pieces and the ratio of weight of those pieces are `1:2:3:4`; If the amount of loss is Rs 70,000, then find out the price of original gem.',1639408631,'','','','','',0,'একটি মূল্যবান মণির দাম তার ওজনের বর্গের সাথে সরলভেদে আছে। এই রত্নটি ভেঙ্গে চারটি টুকরোতে বিভক্ত হয় এবং সেই টুকরোগুলির ওজনের অনুপাত হল `1:2:3:4`; যদি ক্ষতির পরিমাণ 70,000 টাকা হয়, তাহলে আসল রত্নটির দাম খুঁজে বের করুন।',''),(1399,6,104.000,'If on an average 1 vessel in every 10 is wrecked, find the chance that out of 5 vessels expected 4 at least will arrive safely.',1641485247,'`\\frac{45927}{50000}`','','','','',0,'সাধারণভাবে গড়ে 10 টি জাহাজের মধ্যে একটি জাহাজ ডুবে যায়। তাহলে 5 টি জাহাজের মধ্যে কমপক্ষে 4 টি জাহাজের নিরাপদে তীরে পৌছনোর সম্ভাবনা নির্ণয় করো। ',''),(1400,6,130.000,'There are three coins. One is a two headed coin (having head on both faces), another is a biased coin that comes up heads 75% of the time and third is an unbiased coin. One of the three coins is chosen at random and tossed, it shows heads, what is the probability that it was the two headed coin ?',1641808176,'`4/9`','','','','',0,'তিনটি মুদ্রা আছে, যার একটির উভয়দিক Head যুক্ত, দ্বিতীয়টি পক্ষপাতদুষ্ট যেটি 75% Head পড়ে এবং তৃতীয়টি পক্ষপাত শূন্য মুদ্রা। যেকোনো একটি মুদ্রা যথেচ্ছভাবে নির্বাচন করা হল এবং টস করে দেখা গেল যে Head পড়েছে, তাহলে মুদ্রাটি উভয়দিকে Head যুক্ত হওয়ার সম্ভাবনা কত? ',''),(1401,6,25.000,'If three distinct numbers are chosen randomly from the first 100 natural numbers, then the probability that all the three numbers are divisible by both 2 and 3  is --<br />\r\n(a) `4/25`  (b) `4/35`  (c) `4/33`  (d) `4/1155`',1641810069,'(d) `4/1155`','','','','',0,'প্রথম 100 টি স্বাভাবিক সংখ্যার মধ্য থেকে যদৃচ্ছভাবে তিনটি পৃথক সংখ্যা নির্বাচন করা হল। এই সংখ্যাত্রয়ের প্রত্যেকটি 2 ও 3 দ্বারা  বিভাজ্য হওয়ার সম্ভাবনা --<br />\r\n(a) `4/25`  (b) `4/35`  (c) `4/33`  (d) `4/1155`',''),(1402,6,80.000,'If A & B are two independent events, then show that the probability of occurring at least one event is `1 - P(A^c)P(B^c)`',1641811126,NULL,'','','','',0,'যদি A ও B ঘটনাদ্বয় স্বাধীন / অনপেক্ষ হয়, তবে দেখাও যে, A ও B ঘটনাদ্বয়ের মধ্যে অন্ততপক্ষে একটি ঘটার সম্ভাবনা হবে `1 - P(A^c)P(B^c)`',''),(1403,6,132.000,'A and B toss a coin alternately till one of them tosses a head and wins the game. If A starts the game, what are their respective probability of winning?',1641812016,'Probability of A Winning the game is `2/3` and B Winning the game is `1/3`','','','','',0,'দুই ব্যক্তি A ও B একটি মুদ্রাকে একজনের পর অন্যজন এইভাবে ছুঁড়তে থাকে এবং যে আগে হেড ফেলবে সে খেলাটি জিতবে। যদি A খেলাটি শুরু করে, তবে প্রত্যেকের জেতার সম্ভাবনা কত? ',''),(1404,6,27.000,'Two events A & B associated with a random experiment, are such that `P(B)=0.35`, `P(A\\cupB)=0.85` & `P(A\\capB)=0.15`, then find `P(A)`',1641970654,'0.65','1404','','','',0,'কোনো সমসম্ভব পরীক্ষার সঙ্গে সংশ্লিষ্ট দুটি ঘটনা A ও B যদি এমন হয় যে, `P(B)=0.35`, `P(AcupB)=0.85` ও `P(AcapB)=0.15`, তবে `P(A)` এর মান নির্ণয় করো। ',''),(1405,6,131.000,'There are 3 coins in a box. The probability of getting head is `2/3` for two of them, while for remaining it is `1/2`. A coin is chosen at random and tossed 3 times, showing head in each case. What is the probability that the coin is unbiased ?',1641972853,'`27/155`','','','','',0,'একটি বাক্সে 3 টি মুদ্রা আছে। তাদের মধ্যে দুটির ক্ষেত্রে হেড পাবার সম্ভাবনা `2/3` এবং অন্য মুদ্রাটির ক্ষেত্রে এই সম্ভাবনা `1/2`। বাক্সটি থেকে উদ্দেশ্যহীনভাবে একটি মুদ্রা নেওয়া হয় এবং তিনবার টস করে প্রতিবারেই হেড পাওয়া যায় । বাক্সটি থেকে নেওয়া মুদ্রাটি ঝোঁকশূন্য হওয়ার সম্ভাবনা কত? ',''),(1406,6,77.000,'If `P(A | B)=1`, then prove that `P(ABC)=P(BC)`',1641973555,NULL,'','','','',0,'`P(A | B)=1` হলে, প্রমাণ করো যে `P(ABC)=P(BC)`',''),(1407,6,87.000,'The probability that a teacher will give an unannounced test during any class meeting is `1/5`. If a student is absent twice, then find the probability that the student will miss at least one test.',1641975179,'`9/25`','','','','',0,'ছাত্রদের সঙ্গে মিলিত হয়ে একজন শিক্ষকের হঠাৎ পরীক্ষা নেওয়ার সম্ভাবনা `1/5`। যদি একজন ছাত্র দুদিন অনুপস্থিত থাকে, তবে তার অন্ততপক্ষে একটি পরীক্ষা দেওয়ার সুযোগ নষ্ট হওয়ার সম্ভাবনা কত?',''),(1408,6,118.000,'An urn A contains 3 white and 5 red balls, a second urn B contains 5 white and 3 red balls. Two balls are transferred from urn A to urn B, then a ball is drawn from the urn B. What the probability that it is red?',1642595325,NULL,'','','','',0,'একটি A পাত্রের মধ্যে 3টি সাদা ও 5টি লাল মার্বেল আছে । অন্য একটি পাত্র B-এর মধ্যে 5টি সাদা এবং 3টি লাল মার্বেল আছে। A পাত্র থেকে B পাত্রে 2টি মার্বেল স্থানান্তরিত করা হলো ও তারপর B পাত্র থেকে 1টি মার্বেল তোলা হলো । উত্তোলিত মার্বেলটি লাল হওয়ার সম্ভাবনা নির্ণয় করো।',''),(1409,6,110.000,'There are 2 white, 3 red & 4 black balls. One ball is drawn from the bag and then again one ball is drawn from the bag. What is the probability of at least one red ball among 2 drawn balls?',1642602643,'`7/12`','','','','',0,'একটি থলিতে 2টি সাদা, 3টি লাল ও 4টি কালো বল আছে। থলিটি থেকে প্রথমে একটি এবং তারপর আরো একটি বল তুলে নেওয়া হল। তুলে নেওয়া দুটি বলের মধ্যে অন্ততপক্ষে একটি বল লাল হওয়ার সম্ভাবনা নির্ণয় করো।',''),(1410,6,134.000,'Suppose 5% of men and 0.25% of women have grey hair. A grey haired person is selected at random. What is the probability of this person being male? Assume that there are equal number of males and females.',1642607222,'`20/21`','','','','',0,'ধরা যাক 5% পুরুষের এবং 0.25% মহিলার ধূসর রঙের চুল। ধূসর রঙের চুলের একজনকে যথেচ্ছভাবে বাছাই করা হলো। ঐ বাছাই জন পুরুষ হওয়ার সম্ভাবনা নির্ণয় করো। ধরে নাও পুরুষ ও মহিলার সংখ্যা সমান।',''),(1411,6,54.000,'A and B are two independent events such that `P(A)=1/2`, `P(B)=1/5`. Then value of `P(A | A\\cupB)` is --<br />\r\n(a) `1/5`,  (b) `2/5`,  (c) `5/6`,  (d) `5/7`',1642695815,'(c) `5/6`','','','','',0,'A ও B দুটি স্বাধীন ঘটনা এমন যে, `P(A)=1/2`, `P(B)=1/5`। তাহলে `P(A | AcupB)` এর মান হল --<br />\r\n(a) `1/5`,  (b) `2/5`,  (c) `5/6`,  (d) `5/7`',''),(1412,6,47.000,'If `P(A)=0.6`, `P(B)=0.75` and `P(A | B)=0.45`, then find the value of `P(B^c | A)`',1642696512,'`0.4375`','','','','',0,'যদি `P(A)=0.6`, `P(B)=0.75` এবং `P(A | B)=0.45` হয়, তাহলে `P(B^c | A)` এর মান নির্ণয় করো। ',''),(1413,6,109.000,'A bag contains 4 balls.Two balls drawn at random without replacement and are found to be white. What is the probability that all balls are white?',1642779775,'`3/5`','','','','',0,'একটি ব্যাগে 4টি বল আছে। ব্যাগটি থেকে যদৃচ্ছভাবে 2টি বল তোলা হল এবং দেখা গেল যে দুটি বলই সাদা । ব্যাগের সবকটি বল সাদা হওয়ার সম্ভাবনা কত?',''),(1414,6,102.000,'A bag contains 6 balls of different colours and a ball is drawn from it. A\'s probability of speaking the truth is `\\frac{2}{3}` and B\'s probability of speaking thetruth is `\\frac{4}{5}`. If both A and B say that a red ball has been drawn, then find the probability that actually a red ball has been drawn.',1643384680,'`40/41`','','','','',0,'A, 3 বারের মধ্যে 2 বার সত্য কথা বলে এবং B, 5 বারের মধ্যে 4 বার সত্য কথা বলে। তারা নিশ্চিতভাবে বলতে সম্মত হলো যে, একটি ব্যাগে ছয় রকমের বিভিন্ন রঙের বল আছে, যার থেকে একটি বল নেওয়া হলো এবং সেটি লাল রঙের বল । বক্তব্যটি সত্য হওয়ার সম্ভাবনা কত?',''),(1415,15,1.000,'If simple interest for 5 years is `\\frac{1}{5}` of principal, what is the annual simple interest rate -<br />\r\n(a) 4%, (b) 5%, (c) 10%, (d) 25%\r\n',1682000339,'(a) 4%','','','','',0,'5 বছরের মোট সুদ, আসলের `frac{1}{5}` অংশ হলে বার্ষিক সরল সুদের হার কত - <br />\r\n(a) 4%, (b) 5%, (c) 10%, (d) 25%',''),(1416,15,18.000,'At what rate of simple interest per annum, interest of a sum of money, will be `\\frac{2}{5}` of amount in 10 years?\r\n',1682001285,'6`\\frac{2}{3}`%','','','','',0,'শতকরা বার্ষিক সরল সুদের হার কত হলে কোনো টাকার 10 বছরের সুদ, সুদ-আসলের `frac{2}{5}` অংশ হবে?',''),(1417,15,13.000,'Simple interest for `z` months of Rs. `y` at the rate of simple interest of `x`% per annum is Rs. _____.\r\n',1682001705,'`\\frac{xyz}{1200}`','','','','',0,'বার্ষিক `x`% সরল সুদের হারে `y` টাকার `z` মাসের সরল সুদ ___ টাকা।',''),(1418,15,33.000,'Amal Babu deposited Rs 93,000 at 5% simple interest per annum for 3 years in a post office, 5 years in a state-owned bank and 2 years in a cooperative bank in such a way that the amount of interest received from each place is equal. How much money did Amal babu deposit at each place?',1682002374,'Rs. 30,000/- in post-office, Rs. 18,000/- in state-owned bank & Rs. 45,000 in cooperative bank.','1418','','','',0,'অমলবাবু 93,000 টাকা বার্ষিক 5% সরল সুদের হারে পোস্ট অফিসে 3 বছর, রাষ্ট্রায়ত্ব ব্যাংকে 5 বছর ও সমবায় ব্যাংকে 2 বছরের জন্য এমন ভাবে ভাগ করে রাখলেন যাতে প্রত্যেক জায়গা থেকে প্রাপ্ত সুদের পরিমাণ সমান হয়। অমলবাবু কোথায় কত টাকা জমা রেখেছিলেন?',''),(1419,15,14.000,'If the principal is Rs. `x`, amount is Rs. `3x`, and the rate of interest is `x`%, the time will be ___.',1682002650,NULL,'','','','',0,'আসল `x` টাকা, সুদাসল `3x` টাকা এবং সুদের হার `x`% হলে, সময় হবে ___।',''),(1420,15,28.000,'A businessman took a loan of Rs 50,000 from a bank at a simple interest rate of 12% per annum. After 3 years he repaid a total of Rs.25,000 to the bank. How much money will he have to pay to the bank if he wants to repay his loan in full after 5 more years?',1682003302,'Rs. 53320','1420','','','',0,'এক ব্যবসায়ী কোন ব্যাংক থেকে বার্ষিক 12% সরল সুদের হারে 50,000 টাকা ঋণ গ্রহণ করলেন। 3 বছর পর তিনি ব্যাংকে মোট 25,000 টাকা শোধ করলেন। আরো 5 বছর পর তিনি তার ধার সম্পূর্ণভাবে পরিশোধ করতে ইচ্ছুক হলে তাকে কত টাকা ব্যাংকে দিতে হবে?',''),(1421,15,3.000,'If simple interest in 1 month on 1 rupee is 1 paise, the annual interest rate will be - <br />\r\n(a) 1%, (b) 6%, (c) 8%, (d) 12%',1682003411,'(d) 12%','','','','',0,'1 টাকার 1 মাসের সুদ 1 পয়সা হলে বার্ষিক সুদের হার হবে -<br />\r\n(a) 1%, (b) 6%, (c) 8%, (d) 12%',''),(1422,15,27.000,'A fisheries co-operative society borrowed some money from a co-operative bank for fish farming on the condition that interest and `\\frac{1}{5}` of the principal would be repaid every two years at the rate of 9% simple interest per annum. If the first installment is Rs.19000 after two years, calculate how much the society borrowed.',1682004139,'Rs. 50,000','','','','',0,'একটি মৎস্য সমবায় সমিতি মাছ চাষের জন্য একটি সমবায় ব্যাংক থেকে এই শর্তে কিছু টাকা ধার করেছিলেন যে, প্রতি দুই বছর অন্তর বার্ষিক 9% সরল সুদের হারে সুদ ও আসলের `frac{1}{5}` অংশ পরিশোধ করবে। দুই বছর বাদে প্রথম কিস্তির পরিমাণ 19000 টাকা হলে, ওই সমিতি কত টাকা ধার করেছিলেন হিসেব করে লেখো।\r\n',''),(1423,15,9.000,'State true or false.<br />\r\ni) If amount of certain principal for `n` years, at simple interest rate `r`% per annum is Rs. `m`, the principal will be Rs. `\\frac{25m}{(25+n)}`<br />\r\nii) If the original and annual simple interest rates are the same, the interest varies over time in simple proportion.',1682004714,'i) False; ii) True','','','','',0,'সত্য না মিথ্যা উল্লেখ করো।<br />\r\ni) বার্ষিক `r`% সরল সুদের হারে `n` বছরের সবৃদ্ধিমূল `m` টাকা হলে, আসল হবে `frac{25m}{(25+n)}` টাকা।<br />\r\nii) আসল ও বার্ষিক সরল সুদের হার একই থাকলে, সময়ের সঙ্গে সুদ সরল অনুপাতে পরিবর্তিত হয়।',''),(1424,15,22.000,'For a certain principal, amount is Rs. 944 in 3 years at certain rate simple interest. If the annual interest rate increases by 25%, that amount is Rs.980 over the same period. Determine the principal and the annual simple interest rate.\r\n',1682007905,'Principal: Rs. 800, Rate of simple interest: 6%','','','','',0,'কিছু টাকা সরল সুদে 3 বছরে সুদে-মূলে 944 টাকা হয়। যদি বার্ষিক সুদের হার 25% বৃদ্ধি পায়, তবে ওই টাকা একই সময়ে সুদে-মূলে 980 টাকা হয়। টাকার পরিমান ও বার্ষিক সুদের হার নির্ণয় করো। ',''),(1425,15,11.000,'In how many years will the interest on 1200 rupees be 64 rupees at the rate of 8% simple interest per annum?',1682008136,'`\\frac{2}{3}` years','','','','',0,'বার্ষিক 8% সরল সুদের হারে কত বছরে 1200 টাকার সুদ 64 টাকা হবে?',''),(1426,15,4.000,'If the simple interest on Rs. `p` is Rs. `q` and simple interest on Rs. `q` is Rs. `r` (both time and annual simple interest rate are equal), which of the following relations is correct?<br />\r\n(a) `pqr=1`, (b) `p^2 = qr`, (c) `q^2 = pr`, (d) `r^2 = pq`',1682008534,'(c) `q^2 = pr`','','','','',0,'`p` টাকার সরল সুদ `q` টাকা এবং `q` টাকার সরল সুদ `r` টাকা (সময় এবং বার্ষিক সরল সুদের হার উভয় ক্ষেত্রে সমান) হলে নিচের কোন সম্পর্কটি সঠিক ?<br />\r\n(a) `pqr=1`, (b) `p^2 = qr`, (c) `q^2 = pr`, (d) `r^2 = pq`',''),(1427,15,31.000,'A man wants to divide a sum of Rs 37,500 among his son and daughter aged 12 years and 14 years respectively in such a way that on attaining the age of 18 years each gets the same amount at 5% p.a. simple interest. Find the share of son.',1682009478,'Rs. 18,000','1427','','','',0,'এক ব্যক্তি 37,500 টাকা পুত্র ও কন্যার  জন্য এমন ভাবে রাখেন যে, তাদের প্রত্যেকের বয়স যখন 18 বছর হবে, তখন তারা সুদে-আসলে সম পরিমাণ টাকা পাবে। পুত্রের বয়স 12 বছর এবং কন্যার বয়স 14 বছর হলে ওই ব্যক্তি পুত্রের নামে কত টাকা রাখেন?',''),(1428,15,30.000,'Rahim Babu deposits money in the bank for each of his two daughters in such a way that when each daughter turns 18, each daughter will get Rs 1,20,000. The annual simple interest rate of the bank is 10% and the current ages of the girls are 13 years and 8 years respectively. How much money did he keep in the bank for each girl?',1682009685,'Rs. 80,000; Rs. 60,000','1428','','','',0,'রহিমবাবু তাঁর দুই মেয়ের প্রত্যেকের জন্য ব্যাংকে এমনভাবে টাকা জমা রাখেন যাতে প্রত্যেক মেয়ের বয়স যখন 18 বছর হবে, তখন প্রত্যেক মেয়ে 1,20,000 টাকা করে পাবে। ব্যাংকের বার্ষিক সরল সুদের হার 10% এবং মেয়েদের বর্তমান বয়স যথাক্রমে 13 বছর ও 8 বছর। তিনি কোন্ মেয়ের জন্য কত টাকা ব্যাংকে রেখেছিলেন?',''),(1429,15,5.000,'If the ratio of principal and interest is `25:24` in `x` years at a simple interest rate of 12% per annum, the value of `x` will be -<br />\r\n(a) 5,  (b) 8,  (c) 10,  (d) 12.',1682010138,'(b) 8','','','','',0,'বার্ষিক 12% সরল সুদের হারে `x` বছরে কোনো আসল ও সুদের অনুপাত `25:24` হলে `x`-এর মান হবে -<br />\r\n(a) 5,  (b) 8,  (c) 10,  (d) 12.',''),(1430,15,20.000,'As the annual simple interest rate goes from 4% to 3`\\frac{3}{4}`%, the annual income of a person is reduced by Rs.60. Find the person\'s capital.\r\n',1682010749,'24,000/-','','','','',0,'বার্ষিক সরল সুদের হার 4% থেকে 3`frac{3}{4}`% হওয়ায় এক ব্যক্তির বার্ষিক আয় 60 টাকা কম হয়। ওই ব্যক্তির মূলধন নির্ণয় কর।',''),(1431,15,32.000,'Ramen Babu deposited a total of Rs 3,70,000 in three banks. The annual simple interest rates of the three banks are 4%, 5% and 6% respectively. After 1 year interest received from each of the three banks is equal. How much money did he deposit in three banks?',1682159884,'1,50,000; 1,20,000; 1,00,000','1431','','','',0,'রমেনবাবু মোট 3,70,000 টাকা তিনটি ব্যাংকে জমা রাখেন। তিনটি ব্যাংকের বার্ষিক সরল সুদের হার যথাক্রমে 4%, 5% এবং 6%। 1 বছর পর তার তিনটি ব্যাংক থেকে প্রাপ্ত সুদের পরিমাণ সমান হয়। তিনি তিনটি ব্যাংকে কত টাকা করে জমা রেখেছিলেন?',''),(1432,15,29.000,'A borrows Rs.960 from B at a simple interest rate of 6% per annum on condition that he repays the loan in 4 annual installments from the date of borrowing. Pay only ¼ of the principal in each of the first three installments and repay the entire loan in the last installment. How much will A pay in the fourth installment?',1682160243,'Rs. 384','1432','','','',0,'A বার্ষিক 6% হারে সরল সুদে B-এর কাছ থেকে 960 টাকা ধার নিল এই শর্তে যে, সে ধার নেওয়ার পর থেকে পরবর্তী 4টি বার্ষিক কিস্তিতে ধার পরিশোধ করবে। প্রথম তিন কিস্তির প্রত্যেকটিতে কেবল আসলের ¼ অংশ করে দেবে এবং শেষ কিস্তিতে সম্পূর্ণ ধার পরিশোধ করবে। চতুর্থ কিস্তিতে A কত টাকা দেবে?',''),(1433,15,26.000,'A man takes a loan of Rs 4000 at 7% simple interest per annum and exactly one year after the first loan takes a second loan of Rs 4000 at 8% simple interest per annum. How many years after taking the first loan will the interest on his two loans be equal?',1682160535,'8 years','','','','',0,'একজন লোক বার্ষিক 7% সরল সুদে 4000 টাকা ঋণ নেন এবং প্রথম ঋণের ঠিক এক বছর পরে বার্ষিক 8% সরল সুদের হারে 4000 টাকার দ্বিতীয় ঋণ নেন। প্রথম ঋণ নেওয়ার কত বছর পরে তার দুটি ঋণের সুদ সমান হবে?',''),(1434,15,12.000,'In how many years will a capital double at a simple interest rate of 6¼% per annum?\r\n',1682160934,'16 years','','','','',0,'কোনো মূলধন বার্ষিক 6¼% সরল সুদের হারে কত বছরে দ্বিগুণ হবে? ',''),(1435,15,6.000,'In how many years will the ratio of interest to principal be 3:2 at a simple interest rate of 10% per annum -<br />\r\n(a) 12 years, (b) 15 years, (c) 16 years, (d) 18 years',1682161473,'(b) 15 years','','','','',0,'বার্ষিক 10% সরল সুদের হারে কত বছরে সুদ ও আসলের অনুপাত 3:2 হবে -<br />\r\n(a) 12 বছর, (b) 15 বছর, (c) 16 বছর, (d) 18 বছর\r\n',''),(1436,15,19.000,'If the annual interest is 1/16 of the original, how much will be the interest on 690 rupees in 8 months?',1682161744,'Rs. 198.375','','','','',0,'বার্ষিক সুদ, আসলের 1/16 অংশ হলে 8 মাসে 690 টাকার সুদ কত হবে?',''),(1437,15,7.000,'If the ratio of a principal to its annual amount is 25:27, the rate of interest is -<br />\r\n(a) 5%, (b) 6%, (c) 7%, (d) 8%',1682162236,NULL,'','','','',0,'কোনো আসল ও তার বার্ষিক সবৃদ্ধিমূলের অনুপাত 25:27 হলে, সুদের হার -<br />\r\n(a) 5%, (b) 6%, (c) 7%, (d) 8%',''),(1438,15,15.000,'What is the amount of Rs. `a` in `a` month at `a`% annual simple interest?',1682162583,NULL,'','','','',0,'`a`% বার্ষিক সরল সুদের হারে `a` টাকার `a` মাসের সুদ-আসল কতো?',''),(1439,15,16.000,'If the interest on a capital for `x` years at the rate of simple interest `x`% per annum is Rs. `x`, then the capital is -<br />\r\n(a) Rs. `x`, (b) Rs. `100x`, (c) Rs. `100/x`, (d) Rs. `100/x^2`',1682162887,NULL,'','','','',0,'`x`% বার্ষিক সরল সুদের হারে কোনো মূলধনের `x` বছরের সুদ `x` টাকা হলে, মূলধনের পরিমাণ -<br />\r\n(a) `x` টাকা, (b) `100x` টাকা, (c) `100/x` টাকা, (d) `100/x^2` টাকা',''),(1440,15,10.000,'If the monthly interest is Rs. 1 at the simple interest rate of 5% per annum, find the capital.',1682163294,'Rs. 240','','','','',0,'বার্ষিক 5% সরল সুদের হারে কত টাকার মাসিক সুদ 1 টাকা হবে নির্ণয় করো।',''),(1441,15,8.000,'If the total interest on a capital for `n` years at the rate of `r`% simple interest per annum is Rs. `\\frac{Pnr}{25}`, then capital is -<br />\r\n(a) Rs. `2P`, (b) Rs. `4P`, (c) Rs. `P/2`, (d) Rs. `P/4`',1682163792,NULL,'','','','',0,'বার্ষিক `r`% সরল সুদের হারে কোনো মূলধনের `n` বছরের মোট সুদ `\\frac{Pnr}{25}` টাকা হলে মূলধনের পরিমাণ -<br />\r\n(a) `2P` টাকা, (b) `4P` টাকা, (c) `P/2` টাকা, (d) `P/4` টাকা',''),(1442,15,21.000,'As the annual simple interest rate of a bank fell from 5% to 4½%, the annual income of a person decreased by Rs.750. How much money did that person put in the bank?',1682163978,'Rs. 15,000 ','','','','',0,'কোনো ব্যাংকের বার্ষিক সরল সুদের হার 5% থেকে কমে 4½% হওয়ায় এক ব্যক্তির বার্ষিক আয় 750 টাকা কমে গেল। ঐ ব্যক্তি ব্যাংকে কত টাকা রেখেছিলেন?',''),(1443,15,2.000,'If a capital doubles in 20 years, the annual simple interest rate - <br />\r\n(a) 5%, (b) 10%, (c) 15%, (d) 20%',1682164229,NULL,'','','','',0,'কোনো মূলধন 20 বছরে দ্বিগুণ হলে, বার্ষিক সরল সুদের হার -<br />\r\n(a) 5%, (b) 10%, (c) 15%, (d) 20%',''),(1444,15,34.000,'A person deposits part of his savings in the bank and the rest in the post office. He would earn him 50 rupees more at the end of the year if he had kept all the money in the bank. If the annual simple interest rates in the bank and post office are 10% and 8% respectively, where did he keep the money? His savings amount was Rs.5700.',1686634553,'Post office = Rs. 2500, Bank = Rs. 3200','','','','',0,'এক ব্যক্তি তাঁর সঞ্চয়ের কিছু অংশ ব্যাংকে ও বাকি অংশ পোস্ট অফিসে জমা রাখায় তাঁর বছরের শেষে যা আয় হয়, সমস্ত টাকা ব্যাংকে রাখলে তার চেয়ে 50 টাকা বেশি আয় হতো। ব্যাঙ্ক ও পোস্ট অফিসে বার্ষিক সরল সুদের হার যদি যথাক্রমে 10% এবং 8% হয়, তবে তিনি কোথায় কত টাকা রেখেছিলেন? তাঁর সঞ্চয়ের পরিমাণ ছিল 5700 টাকা।',''),(1445,15,24.000,'A person borrows Rs 20,000 at 15% simple interest per annum while building a house and borrows Rs 15,000 from the cooperative society at 12% simple interest per annum at the time of his daughter\'s marriage one year later. How much cash will the person have, if he repaid both loans with interest by getting a retirement gratuity of Rs.56,000 after three years of his daughter\'s marriage?',1686636097,'Rs. 3600','','','','',0,'এক ব্যক্তি বাড়ি তৈরি করার সময় বার্ষিক 15% সরল সুদের হারে 20,000 টাকা ধার করার এক বছর পর মেয়ের বিয়ের সময় সমবায় সমিতি থেকে বার্ষিক 12% সরল সুদের হারে 15,000 টাকা ধার করেন। মেয়ের বিয়ের তিন বছর পর 56,000 টাকা অবসরকালীন গ্র্যাচুইটি পেয়ে সুদ সহ উভয় ঋণ শোধ করার পর তাঁর কাছে নগদ কত টাকা থাকবে?',''),(1446,15,25.000,'Find the capital and annual simple interest if a capital yields Rs. 7100 in 7 years and Rs. 6200 in 4 years at the same annual percentage rate.',1686636878,'Rs. 5000, 6%','','','','',0,'কোন মূলধন একই বার্ষিক শতকরা সরল সুদের হারে 7 বছরে সুদে-আসলে 7100 টাকা এবং 4 বছরে সুদে-আসলে 6200 টাকা হলে মূলধন এবং বার্ষিক শতকরা সরল সুদের হার নির্ণয় করো।',''),(1447,15,17.000,'At what rate of simple interest per annum will a principal double in 16⅔ years?',1686637231,'6%','','','','',0,'বার্ষিক শতকরা কত হার সুদে কোনো মূলধন 16⅔ বছরে সুদেমূলে দ্বিগুণ হবে ?',''),(1448,15,23.000,'A bank pays simple interest at the rate of 5% per annum. In that bank Mukesh Babu deposited Rs 15,000 in the beginning of the year, after 3 months he withdrew Rs 3000 and after another three months of withdrawal he again deposited Rs 8000. Determine how much money Mukesh Babu will receive with interest at the end of that year.',1686638050,'Rs 20,837.50','','','','',0,'কোন ব্যাংক বার্ষিক 5% হারে সরল সুদ দেয়। ওই ব্যাংকে মুকেশ বাবু বছরের প্রথমে 15,000 টাকা জমা দেওয়ার 3 মাস পরে 3000 টাকা তুলে নিলেন এবং টাকা তুলে নেওয়ার আরও তিন মাস পরে আবার তিনি 8000 টাকা জমা দিলেন। ওই বছরের শেষে মুকেশ বাবু সুদে-আসলে কত টাকা পাবেন নির্ণয় করো।',''),(1449,16,1.000,'The general solution of `|cos x| = cos x - 2 sin x` is - <br />\r\n(a) `x = n\\pi`, (b) `x = n\\pi + \\pi/4`, (c) `x = n\\pi + (-1)^n (\\pi/4)` where `n \\in I`, (d) `x = (2n+1)\\pi + \\pi/4`',1687073002,'(d) `x = (2n+1)\\pi + \\pi/4`','1449','','','',0,'',''),(1450,16,2.000,'Find the general solution of `8^{1+|cos x|+cos^2x+|cos^3x|+\\cdots \\infty}=4^3`',1687080733,'`x = 2n\\pi \\pm \\pi/3, 2n\\pi \\pm \\frac{2\\pi}{3}`  where `n \\in mathbb{Z}`','1450','','','',0,'',''),(1451,13,3.000,'If `y \\propto x^2` and `y=9` when `x=9`. Then find the value of `x` when `y=4` ',1687260258,'`\\pm 6`','','','','',0,'যদি `y propto x^2` এবং `y=9` হলে `x=9` হয়, তবে `y=4` হলে `x` এর মান নির্ণয় করো।  ',''),(1452,13,10.000,'`x` varies directly as square of `y` and inversely as `z`. If `x=2` when `y=4` and `z=2`, then find the value of `y` when `x=3` and `z=3`.',1687261526,'`\\pm 6`','','','','',0,'`x`, `y`-এর বর্গের সাথে সরলভেদে এবং `z`-এর সঙ্গে ব্যস্তভেদে আছে। যদি `y=4` ও `z=2` হলে `x=2` হয়, তবে `x=3` ও `z=3` হলে `y` এর মান নির্ণয় করো। ',''),(1453,13,1.000,'If `x^2 \\propto yz`, `y^2 \\propto zx` and `z^2 \\propto xy`, then find the relation among these variation constants. ',1687262296,'Relation - product of three variation constants is 1','','','','',0,'যদি `x^2 propto yz`, `y^2 propto zx` এবং `z^2 propto xy` হয়, তবে ভেদ ধ্রুবক তিনটির মধ্যে সম্পর্ক নির্ণয় করো। ',''),(1454,13,2.000,'If `x/y \\propto (x+y)` and `y/x \\propto (x-y)` then prove that `(x^2-y^2)` is constant.',1687263226,NULL,'','','','',0,'`x/y \\propto (x+y)` এবং `y/x \\propto (x-y)` হলে প্রমাণ করো যে, `(x^2-y^2)` ধ্রুবক।',''),(1455,13,4.000,'If `x \\propto yz` and `y \\propto zx`, then show that `z` is a non-zero constant. ',1687269324,NULL,'','','','',0,'যদি `x \\propto yz` এবং `y \\propto zx` হয়, তাহলে দেখাও যে `z` একটি অশূন্য ধ্রুবক। ',''),(1456,13,11.000,'If `(1/x - 1/y) \\propto \\frac{1}{x-y}`, then show that `(x^2+y^2) \\propto xy`',1687269521,NULL,'','','','',0,'যদি `(1/x - 1/y) \\propto \\frac{1}{x-y}` হয়, তাহলে দেখাও যে `(x^2+y^2) \\propto xy`',''),(1457,13,13.000,'Given that, `A = B + C`, where `B \\propto x^2` and `C \\propto x^3`. If `A=0` when `x=1` and `A=2` when `x=-1`, then express `A` in terms of `x`.',1687269897,NULL,'','','','',0,'প্রদত্ত যে, `A = B + C`, যেখানে `B \\propto x^2` এবং `C \\propto x^3` । যদি `A=0` যখন `x=1` এবং `A=2` যখন `x=-1` হয়, তাহলে `A`-কে `x` দ্বারা প্রকাশ করো। ',''),(1458,13,6.000,'If `(a^2+b^2) \\propto ab`, then prove that, `(a+b) \\propto (a-b)`.',1687270157,NULL,'','','','',0,'যদি `(a^2+b^2) \\propto ab` হয়, তাহলে প্রমাণ করো যে, `(a+b) \\propto (a-b)` ।',''),(1459,13,5.000,'If `(a+b) \\propto (a-b)`, then prove that `a^2+b^2 \\propto ab`.',1687270343,NULL,'','','','',0,'যদি `(a+b) \\propto (a-b)` হয়, তাহলে প্রমাণ করো যে `a^2+b^2 \\propto ab` । ',''),(1460,13,9.000,'If `(x^3 - 1/y^3) \\propto (x^3 + 1/y^3)`, then prove that `x \\propto 1/y`.',1687270526,NULL,'','','','',0,'যদি `(x^3 - 1/y^3) \\propto (x^3 + 1/y^3)` হয়, তাহলে প্রমাণ করো যে, `x \\propto 1/y` । ',''),(1461,13,7.000,'If `x \\propto y` and `y \\propto z`, then prove that `x^2+y^2+z^2 \\propto xy+yz+zx`.',1687271950,NULL,'','','','',0,'যদি `x \\propto y` এবং `y \\propto z` হয়, তাহলে প্রমাণ করো যে, `x^2+y^2+z^2 \\propto xy+yz+zx` । ',''),(1462,13,15.000,'The volume of a sphere varies directly as the cube of its radius. Three inert spheres with diameters of 1½, 2 and 2½ meters were melted and a new inert sphere was prepared. What is the diameter of the new sphere?',1687273026,'3 meter','','','','',0,'গোলকের আয়তন তার ব্যাসার্ধের ত্রিঘাতের সঙ্গে সরল ভেদে পরিবর্তিত হয়। 1½, 2 এবং 2½ মিটার ব্যাসবিশিষ্ট তিনটি নিরেট গোলককে গলিয়ে একটি নতুন নিরেট গোলক প্রস্তুত করা হলো। নতুন গোলকটির ব্যাস কত ?',''),(1463,13,16.000,'The volume of a cone varies jointly as the square of its radius and its height. If the ratio of the radii of two cones is 3:4 and the ratio of their heights is 6:5, find the ratio of their volumes.',1687274022,NULL,'','','','',0,'একটি শঙ্কুর আয়তন উহার ব্যাসার্ধের বর্গ এবং উচ্চতার সঙ্গে সরল ভেদে আছে। যদি দুটি শঙ্কুর ব্যাসার্ধের অনুপাত 3:4 এবং ইহাদের উচ্চতার অনুপাত 6:5 হয়, তাহলে উহাদের আয়তনের অনুপাত নির্ণয় করো। ',''),(1464,13,14.000,'`y` is equal to the sum of two variables, one varies directly as `x` and other varies inversely as `x`. If `x=1, y=-1` and `x=3, y=5`, then find the relationship between `x` and `y`.',1687274767,'`y = 2x - 3/x`','','','','',0,'`y` দুটি চলরাশির সমষ্টির সঙ্গে সমান, যাদের একটি `x` এর সহিত সমানুপাতী এবং অপরটি `x` এর সঙ্গে ব্যস্তানুপাতী। যদি `x=1, y=-1` এবং `x=3, y=5` হয়, তাহলে `x` এবং `y` এর মধ্যে সম্পর্ক নির্ণয় করো।',''),(1465,13,8.000,'If `a \\propto b` and `b \\propto c`, then prove that `a^3+b^3+c^3 \\propto abc`.',1687275085,NULL,'','','','',0,'যদি `a \\propto b` এবং `b \\propto c` হয়, তবে প্রমাণ করো যে `a^3+b^3+c^3 \\propto abc` ।',''),(1466,13,12.000,'If `(a+b) \\propto \\sqrt{ab}`, then show that `(\\sqrt{a}+\\sqrt{b}) \\propto (\\sqrt{a}-\\sqrt{b})`.',1687275458,NULL,'','','','',0,'যদি `(a+b) \\propto \\sqrt{ab}` হয়, তবে দেখাও যে `(\\sqrt{a}+\\sqrt{b}) \\propto (\\sqrt{a}-\\sqrt{b})` ।',''),(1467,17,1.000,'Find the value of `lim_{x \\to 3} \\frac{x-3}{|x-3|}`',1688301494,'Limit does not exist','','','','',0,'',''),(1468,17,2.000,'Find the value of `lim_{x \\to 3} \\frac{\\sqrt{x-3}+\\sqrt{x}-\\sqrt{3}}{\\sqrt{x^2-9}}`',1688301866,'`1/\\sqrt{6}`','1468','','','',0,'',''),(1469,17,6.000,'Show that the following limit does not exist.<br /> `lim_{x \\to 4}\\frac{[x]+|x-4|}{x^2+5x}`',1688302121,'N.A.','1469','','','',0,'',''),(1470,17,3.000,'Find the value of `lim_{x \\to 1} (|x+1|+|x-1|)`',1688302263,'2','1470','','','',0,'',''),(1471,17,4.000,'A function `f(x)` defined in the domain `0 \\le x \\le 3` by following.<br />\r\n$f(x)=\\begin{cases} 2x-3,& \\text{when } 0 \\le x \\le 2\\\\ x^2,   & \\text{when } x=2 \\\\\\frac{x}{2}, & \\text{when }2 \\le x \\le 3 \\end{cases}$ <br />Find the value of `lim_{x \\to 2} f(x)` if exists. ',1688302846,'1','','','','',0,'',''),(1472,17,7.000,'If `G(x) = -\\sqrt{25-x^2}`, then find the value of `lim_{x \\to 1} \\frac{G(x)-G(1)}{x-1}`',1688303034,'`\\frac{1}{2\\sqrt{6}}`','1472','','','',0,'',''),(1473,17,8.000,'If `lim_{x \\to a} f(x) = lim_{x \\to a} g(x)`, then will `f(x) = g(x)` always hold true?',1688303282,'No','1473','','','',0,'',''),(1474,17,9.000,'If `lim_{x \\to 2} \\frac{ax^2-b}{x-2} = 4`, then find the value of `a` and `b`.',1688303482,'`a=1, b=4`','1474','','','',0,'',''),(1475,17,10.000,'If `lim_{x \\to 3} \\frac{ax^2-b}{x-3} = 6`, then find the value of `a` and `b`.',1688303541,'`a=1, b=9`','','','','',0,'',''),(1476,17,5.000,'If $f(x)=\\begin{cases} a+bx,& \\text{when } x \\lt 1\\\\ 4,   & \\text{when } x=1 \\\\b-ax, & \\text{when } x \\gt 1 \\end{cases}$ <br />Find the value of `a` and `b` if `lim_{x \\to 1} f(x) = f(1)`. ',1688304711,'`a=0, b=4`','1476','','','',0,'',''),(1477,17,11.000,'Find the value of `lim_{x \\to 3} \\frac{x-3}{\\sqrt{x-2} - \\sqrt{4-x}}`',1688305007,'1','1477','','','',0,'',''),(1478,17,12.000,'Find the value of `lim_{x \\to 2} \\frac{\\sqrt{1+2x}-\\sqrt{1+x^2}}{2-x}`',1688305186,'`\\frac{1}{\\sqrt{5}}`','1478','','','',0,'',''),(1479,17,13.000,'Find the value of `lim_{x \\to 2} \\frac{x - \\sqrt{3x-2}}{x^2 - 4}`',1688305476,'`1/16`','','','','',0,'',''),(1480,17,14.000,'Find the value of `lim_{x \\to 0} \\frac{(1+x)^{1/3}-(1-x)^{1/3}}{x}`',1688305990,'`2/3`','1480','','','',0,'',''),(1481,17,15.000,'Find the value of `lim_{x \\to 2} \\frac{2 - \\sqrt{2+x}}{2^{1/3} - (4-x)^{1/3}}`',1688306959,'$-\\frac{3}{4} \\sqrt[3]{4}$','1481,1481.02','','','',0,'',''),(1482,18,1.000,'Find the equation of the straight line through the origin which divides internally the connecting segment of the two points `(3, 6)` and `(-2, -14)` in the ratio `2:3`.',1694760479,'`2x+y=0`','','','','',0,'মূলবিন্দুগামী যে সরলরেখা `(3, 6)` ও `(-2, -14)` বিন্দু দুটির সংযোজক রেখাংশকে `2:3` অনুপাতে অন্তর্বিভক্ত করেছে, তার সমীকরণ নির্ণয় করো।',''),(1483,18,2.000,'Find the ratio in which the straight line `3x+4y=21` divides the line segment joining the points `(-9, 5)` and `(7, 9)`.',1694770198,'Internally in the ratio `7:9`','','','','',0,'`3x+4y=21` সরলরেখাটি `(-9, 5)` ও `(7, 9)` বিন্দুদ্বয়ের সংযোজক রেখাংশকে কী অনুপাতে বিভক্ত করেছে তা নির্ণয় করো।',''),(1484,18,3.000,'Find the equation of a straight line whose slope is `-2` and which cuts off `3` units of length in the positive direction of the `x`-axis.',1694776065,'`2x+y=6`','','','','',0,'এরূপ সরলরেখার সমীকরণ নির্ণয় করো যার প্রবণতা `-2` এবং যা `x` অক্ষের ধনাত্মক দিক থেকে `3` একক দৈর্ঘ্য ছিন্ন করে।',''),(1485,18,4.000,'A straight line through the point `(3,4)` makes an angle of `45°` with the positive direction of the `x`-axis. Find the distance from the point `(3,4)` to the point where this straight line intersects the straight line `y=5x+1`.',1694777466,'`3\\sqrt{2}` unit','','','','',0,'`(3,4)` বিন্দুগামী একটি সরলরেখা `x` অক্ষের ধনাত্মক দিকের সঙ্গে `45°` কোণ উৎপন্ন করে। এই সরলরেখাটি `y=5x+1` সরলরেখাকে যে বিন্দুতে ছেদ করে, `(3,4)` বিন্দু থেকে ওই বিন্দুর দূরত্ব নির্ণয় করো।',''),(1486,18,5.000,'Find the distance of the straight line `4x-y=0` along the straight line that makes an angle of `135°` with the positive direction of `x`-axis from the point `P(4,1)`.',1694778497,'`3\\sqrt{2}` unit','','','','',0,'`P(4,1)` বিন্দু থেকে `x` অক্ষের ধনাত্মক দিকের সঙ্গে যে সরলরেখা `135°` কোণ উৎপন্ন করে, সেই সরলরেখা বরাবর  `4x-y=0` সরলরেখার দূরত্ব নির্ণয় করো।',''),(1487,18,6.000,'Find the distance from the point `(1,1)` on the straight line `4x-3y=1` along the straight line `2x-y=3`.',1694779162,'5 unit ','','','','',0,'`4x-3y=1` সরলরেখাস্থিত `(1,1)` বিন্দু থেকে ঐ সরলরেখা বরাবর `2x-y=3` সরলরেখার দূরত্ব নির্ণয় করো।',''),(1488,12,41.000,'Find the fourth root of `7-24i`',1699622111,'`\\pm 1/sqrt{2}(3-i)`, `\\pm 1/sqrt{2}(1+3i)`','1488','','','',0,'',''),(1489,19,1.000,'If in a triangle, `a=3`, `b=5` and `c=7`, then show that the triangle is obtuse.',1705837630,NULL,'','','','',0,'কোনো ত্রিভুজে `a=3`, `b=5` এবং `c=7` হলে দেখাও যে, ত্রিভুজটি স্থূলকোণী।',''),(1490,19,2.000,'If in a triangle, `R=2r`, then show that the triangle is equilateral.',1705840506,NULL,'','','','',0,'কোনো ত্রিভুজে `R=2r` হলে দেখাও যে ত্রিভুজটি সমবাহু।',''),(1491,19,3.000,'If in a triangle `8R^2=a^2+b^2+c^2` (in conventional notation), then show that the triangle is right-angled.',1705840830,'','','','','',0,'যদি কোনো ত্রিভুজে `8R^2=a^2+b^2+c^2` হয় (প্রচলিত সংকেতে) তবে দেখাও যে ত্রিভুজটি সমকোণী।',''),(1492,20,2.000,'The equation `2(x^2+3x-2)=2x^2+x-9` is - (a) Liner, (b) Quadric, (c) Cubic, (d) None of these',1706363875,NULL,'','','','',0,'`2(x^2+3x-2)=2x^2+x-9` সমীকরণটি - (a) রৈখিক, (b) দ্বিঘাত, (c) ত্রিঘাত, (d) কোনোটিই নয়।',''),(1493,20,4.000,'i) Number of roots of the equation `x^2=x` is - (a) 0, (b) 1, (c) 2, (d) 3 <br />\r\nii) Number of roots of the equation `x^2/x=5` is - (a) 0, (b) 1, (c) 2, (d) 3',1706364036,'','','','','',0,'i) `x^2=x` সমীকরণের বীজের সংখ্যা - (a) 0, (b) 1, (c) 2, (d) 3 <br />\r\nii) `x^2/x=5` সমীকরণের বীজের সংখ্যা - (a) 0, (b) 1, (c) 2, (d) 3',''),(1494,20,6.000,'If the roots of the equation `9x^2+6x+a=0` are real and equal, then the valie of `a` is - (a) `- 2/3` , (b) `1/9` , (c) `1`, (d) `9`',1706364329,NULL,'','','','',0,'`9x^2+6x+a=0` সমীকরণটির বীজদ্বয় বাস্তব ও সমান হলে `a` এর মান হবে - (a) `- 2/3` , (b) `1/9` , (c) `1`, (d) `9`',''),(1495,20,1.000,'For which value of `a`, the equation `(a-2)x^2+3x+5=0` will not be a quadratic equation ?',1706364706,NULL,'','','','',0,'`a` এর কোন্ মানের জন্য `(a-2)x^2+3x+5=0` সমীকরণটি দ্বিঘাত সমীকরণ হবে না ?',''),(1496,20,22.000,'If one of the roots of the equation `x^2-(2+b)x+6=0` is `2`, then what will be its other root?',1706365395,NULL,'','','','',0,'`x^2-(2+b)x+6=0` সমীকরণের একটি বীজ `2` হলে, অপর বীজটি কত হবে?',''),(1497,20,21.000,'State True or False. The roots of the equation `x^2+x+1=0` are real.',1706365703,NULL,'','','','',0,'সত্য অথবা মিথ্যা লেখো। `x^2+x+1=0` সমীকরণের বীজদ্বয় বাস্তব।',''),(1498,20,19.000,'If the numerical value of the roots of the equation `3x^2+(2m-1)x+(m+3)=0` are equal but opposite sign, then what will be the value of `m` ?',1706366057,NULL,'','','','',0,'`3x^2+(2m-1)x+(m+3)=0` সমীকরণটির বীজদ্বয়ের সাংখ্যমান সমান কিন্তু বিপরীত চিহ্ন যুক্ত হলে `m` এর মান কত হবে?',''),(1499,20,33.000,'Show that, for `a=b`, the equation `2(a^2+b^2)x^2+2(a+b)x+1=0` has no real roots.',1706366335,NULL,'','','','',0,'`a=b` হলে, দেখাও যে `2(a^2+b^2)x^2+2(a+b)x+1=0` দ্বিঘাত সমীকরণের কোনো বাস্তব বীজ থাকবে না।',''),(1500,20,17.000,'i) If the roots of the equation are `2, -3`, then write its quadratic equation. <br />\r\nii) If the roots of the equation are `4\\pm\\sqrt{3}`, then write its quadratic equation. ',1706366625,'','','','','',0,'i) কোনো দ্বিঘাত সমীকরণের বীজদ্বয় `2, -3` হলে দ্বিঘাত সমীকরণটি লেখো।<br />\r\nii) কোনো দ্বিঘাত সমীকরণের বীজদ্বয় `4\\pm\\sqrt{3}` হলে দ্বিঘাত সমীকরণটি লেখো।',''),(1501,20,20.000,'Find the condition for which the roots of the equation `ax^2+bx+c=0` `(a\\ne 0)` are reciprocal and having opposite sign to each other.',1706366906,'','','','','',0,'`ax^2+bx+c=0` `(a\\ne 0)` সমীকরণের বীজদ্বয় অন্যোন্যক ও বিপরীত চিহ্নযুক্ত হওয়ার শর্ত নির্ণয় করো। ',''),(1502,20,27.000,'If the roots of the equation `x^2-22x+105=0` are `\\alpha` and `\\beta` , then find the value of `\\alpha - \\beta`.',1706367698,NULL,'','','','',0,'`x^2-22x+105=0` সমীকরণের দুটি বীজ `\\alpha` ও `\\beta` হলে, `\\alpha - \\beta` এর মান নির্ণয় করো।',''),(1503,20,26.000,'Find the ratio of sum and product of the roots of the equation `7x^2-66x+27=0`.',1706367957,NULL,'','','','',0,'`7x^2-66x+27=0` সমীকরণটির বীজদ্বয়ের যোগফল ও গুণফল এর অনুপাত নির্ণয় করো।',''),(1504,20,25.000,'If the sum of the roots of the equation `x^2-x=K(2x-1)` is 2, then find the value of `K`.',1706445330,NULL,'','','','',0,'`x^2-x=K(2x-1)` সমীকরণের বীজদ্বয়ের সমষ্টি 2 হলে, `K` এর মান নির্ণয় করো।',''),(1505,20,28.000,'If the roots of the quadratic equation `5x^2-2x+3=0` are `\\alpha` and `\\beta`, then find the value of  `1/\\alpha + 1/\\beta`.',1706445872,'','','','','',0,'`5x^2-2x+3=0` দ্বিঘাত সমীকরণের বীজদ্বয় `\\alpha` ও `\\beta` হলে, `1/\\alpha + 1/\\beta` এর মান নির্ণয় করো।',''),(1506,20,36.000,'In the quadratic equation `ax^2+bx+c=0` (`a\\ne0`), if one root is double to other root, then prove that `2b^2=9ac`.',1706446303,'','','','','',0,'`ax^2+bx+c=0` (`a\\ne0`) দ্বিঘাত সমীকরণের একটি বীজ অপরটির দ্বিগুণ হলে, প্রমাণ করো `2b^2=9ac` ।',''),(1507,20,39.000,'Find the equation whose roots are square of the roots of `x^2+x+1=0`.',1706446625,NULL,'','','','',0,'`x^2+x+1=0` সমীকরণের বীজগুলির বর্গ যে সমীকরণের বীজ, সেই সমীকরণটি নির্ণয় করো।',''),(1508,20,32.000,'If the roots of the quadratic equation `(1+m^2)x^2+2mcx+(c^2-p^2)=0` are real and equal, then prove that `c^2=p^2(1+m^2)`.',1706446891,NULL,'','','','',0,'`(1+m^2)x^2+2mcx+(c^2-p^2)=0` দ্বিঘাত সমীকরণের বীজদুটি বাস্তব ও সমান হলে প্রমাণ করো যে, `c^2=p^2(1+m^2)`',''),(1509,20,31.000,'Find the values of `p`, for which the quadratic equation `x^2+(p-3)x+p=0` has real and equal roots.',1706447255,NULL,'','','','',0,'`p` এর যে সকল মানের জন্য `x^2+(p-3)x+p=0` সমীকরণের বীজদ্বয় বাস্তব ও সমান তা নির্ণয় করো।',''),(1510,20,34.000,'If the equations `x^2+px+r=0` and `x^2+rx+p=0` (`p \\ne r`) have a common root, then find the value of  `1+p+r`.',1706447794,'','','','','',0,'`x^2+px+r=0` এবং `x^2+rx+p=0` (`p \\ne r`) সমীকরণ দ্বয়ের একটি সাধারণ বীজ থাকলে `1+p+r` এর মান নির্ণয় করো।',''),(1511,20,3.000,'If one of the roots of the quadratic equation `kx^2+2x-3=0` is 2, then the value of `k` is - (a) `- 3/2` , (b) `1/2`, (c) `- 1/4`, (d) `1/4`',1706448346,NULL,'','','','',0,'`kx^2+2x-3=0` দ্বিঘাত সমীকরণের একটি বীজ 2 হলে, `k` এর মান - (a) `- 3/2` , (b) `1/2`, (c) `- 1/4`, (d) `1/4`',''),(1512,20,23.000,'If two roots of the equation `x^2+px+q=0` are `p` and `q`, then find the value of `p` and `q`.',1707715429,'','','','','',0,'যদি `x^2+px+q=0`-এর বীজ দুটি `p` ও `q` হয়, তবে `p` ও `q`-এর মান নির্ণয় করো।',''),(1513,20,40.000,'Find the equation whose roots are square of the roots of `2x^2+3x+1=0`',1707810221,'','','','','',0,'যে সমীকরণের বীজ দুটি `2x^2+3x+1=0` -এর বীজদ্বয়ের বর্গ, সেই সমীকরণটি নির্ণয় করো।',''),(1514,20,35.000,'If one of the roots of the equation `x^2+px+12=0` is `2` and two roots of the equation `x^2+px+q=0` are equal, then find the value of `q`.',1707811103,NULL,'','','','',0,'যদি `x^2+px+12=0`-এর একটি বীজ `2` হয়, এবং `x^2+px+q=0` -এর বীজ দুটি সমান হয়, তবে `q`-এর মান নির্ণয় করো।',''),(1515,20,24.000,'Find the quadratic equation whose sum and product of the roots are `b/a` and `c^2/b` respectively.',1707811398,NULL,'','','','',0,'যে দ্বিঘাত সমীকরণের বীজদ্বয়ের সমষ্টি `b/a` এবং গুণফল `c^2/b`, সেই সমীকরণটি নির্ণয় করো।',''),(1516,20,41.000,'If the roots of the equation `x^2-2x+2=0` are `\\alpha` and `\\beta`, find the equation whose roots are `(\\alpha+2)` and `(\\beta+2)`.',1707811718,'','','','','',0,'`x^2-2x+2=0` সমীকরণের বীজ দুটি `\\alpha` এবং `\\beta`, যে সমীকরণের বীজদুটি `(\\alpha+2)` এবং `(\\beta+2)` তার সমীকরণ নির্ণয় করো।',''),(1517,20,38.000,'Find the equation whose roots are reciprocal of the roots of `2x^2+5x+3=0`.',1707812515,NULL,'','','','',0,'যে সমীকরণের বীজ দুটি `2x^2+5x+3=0` সমীকরণের বীজ দুটির অন্যোন্যক, তার সমীকরণ নির্ণয় করো।',''),(1518,20,9.000,'If the ratio of two roots of the equation `2x^2-5x+p=0` is `2:3`, then value of `p` is - (a) `2`, (b) `3`, (c) `4`, (d) `1/4`',1707813041,NULL,'','','','',0,'যদি `2x^2-5x+p=0` সমীকরণের বীজদুটির অনুপাত `2:3` হয়, তবে `p` -এর মান হবে - (a) `2`, (b) `3`, (c) `4`, (d) `1/4`',''),(1519,20,7.000,'The roots of the equal are `3x^2-12x+10=0` - (a) irrational and unequal, (b) rational and equal, (c) real and unequal, (d) not real',1707813545,NULL,'','','','',0,'`3x^2-12x+10=0` সমীকরণের বীজদুটি - (a) অমূলদ ও অসমান, (b) মূলদ ও সমান, (c) বাস্তব এবং অসমান, (d) অবাস্তব',''),(1520,20,37.000,'If two roots of the equation `(b-c)x^2+(c-a)x+(a-b)=0` are equal, then prove that `2b=a+c`.',1707814086,NULL,'','','','',0,'`(b-c)x^2+(c-a)x+(a-b)=0` দ্বিঘাত সমীকরণের বীজদ্বয় সমান হলে, প্রমাণ করো `2b=a+c`',''),(1521,20,18.000,'By applying Sridhar Acharya\'s formula, if the roots of the equation `5x^2+2x-7=0` are `x=\\frac{k\\pm12}{10}`, then find the value of `k`.',1707814463,'','','','','',0,'`5x^2+2x-7=0` সমীকরণে শ্রীধর আচার্যের সূত্র প্রয়োগ করে `x=\\frac{k\\pm12}{10}` পাওয়া গেলে `k` -এর মান নির্ণয় করো।',''),(1522,20,42.000,'If `a+b+c=0`, then show that one of the roots of the equation `ax^2+bx+c=0` is `1`.',1707815070,NULL,'','','','',0,'যদি `a+b+c=0` হয়, তবে দেখাও যে `ax^2+bx+c=0` সমীকরণের একটি বীজ `1` হবে।',''),(1523,20,30.000,'If the `x` excluded term of the equation `x^2+ax+b=0` is misprinted as 36 instead of 30, it turns out that roots are 10 and 3; then find the correct quadratic equation.',1707815761,'','','','','',0,'`x^2+ax+b=0` সমীকরণের `x`-বর্জিত পদটি 36 -এর স্থানে 30 ভুল ছাপা হয় দেখা গেল যে, ওর বীজদ্বয় 10 এবং 3; শুদ্ধ সমীকরণটি নির্ণয় করো।',''),(1524,20,10.000,'If the ratio of two roots of the equation `px^2+qx+r=0 (p\\ne0)` is `m:n`, then the value of `\\frac{mn}{(m+n)^2}` is - (a) `q^2/{pr}`, (b) `{pr}/q^2`, (c) `pqr`, (d) `{pq}/r^2`',1707824746,'','','','','',0,'`px^2+qx+r=0 (p\\ne0)` -এর বীজদ্বয়ের অনুপাত `m:n` হলে `\\frac{mn}{(m+n)^2}` এর মান - (a) `q^2/{pr}`, (b) `{pr}/q^2`, (c) `pqr`, (d) `{pq}/r^2`',''),(1525,20,11.000,'In the equation `x^2+px+q=0`, coefficient of `x` is misprinted as `13` instead of `14`, and it turns out that roots are `-5` and `-8`. Then roots of the original equation are - (a) `-4, -10`, (b) `4, 10`, (c) `-4, 10`, (d) `4, -10`',1707825952,'','','','','',0,'`x^2+px+q=0` সমীকরণে `x`-এর সহগ `14` এর পরিবর্তে `13` লেখায়, বীজদুটি `-5` এবং `-8` হল। মূল সমীকরণের বীজ দুটি হবে - (a) `-4, -10`, (b) `4, 10`, (c) `-4, 10`, (d) `4, -10`',''),(1526,20,12.000,'If one root of the equation `x^2-x-k=0` is square of its other root, then value of `k` is - (a) `2\\pm\\sqrt{3}`, (b) `3\\pm\\sqrt{2}`, (c) `2\\pm\\sqrt{5}`, (d) `5\\pm\\sqrt{2}`',1707826433,'','','','','',0,'`x^2-x-k=0` সমীকরণের একটি বীজ অপরটির বর্গ হলে `k`-এর মান - (a) `2\\pm\\sqrt{3}`, (b) `3\\pm\\sqrt{2}`, (c) `2\\pm\\sqrt{5}`, (d) `5\\pm\\sqrt{2}`',''),(1527,20,13.000,'If two roots of the equation `x(x-3)=4` are `\\alpha` and `\\beta`, then value of `\\alpha^2+\\beta^2` is - (a) 14, (b) 15, (c) 16, (d) 17',1707826830,'','','','','',0,'`x(x-3)=4` এর বীজ দুটি `\\alpha` ও `\\beta` হলে `\\alpha^2+\\beta^2`-এর মান - (a) 14, (b) 15, (c) 16, (d) 17',''),(1528,20,14.000,'If `p, q, r` rational and `p+q+r=0`, then the roots of the quadratic equation `px^2+qx+r=0` are - (a) rational, (b) irrational, (c) real, (d) none of these.',1707827259,NULL,'','','','',0,'যদি `p, q, r` মূলদ এবং `p+q+r=0` হয়, তবে `px^2+qx+r=0` দ্বিঘাত সমীকরণের বীজগুলি - (a) মূলদ, (b) অমূলদ, (c) বাস্তব, (d) এদের কোনোটিই নয়।',''),(1529,20,15.000,'If the roots of `x^2+px+12=0` are `\\alpha`, `\\beta` and are such that `\\alpha-\\beta=1`, then value of  `p` is - (a) `\\pm2`, (b) `\\pm5`, (c) `\\pm6`, (d) `\\pm7`',1707827616,'','','','','',0,'`x^2+px+12=0` এর বীজদ্বয় `\\alpha` ও `\\beta` এরূপ যে `\\alpha-\\beta=1`, তাহলে `p` এর মান - (a) `\\pm2`, (b) `\\pm5`, (c) `\\pm6`, (d) `\\pm7`',''),(1530,20,16.000,'If the roots of the equation `x^2-5x+6=0` are `\\alpha` and `\\beta`, then value of `\\alpha/\\beta + \\beta/\\alpha` is - (a) `15/2`, (b) `2/15`, (c) `6/13`, (d) `13/6`',1707828550,NULL,'','','','',0,'`x^2-5x+6=0` সমীকরণের বীজদুটি `\\alpha` ও `\\beta` হলে `\\alpha/\\beta + \\beta/\\alpha` এর মান - (a) `15/2`, (b) `2/15`, (c) `6/13`, (d) `13/6`',''),(1531,20,8.000,'If the roots of the quadratic equation `px^2+pqrx+qr=0` are reciprocal to each other, then (a) `pqr=1`, (b) `q=pr`, (c) `qr=1`, (d) `p=qr`',1707829014,NULL,'','','','',0,'`px^2+pqrx+qr=0` দ্বিঘাত সমীকরণের একটি বীজ অপর বীজের অন্যোন্যক হয়, তবে (a) `pqr=1`, (b) `q=pr`, (c) `qr=1`, (d) `p=qr`',''),(1532,20,29.000,'If the roots of quadratic equation `5x^2+2x-3=0` are `\\alpha` and `\\beta`, then find the value of `(\\frac{\\alpha^2}{\\beta}+\\frac{\\beta^2}{\\alpha})`.',1707829778,'','','','','',0,'`5x^2+2x-3=0` দ্বিঘাত সমীকরণের বীজদ্বয় `\\alpha` ও `\\beta` হলে, `(\\frac{\\alpha^2}{\\beta}+\\frac{\\beta^2}{\\alpha})`-এর মান নির্ণয় করো।',''),(1533,20,47.000,'If the speed of a boat in still water is 8 km/h, the boat travels 15 km downstream and 22 km upstream in 5 hours, find the speed of the current.',1707830392,'','','','','',0,'স্থির জলে একটি নৌকার গতিবেগ 8 কিমি /ঘন্টা, নৌকাটি 5 ঘণ্টায় স্রোতের অনুকূলে 15 কিমি এবং স্রোতের প্রতিকূলে 22 কিমি গেলে, স্রোতের বেগ কত ছিল নির্ণয় করো।',''),(1534,20,45.000,'If the product of two consecutive positive odd numbers is 143, form the equation and find the two numbers using Sridhar Acharya\'s formula.',1707830726,NULL,'','','','',0,'দুটি ক্রমিক ধনাত্মক অযুগ্ম সংখ্যার গুণফল 143 হলে সমীকরণটি গঠন করো এবং শ্রীধর আচার্যের সূত্র প্রয়োগ করে সংখ্যা দুটি নির্ণয় করো।',''),(1535,20,46.000,'If the price of pens is reduced by Rs 6 per dozen, 3 more pens are available at Rs 30. Determine the cost per dozen pens before price reduction.',1707831026,NULL,'','','','',0,'কলমের মূল্য প্রতি ডজনে 6 টাকা কমলে 30 টাকায় আরো 3টি কলম বেশি পাওয়া যায়। কমার পূর্বে প্রতি ডজন কলমের মূল্য নির্ণয় করো।',''),(1536,20,43.000,'A rectangular field has an area of ​​2000 square meters and perimeter of 180 meters. Find the length and width of the rectangular area.',1707831298,NULL,'','','','',0,'একটি আয়তাকার ক্ষেত্রের ক্ষেত্রফল 2000 বর্গমিটার এবং পরিসীমা 180 মিটার। আয়তাকার ক্ষেত্রটির দৈর্ঘ্য ও প্রস্থ নির্ণয় করো।',''),(1537,20,44.000,'The units digit of a two-digit number is 6 more than the tens digit and product of the two digits is 12 less than the number. What could be the units digit of the number?',1707845589,NULL,'','','','',0,'দুই অংকবিশিষ্ট একটি সংখ্যার একক স্থানীয় অংকটি দশক স্থানীয় অংক অপেক্ষা 6 বেশি এবং অঙ্কদ্বয়ের গুণফল সংখ্যাটির চেয়ে 12 কম। সংখ্যাটির এককের অংক কি কি হতে পারে?',''),(1538,20,51.000,'Solve: `\\frac{1}{(x-2)(x-4)} + \\frac{1}{(x-4)(x-6)} + \\frac{1}{(x-6)(x-8)} + 1/3 = 0` (`x \\ne 2,4,6,8`)',1707845918,'','','','','',0,'সমাধান করো। `\\frac{1}{(x-2)(x-4)} + \\frac{1}{(x-4)(x-6)} + \\frac{1}{(x-6)(x-8)} + 1/3 = 0` (`x \\ne 2,4,6,8`)',''),(1539,20,52.000,'Solve: `\\frac{1}{a+b+x} = 1/a + 1/b + 1/x` [`x \\ne 0, -(a+b)`]',1707846141,NULL,'','','','',0,'সমাধান করো। `\\frac{1}{a+b+x} = 1/a + 1/b + 1/x` [`x \\ne 0, -(a+b)`]',''),(1540,20,53.000,'Solve: `\\frac{a}{ax-1} + \\frac{b}{bx-1} = a+b` [`x \\ne 1/a,1/b`]',1707846273,NULL,'','','','',0,'সমাধান করো। `\\frac{a}{ax-1} + \\frac{b}{bx-1} = a+b` [`x \\ne 1/a,1/b`]',''),(1541,20,49.000,'Solve: `\\frac{x}{x+1} + \\frac{x+1}{x} = 2\\frac{1}{12}` [`x \\ne 0, -1`]',1707846441,NULL,'','','','',0,'সমাধান করো। `\\frac{x}{x+1} + \\frac{x+1}{x} = 2\\frac{1}{12}` [`x \\ne 0, -1`]',''),(1542,20,48.000,'Solve: `(\\frac{x+4}{x-4})^2 - 5(\\frac{x+4}{x-4}) + 6 = 0` [`x \\ne 4`]',1707846581,'','','','','',0,'সমাধান করো। `(\\frac{x+4}{x-4})^2 - 5(\\frac{x+4}{x-4}) + 6 = 0` [`x \\ne 4`]',''),(1543,20,50.000,'Solve: `\\frac{1}{x} - \\frac{1}{x+b} = \\frac{1}{a} - \\frac{1}{a+b}` [`x \\ne 0,-b`]',1707846794,NULL,'','','','',0,'সমাধান করো। `\\frac{1}{x} - \\frac{1}{x+b} = \\frac{1}{a} - \\frac{1}{a+b}` [`x \\ne 0,-b`]',''),(1544,20,5.000,'The roots of the quadratic equation `1/a x^2 - (b+c)x + abc = 0` (`a\\ne0`) are - (a) `b,c`, (b) `b/a, c/a`, (c) `ab, ac`, (d) `a, bc`',1707847182,NULL,'','','','',0,'`1/a x^2 - (b+c)x + abc = 0` (`a\\ne0`) দ্বিঘাত সমীকরণের বীজদ্বয় - (a) `b,c`, (b) `b/a, c/a`, (c) `ab, ac`, (d) `a, bc`',''),(1545,5,1.000,'Which of the given statements is true –\r\n(a) subset of an infinite set is an infinite set, (b) the set of even numbers greater than 889 is an infinite set, (c) the set of odd negative numbers greater than -150 is an infinite set, (d) A={`x : x` real number and `0\\ltx\\le1`} is a finite set.',1711599012,'(b)','','','','',0,'প্রদত্ত কোন্ বিবৃতিটি সত্য - \r\n(a) কোনো অসীম সংখ্যার উপসেট একটি অসীম সেট, (b) 889 অপেক্ষা বড় যুগ্ম সংখ্যাগুলির একটি অসীম সেট, (c) -150 অপেক্ষা বড় ঋণাত্মক অযুগ্ম সংখ্যাগুলোর সেট একটি অসীম সেট, (d) A={`x : x` বাস্তব সংখ্যা এবং `0ltxle1`} হলো একটি সসীম সেট।',''),(1546,5,31.000,'If `A={1, 2, 3}`, `B = {3, 4}`, `C={4, 5, 6}`, then `(A \\times C) \\cap (B \\times C)` = --<br />\r\n(a) {(1,4), (3,4)}, (b) {(3,4)}, (c) {3,4}, (d) none of these.',1711612367,'(b) {(3,4)}','','','','',0,'`A={1, 2, 3}`, `B = {3, 4}`, `C={4, 5, 6}` হলে `(A 	imes C) cap (B 	imes C)` = --<br />\r\n(a) {(1,4), (3,4)}, (b) {(3,4)}, (c) {3,4}, (d) এদের কোনোটিই নয়',''),(1547,5,5.000,'Which of the following set is an infinite set ? <br />\r\n(a) A = {`x : x` is an integer and `-1 \\le x\\lt 1`}, (b) B = set of even negative integers greater than `-100`, (c) C = set of positive integers less than 100, (d) D = {`x : x` is real and `-1 \\le x\\lt 1`}',1711613611,'(d) D = {`x : x` is real and `-1 \\le x\\lt 1`}','','','','',0,'প্রদত্ত সংজ্ঞাত সেট্ গুলির মধ্যে কোন্ টি অসীম সেট? <br />\r\n(a) A = {`x : x` -এর মান একটি পূর্ণসংখ্যা এবং `-1 le xlt 1`}, (b) B = `-100` অপেক্ষা বড় ঋণাত্মক যুগ্ম সংখ্যাসমূহের সেট, (c) C = 100 এর চেয়ে ছোট ধনাত্মক পূর্ণ সংখ্যাসমূহের সেট, (d) D = {`x : x` বাস্তব ও `-1 le xlt 1`}',''),(1548,5,26.000,'If `B` is a power set of `A`, then which of the following is correct? <br />\r\n(a) `A \\supset B`, (b) `B \\supset A`, (c) `A \\in B`, (d) `A = B`',1711623324,'(c) `A \\in B`','','','','',0,'`A` সেটের সূচক সেট `B` হলে প্রদত্ত কোনটি সঠিক ? <br />\r\n(a) `A supset B`, (b) `B supset A`, (c) `A in B`, (d) `A = B`',''),(1549,5,13.000,'`A \\cap (B - A) = -` (a) `A`, (b) `B`, (c) `\\phi`, (d) `S`',1711624383,'(c) `\\phi`','','','','',0,'`A cap (B - A) = -` (a) `A`, (b) `B`, (c) `phi`, (d) `S`',''),(1550,5,25.000,'If `A={2,4,6,8}`, then which of the following is correct -<br />\r\n(a) `{2,4} \\in A`, (b) `{2,4} \\subseteq A`, (c) `{2,4} \\subset A`,  (d) `{2,4} \\in A^c`',1711627830,'(c) `{2,4} \\subset A`','','','','',0,'`A={2,4,6,8}` হলে কোনটি সঠিক -<br />\r\n(a) `{2,4} in A`, (b) `{2,4} subseteq A`, (c) `{2,4} subset A`,  (d) `{2,4} in A^c`',''),(1551,5,28.000,'Power set of `A={\\phi,{\\phi}}` is -<br />\r\n(a) `{\\phi,{\\phi},{{\\phi}}}`, (b) `{\\phi,{\\phi},A}`, (c) `{\\phi,{\\phi},{{\\phi}},A}`, (d) none of these.',1711628702,'(c) `{\\phi,{\\phi},{{\\phi}},A}`','','','','',0,'`A={phi,{phi}}` সেটের সূচক সেট (power set) হবে -<br />\r\n(a) `{phi,{phi},{{phi}}}`, (b) `{phi,{phi},A}`, (c) `{phi,{phi},{{phi}},A}`, (d) এদের কোনোটিই নয়',''),(1552,5,23.000,'Which of the following is true? -<br />\r\n(a) `\\phi \\in {0}`, (b) `0 \\in \\phi`, (c) `\\phi \\subseteq {0}`, (d) `{0} \\subseteq \\phi`',1711629533,'(c) `\\phi \\subseteq {0}`','','','','',0,'প্রদত্ত কোনটি সত্য হবে -<br />\r\n(a) `phi in {0}`, (b) `0 in phi`, (c) `phi subseteq {0}`, (d) `{0} subseteq phi`',''),(1553,5,16.000,'For two unequal sets `A` and `B`, if `A \\cup B = A`, then the correct option is -<br />\r\n(a) `A \\cap B = A`, (b) `A \\cap B = B`, (c) `A \\cap B = B^c`, (d) `A \\cap B = \\phi`',1711634771,'(b) `A \\cap B = B`','','','','',0,'`A` ও `B` দুটি অসমান সেটের জন্য `A cup B = A` হলে, সঠিক সম্পর্কটি হল -<br />\r\n(a) `A cap B = A`, (b) `A cap B = B`, (c) `A cap B = B^c`, (d) `A cap B = phi`',''),(1554,5,18.000,'For two non-empty sets `A` ও `B`, if `A \\subset B` then correct option is -<br /> (a) `A \\cap B = A^c`, (b) `A \\cup B = A`, (c) `A \\cap B = A`, (d) none of these.',1711635516,'(c) `A \\cap B = A`','','','','',0,'`A` ও `B` দুটি অশূন্য সেটের জন্য `A subset B` হলে, সঠিক সম্পর্কটি হল -<br /> (a) `A cap B = A^c`, (b) `A cup B = A`, (c) `A cap B = A`, (d) এদের কোনোটিই নয়।',''),(1555,5,22.000,'The number of subsets of `\\phi` is -<br />\r\n(a) 0, (b) 1, (c) 2, (d) none of these.',1712036965,'(b) 1','','','','',0,'`phi` সেটের সম্ভাব্য উপসেটের সংখ্যা -<br />\r\n(a) 0, (b) 1, (c) 2, (d) এদের কোনোটিই নয়।',''),(1556,5,27.000,'A is a non-empty set and P(A) is power set of A.  Then number of terms in  P{P{P{P(A)}}} is -<br />\r\n(a) 0, (b) 1, (c) 4, (d) 16',1712038954,'(d) 16','','','','',0,'A একটি শূন্য সেট এবং P(A) হলো A সেটের সূচক সেট। তবে P{P{P{P(A)}}} সেটের পদসংখ্যা -<br />\r\n(a) 0, (b) 1, (c) 4, (d) 16',''),(1557,5,7.000,'If the number of members of two sets `A` and `B` are 3 and 6 respectively, then minimum number of members in the set `A \\cup B` is -<br />\r\n(a) 3, (b) 6, (c) 9, (d) 18',1712044822,'(b) 6','','','','',0,'`A` এবং `B` দুটি সেটের পদসংখ্যা যথাক্রমে 3 এবং 6 হলে `A cup B` সেটের ন্যূনতম পদসংখ্যা হবে -<br />\r\n(a) 3, (b) 6, (c) 9, (d) 18',''),(1558,5,10.000,'If `B \\subseteq A`, then `B - A` is -<br />\r\n(a) `B`, (b) `A`, (c) `\\phi`, (d) `A^c`',1712054543,' (c) `\\phi`','','','','',0,'যদি `B subseteq A` হয়, তবে `B - A` সেটটি হবে -<br />\r\n(a) `B`, (b) `A`, (c) `phi`, (d) `A^c`',''),(1559,5,32.000,'`(A \\cup B \\cup C) \\cap (A \\cap B^c \\cap C^c) \\cap C^c` equals to the set -<br />\r\n(a) `B \\cap C^c`, (b) `A \\cap C`, (c) `B^c \\cap C^c`, (d) none of these.',1712057349,'(a) `B \\cap C^c`','','','','',0,'`(A cup B cup C) cap (A cap B^c cap C^c) cap C^c` এর সমান হলো-<br />\r\n(a) `B cap C^c`, (b) `A cap C`, (c) `B^c cap C^c`, (d) কোনোটিই নয়।',''),(1560,5,30.000,'For any three sets `A`, `B`, `C`, if `A= B \\cap C` and `B = C \\cap A` then -<br />\r\n(a) `A \\subset B`, (b) `B \\subset A`, (c) `A = B`, (d) `A \\subset B^c`',1712058765,'(c) `A = B`','','','','',0,'তিনটি সেট `A`, `B`, `C` এমন যে `A= B cap C` এবং `B = C cap A` হলে -<br />\r\n(a) `A subset B`, (b) `B subset A`, (c) `A = B`, (d) `A subset B^c`',''),(1561,5,33.000,'`A`, `B` are `C` three sets and `A \\cup B \\cup C = S` (`S` = universal set). Then `{(A - B) \\cup (B - C) \\cup (C - A)}^c` equals to the set -- <br />\r\n(a) `A \\cup B \\cup C`, (b) `A \\cup (B \\cap C)`, (c) `A \\cap B \\cap C`, (d) `A \\cap (B \\cup C)`',1712145098,'(c) `A \\cap B \\cap C`','','','','',0,'`A`, `B` এবং `C` তিনটি সেট এবং `A cup B cup C = S` (`S` = সার্বিক সেট)। তবে `{(A - B) cup (B - C) cup (C - A)}^c` -এর সমান হবে -- <br />\r\n(a) `A cup B cup C`, (b) `A cup (B cap C)`, (c) `A cap B cap C`, (d) `A cap (B cup C)`',''),(1562,5,6.000,'If `A = {5k +1 : k \\in \\mathbb{Z}}` and `B = { x : x \\in \\mathbb{Z}` and `x+4` is divisible by 5`}`, then which of the following is true?-- <br />\r\n(a) `A \\subset B`, (b) `B \\subset A`, (c) `A = B`, (d) `A \\cap B = \\phi`',1712462920,'(c) `A = B`','','','','',0,'যদি `A = {5k +1 : k in mathbb{Z}}` এবং `B = { x : x in mathbb{Z}` এবং `x+4` সংখ্যাটি 5 দ্বারা বিভাজ্য`}` হয়, তবে নীচের কোন্ টি সত্য ?-- <br />\r\n(a) `A subset B`, (b) `B subset A`, (c) `A = B`, (d) `A cap B = phi`',''),(1563,5,4.000,'Which of the following set is a null set?- <br />\r\n(a) `A= {x : x` is prime number and `31 \\lt x \\lt 37}`, (b) `B = {x : x` is integer and `0 \\lt x \\le 1}`, (c) `C = {\\phi}`, (d) `D = {x : x+1=1}`',1712463505,'(a) A set','','','','',0,'প্রদত্ত সেটগুলির মধ্যে কোনটি শূন্য সেট?- <br />\r\n(a) `A= {x : x` মৌলিক সংখ্যা এবং `31 lt x lt 37}`, (b) `B = {x : x` পূর্ণসংখ্যা এবং `0 lt x le 1}`, (c) `C = {phi}`, (d) `D = {x : x+1=1}`',''),(1564,5,3.000,'If `A = {(x,y) : y = 1/x, x\\ne0, x \\in mathbb{R}}` and `B = {(x,y) : y = -x, x \\in mathbb{R}}`, then which of the following is true?- <br />\r\n(a) `A \\cap B = A`, (b) `A \\cap B = B`, (c) `A \\cap B = \\phi`, (d) None of these.',1712464536,'(c) `A \\cap B = \\phi`','','','','',0,'`A = {(x,y) : y = 1/x, x\ne0, x in mathbb{R}}` এবং `B = {(x,y) : y = -x, x in mathbb{R}}` হলে নীচের কোনটি সত্য?- <br />\r\n(a) `A cap B = A`, (b) `A cap B = B`, (c) `A cap B = phi`, (d) কোনোটিই নয়।',''),(1565,5,2.000,'If `A = {x : x \\in \\mathbb{Z}` and `0 \\lt x \\lt 4}` and `B = {x : x \\in \\mathbb{Z}` and `3 \\le x \\le 6}`, then `A \\cap B` =-?- <br />\r\n(a) `{2}`, (b) `{3}`, (c) `3`, (d) `\\phi`',1712465957,'(b) `{3}`','','','','',0,'যদি `A = {x : x in mathbb{Z}` এবং `0 lt x lt 4}` এবং `B = {x : x in mathbb{Z}` এবং `3 le x le 6}`, তাহলে `A cap B` =-?- <br />\r\n(a) `{2}`, (b) `{3}`, (c) `3`, (d) `phi`',''),(1566,2,47.000,'If `x = Ae^{-\\frac{kt}{2}} cos (pt + \\epsilon)` where `A, k, p, \\epsilon` are constants, then show that `\\frac{d^2x}{dt^2}+k \\frac{dx}{dt} + n^2x=0` where `n^2 = p^2 + k^2/4`',1712494469,'N.A.','1566','','','',0,'',''),(1567,10,9.000,'A relation `R` defined on the set `A = {2, 5, 7, 10}` by `x R y \\iff x \\equiv y (mod 5)`. Express `R` as a set of ordered pair.',1712578935,'R ={(2, 7), (7, 2), (5, 10), (10, 5), (2, 2), (5, 5), (7, 7) , (10, 10)}','','','','',0,'`A = {2, 5, 7, 10}` সেটের উপর একটি সম্বন্ধ `R` নিম্নলিখিত রূপে সংজ্ঞাত `x R y iff x equiv y (mod 5)` । ক্রমযুগলের সেটরূপে `R` কে প্রকাশ করো। ',''),(1568,10,11.000,'A relation `R = {(x, y) : x+2y=19, x,y \\in A}` defined on the set `A = {1, 2, 3, ...., 10}`. Then find `R^{-1}`.',1712656710,'{(17, 1), (15, 2), (13, 3), (11, 4), (9, 5), (7, 6), (5, 7), (3, 8), (1, 9)}','','','','',0,'`A = {1, 2, 3, ...., 10}` এবং `R = {(x, y) : x+2y=19, x,y in A}` একটি সম্বন্ধ হলে `R^{-1}` নির্ণয় করো।',''),(1569,10,19.000,'A relation `R` defined by : `R ={(2x-5, x-3) : x \\lt 12, x` even and `x \\in \\mathbb{N}}`, find the domain of definition and range of `R`.',1712658230,'{-1, 3, 7, 11, 15}, {-1, 1, 3, 5, 7}','','','','',0,'একটি সম্বন্ধ `R` নিম্নরূপে সংজ্ঞাত : `R ={(2x-5, x-3) : x lt 12, x` যুগ্ম এবং `x in mathbb{N}}`, `R` সম্বন্ধটির সংজ্ঞার অঞ্চল এবং প্রসার নির্ণয় করো।',''),(1570,10,8.000,'A relation `R = {(x,y) : x \\gt y}` defined on the set `A = {x : x \\in N, 1 \\le x \\le 5}`. Express `R` as the set of ordered pair.',1712662026,'{(2,1), (3,1), (3,2), (4,1), (4,2), (4,3), (5,1), (5,2), (5,3), (5,4)}','','','','',0,'`A = {x : x in N, 1 le x le 5}` সেটের উপর একটি সম্বন্ধ `R = {(x,y) : x gt y}` হলে ক্রমযুগলের সেটরূপে `R` কে প্রকাশ করো। ',''),(1571,10,10.000,'For the relation `R={(a,b) : a+b \\le 7` and `a=1, 2, 3, 4` and `b \\in \\mathbb{N}}`, find domain of definition of `R^{-1}`.',1712663206,'{1, 2, 3, 4, 5, 6}','','','','',0,'একটি সম্বন্ধ `R={(a,b) : a+b le 7` এবং `a=1, 2, 3, 4` ও `b in mathbb{N}}` হলে, `R^{-1}` এর ক্ষেত্র নির্ণয় করো। ',''),(1572,10,7.000,'A relation `R` on the set `A = {1, 2, 3}` is defined by,  `a R b \\implies |a-b| \\le 5`, then which of the following statement is wrong- <br />\r\n(a) `R = {(1, 1), (2, 2), (3, 3), (2, 1), (1, 2), (2, 3), (3, 2)}`, (b) `R^{-1} = R`, (c) domain of definition of `R` is `{1, 2, 3}`, (d) range of `R` is `{5}`',1713185188,'(d) range of `R` is `{5}`','','','','',0,'`A = {1, 2, 3}` সেটে `R` সম্বন্ধটি এরূপে সংজ্ঞাত যে,  `a R b implies |a-b| le 5`, তবে ভুল বিবৃতিটি হলো - <br />\r\n(a) `R = {(1, 1), (2, 2), (3, 3), (2, 1), (1, 2), (2, 3), (3, 2)}`, (b) `R^{-1} = R`, (c) `R` সম্বন্ধটির ক্ষেত্র বা সংজ্ঞার অঞ্চল `{1, 2, 3}`, (d) `R` সম্বন্ধটির পাল্লা বা প্রসার `{5}`','');
/*!40000 ALTER TABLE `js_maths` ENABLE KEYS */;
UNLOCK TABLES;
/*!40103 SET TIME_ZONE=@OLD_TIME_ZONE */;

/*!40101 SET SQL_MODE=@OLD_SQL_MODE */;
/*!40014 SET FOREIGN_KEY_CHECKS=@OLD_FOREIGN_KEY_CHECKS */;
/*!40014 SET UNIQUE_CHECKS=@OLD_UNIQUE_CHECKS */;
/*!40101 SET CHARACTER_SET_CLIENT=@OLD_CHARACTER_SET_CLIENT */;
/*!40101 SET CHARACTER_SET_RESULTS=@OLD_CHARACTER_SET_RESULTS */;
/*!40101 SET COLLATION_CONNECTION=@OLD_COLLATION_CONNECTION */;
/*!40111 SET SQL_NOTES=@OLD_SQL_NOTES */;

-- Dump completed on 2024-07-08 16:46:52
