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
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2020
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2019
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2018
08apr 30 15apr 28 15apr_shift1 28 15apr_shift2 6 16apr 19
2017
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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_02apr_shift2

36 maths questions

Q1 Dimensional Analysis View
Q1. Applying the principle of homogeneity of dimensions, determine which one is correct, where $T$ is time period, $G$ is gravitational constant, $M$ is mass, $r$ is radius of orbit.
(1) $T ^ { 2 } = \frac { 4 \pi ^ { 2 } r ^ { 2 } } { G M }$
(2) $T ^ { 2 } = \frac { 4 \pi ^ { 2 } r } { G M ^ { 2 } }$
(3) $T ^ { 2 } = \frac { 4 \pi ^ { 2 } r ^ { 3 } } { G M }$
(4) $T ^ { 2 } = 4 \pi ^ { 2 } r ^ { 3 }$
Q2 Vectors Introduction & 2D Vector Word Problem / Physical Application View
Q2. A cyclist starts from the point $P$ of a circular ground of radius 2 km and travels along its circumference to the [Figure] point S . The displacement of a cyclist is:
(1) $\sqrt { 8 } \mathrm {~km}$
(2) 8 km
(3) 6 km
(4) 4 km
Q3 Friction Equilibrium on Inclined Plane (Static) View
Q3. A 2 kg brick begins to slide over a surface which is inclined at an angle of $45 ^ { \circ }$ with respect to horizontal axis. The co-efficient of static friction between their surfaces is:
(1) 1.7
(2) $\frac { 1 } { \sqrt { 3 } }$
(3) 0.5
(4) 1
Q4 Work done and energy Conservation of energy on frictionless tracks and pendulums View
Q4. A body of $m \mathrm {~kg}$ slides from rest along the curve of vertical circle from point $A$ to $B$ in friction less path. The [Figure] velocity of the body at $B$ is: (given, $R = 14 \mathrm {~m} , g = 10 \mathrm {~m} / \mathrm { s } ^ { 2 }$ and $\sqrt { 2 } = 1.4$ )
(1) $16.7 \mathrm {~m} / \mathrm { s }$
(2) $19.8 \mathrm {~m} / \mathrm { s }$
(3) $10.6 \mathrm {~m} / \mathrm { s }$
(4) $21.9 \mathrm {~m} / \mathrm { s }$
Q21 Constant acceleration (SUVAT) Braking and stopping distance View
Q21. A bus moving along a straight highway with speed of $72 \mathrm {~km} / \mathrm { h }$ is brought to halt within $4 s$ after applying the brakes. The distance travelled by the bus during this time (Assume the retardation is uniform) is $\_\_\_\_$ m.
Q22 Centre of Mass 1 View
Q22. In a system two particles of masses $m _ { 1 } = 3 \mathrm {~kg}$ and $m _ { 2 } = 2 \mathrm {~kg}$ are placed at certain distance from each other. The particle of mass $m _ { 1 }$ is moved towards the center of mass of the system through a distance 2 cm . In order to keep the center of mass of the system at the original position, the particle of mass $m _ { 2 }$ should move towards the center of mass by the distance $\_\_\_\_$ cm.
Q24 Simple Harmonic Motion View
Q24. The displacement of a particle executing SHM is given by $x = 10 \sin \left( w t + \frac { \pi } { 3 } \right) m$. The time period of motion is 3.14 s . The velocity of the particle at $t = 0$ is $\_\_\_\_$ $\mathrm { m } / \mathrm { s }$.
Q61 Complex Numbers Argand & Loci Intersection of Loci and Simultaneous Geometric Conditions View
Q61. The area (in sq. units) of the region $S = \{ z \in \mathbb { C } : | z - 1 | \leq 2 ; ( z + \bar { z } ) + i ( z - \bar { z } ) \leq 2 , \operatorname { Im } ( z ) \geq 0 \}$ is
(1) $\frac { 7 \pi } { 3 }$
(2) $\frac { 7 \pi } { 4 }$
(3) $\frac { 17 \pi } { 8 }$
(4) $\frac { 3 \pi } { 2 }$
Q62 Arithmetic Sequences and Series Telescoping or Non-Standard Summation Involving an AP View
Q62. The value of $\frac { 1 \times 2 ^ { 2 } + 2 \times 3 ^ { 2 } + \ldots + 100 \times ( 101 ) ^ { 2 } } { 1 ^ { 2 } \times 2 + 2 ^ { 2 } \times 3 + \ldots + 100 ^ { 2 } \times 101 }$ is
(1) $\frac { 32 } { 31 }$
(2) $\frac { 31 } { 30 }$
(3) $\frac { 306 } { 305 }$
(4) $\frac { 305 } { 301 }$
Q63 Arithmetic Sequences and Series Arithmetic-Geometric Hybrid Problem View
Q63. Let three real numbers $a , b , c$ be in arithmetic progression and $a + 1 , b , c + 3$ be in geometric progression. If $a > 10$ and the arithmetic mean of $a , b$ and $c$ is 8 , then the cube of the geometric mean of $a , b$ and $c$ is
(1) 128
(2) 316
(3) 120
(4) 312
Q64 Binomial Theorem (positive integer n) Determine Parameters from Conditions on Coefficients or Terms View
Q64. If the coefficients of $x ^ { 4 } , x ^ { 5 }$ and $x ^ { 6 }$ in the expansion of $( 1 + x ) ^ { n }$ are in the arithmetic progression, then the maximum value of $n$ is:
(1) 7
(2) 21
(3) 28
(4) 14
Q65 Circles Chord Length and Chord Properties View
Q65. Let $C$ be a circle with radius $\sqrt { 10 }$ units and centre at the origin. Let the line $x + y = 2$ intersects the circle C at the points P and Q . Let MN be a chord of C of length 2 unit and slope - 1 . Then, a distance (in units) between the chord PQ and the chord MN is
(1) $3 - \sqrt { 2 }$
(2) $\sqrt { 2 } + 1$
(3) $\sqrt { 2 } - 1$
(4) $2 - \sqrt { 3 }$
Q66 Conic sections Chord Properties and Midpoint Problems View
Q66. Let PQ be a chord of the parabola $y ^ { 2 } = 12 x$ and the midpoint of PQ be at $( 4,1 )$. Then, which of the following point lies on the line passing through the points P and Q ?
(1) $( 3 , - 3 )$
(2) $( 2 , - 9 )$
(3) $\left( \frac { 3 } { 2 } , - 16 \right)$
(4) $\left( \frac { 1 } { 2 } , - 20 \right)$
Q67 Conic sections Circle-Conic Interaction with Tangency or Intersection View
Q67. Consider a hyperbola H having centre at the origin and foci on the x -axis. Let $\mathrm { C } _ { 1 }$ be the circle touching the hyperbola H and having the centre at the origin. Let $\mathrm { C } _ { 2 }$ be the circle touching the hyperbola H at its vertex and having the centre at one of its foci. If areas (in sq units) of $C _ { 1 }$ and $C _ { 2 }$ are $36 \pi$ and $4 \pi$, respectively, then the length (in units) of latus rectum of H is
(1) $\frac { 14 } { 3 }$
(2) $\frac { 28 } { 3 }$
(3) $\frac { 11 } { 3 }$
(4) $\frac { 10 } { 3 }$
Q68 Standard Integrals and Reverse Chain Rule Limit Involving an Integral (FTC Application) View
Q68. Let $f ( x ) = \int _ { 0 } ^ { x } \left( t + \sin \left( 1 - e ^ { \prime } \right) \right) d t , x \in \mathbb { R }$. Then, $\lim _ { x \rightarrow 0 } \frac { f ( x ) } { x ^ { 3 } }$ is equal to
(1) $- \frac { 1 } { 6 }$
(2) $\frac { 2 } { 3 }$
(3) $- \frac { 2 } { 3 }$
(4) $\frac { 1 } { 6 }$
Q69 Measures of Location and Spread View
Q69. If the mean of the following probability distribution of a random variable $X$ :
X02468
$\mathrm { P } ( \mathrm { X } )$$a$$2 a$$a + b$$2 b$$3 b$

is $\frac { 46 } { 9 }$, then the variance of the distribution is
(1) $\frac { 173 } { 27 }$
(2) $\frac { 566 } { 81 }$
(3) $\frac { 151 } { 27 }$
(4) $\frac { 581 } { 81 }$
Q71 Matrices Matrix Power Computation and Application View
Q71. Let $A = \left[ \begin{array} { l l } 1 & 2 \\ 0 & 1 \end{array} \right]$ and $B = I + \operatorname { adj } ( A ) + ( \operatorname { adj } A ) ^ { 2 } + \ldots + ( \operatorname { adj } A ) ^ { 10 }$. Then, the sum of all the elements of the matrix $B$ is:
(1) - 124
(2) 22
(3) - 88
(4) - 110
Q72 Trig Graphs & Exact Values View
Q72. Given that the inverse trigonometric function assumes principal values only. Let $x , y$ be any two real numbers in $[ - 1,1 ]$ such that $\cos ^ { - 1 } x - \sin ^ { - 1 } y = \alpha , \frac { - \pi } { 2 } \leq \alpha \leq \pi$. Then, the minimum value of $x ^ { 2 } + y ^ { 2 } + 2 x y \sin \alpha$ is
(1) 0
(2) - 1
(3) $\frac { 1 } { 2 }$
(4) $- \frac { 1 } { 2 }$
Q73 Differentiating Transcendental Functions Limit involving transcendental functions View
Q73. If the function $f ( x ) = \left\{ \begin{array} { l l } \frac { 72 ^ { x } - 9 ^ { x } - 8 ^ { x } + 1 } { \sqrt { 2 } - \sqrt { 1 + \cos x } } , & x \neq 0 \\ a \log _ { e } 2 \log _ { e } 3 & , x = 0 \end{array} \right.$ is continuous at $x = 0$, then the value of $a ^ { 2 }$ is equal to
(1) 968
(2) 1152
(3) 746
(4) 1250
Q74 Stationary points and optimisation Find absolute extrema on a closed interval or domain View
Q74. Let $f ( x ) = 3 \sqrt { x - 2 } + \sqrt { 4 - x }$ be a real valued function. If $\alpha$ and $\beta$ are respectively the minimum and the maximum values of $f$, then $\alpha ^ { 2 } + 2 \beta ^ { 2 }$ is equal to
(1) 42
(2) 38
(3) 24
(4) 44
Q75 Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution View
Q75. If the value of the integral $\int _ { - 1 } ^ { 1 } \frac { \cos \alpha x } { 1 + 3 ^ { x } } d x$ is $\frac { 2 } { \pi }$. Then, a value of $\alpha$ is
(1) $\frac { \pi } { 3 }$
(2) $\frac { \pi } { 6 }$
(3) $\frac { \pi } { 4 }$
(4) $\frac { \pi } { 2 }$
Q76 Areas by integration View
Q76. The area (in sq. units) of the region described by $\left\{ ( x , y ) : y ^ { 2 } \leq 2 x \right.$, and $\left. y \geq 4 x - 1 \right\}$ is
(1) $\frac { 11 } { 32 }$
(2) $\frac { 8 } { 9 }$
(3) $\frac { 11 } { 12 }$
(4) $\frac { 9 } { 32 }$
Q77 First order differential equations (integrating factor) View
Q77. Let $y = y ( x )$ be the solution of the differential equation $\left( x ^ { 2 } + 4 \right) ^ { 2 } d y + \left( 2 x ^ { 3 } y + 8 x y - 2 \right) d x = 0$. If $y ( 0 ) = 0$, then $y ( 2 )$ is equal to
(1) $\frac { \pi } { 32 }$
(2) $2 \pi$
(3) $\frac { \pi } { 8 }$
(4) $\frac { \pi } { 16 }$
Q78 Vectors: Cross Product & Distances View
Q78. Let $\vec { a } = \hat { i } + \hat { j } + \hat { k } , \vec { b } = 2 \hat { i } + 4 \hat { j } - 5 \hat { k }$ and $\vec { c } = x \hat { i } + 2 \hat { j } + 3 \hat { k } , x \in \mathbb { R }$. If $\vec { d }$ is the unit vector in the direction of $\vec { b } + \vec { c }$ such that $\vec { a } \cdot \vec { d } = 1$, then $( \vec { a } \times \vec { b } ) \cdot \vec { c }$ is equal to
(1) 11
(2) 3
(3) 9
(4) 6
Q79 Vectors 3D & Lines Vector Algebra and Triple Product Computation View
Q79. For $\lambda > 0$, let $\theta$ be the angle between the vectors $\vec { a } = \hat { i } + \lambda \hat { j } - 3 \hat { k }$ and $\vec { b } = 3 \hat { i } - \hat { j } + 2 \hat { k }$. If the vectors $\vec { a } + \vec { b }$ and $\vec { a } - \vec { b }$ are mutually perpendicular, then the value of $( 14 \cos \theta ) ^ { 2 }$ is equal to
(1) 50
(2) 40
(3) 25
(4) 20
Q80 Vectors 3D & Lines Distance from a Point to a Line (Show/Compute) View
Q80. Let P be the point of intersection of the lines $\frac { x - 2 } { 1 } = \frac { y - 4 } { 5 } = \frac { z - 2 } { 1 }$ and $\frac { x - 3 } { 2 } = \frac { y - 2 } { 3 } = \frac { z - 3 } { 2 }$. Then, the shortest distance of P from the line $4 x = 2 y = z$ is
(1) $\frac { 5 \sqrt { 14 } } { 7 }$
(2) $\frac { 3 \sqrt { 14 } } { 7 }$
(3) $\frac { \sqrt { 14 } } { 7 }$
(4) $\frac { 6 \sqrt { 14 } } { 7 }$
Q81 Combinations & Selection Selection with Group/Category Constraints View
Q81. There are 4 men and 5 women in Group A, and 5 men and 4 women in Group B. If 4 persons are selected from each group, then the number of ways of selecting 4 men and 4 women is $\_\_\_\_$
Q82 Discriminant and conditions for roots Parameter range for specific root conditions (location/count) View
Q82. Let $S = \left\{ \sin ^ { 2 } 2 \theta : \left( \sin ^ { 4 } \theta + \cos ^ { 4 } \theta \right) x ^ { 2 } + ( \sin 2 \theta ) x + \left( \sin ^ { 6 } \theta + \cos ^ { 6 } \theta \right) = 0 \right.$ has real roots $\}$. If $\alpha$ and $\beta$ be the smallest and largest elements of the set $S$, respectively, then $3 \left( ( \alpha - 2 ) ^ { 2 } + ( \beta - 1 ) ^ { 2 } \right)$ equals $\_\_\_\_$
Q83 Straight Lines & Coordinate Geometry Triangle Properties and Special Points View
Q83. Consider a triangle ABC having the vertices $\mathrm { A } ( 1,2 ) , \mathrm { B } ( \alpha , \beta )$ and $\mathrm { C } ( \gamma , \delta )$ and angles $\angle A B C = \frac { \pi } { 6 }$ and $\angle B A C = \frac { 2 \pi } { 3 }$. If the points B and C lie on the line $y = x + 4$, then $\alpha ^ { 2 } + \gamma ^ { 2 }$ is equal to $\_\_\_\_$
Q84 Matrices Linear System and Inverse Existence View
Q84. Let $A$ be a $2 \times 2$ symmetric matrix such that $A \left[ \begin{array} { l } 1 \\ 1 \end{array} \right] = \left[ \begin{array} { l } 3 \\ 7 \end{array} \right]$ and the determinant of $A$ be 1 . If $A ^ { - 1 } = \alpha A + \beta I$, where $I$ is an identity matrix of order $2 \times 2$, then $\alpha + \beta$ equals $\_\_\_\_$
Q85 Composite & Inverse Functions Evaluate Composition from Algebraic Definitions View
Q85. Consider the function $f : \mathbb { R } \rightarrow \mathbb { R }$ defined by $f ( x ) = \frac { 2 x } { \sqrt { 1 + 9 x ^ { 2 } } }$. If the composition of $f , \underbrace { ( f \circ f \circ f \circ \cdots \circ f ) } _ { 10 \text { times } } ( x ) = \frac { 2 ^ { 10 } x } { \sqrt { 1 + 9 \alpha x ^ { 2 } } }$, then the value of $\sqrt { 3 \alpha + 1 }$ is equal to $\_\_\_\_$
Q86 Stationary points and optimisation Count or characterize roots using extremum values View
Q86. Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a thrice differentiable function such that $f ( 0 ) = 0 , f ( 1 ) = 1 , f ( 2 ) = - 1 , f ( 3 ) = 2$ and $f ( 4 ) = - 2$. Then, the minimum number of zeros of $\left( 3 f ^ { \prime } f ^ { \prime \prime } + f f ^ { \prime \prime \prime } \right) ( x )$ is $\_\_\_\_$
Q87 Integration by Parts Indefinite Integration by Parts View
Q87. If $\int \operatorname { cosec } ^ { 5 } x d x = \alpha \cot x \operatorname { cosec } x \left( \operatorname { cossc } ^ { 2 } x + \frac { 3 } { 2 } \right) + \beta \log _ { \epsilon } \left| \tan \frac { x } { 2 } \right| + C$ where $\alpha , \beta \in \mathbb { R }$ and C is the constant of integration, then the value of $8 ( \alpha + \beta )$ equals $\_\_\_\_$
Q88 Differential equations Solving Separable DEs with Initial Conditions View
Q88. Let $y = y ( x )$ be the solution of the differential equation $( x + y + 2 ) ^ { 2 } d x = d y , y ( 0 ) = - 2$. Let the maximum and minimum values of the function $y = y ( x )$ in $\left[ 0 , \frac { \pi } { 3 } \right]$ be $\alpha$ and $\beta$, respectively. If $( 3 \alpha + \pi ) ^ { 2 } + \beta ^ { 2 } = \gamma + \delta \sqrt { 3 } , \gamma , \delta \in \mathbb { Z }$, then $\gamma + \delta$ equals $\_\_\_\_$
Q89 Vectors 3D & Lines Distance from a Point to a Line (Show/Compute) View
Q89. Consider a line L passing through the points $\mathrm { P } ( 1,2,1 )$ and $\mathrm { Q } ( 2,1 , - 1 )$. If the mirror image of the point $\mathrm { A } ( 2,2,2 )$ in the line L is $( \alpha , \beta , \gamma )$, then $\alpha + \beta + 6 \gamma$ is equal to $\_\_\_\_$
Q90 Binomial Distribution Compute Cumulative or Complement Binomial Probability View
Q90. In a tournament, a team plays 10 matches with probabilities of winning and losing each match as $\frac { 1 } { 3 }$ and $\frac { 2 } { 3 }$ respectively. Let $x$ be the number of matches that the team wins, and $y$ be the number of matches that team loses. If the probability $\mathrm { P } ( | x - y | \leq 2 )$ is $p$, then $3 ^ { 9 } p$ equals $\_\_\_\_$
ANSWER KEYS

\begin{tabular}{|l|l|l|} \hline 1. (3) & 2. (1) & 3. (4) \hline 9. (2) & 10. (1) & 11. (1) \hline 17. (3) & 18. (4) & 19. (4) \hline 25. (750) & 26. (32) & 27. (5) \hline 33. (4) & 34. (2) & 35. (4) \hline