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
session1_24feb_shift1 10 session1_24feb_shift2 7 session1_25feb_shift1 29 session1_25feb_shift2 29 session1_26feb_shift2 17 session2_16mar_shift1 29 session2_16mar_shift2 15 session2_17mar_shift1 20 session2_17mar_shift2 24 session2_18mar_shift1 12 session2_18mar_shift2 11 session3_20jul_shift1 30 session3_20jul_shift2 29 session3_22jul_shift1 7 session3_25jul_shift1 2 session3_25jul_shift2 15 session3_27jul_shift1 3 session3_27jul_shift2 4 session4_01sep_shift2 11 session4_26aug_shift1 5 session4_26aug_shift2 2 session4_27aug_shift1 3 session4_27aug_shift2 28 session4_31aug_shift1 28 session4_31aug_shift2 4
2020
session1_07jan_shift1 26 session1_07jan_shift2 17 session1_08jan_shift1 5 session1_08jan_shift2 12 session1_09jan_shift1 22 session1_09jan_shift2 18 session2_02sep_shift1 19 session2_02sep_shift2 17 session2_03sep_shift1 21 session2_03sep_shift2 9 session2_04sep_shift1 10 session2_04sep_shift2 24 session2_05sep_shift1 23 session2_05sep_shift2 27 session2_06sep_shift1 13 session2_06sep_shift2 10
2019
session1_09jan_shift1 6 session1_09jan_shift2 29 session1_10jan_shift1 30 session1_10jan_shift2 12 session1_11jan_shift1 6 session1_11jan_shift2 5 session1_12jan_shift1 10 session1_12jan_shift2 20 session2_08apr_shift1 29 session2_08apr_shift2 29 session2_09apr_shift1 29 session2_09apr_shift2 29 session2_10apr_shift1 2 session2_10apr_shift2 3 session2_12apr_shift1 3 session2_12apr_shift2 9
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
2025 session1_22jan_shift2

25 maths questions

Q1 3x3 Matrices Determinant of Parametric or Structured Matrix View
For a $3 \times 3$ matrix $M$, let trace ( $M$ ) denote the sum of all the diagonal elements of $M$. Let $A$ be a $3 \times 3$ matrix such that $| A | = \frac { 1 } { 2 }$ and trace $( A ) = 3$. If $B = \operatorname { adj } ( \operatorname { adj } ( 2 A ) )$, then the value of $| B | +$ trace $( \mathrm { B } )$ equals :
(1) 56
(2) 132
(3) 174
(4) 280
Q2 Permutations & Arrangements Linear Arrangement with Constraints View
In a group of 3 girls and 4 boys, there are two boys $B _ { 1 }$ and $B _ { 2 }$. The number of ways, in which these girls and boys can stand in a queue such that all the girls stand together, all the boys stand together, but $B _ { 1 }$ and $B _ { 2 }$ are not adjacent to each other, is :
(1) 96
(2) 144
(3) 120
(4) 72
Q3 Binomial Theorem (positive integer n) Find a Specific Coefficient in a Single Binomial Expansion View
Let $\alpha , \beta , \gamma$ and $\delta$ be the coefficients of $x ^ { 7 } , x ^ { 5 } , x ^ { 3 }$ and $x$ respectively in the expansion of $\left( x + \sqrt { x ^ { 3 } - 1 } \right) ^ { 5 } + \left( x - \sqrt { x ^ { 3 } - 1 } \right) ^ { 5 } , x > 1$. If u and v satisfy the equations $\begin{aligned} & \alpha u + \beta v = 18 \\ & \gamma u + \delta v = 20 \end{aligned}$ then $u + v$ equals :
(1) 5
(2) 3
(3) 4
(4) 8
Q4 Vectors 3D & Lines Distance from a Point to a Line (Show/Compute) View
Let a line pass through two distinct points $P ( - 2 , - 1,3 )$ and $Q$, and be parallel to the vector $3 \hat { i } + 2 \hat { j } + 2 \hat { k }$. If the distance of the point Q from the point $\mathrm { R } ( 1,3,3 )$ is 5 , then the square of the area of $\triangle P Q R$ is equal to :
(1) 148
(2) 136
(3) 144
(4) 140
Q5 Conditional Probability Direct Conditional Probability Computation from Definitions View
If $A$ and $B$ are two events such that $P ( A \cap B ) = 0.1$, and $P ( A \mid B )$ and $P ( B \mid A )$ are the roots of the equation $12 x ^ { 2 } - 7 x + 1 = 0$, then the value of $\frac { \mathrm { P } ( \overline { \mathrm { A } } \cup \overline { \mathrm { B } } ) } { \mathrm { P } ( \overline { \mathrm { A } } \cap \overline { \mathrm { B } } ) }$ is :
(1) $\frac { 4 } { 3 }$
(2) $\frac { 7 } { 4 }$
(3) $\frac { 5 } { 3 }$
(4) $\frac { 9 } { 4 }$
Q6 Standard Integrals and Reverse Chain Rule Integral Equation to Determine a Function Value View
If $\int \mathrm { e } ^ { x } \left( \frac { x \sin ^ { - 1 } x } { \sqrt { 1 - x ^ { 2 } } } + \frac { \sin ^ { - 1 } x } { \left( 1 - x ^ { 2 } \right) ^ { 3 / 2 } } + \frac { x } { 1 - x ^ { 2 } } \right) \mathrm { d } x = \mathrm { g } ( x ) + \mathrm { C }$, where C is the constant of integration, then $g \left( \frac { 1 } { 2 } \right)$ equals :
(1) $\frac { \pi } { 4 } \sqrt { \frac { e } { 3 } }$
(2) $\frac { \pi } { 6 } \sqrt { \frac { e } { 3 } }$
(3) $\frac { \pi } { 4 } \sqrt { \frac { e } { 2 } }$
(4) $\frac { \pi } { 6 } \sqrt { \frac { e } { 2 } }$
Q7 Areas Between Curves Compute Area Directly (Numerical Answer) View
The area of the region enclosed by the curves $y = x ^ { 2 } - 4 x + 4$ and $y ^ { 2 } = 16 - 8 x$ is :
(1) $\frac { 8 } { 3 }$
(2) $\frac { 4 } { 3 }$
(3) 8
(4) 5
Q8 Stationary points and optimisation Find critical points and classify extrema of a given function View
Let $f ( x ) = \int _ { 0 } ^ { x ^ { 2 } } \frac { \mathrm { t } ^ { 2 } - 8 \mathrm { t } + 15 } { \mathrm { e } ^ { t } } \mathrm { dt } , x \in \mathbf { R }$. Then the numbers of local maximum and local minimum points of $f$, respectively, are :
(1) 2 and 3
(2) 2 and 2
(3) 3 and 2
(4) 1 and 3
Q9 Conic sections Triangle or Quadrilateral Area and Perimeter with Foci View
Let $\mathrm { P } ( 4,4 \sqrt { 3 } )$ be a point on the parabola $y ^ { 2 } = 4 \mathrm { a } x$ and PQ be a focal chord of the parabola. If M and $N$ are the foot of perpendiculars drawn from $P$ and $Q$ respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to :
(1) $17 \sqrt { 3 }$
(2) $\frac { 263 \sqrt { 3 } } { 8 }$
(3) $\frac { 34 \sqrt { 3 } } { 3 }$
(4) $\frac { 343 \sqrt { 3 } } { 8 }$
Q10 Vectors Introduction & 2D Perpendicularity or Parallel Condition View
Let $\vec { a }$ and $\vec { b }$ be two unit vectors such that the angle between them is $\frac { \pi } { 3 }$. If $\lambda \vec { a } + 2 \vec { b }$ and $3 \vec { a } - \lambda \vec { b }$ are perpendicular to each other, then the number of values of $\lambda$ in $[ - 1,3 ]$ is :
(1) 2
(2) 1
(3) 0
(4) 3
Q11 Exponential Functions Limit Evaluation View
If $\lim _ { x \rightarrow \infty } \left( \left( \frac { \mathrm { e } } { 1 - \mathrm { e } } \right) \left( \frac { 1 } { \mathrm { e } } - \frac { x } { 1 + x } \right) \right) ^ { x } = \alpha$, then the value of $\frac { \log _ { \mathrm { e } } \alpha } { 1 + \log _ { \mathrm { e } } \alpha }$ equals:
(1) $e ^ { - 1 }$
(2) $\mathrm { e } ^ { 2 }$
(3) $e ^ { - 2 }$
(4) e
Q12 Composite & Inverse Functions Counting Functions with Composition or Mapping Constraints View
Let $\mathrm { A } = \{ 1,2,3,4 \}$ and $\mathrm { B } = \{ 1,4,9,16 \}$. Then the number of many-one functions $f : \mathrm { A } \rightarrow \mathrm { B }$ such that $1 \in f ( \mathrm {~A} )$ is equal to :
(1) 151
(2) 139
(3) 163
(4) 127
Q13 Arithmetic Sequences and Series Counting or Combinatorial Problems on APs View
Suppose that the number of terms in an A.P. is $2k , k \in N$. If the sum of all odd terms of the A.P. is 40 , the sum of all even terms is 55 and the last term of the A.P. exceeds the first term by 27 , then k is equal to :
(1) 6
(2) 5
(3) 8
(4) 4
Q14 Vectors 3D & Lines Distance from a Point to a Line (Show/Compute) View
The perpendicular distance, of the line $\frac { x - 1 } { 2 } = \frac { y + 2 } { - 1 } = \frac { z + 3 } { 2 }$ from the point $\mathrm { P } ( 2 , - 10,1 )$, is :
(1) 6
(2) $5 \sqrt { 2 }$
(3) $4 \sqrt { 3 }$
(4) $3 \sqrt { 5 }$
Q15 Matrices Linear System and Inverse Existence View
If the system of linear equations : $$x + y + 2z = 6$$ $$2x + 3y + \mathrm { a } z = \mathrm { a } + 1$$ $$- x - 3 y + \mathrm { b } z = 2 \mathrm {~b}$$ where $a , b \in \mathbf { R }$, has infinitely many solutions, then $7a + 3b$ is equal to :
(1) 16
(2) 12
(3) 22
(4) 9
Q16 First order differential equations (integrating factor) View
If $x = f ( y )$ is the solution of the differential equation $\left( 1 + y ^ { 2 } \right) + \left( x - 2 \mathrm { e } ^ { \tan ^ { - 1 } y } \right) \frac { \mathrm { d } y } { \mathrm {~d} x } = 0 , y \in \left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$ with $f ( 0 ) = 1$, then $f \left( \frac { 1 } { \sqrt { 3 } } \right)$ is equal to :
(1) $e ^ { \pi / 12 }$
(2) $e ^ { \pi / 4 }$
(3) $e ^ { \pi / 3 }$
(4) $e ^ { \pi / 6 }$
Q17 Solving quadratics and applications Finding roots or coefficients of a quadratic using Vieta's relations View
Let $\alpha _ { \theta }$ and $\beta _ { \theta }$ be the distinct roots of $2 x ^ { 2 } + ( \cos \theta ) x - 1 = 0 , \theta \in ( 0,2 \pi )$. If m and M are the minimum and the maximum values of $\alpha _ { \theta } ^ { 4 } + \beta _ { \theta } ^ { 4 }$, then $16 ( M + m )$ equals :
(1) 24
(2) 25
(3) 17
(4) 27
Q18 Standard trigonometric equations Solve trigonometric equation for solutions in an interval View
The sum of all values of $\theta \in [ 0,2 \pi ]$ satisfying $2 \sin ^ { 2 } \theta = \cos 2 \theta$ and $2 \cos ^ { 2 } \theta = 3 \sin \theta$ is
(1) $4 \pi$
(2) $\frac { 5 \pi } { 6 }$
(3) $\pi$
(4) $\frac { \pi } { 2 }$
Q19 Complex Numbers Argand & Loci Distance and Region Optimization on Loci View
Let the curve $z ( 1 + i ) + \bar { z } ( 1 - i ) = 4 , z \in \mathrm { C }$, divide the region $| z - 3 | \leq 1$ into two parts of areas $\alpha$ and $\beta$. Then $| \alpha - \beta |$ equals :
(1) $1 + \frac { \pi } { 2 }$
(2) $1 + \frac { \pi } { 3 }$
(3) $1 + \frac { \pi } { 6 }$
(4) $1 + \frac { \pi } { 4 }$
Q20 Conic sections Confocal or Related Conic Construction View
Let $\mathrm { E } : \frac { x ^ { 2 } } { \mathrm { a } ^ { 2 } } + \frac { y ^ { 2 } } { \mathrm {~b} ^ { 2 } } = 1 , \mathrm { a } > \mathrm { b }$ and $\mathrm { H } : \frac { x ^ { 2 } } { \mathrm {~A} ^ { 2 } } - \frac { y ^ { 2 } } { \mathrm {~B} ^ { 2 } } = 1$. Let the distance between the foci of E and the foci of $H$ be $2 \sqrt { 3 }$. If $a - A = 2$, and the ratio of the eccentricities of $E$ and $H$ is $\frac { 1 } { 3 }$, then the sum of the lengths of their latus rectums is equal to:
(1) 10
(2) 9
(3) 8
(4) 7
Q21 Binomial Theorem (positive integer n) Evaluate a Summation Involving Binomial Coefficients View
If $\sum _ { r = 1 } ^ { 30 } \frac { r ^ { 2 } \left( { } ^ { 30 } C _ { r } \right) ^ { 2 } } { { } ^ { 30 } C _ { r - 1 } } = \alpha \times 2 ^ { 29 }$, then $\alpha$ is equal to $\_\_\_\_$
Q22 Proof Computation of a Limit, Value, or Explicit Formula View
Let $A = \{ 1,2,3 \}$. The number of relations on $A$, containing $( 1,2 )$ and $( 2,3 )$, which are reflexive and transitive but not symmetric, is $\_\_\_\_$
Q23 Straight Lines & Coordinate Geometry Triangle Properties and Special Points View
Let $A ( 6,8 ) , B ( 10 \cos \alpha , - 10 \sin \alpha )$ and $C ( - 10 \sin \alpha , 10 \cos \alpha )$, be the vertices of a triangle. If $L ( a , 9 )$ and $G ( h , k )$ be its orthocenter and centroid respectively, then $( 5 a - 3 h + 6 k + 100 \sin 2 \alpha )$ is equal to $\_\_\_\_$
Q24 Differential equations First-Order Linear DE: General Solution View
Let $y = f ( x )$ be the solution of the differential equation $\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { x y } { x ^ { 2 } - 1 } = \frac { x ^ { 6 } + 4 x } { \sqrt { 1 - x ^ { 2 } } } , - 1 < x < 1$ such that $f ( 0 ) = 0$. If $6 \int _ { - 1 / 2 } ^ { 1 / 2 } f ( x ) \mathrm { d } x = 2 \pi - \alpha$ then $\alpha ^ { 2 }$ is equal to $\_\_\_\_$
Q25 Straight Lines & Coordinate Geometry Geometric Figure on Coordinate Plane View
Let the distance between two parallel lines be 5 units and a point $P$ lie between the lines at a unit distance from one of them. An equilateral triangle $PQR$ is formed such that $Q$ lies on one of the parallel lines, while $R$ lies on the other. Then $( QR ) ^ { 2 }$ is equal to $\_\_\_\_$