jee-main

Papers (191)
2026
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2025
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2024
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2023
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2022
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2021
session1_24feb_shift1 9 session1_24feb_shift2 4 session1_25feb_shift1 29 session1_25feb_shift2 29 session1_26feb_shift2 15 session2_16mar_shift1 29 session2_16mar_shift2 18 session2_17mar_shift1 21 session2_17mar_shift2 27 session2_18mar_shift1 18 session2_18mar_shift2 9 session3_20jul_shift1 29 session3_20jul_shift2 29 session3_22jul_shift1 9 session3_25jul_shift1 8 session3_25jul_shift2 14 session3_27jul_shift1 4 session3_27jul_shift2 7 session4_01sep_shift2 14 session4_26aug_shift1 5 session4_26aug_shift2 2 session4_27aug_shift1 3 session4_27aug_shift2 29 session4_31aug_shift1 28 session4_31aug_shift2 4
2020
session1_07jan_shift1 28 session1_07jan_shift2 20 session1_08jan_shift1 5 session1_08jan_shift2 11 session1_09jan_shift1 26 session1_09jan_shift2 16 session2_02sep_shift1 18 session2_02sep_shift2 16 session2_03sep_shift1 23 session2_03sep_shift2 8 session2_04sep_shift1 14 session2_04sep_shift2 27 session2_05sep_shift1 22 session2_05sep_shift2 29 session2_06sep_shift1 11 session2_06sep_shift2 10
2019
session1_09jan_shift1 6 session1_09jan_shift2 29 session1_10jan_shift1 29 session1_10jan_shift2 14 session1_11jan_shift1 6 session1_11jan_shift2 5 session1_12jan_shift1 10 session1_12jan_shift2 29 session2_08apr_shift1 29 session2_08apr_shift2 29 session2_09apr_shift1 29 session2_09apr_shift2 29 session2_10apr_shift1 2 session2_10apr_shift2 5 session2_12apr_shift1 3 session2_12apr_shift2 9
2018
08apr 30 15apr 28 15apr_shift1 28 15apr_shift2 6 16apr 19
2017
02apr 30 08apr 30 09apr 34
2016
03apr 28 09apr 29 10apr 30
2015
04apr 29 10apr 29 11apr 8
2014
06apr 28 09apr 28 11apr 4 12apr 5 19apr 29
2013
07apr 29 09apr 12 22apr 5 23apr 14 25apr 13
2012
07may 17 12may 21 19may 14 26may 17 offline 30
2011
jee-main_2011.pdf 18
2010
jee-main_2010.pdf 6
2009
jee-main_2009.pdf 2
2008
jee-main_2008.pdf 4
2007
jee-main_2007.pdf 38
2006
jee-main_2006.pdf 15
2005
jee-main_2005.pdf 25
2004
jee-main_2004.pdf 22
2003
jee-main_2003.pdf 8
2002
jee-main_2002.pdf 12
2025 session2_04apr_shift1

37 maths questions

Q1 Dimensional Analysis View
Q1. To find the spring constant $( k )$ of a spring experimentally, a student commits $2 \%$ positive error in the measurement of time and $1 \%$ negative error in measurement of mass. The percentage error in determining value of $k$ is :
(1) $5 \%$
(2) $1 \%$
(3) $3 \%$
(4) $4 \%$
Q2 Dimensional Analysis View
Q2.
LIST ILIST II
A.TorqueI.$\left[ M ^ { 1 } L ^ { 1 } T ^ { - 2 } A ^ { - 2 } \right]$
B.Magnetic fieldII.$\left[ L ^ { 2 } A ^ { 1 } \right]$
C.Magnetic momentIII.$\left[ M ^ { 1 } T ^ { - 2 } A ^ { - 1 } \right]$
D.Permeability of free spaceIV.$\left[ M ^ { 1 } L ^ { 2 } T ^ { - 2 } \right]$

Choose the correct answer from the options given below:
(1) A-III, B-I, C-II, D-IV
(2) A-IV, B-II, C-III, D-I
(3) A-IV, B-III, C-II, D-I
(4) A-I, B-III, C-II, D-IV
Q3 Constant acceleration (SUVAT) Acceleration then deceleration (two-phase motion) View
Q3. A train starting from rest first accelerates uniformly up to a speed of $80 \mathrm {~km} / \mathrm { h }$ for time $t$, then it moves with a constant speed for time $3 t$. The average speed of the train for this duration of journey will be (in $\mathrm { km } / \mathrm { h }$ ) :
(1) 40
(2) 80
(3) 30
(4) 70
Q4 Newton's laws and connected particles Atwood machine and pulley systems View
Q4. A light string passing over a smooth light pulley connects two blocks of masses $m _ { 1 }$ and $m _ { 2 }$ (where $m _ { 2 } > m _ { 1 }$ ). If the acceleration of the system is $\frac { g } { \sqrt { 2 } }$, then the ratio of the masses $\frac { m _ { 1 } } { m _ { 2 } }$ is:
(1) $\frac { 1 + \sqrt { 5 } } { \sqrt { 5 } - 1 }$
(2) $\frac { \sqrt { 2 } - 1 } { \sqrt { 2 } + 1 }$
(3) $\frac { 1 + \sqrt { 5 } } { \sqrt { 2 } - 1 }$
(4) $\frac { \sqrt { 3 } + 1 } { \sqrt { 2 } - 1 }$
Q5 Work done and energy Work-energy theorem: finding speed or kinetic energy from net work View
Q5. A bullet of mass 50 g is fired with a speed $100 \mathrm {~m} / \mathrm { s }$ on a plywood and emerges with $40 \mathrm {~m} / \mathrm { s }$. The percentage loss of kinetic energy is :
(1) $84 \%$
(2) $16 \%$
(3) $32 \%$
(4) $44 \%$
Q6 Work done and energy Kinetic energy and momentum relationships View
Q6. Four particles $A , B , C , D$ of mass $\frac { m } { 2 } , m , 2 m , 4 m$, have same momentum, respectively. The particle with maximum kinetic energy is :
(1) $B$
(2) $A$
(3) $D$
(4) $C$
Q7 Work done and energy Work done by gravity in specific scenarios View
Q7. To project a body of mass $m$ from earth's surface to infinity, the required kinetic energy is (assume, the radius of earth is $R _ { E } , g =$ acceleration due to gravity on the surface of earth):
(1) $2 m g R _ { E }$
(2) $4 m g R _ { E }$
(3) $m g R _ { E }$
(4) $1 / 2 m g R _ { E }$
Q8 Forces, equilibrium and resultants View
Q8. A small ball of mass $m$ and density $\rho$ is dropped in a viscous liquid of density $\rho _ { 0 }$. After sometime, the ball falls with constant velocity. The viscous force on the ball is :
(1) $m g \left( 1 - \rho \rho _ { 0 } \right)$
(2) $m g \left( 1 + \frac { \rho } { \rho _ { 0 } } \right)$
(3) $m g \left( \frac { \rho _ { 0 } } { \rho } - 1 \right)$
(4) $m g \left( 1 - \frac { \rho _ { 0 } } { \rho } \right) _ { \nabla }$
Q21 Vectors Introduction & 2D Perpendicularity or Parallel Condition View
Q21. For three vectors $\vec { A } = ( - x \hat { i } - 6 \hat { j } - 2 \hat { k } ) , \vec { B } = ( - \hat { i } + 4 \hat { j } + 3 \hat { k } )$ and $\vec { C } = ( - 8 \hat { i } - \hat { j } + 3 \hat { k } )$, if $\vec { A } \cdot ( \vec { B } \times \vec { C } ) = 0$, then value of $x$ is $\_\_\_\_$
Q61 Sequences and series, recurrence and convergence Multiple-choice on sequence properties View
Q61. Let $\alpha , \beta$ be the distinct roots of the equation $x ^ { 2 } - \left( t ^ { 2 } - 5 t + 6 \right) x + 1 = 0 , t \in \mathbb { R }$ and $a _ { n } = \alpha ^ { n } + \beta ^ { n }$. Then the minimum value of $\frac { a _ { 2023 } + a _ { 2025 } } { a _ { 2024 } }$ is
(1) $- 1 / 4$
(2) $- 1 / 4$
(3) $- 1 / 2$
(4) $1 / 4$
Q62 Combinations & Selection Geometric Combinatorics View
Q62. The number of triangles whose vertices are at the vertices of a regular octagon but none of whose sides is a side of the octagon is
(1) 48
(2) 56
(3) 24
(4) 16
Q63 Principle of Inclusion/Exclusion View
Q63. Let $A = \{ n \in [ 100,700 ] \cap \mathbb { N } : n$ is neither a multiple of 3 nor a multiple of $4 \}$. Then the number of elements in $A$ is
(1) 290
(2) 280
(3) 300
(4) 310
Q64 Straight Lines & Coordinate Geometry Line Equation and Parametric Representation View
Q64. Let a variable line of slope $m > 0$ passing through the point $( 4 , - 9 )$ intersect the coordinate axes at the points $A$ and $B$. The minimum value of the sum of the distances of $A$ and $B$ from the origin is
(1) 30
(2) 25
(3) 15
(4) 10
Q65 Vectors 3D & Lines Vector Algebra and Triple Product Computation View
Q65. If $A ( 3,1 , - 1 ) , B \left( \frac { 5 } { 3 } , \frac { 7 } { 3 } , \frac { 1 } { 3 } \right) , C ( 2,2,1 )$ and $D \left( \frac { 10 } { 3 } , \frac { 2 } { 3 } , \frac { - 1 } { 3 } \right)$ are the vertices of a quadrilateral $A B C D$, then its area is
(1) $\frac { 2 \sqrt { 2 } } { 3 }$
(2) $\frac { 5 \sqrt { 2 } } { 3 }$
(3) $2 \sqrt { 2 }$
(4) $\frac { 4 \sqrt { 2 } } { 3 }$
Q66 Circles Inscribed/Circumscribed Circle Computations View
Q66. A circle is inscribed in an equilateral triangle of side of length 12 . If the area and perimeter of any square inscribed in this circle are $m$ and $n$, respectively, then $m + n ^ { 2 }$ is equal to
(1) 408
(2) 414
(3) 396
(4) 312
Q67 Circles Circle Equation Derivation View
Q67. Let $C$ be the circle of minimum area touching the parabola $y = 6 - x ^ { 2 }$ and the lines $y = \sqrt { 3 } | x |$. Then, which one of the following points lies on the circle $C$ ?
(1) $( 1,2 )$
(2) $( 1,1 )$
(3) $( 2,2 )$
(4) $( 2,4 )$
Q68 Chain Rule Limit Involving Derivative Definition of Composed Functions View
Q68. Let $f : ( - \infty , \infty ) - \{ 0 \} \rightarrow \mathbb { R }$ be a differentiable function such that $f ^ { \prime } ( 1 ) = \lim _ { a \rightarrow \infty } a ^ { 2 } f \left( \frac { 1 } { a } \right)$. Then $\lim _ { a \rightarrow \infty } \frac { a ( a + 1 ) } { 2 } \tan ^ { - 1 } \left( \frac { 1 } { a } \right) + a ^ { 2 } - 2 \log _ { e } a$ is equal to
(1) $\frac { 3 } { 2 } + \frac { \pi } { 4 }$
(2) $\frac { 3 } { 4 } + \frac { \pi } { 8 }$
(3) $\frac { 3 } { 8 } + \frac { \pi } { 4 }$
(4) $\frac { 5 } { 2 } + \frac { \pi } { 8 }$
Q69 Measures of Location and Spread View
Q69. The mean and standard deviation of 20 observations are found to be 10 and 2 . respectively. On rechecking, it was found that an observation by mistake was taken 8 instead of 12 . The correct standard deviation is
(1) 1.8
(2) 1.94
(3) $\sqrt { 3.96 }$
(4) $\sqrt { 3.86 }$
Q70 Probability Definitions Combinatorial Counting (Non-Probability) View
Q70. Let the relations $R _ { 1 }$ and $R _ { 2 }$ on the set $X = \{ 1,2,3 , \ldots , 20 \}$ be given by $R _ { 1 } = \{ ( x , y ) : 2 x - 3 y = 2 \}$ and $R _ { 2 } = \{ ( x , y ) : - 5 x + 4 y = 0 \}$. If $M$ and $N$ be the minimum number of elements required to be added in $R _ { 1 }$ and $R _ { 2 }$, respectively, in order to make the relations symmetric, then $M + N$ equals
(1) 12
(2) 16
(3) 8
(4) 10
Q71. For $\alpha , \beta \in \mathbb { R }$ and a natural number $n$, let $A _ { r } = \left| \begin{array} { c c c } r & 1 & \frac { n ^ { 2 } } { 2 } + \alpha \\ 2 r & 2 & n ^ { 2 } - \beta \\ 3 r - 2 & 3 & \frac { n ( 3 n - 1 ) } { 2 } \end{array} \right|$. Then
(1) 0
(2) $4 \alpha + 2 \beta$
(3) $2 \alpha + 4 \beta$
(4) $2 n$
Q72 Composite & Inverse Functions Injectivity, Surjectivity, or Bijectivity Classification View
Q72. The function $\mathrm { f } : \mathrm { R } - > \mathrm { R } , f ( x ) = \frac { x ^ { 2 } + 2 x - 15 } { x ^ { 2 } - 4 x + 9 } , x \in \mathbb { R }$ is
(1) one-one but not onto.
(2) both one-one and onto.
(3) onto but not one-one.
(4) neither one-one nor onto.
Q73 Chain Rule Piecewise Function Differentiability Analysis View
Q73. If $f ( x ) = \left\{ \begin{array} { l } x ^ { 3 } \sin \left( \frac { 1 } { x } \right) , x \neq 0 \\ 0 \quad , x = 0 \end{array} \right.$ then
(1) $f ^ { \prime \prime } \left( \frac { 2 } { \pi } \right) = \frac { 24 - \pi ^ { 2 } } { 2 \pi }$
(2) $f ^ { \prime \prime } \left( \frac { 2 } { \pi } \right) = \frac { 12 - \pi ^ { 2 } } { 2 \pi }$
(3) $f ^ { \prime \prime } ( 0 ) = 1$
(4) $f ^ { \prime \prime } ( 0 ) = 0$
Q74 Stationary points and optimisation Determine intervals of increase/decrease or monotonicity conditions View
Q74. The interval in which the function $f ( x ) = x ^ { x } , x > 0$, is strictly increasing is
(1) $\left( 0 , \frac { 1 } { e } \right]$
(2) $( 0 , \infty )$
(3) $\left. \left[ \frac { 1 } { e } , \infty \right) \right] _ { V }$
(4) $\left[ \frac { 1 } { e ^ { 2 } } , 1 \right)$
Q75 Integration by Substitution Substitution to Evaluate a Definite Integral (Numerical Answer) View
Q75. $\int _ { 0 } ^ { \pi / 4 } \frac { \cos ^ { 2 } x \sin ^ { 2 } x } { \left( \cos ^ { 3 } x + \sin ^ { 3 } x \right) ^ { 2 } } d x$ is equal to
(1) $1 / 6$
(2) $1 / 3$
(3) $1 / 12$
(4) $1 / 9$
Q76 Areas by integration View
Q76. Let the area of the region enclosed by the curves $y = 3 x , 2 y = 27 - 3 x$ and $y = 3 x - x \sqrt { x }$ be $A$. Then $10 A$ is equal to
(1) 172
(2) 162
(3) 154
(4) 184
Q77 First order differential equations (integrating factor) View
Q77. Let $y = y ( x )$ be the solution of the differential equation $\left( 1 + x ^ { 2 } \right) \frac { d y } { d x } + y = e ^ { \tan ^ { - 1 } x } , y ( 1 ) = 0$. Then $y ( 0 )$ is
(1) $\frac { 1 } { 2 } \left( e ^ { \pi / 2 } - 1 \right)$
(2) $\frac { 1 } { 2 } \left( 1 - e ^ { \pi / 2 } \right)$
(3) $\frac { 1 } { 4 } \left( 1 - e ^ { \pi / 2 } \right)$
(4) $\frac { 1 } { 4 } \left( e ^ { \pi / 2 } - 1 \right)$
Q78 First order differential equations (integrating factor) View
Q78. Let $y = y ( x )$ be the solution of the differential equation $\left( 2 x \log _ { e } x \right) \frac { d y } { d x } + 2 y = \frac { 3 } { x } \log _ { e } x , x > 0$ and $y \left( e ^ { - 1 } \right) = 0$. Then, $y ( e )$ is equal to
(1) $- \frac { 3 } { \mathrm { e } }$
(2) $- \frac { 3 } { 2 e }$
(3) $- \frac { 2 } { 3 e }$
(4) $- \frac { 2 } { \mathrm { e } }$
Q79 Vectors 3D & Lines Shortest Distance Between Two Lines View
Q79. The shortest distance between the lines $\frac { x - 3 } { 2 } = \frac { y + 15 } { - 7 } = \frac { z - 9 } { 5 }$ and $\frac { x + 1 } { 2 } = \frac { y - 1 } { 1 } = \frac { z - 9 } { - 3 }$ is
(1) $8 \sqrt { 3 }$
(2) $4 \sqrt { 3 }$
(3) $5 \sqrt { 3 }$
(4) $6 \sqrt { 3 }$
Q80 Conditional Probability Bayes' Theorem with Production/Source Identification View
Q80. A company has two plants $A$ and $B$ to manufacture motorcycles. $60 \%$ motorcycles are manufactured at plant $A$ and the remaining are manufactured at plant $B .80 \%$ of the motorcycles manufactured at plant $A$ are rated of the standard quality, while $90 \%$ of the motorcycles manufactured at plant $B$ are rated of the standard quality. A motorcycle picked up randomly from the total production is found to be of the standard quality. If $p$ is the probability that it was manufactured at plant $B$, then $126 p$ is
(1) 54
(2) 66
(3) 64
(4) 56
Q81.Let $x _ { 1 } , x _ { 2 } , x _ { 3 } , x _ { 4 }$ be the solution of the equation $4 x ^ { 4 } + 8 x ^ { 3 } - 17 x ^ { 2 } - 12 x + 9 = 0$ and $\left( 4 + x _ { 1 } ^ { 2 } \right) \left( 4 + x _ { 2 } ^ { 2 } \right) \left( 4 + x _ { 3 } ^ { 2 } \right) \left( 4 + x _ { 4 } ^ { 2 } \right) = \frac { 125 } { 16 } m$. Then the value of $m$ is
Q82 Sequences and series, recurrence and convergence Summation of sequence terms View
Q82. Let the first term of a series be $T _ { 1 } = 6$ and its $r ^ { \text {th } }$ term $T _ { r } = 3 T _ { r - 1 } + 6 ^ { r } , r = 2,3 , \quad n$. If the sum of the first $n$ terms of this series is $\frac { 1 } { 5 } \left( n ^ { 2 } - 12 n + 39 \right) \left( 4 \cdot 6 ^ { n } - 5 \cdot 3 ^ { n } + 1 \right)$, then $n$ is equal to $\_\_\_\_$
Q83 Binomial Theorem (positive integer n) Determine Parameters from Conditions on Coefficients or Terms View
Q83. If the second, third and fourth terms in the expansion of $( x + y ) ^ { n }$ are 135,30 and $\frac { 10 } { 3 }$, respectively, then $6 \left( n ^ { 3 } + x ^ { 2 } + y \right)$ is equal to $\_\_\_\_$
Q84 Conic sections Tangent and Normal Line Problems View
Q84. Let a conic $C$ pass through the point $( 4 , - 2 )$ and $P ( x , y ) , x \geq 3$, be any point on $C$. Let the slope of the line touching the conic $C$ only at a single point $P$ be half the slope of the line joining the points $P$ and $( 3 , - 5 )$. If the focal distance of the point $( 7,1 )$ on $C$ is $d$, then $12 d$ equals $\_\_\_\_$
Q85 Conic sections Tangent and Normal Line Problems View
Q85. Let $L _ { 1 } , L _ { 2 }$ be the lines passing through the point $P ( 0,1 )$ and touching the parabola $9 x ^ { 2 } + 12 x + 18 y - 14 = 0$. Let $Q$ and $R$ be the points on the lines $L _ { 1 }$ and $L _ { 2 }$ such that the $\triangle P Q R$ is an isosceles triangle with base $Q R$. If the slopes of the lines $Q R$ are $m _ { 1 }$ and $m _ { 2 }$, then $16 \left( m _ { 1 } ^ { 2 } + m _ { 2 } ^ { 2 } \right)$ is equal to $\_\_\_\_$
Q86 Matrices Linear System and Inverse Existence View
Q86. Let $\alpha \beta \gamma = 45 ; \alpha , \beta , \gamma \in \mathbb { R }$. If $x ( \alpha , 1,2 ) + y ( 1 , \beta , 2 ) + z ( 2,3 , \gamma ) = ( 0,0,0 )$ for some $x , y , z \in \mathbb { R } , x y z \neq 0$, then $6 \alpha + 4 \beta + \gamma$ is equal to $\_\_\_\_$
Q87 Standard trigonometric equations Inverse trigonometric equation View
Q87. For $n \in \mathrm {~N}$, if $\cot ^ { - 1 } 3 + \cot ^ { - 1 } 4 + \cot ^ { - 1 } 5 + \cot ^ { - 1 } n = \frac { \pi } { 4 }$, then $n$ is equal to $\_\_\_\_$
Q88 Standard Integrals and Reverse Chain Rule Integral Equation to Determine a Function Value View
Q88. Let $r _ { k } = \frac { \int _ { 0 } ^ { 1 } \left( 1 - x ^ { 7 } \right) ^ { k } d x } { \int _ { 0 } ^ { 1 } \left( 1 - x ^ { 7 } \right) ^ { k + 1 } d x } , k \in \mathbb { N }$. Then the value of $\sum _ { k = 1 } ^ { 10 } \frac { 1 } { 7 \left( r _ { k } - 1 \right) }$ is equal to $\_\_\_\_$
Q89 Vectors 3D & Lines Vector Algebra and Triple Product Computation View
Q89. Let $\vec { a } = 2 \hat { i } - 3 \hat { j } + 4 \hat { k } , \vec { b } = 3 \hat { i } + 4 \hat { j } - 5 \hat { k }$ and a vector $\vec { c }$ be such that $\vec { a } \times ( \vec { b } + \vec { c } ) + \vec { b } \times \vec { c } = \hat { i } + 8 \hat { j } + 13 \hat { k }$. If $\vec { a } \cdot \vec { c } = 13$, then $( 24 - \vec { b } \cdot \vec { c } )$ is equal to $\_\_\_\_$
Q90 Vectors 3D & Lines Distance from a Point to a Line (Show/Compute) View
Q90. Let $P$ be the point $( 10 , - 2 , - 1 )$ and $Q$ be the foot of the perpendicular drawn from the point $R ( 1,7,6 )$ on the line passing through the points $( 2 , - 5,11 )$ and $( - 6,7 , - 5 )$. Then the length of the line segment $P Q$ is equal to
ANSWER KEYS

\begin{tabular}{|l|l|l|l|l|l|l|l|} \hline 1. (1) & 2. (3) & 3. (4) & 4. (2) & 5. (1) & 6. (2) & 7. (3) & 8. (4) \hline 9. (2) & 10. (3) & 11. (2) & 12. (4) & 13. (1) & 14. (3) & 15. (1) & 16. (3) \hline 17. (3) & 18. (1) & 19. (1) & 20. (3) & 21. (4) & 22. (13) & 23. (1) & 24. (12) \hline 25. (2) & 26. (16) & 27. (5) & 28. (250) & 29. (60) & 30. (16) & 31. (4) & 32. (2) \hline 33. (4) & 34. (2) & 35. (3) & 36. (2) & 37. (2) & 38. (2) & 39. (2) & 40. (1) \hline 41. (4) & 42. (3) & 43. (3) & 44. (2) & 45. (4) & 46. (1) & 47. (1) & 48. (4) \hline 49. (1) & 50. (2) & 51. (661) & 52. (5) & 53. (274) & 54. (2) & 55. (76) & 56. (3) \hline 57. (877) & 58. (6) & 59. (8) & 60. (20) & 61. (2) & 62. (4) & 63. (3) & 64. (2) \hline 65. (4) & 66. (1) & 67. (4) & 68. (4) & 69. (3) & 70. (4) & 71. (2) & 72. (4) \hline 73. (1) & 74. (3) & 75. (1) & 76. (2) & 77. (2) & 78. (1) & 79. (2) & 80. (1) \hline