jee-main

Papers (169)
2025
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2024
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2023
session1_01feb_shift1 24 session1_01feb_shift2 3 session1_24jan_shift1 13 session1_24jan_shift2 12 session1_25jan_shift1 28 session1_25jan_shift2 27 session1_29jan_shift1 29 session1_29jan_shift2 28 session1_30jan_shift1 2 session1_30jan_shift2 29 session1_31jan_shift1 28 session1_31jan_shift2 17 session2_06apr_shift1 5 session2_06apr_shift2 17 session2_08apr_shift1 29 session2_08apr_shift2 14 session2_10apr_shift1 29 session2_10apr_shift2 15 session2_11apr_shift1 5 session2_11apr_shift2 4 session2_12apr_shift1 26 session2_13apr_shift1 25 session2_13apr_shift2 20 session2_15apr_shift1 20
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
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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
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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_28jan_shift2

25 maths questions

Q1 Matrices Matrix Power Computation and Application View
Let $A = \left[ \begin{array} { c c } \frac { 1 } { \sqrt { 2 } } & - 2 \\ 0 & 1 \end{array} \right]$ and $P = \left[ \begin{array} { c c } \cos \theta & - \sin \theta \\ \sin \theta & \cos \theta \end{array} \right] , \theta > 0$. If $\mathrm { B } = \mathrm { PAP } ^ { \mathrm { T } } , \mathrm { C } = \mathrm { P } ^ { \mathrm { T } } \mathrm { B } ^ { 10 } \mathrm { P }$ and the sum of the diagonal elements of $C$ is $\frac { \mathrm { m } } { \mathrm { n } }$, where $\operatorname { gcd } ( \mathrm { m } , \mathrm { n } ) = 1$, then $\mathrm { m } + \mathrm { n }$ is :
(1) 127
(2) 258
(3) 65
(4) 2049
Q2 Vectors Introduction & 2D Magnitude of Vector Expression View
If the components of $\overrightarrow { \mathrm { a } } = \alpha \hat { i } + \beta \hat { j } + \gamma \hat { k }$ along and perpendicular to $\overrightarrow { \mathrm { b } } = 3 \hat { i } + \hat { j } - \hat { k }$ respectively, are $\frac { 16 } { 11 } ( 3 \hat { i } + \hat { j } - \hat { k } )$ and $\frac { 1 } { 11 } ( - 4 \hat { i } - 5 \hat { j } - 17 \hat { k } )$, then $\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }$ is equal to :
(1) 26
(2) 18
(3) 23
(4) 16
Q3 Straight Lines & Coordinate Geometry Point-to-Line Distance Computation View
Let $\mathrm { A } , \mathrm { B } , \mathrm { C }$ be three points in $xy$-plane, whose position vector are given by $\sqrt { 3 } \hat { i } + \hat { j } , \hat { i } + \sqrt { 3 } \hat { j }$ and $\mathrm { a } \hat { i } + ( 1 - \mathrm { a } ) \hat { j }$ respectively with respect to the origin O. If the distance of the point C from the line bisecting the angle between the vectors $\overrightarrow { \mathrm { OA } }$ and $\overrightarrow { \mathrm { OB } }$ is $\frac { 9 } { \sqrt { 2 } }$, then the sum of all the possible values of $a$ is :
(1) 2
(2) $9/2$
(3) 1
(4) 0
Q4 Binomial Theorem (positive integer n) Count Integral or Rational Terms in a Binomial Expansion View
Let the coefficients of three consecutive terms $T _ { r } , T _ { r + 1 }$ and $T _ { r + 2 }$ in the binomial expansion of $( a + b ) ^ { 12 }$ be in a G.P. and let $p$ be the number of all possible values of $r$. Let $q$ be the sum of all rational terms in the binomial expansion of $( \sqrt [ 4 ] { 3 } + \sqrt [ 3 ] { 4 } ) ^ { 12 }$. Then $\mathrm { p } + \mathrm { q }$ is equal to :
(1) 283
(2) 287
(3) 295
(4) 299
Q5 Function Transformations View
Let $[ x ]$ denote the greatest integer less than or equal to $x$. Then the domain of $f ( x ) = \sec ^ { - 1 } ( 2 [ x ] + 1 )$ is :
(1) $( - \infty , - 1 ] \cup [ 0 , \infty )$
(2) $( - \infty , - 1 ] \cup [ 1 , \infty )$
(3) $( - \infty , \infty )$
(4) $( - \infty , \infty ) - \{ 0 \}$
Q6 Permutations & Arrangements Probability via Permutation Counting View
Let S be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set S, one word is selected at random. The probability that the selected word will NOT have vowels in alphabetical order is :
(1) $\frac { 1 } { 2 }$
(2) $\frac { 1 } { 4 }$
(3) $\frac { 2 } { 3 }$
(4) $\frac { 1 } { 3 }$
Q7 Addition & Double Angle Formulae Telescoping Sum of Trigonometric Terms View
If $\sum _ { r = 1 } ^ { 13 } \left\{ \frac { 1 } { \sin \left( \frac { \pi } { 4 } + ( r - 1 ) \frac { \pi } { 6 } \right) \sin \left( \frac { \pi } { 4 } + \frac { r \pi } { 6 } \right) } \right\} = a \sqrt { 3 } + b , a , b \in \mathbf { Z }$, then $a ^ { 2 } + b ^ { 2 }$ is equal to :
(1) 10
(2) 4
(3) 2
(4) 8
Q8 Indefinite & Definite Integrals Finding a Function from an Integral Equation View
Let $f$ be a real valued continuous function defined on the positive real axis such that $g ( x ) = \int _ { 0 } ^ { x } \mathrm { t } f ( \mathrm { t } ) \mathrm { dt }$. If $\mathrm { g } \left( x ^ { 3 } \right) = x ^ { 6 } + x ^ { 7 }$, then value of $\sum _ { r = 1 } ^ { 15 } f \left( \mathrm { r } ^ { 3 } \right)$ is :
(1) 270
(2) 340
(3) 320
(4) 310
Q9 Composite & Inverse Functions Determine Domain or Range of a Composite Function View
Let $f : [ 0,3 ] \rightarrow \mathrm { A }$ be defined by $f ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 36 x + 7$ and $g : [ 0 , \infty ) \rightarrow B$ be defined by $\mathrm { g } ( x ) = \frac { x ^ { 2025 } } { x ^ { 2025 } + 1 }$. If both the functions are onto and $\mathrm { S } = \{ x \in \mathbf { Z } : x \in \mathrm {~A}$ or $x \in \mathrm {~B} \}$, then $\mathrm { n } ( \mathrm { S } )$ is equal to :
(1) 29
(2) 30
(3) 31
(4) 36
Q10 Conditional Probability Bayes' Theorem with Production/Source Identification View
Bag $B _ { 1 }$ contains 6 white and 4 blue balls, Bag $B _ { 2 }$ contains 4 white and 6 blue balls, and Bag $B _ { 3 }$ contains 5 white and 5 blue balls. One of the bags is selected at random and a ball is drawn from it. If the ball is white, then the probability, that the ball is drawn from Bag $B _ { 2 }$, is :
(1) $\frac { 4 } { 15 }$
(2) $\frac { 1 } { 3 }$
(3) $\frac { 2 } { 5 }$
(4) $\frac { 2 } { 3 }$
Q11 Integration by Parts Definite Integral Evaluation by Parts View
Let $\mathrm { f } : \mathbf { R } \rightarrow \mathbf { R }$ be a twice differentiable function such that $f ( 2 ) = 1$. If $\mathrm { F } ( x ) = x f ( x )$ for all $x \in \mathbf { R }$, $\int _ { 0 } ^ { 2 } x \mathrm {~F} ^ { \prime } ( x ) \mathrm { d } x = 6$ and $\int _ { 0 } ^ { 2 } x ^ { 2 } \mathrm {~F} ^ { \prime \prime } ( x ) \mathrm { d } x = 40$, then $\mathrm { F } ^ { \prime } ( 2 ) + \int _ { 0 } ^ { 2 } \mathrm {~F} ( x ) \mathrm { d } x$ is equal to :
(1) 11
(2) 13
(3) 15
(4) 9
Q12 Sequences and Series Evaluation of a Finite or Infinite Sum View
For positive integers $n$, if $4 a _ { n } = \left( n ^ { 2 } + 5 n + 6 \right)$ and $S _ { n } = \sum _ { k = 1 } ^ { n } \left( \frac { 1 } { a _ { k } } \right)$, then the value of $507 S _ { 2025 }$ is :
(1) 540
(2) 675
(3) 1350
(4) 135
Q13 Composite & Inverse Functions Recover a Function from a Composition or Functional Equation View
Let $f : \mathbf { R } - \{ 0 \} \rightarrow ( - \infty , 1 )$ be a polynomial of degree 2, satisfying $f ( x ) f \left( \frac { 1 } { x } \right) = f ( x ) + f \left( \frac { 1 } { x } \right)$. If $f ( K ) = - 2 K$, then the sum of squares of all possible values of $K$ is :
(1) 7
(2) 6
(3) 1
(4) 9
Q14 Conic sections Circle-Conic Interaction with Tangency or Intersection View
If $A$ and $B$ are the points of intersection of the circle $x ^ { 2 } + y ^ { 2 } - 8 x = 0$ and the hyperbola $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 4 } = 1$ and a point P moves on the line $2 x - 3 y + 4 = 0$, then the centroid of $\triangle \mathrm { PAB }$ lies on the line :
(1) $x + 9 y = 36$
(2) $4 x - 9 y = 12$
(3) $6 x - 9 y = 20$
(4) $9 x - 9 y = 32$
Q15 Integration by Substitution Substitution to Compute an Indefinite Integral with Initial Condition View
If $f ( x ) = \int \frac { 1 } { x ^ { 1 / 4 } \left( 1 + x ^ { 1 / 4 } \right) } \mathrm { d } x , f ( 0 ) = - 6$, then $f ( 1 )$ is equal to :
(1) $4 \left( \log _ { e } 2 - 2 \right)$
(2) $2 - \log _ { e ^ { 2 } } 2$
(3) $\log _ { e } 2 + 2$
(4) $4 \left( \log _ { e } 2 + 2 \right)$
Q16 Areas Between Curves Compute Area Directly (Numerical Answer) View
The area of the region bounded by the curves $x \left( 1 + y ^ { 2 } \right) = 1$ and $y ^ { 2 } = 2 x$ is:
(1) $2 \left( \frac { \pi } { 2 } - \frac { 1 } { 3 } \right)$
(2) $\frac { \pi } { 2 } - \frac { 1 } { 3 }$
(3) $\frac { \pi } { 4 } - \frac { 1 } { 3 }$
(4) $\frac { 1 } { 2 } \left( \frac { \pi } { 2 } - \frac { 1 } { 3 } \right)$
Q17 Vectors 3D & Lines Distance from a Point to a Line (Show/Compute) View
The square of the distance of the point $\left( \frac { 15 } { 7 } , \frac { 32 } { 7 } , 7 \right)$ from the line $\frac { x + 1 } { 3 } = \frac { y + 3 } { 5 } = \frac { z + 5 } { 7 }$ in the direction of the vector $\hat { i } + 4 \hat { j } + 7 \hat { k }$ is :
(1) 54
(2) 44
(3) 41
(4) 66
Q18 Circles Chord Length and Chord Properties View
If the midpoint of a chord of the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$ is $( \sqrt { 2 } , 4 / 3 )$, and the length of the chord is $\frac { 2 \sqrt { \alpha } } { 3 }$, then $\alpha$ is :
(1) 20
(2) 22
(3) 18
(4) 26
Q19 Complex Numbers Arithmetic Systems of Equations via Real and Imaginary Part Matching View
If $\alpha + i \beta$ and $\gamma + i \delta$ are the roots of $x ^ { 2 } - ( 3 - 2 i ) x - ( 2 i - 2 ) = 0 , i = \sqrt { - 1 }$, then $\alpha \gamma + \beta \delta$ is equal to :
(1) $-2$
(2) 6
(3) $-6$
(4) 2
Q20 Straight Lines & Coordinate Geometry Slope and Angle Between Lines View
Two equal sides of an isosceles triangle are along $- x + 2 y = 4$ and $x + y = 4$. If m is the slope of its third side, then the sum, of all possible distinct values of $m$, is :
(1) $- 2 \sqrt { 10 }$
(2) 12
(3) 6
(4) $-6$
Q21 Conic sections Triangle or Quadrilateral Area and Perimeter with Foci View
Let A and B be the two points of intersection of the line $y + 5 = 0$ and the mirror image of the parabola $y ^ { 2 } = 4 x$ with respect to the line $x + y + 4 = 0$. If d denotes the distance between A and B, and a denotes the area of $\triangle S A B$, where $S$ is the focus of the parabola $y ^ { 2 } = 4 x$, then the value of $( a + d )$ is
Q22 Permutations & Arrangements Forming Numbers with Digit Constraints View
The number of natural numbers, between 212 and 999, such that the sum of their digits is 15, is
Q23 First order differential equations (integrating factor) View
If $y = y ( x )$ is the solution of the differential equation, $\sqrt { 4 - x ^ { 2 } } \frac { \mathrm {~d} y } { \mathrm {~d} x } = \left( \left( \sin ^ { - 1 } \left( \frac { x } { 2 } \right) \right) ^ { 2 } - y \right) \sin ^ { - 1 } \left( \frac { x } { 2 } \right) , - 2 \leq x \leq 2 , y ( 2 ) = \frac { \pi ^ { 2 } - 8 } { 4 }$, then $y ^ { 2 } ( 0 )$ is equal to
Q24 Arithmetic Sequences and Series Counting or Combinatorial Problems on APs View
The interior angles of a polygon with $n$ sides, are in an A.P. with common difference $6 ^ { \circ }$. If the largest interior angle of the polygon is $219 ^ { \circ }$, then n is equal to
Q25 Laws of Logarithms Simplify or Evaluate a Logarithmic Expression View
Let $f ( x ) = \lim _ { \mathrm { n } \rightarrow \infty } \sum _ { \mathrm { r } = 0 } ^ { \mathrm { n } } \left( \frac { \tan \left( x / 2 ^ { r + 1 } \right) + \tan ^ { 3 } \left( x / 2 ^ { r + 1 } \right) } { 1 - \tan ^ { 2 } \left( x / 2 ^ { r + 1 } \right) } \right)$. Then $\lim _ { x \rightarrow 0 } \frac { \mathrm { e } ^ { x } - \mathrm { e } ^ { f ( x ) } } { ( x - f ( x ) ) }$ is equal to