jee-main

Papers (169)
2025
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2024
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2023
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2022
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2021
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2020
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2019
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2018
08apr 29 15apr 28 15apr_shift1 28 15apr_shift2 2 16apr 15
2017
02apr 28 08apr 29 09apr 30
2016
03apr 30 09apr 30 10apr 28
2015
04apr 29 10apr 30
2014
06apr 28 09apr 28 11apr 4 12apr 5 19apr 29
2013
07apr 29 09apr 14 22apr 5 23apr 14 25apr 13
2012
07may 18 12may 22 19may 13 26may 17 offline 30
2011
jee-main_2011.pdf 13
2010
jee-main_2010.pdf 1
2009
jee-main_2009.pdf 1
2008
jee-main_2008.pdf 1
2007
jee-main_2007.pdf 38
2005
jee-main_2005.pdf 19
2004
jee-main_2004.pdf 11
2003
jee-main_2003.pdf 9
2002
jee-main_2002.pdf 8
2018 08apr

29 maths questions

Q61 Inequalities Absolute Value Inequality View
Let $S = \{ x \in R : x \geq 0 \& 2 | \sqrt { x } - 3 | + \sqrt { x } ( \sqrt { x } - 6 ) + 6 = 0 \}$. Then $S$ :
(1) Contains exactly four elements
(2) Is an empty set
(3) Contains exactly one element
(4) Contains exactly two elements
Q62 Complex Numbers Arithmetic Roots of Unity and Cyclotomic Expressions View
If $\alpha , \beta \in C$ are the distinct roots of the equation $x ^ { 2 } - x + 1 = 0$, then $\alpha ^ { 101 } + \beta ^ { 107 }$ is equal to
(1) 2
(2) - 1
(3) 0
(4) 1
Q63 Permutations & Arrangements Selection and Task Assignment View
From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle. The number of such arrangements is:
(1) At least 750 but less than 1000
(2) At least 1000
(3) Less than 500
(4) At least 500 but less than 750
Q64 Arithmetic Sequences and Series Summation of Derived Sequence from AP View
Let $A$ be the sum of the first 20 terms and $B$ be the sum of the first 40 terms of the series $1 ^ { 2 } + 2 \cdot 2 ^ { 2 } + 3 ^ { 2 } + 2 \cdot 4 ^ { 2 } + 5 ^ { 2 } + 2 \cdot 6 ^ { 2 } + \ldots$ If $B - 2 A = 100 \lambda$, then $\lambda$ is equal to :
(1) 496
(2) 232
(3) 248
(4) 464
Q65 Arithmetic Sequences and Series Summation of Derived Sequence from AP View
Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots \ldots , a _ { 49 }$ be in $A.P$. such that $\sum _ { k = 0 } ^ { 12 } a _ { 4 k + 1 } = 416$ and $a _ { 9 } + a _ { 43 } = 66$. If $a _ { 1 } ^ { 2 } + a _ { 2 } ^ { 2 } + \ldots + a _ { 17 } ^ { 2 } = 140 m$, then $m$ is equal to:
(1) 33
(2) 66
(3) 68
(4) 34
Q66 Binomial Theorem (positive integer n) Evaluate a Summation Involving Binomial Coefficients View
The sum of the co-efficient of all odd degree terms in the expansion of $\left( x + \sqrt { x ^ { 3 } - 1 } \right) ^ { 5 } + \left( x - \sqrt { x ^ { 3 } - 1 } \right) ^ { 5 } , ( x > 1 )$ is
(1) 2
(2) - 1
(3) 0
(4) 1
Q67 Standard trigonometric equations Solve trigonometric equation for solutions in an interval View
If sum of all the solutions of the equation $8 \cos x \cdot \left( \cos \left( \frac { \pi } { 6 } + x \right) \cdot \cos \left( \frac { \pi } { 6 } - x \right) - \frac { 1 } { 2 } \right) = 1$ in $[ 0 , \pi ]$ is $k \pi$, then $k$ is equal to:
(1) $\frac { 20 } { 9 }$
(2) $\frac { 2 } { 3 }$
(3) $\frac { 13 } { 9 }$
(4) $\frac { 8 } { 9 }$
Q68 Straight Lines & Coordinate Geometry Locus Determination View
A straight line through a fixed point $( 2,3 )$ intersects the coordinate axes at distinct points $P$ and $Q$. If $O$ is the origin and the rectangle $O P R Q$ is completed, then the locus of $R$ is:
(1) $3 x + 2 y = 6 x y$
(2) $3 x + 2 y = 6$
(3) $2 x + 3 y = x y$
(4) $3 x + 2 y = x y$
Q69 Circles Tangent Lines and Tangent Lengths View
If the tangent at $( 1,7 )$ to the curve $x ^ { 2 } = y - 6$ touch the circle $x ^ { 2 } + y ^ { 2 } + 16 x + 12 y + c = 0$ then the value of $c$ is:
(1) 95
(2) 195
(3) 185
(4) 85
Q70 Circles Inscribed/Circumscribed Circle Computations View
Tangent and normal are drawn at $P ( 16,16 )$ on the parabola $y ^ { 2 } = 16 x$, which intersect the axis of the parabola at $A \& B$, respectively. If $C$ is the center of the circle through the points $P , A \& B$ and $\angle C P B = \theta$, then a value of $\tan \theta$ is:
(1) $\frac { 4 } { 3 }$
(2) $\frac { 1 } { 2 }$
(3) 2
(4) 3
Q71 Circles Circle Identification and Classification View
Two sets $A$ and $B$ are as under: $A = \{ ( a , b ) \in R \times R : | a - 5 | < 1$ and $| b - 5 | < 1 \}$; $B = \left\{ ( a , b ) \in R \times R : 4 ( a - 6 ) ^ { 2 } + 9 ( b - 5 ) ^ { 2 } \leq 36 \right\}$. Then :
(1) neither $A \subset B$ nor $B \subset A$
(2) $B \subset A$
(3) $A \subset B$
(4) $A \cap B = \phi$ (an empty set)
Q72 Conic sections Triangle or Quadrilateral Area and Perimeter with Foci View
Tangents are drawn to the hyperbola $4 x ^ { 2 } - y ^ { 2 } = 36$ at the points $P$ and $Q$. If these tangents intersect at the point $T ( 0,3 )$ then the area (in sq. units) of $\triangle P T Q$ is:
(1) $36 \sqrt { 5 }$
(2) $45 \sqrt { 5 }$
(3) $54 \sqrt { 3 }$
(4) $60 \sqrt { 3 }$
Q73 Curve Sketching Limit Computation from Algebraic Expressions View
For each $t \in R$, let $[ t ]$ be the greatest integer less than or equal to $t$. Then $\lim _ { x \rightarrow 0 ^ { + } } x \left( \left[ \frac { 1 } { x } \right] + \left[ \frac { 2 } { x } \right] + \ldots + \left[ \frac { 15 } { x } \right] \right)$
(1) does not exist (in $R$ )
(2) is equal to 0
(3) is equal to 15
(4) is equal to 120
Q75 Measures of Location and Spread View
If $\sum _ { i = 1 } ^ { 9 } \left( x _ { i } - 5 \right) = 9$ and $\sum _ { i = 1 } ^ { 9 } \left( x _ { i } - 5 \right) ^ { 2 } = 45$, then the standard deviation of the 9 items $x _ { 1 } , x _ { 2 } , \ldots , x _ { 9 }$ is
(1) 3
(2) 9
(3) 4
(4) 2
Q76 Sine and Cosine Rules Heights and distances / angle of elevation problem View
$P Q R$ is a triangular park with $P Q = P R = 200 \mathrm {~m}$. A T.V. tower stands at the mid-point of $Q R$. If the angles of elevation of the top of the tower at $P , Q$ and $R$ are respectively, $45 ^ { \circ } , 30 ^ { \circ }$ and $30 ^ { \circ }$, then the height of the tower (in m) is:
(1) $50 \sqrt { 2 }$
(2) 100
(3) 50
(4) $100 \sqrt { 3 }$
Q77 Straight Lines & Coordinate Geometry Triangle Properties and Special Points View
Let the orthocentre and centroid of a triangle be $A ( - 3,5 )$ and $B ( 3,3 )$ respectively. If $C$ is the circumcentre of this triangle, then the radius of the circle having line segment $A C$ as diameter, is:
(1) $\frac { 3 \sqrt { } 5 } { 2 }$
(2) $\sqrt { 10 }$
(3) $2 \sqrt { 10 }$
(4) $3 \sqrt { \frac { 5 } { 2 } }$
Q78 Matrices Linear System and Inverse Existence View
If the system of linear equations $x + k y + 3 z = 0$ $3 x + k y - 2 z = 0$ $2 x + 4 y - 3 z = 0$ has a non-zero solution $( x , y , z )$, then $\frac { x z } { y ^ { 2 } }$ is equal to:
(1) 30
(2) - 10
(3) 10
(4) - 30
Q79 Matrices Determinant and Rank Computation View
$\left| \begin{array} { c c c } x - 4 & 2 x & 2 x \\ 2 x & x - 4 & 2 x \\ 2 x & 2 x & x - 4 \end{array} \right| = ( A + B x ) ( x - A ) ^ { 2 }$, then the ordered pair $( A , B )$ is equal to
(1) $( 4,5 )$
(2) $( - 4 , - 5 )$
(3) $( - 4,3 )$
(4) $( - 4,5 )$
Q80 Differentiation from First Principles View
Let $S = \left\{ t \in R : f ( x ) = | x - \pi | \cdot \left( e ^ { | x | } - 1 \right) \sin | x |\right.$ is not differentiable at $\left. t \right\}$. Then, the set $S$ is equal to:
(1) $\{ 0 , \pi \}$
(2) $\phi$ (an empty set)
(3) $\{ 0 \}$
(4) $\{ \pi \}$
Q81 Circles Intersection of Circles or Circle with Conic View
If the curves $y ^ { 2 } = 6 x , 9 x ^ { 2 } + b y ^ { 2 } = 16$ intersect each other at right angles, then the value of $b$ is:
(1) $\frac { 9 } { 2 }$
(2) 6
(3) $\frac { 7 } { 2 }$
(4) 4
Q82 Stationary points and optimisation Find critical points and classify extrema of a given function View
Let $f ( x ) = x ^ { 2 } + \frac { 1 } { x ^ { 2 } }$ and $g ( x ) = x - \frac { 1 } { x } , x \in R - \{ - 1,0,1 \}$. If $h ( x ) = \frac { f ( x ) } { g ( x ) }$, then the local minimum value of $h ( x )$ is:
(1) $2 \sqrt { 2 }$
(2) 3
(3) - 3
(4) $- 2 \sqrt { 2 }$
Q83 Integration by Substitution Substitution to Transform Integral Form (Show Transformed Expression) View
The integral $\int \frac { \sin ^ { 2 } x \cos ^ { 2 } x } { \left( \sin ^ { 5 } x + \cos ^ { 3 } x \sin ^ { 2 } x + \sin ^ { 3 } x \cos ^ { 2 } x + \cos ^ { 5 } x \right) ^ { 2 } } d x$, is equal to (where $C$ is the constant of integration).
(1) $\frac { - 1 } { 1 + \cot ^ { 3 } x } + C$
(2) $\frac { 1 } { 3 \left( 1 + \tan ^ { 3 } x \right) } + C$
(3) $\frac { - 1 } { 3 \left( 1 + \tan ^ { 3 } x \right) } + C$
(4) $\frac { 1 } { 1 + \cot ^ { 3 } x } + C$
Q84 Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution View
The values of $\int _ { - \frac { \pi } { 2 } } ^ { \frac { \pi } { 2 } } \frac { \sin ^ { 2 } x } { 1 + 2 ^ { x } } d x$ is
(1) $\frac { \pi } { 4 }$
(2) $\frac { \pi } { 8 }$
(3) $\frac { \pi } { 2 }$
(4) $4 \pi$
Q85 Areas by integration View
Let $g ( x ) = \cos x ^ { 2 } , f ( x ) = \sqrt { x }$, and $\alpha , \beta ( \alpha < \beta )$ be the roots of the quadratic equation $18 x ^ { 2 } - 9 \pi x + \pi ^ { 2 } = 0$. Then the area (in sq. units) bounded by the curve $y = ( g o f ) ( x )$ and the lines $x = \alpha , x = \beta$ and $y = 0$, is
(1) $\frac { 1 } { 2 } ( \sqrt { 2 } - 1 )$
(2) $\frac { 1 } { 2 } ( \sqrt { 3 } - 1 )$
(3) $\frac { 1 } { 2 } ( \sqrt { 3 } + 1 )$
(4) $\frac { 1 } { 2 } ( \sqrt { 3 } - \sqrt { 2 } )$
Q86 First order differential equations (integrating factor) View
Let $y = y ( x )$ be the solution of the differential equation $\sin x \frac { d y } { d x } + y \cos x = 4 x , x \in ( 0 , \pi )$. If $y \left( \frac { \pi } { 2 } \right) = 0$, then $y \left( \frac { \pi } { 6 } \right)$ is equal to
(1) $- \frac { 4 } { 9 } \pi ^ { 2 }$
(2) $\frac { 4 } { 9 \sqrt { 3 } } \pi ^ { 2 }$
(3) $\frac { - 8 } { 9 \sqrt { 3 } } \pi ^ { 2 }$
(4) $- \frac { 8 } { 9 } \pi ^ { 2 }$
Q87 Vectors Introduction & 2D Perpendicularity or Parallel Condition View
Let $\vec { u }$ be a vector coplanar with the vectors $\vec { a } = 2 \hat { i } + 3 \hat { j } - \widehat { k }$ and $\vec { b } = \hat { j } + \widehat { k }$. If $\vec { u }$ is perpendicular to $\vec { a }$ and $\vec { u } \cdot \vec { b } = 24$, then $| \vec { u } | ^ { 2 }$ is equal to:
(1) 84
(2) 336
(3) 315
(4) 256
Q88 Vectors: Lines & Planes Distance Computation (Point-to-Plane or Line-to-Line) View
If $L _ { 1 }$ is the line of intersection of the planes $2 x - 2 y + 3 z - 2 = 0 , x - y + z + 1 = 0$ and $L _ { 2 }$ is the line of intersection of the planes $x + 2 y - z - 3 = 0,3 x - y + 2 z - 1 = 0$, then the distance of the origin from the plane, containing the lines $L _ { 1 }$ and $L _ { 2 }$ is
(1) $\frac { 1 } { \sqrt { 2 } }$
(2) $\frac { 1 } { 4 \sqrt { 2 } }$
(3) $\frac { 1 } { 3 \sqrt { 2 } }$
(4) $\frac { 1 } { 2 \sqrt { 2 } }$
Q89 Vectors 3D & Lines Distance from a Point to a Line (Show/Compute) View
The length of the projection of the line segment joining the points $( 5 , - 1,4 )$ and $( 4 , - 1,3 )$ on the plane, $x + y + z = 7$ is
(1) $\sqrt { \frac { 2 } { 3 } }$
(2) $\frac { 2 } { \sqrt { 3 } }$
(3) $\frac { 2 } { 3 }$
(4) $\frac { 1 } { 3 }$
Q90 Probability Definitions Conditional Probability and Bayes' Theorem View
A bag contains 4 red and 6 black balls. A ball is drawn at random from the bag, its color is observed and this ball along with two additional balls of the same color are returned to the bag. If now a ball is drawn at random from the bag, then the probability that this drawn ball is red, is:
(1) $\frac { 3 } { 4 }$
(2) $\frac { 3 } { 10 }$
(3) $\frac { 2 } { 5 }$
(4) $\frac { 1 } { 5 }$