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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
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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
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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_07apr_shift1

32 maths questions

Q6 Newton's laws and connected particles Force from velocity change (Newton's second law with impulse/momentum) View
Q6. A player caught a cricket ball of mass 150 g moving at a speed of $20 \mathrm {~m} / \mathrm { s }$. If the catching process is completed in 0.1 s , the magnitude of force exerted by the ball on the hand of the player is:
(1) 3 N
(2) 300 N
(3) 150 N
(4) 30 N
Q7 Circular Motion 1 Ratio / Comparison of Circular Motion Quantities View
Q7. Two planets $A$ and $B$ having masses $m _ { 1 }$ and $m _ { 2 }$ move around the sun in circular orbits of $r _ { 1 }$ and $r _ { 2 }$ radii respectively. If angular momentum of $A$ is $L$ and that of $B$ is 3 L , the ratio of time period $\left( \frac { T _ { A } } { T _ { B } } \right)$ is:
(1) $\left( \frac { r _ { 2 } } { r _ { 1 } } \right) ^ { \frac { 3 } { 2 } }$
(2) $\frac { 1 } { 27 } \left( \frac { m _ { 2 } } { m _ { 1 } } \right) ^ { 3 }$
(3) $27 \left( \frac { m _ { 1 } } { m _ { 2 } } \right) ^ { 3 }$
(4) $\left( \frac { r _ { 1 } } { r _ { 2 } } \right) ^ { 3 }$
Q61 Exponential Equations & Modelling Solve Exponential Equation for Unknown Variable View
Q61. The sum of all the solutions of the equation $( 8 ) ^ { 2 x } - 16 \cdot ( 8 ) ^ { x } + 48 = 0$ is :
(1) $1 + \log _ { 8 } ( 6 )$
(2) $1 + \log _ { 6 } ( 8 )$
(3) $\log _ { 8 } ( 6 )$
(4) $\log _ { 8 } ( 4 )$
Q62 Complex Numbers Arithmetic Solving Equations for Unknown Complex Numbers View
Q62. Let $z$ be a complex number such that $| z + 2 | = 1$ and $\operatorname { Im } \left( \frac { z + 1 } { z + 2 } \right) = \frac { 1 } { 5 }$. Then the value of $| \operatorname { Re } ( \overline { z + 2 } ) |$ is
(1) $\frac { 2 \sqrt { 6 } } { 5 }$
(2) $\frac { 24 } { 5 }$
(3) $\frac { 1 + \sqrt { 6 } } { 5 }$
(4) $\frac { \sqrt { 6 } } { 5 }$
Q63 Binomial Theorem (positive integer n) Combinatorial Counting via Binomial Theorem View
Q63. If the set $R = \{ ( a , b ) : a + 5 b = 42 , a , b \in \mathbb { N } \}$ has $m$ elements and $\sum _ { n = 1 } ^ { m } \left( 1 - i ^ { n ! } \right) = x + i y$, where $i = \sqrt { - 1 }$ , then the value of $m + x + y$ is
(1) 12
(2) 4
(3) 8
(4) 5
Q64 Trig Proofs Trigonometric Equation Constraint Deduction View
Q64. If $\sin x = - \frac { 3 } { 5 }$, where $\pi < x < \frac { 3 \pi } { 2 }$, then $80 \left( \tan ^ { 2 } x - \cos x \right)$ is equal to
(1) 108
(2) 109
(3) 18
(4) 19
Q65 Straight Lines & Coordinate Geometry Section Ratio and Division of Segments View
Q65. The equations of two sides AB and AC of a triangle ABC are $4 x + y = 14$ and $3 x - 2 y = 5$, respectively. The point $\left( 2 , - \frac { 4 } { 3 } \right)$ divides the third side BC internally in the ratio $2 : 1$. the equation of the side BC is
(1) $x + 3 y + 2 = 0$
(2) $x - 6 y - 10 = 0$
(3) $x - 3 y - 6 = 0$
(4) $x + 6 y + 6 = 0$
Q66 Circles Circles Tangent to Each Other or to Axes View
Q66. Let the circles $C _ { 1 } : ( x - \alpha ) ^ { 2 } + ( y - \beta ) ^ { 2 } = r _ { 1 } ^ { 2 }$ and $C _ { 2 } : ( x - 8 ) ^ { 2 } + \left( y - \frac { 15 } { 2 } \right) ^ { 2 } = r _ { 2 } ^ { 2 }$ touch each other externally at the point $( 6,6 )$. If the point $( 6,6 )$ divides the line segment joining the centres of the circles $C _ { 1 }$ and $C _ { 2 }$ internally in the ratio $2 : 1$, then $( \alpha + \beta ) + 4 \left( r _ { 1 } ^ { 2 } + r _ { 2 } ^ { 2 } \right)$ equals
(1) 125
(2) 130
(3) 110
(4) 145
Q67 Conic sections Focal Distance and Point-on-Conic Metric Computation View
Q67. Let $H : \frac { - x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ be the hyperbola, whose eccentricity is $\sqrt { 3 }$ and the length of the latus rectum is $4 \sqrt { 3 }$. Suppose the point $( \alpha , 6 ) , \alpha > 0$ lies on $H$. If $\beta$ is the product of the focal distances of the point $( \alpha , 6 )$, then $\alpha ^ { 2 } + \beta$ is equal to
(1) 172
(2) 171
(3) 169
(4) 170
Q68 3x3 Matrices Determinant of Parametric or Structured Matrix View
Q68. Let $A = \left[ \begin{array} { l l l } 2 & a & 0 \\ 1 & 3 & 1 \\ 0 & 5 & b \end{array} \right]$. If $A ^ { 3 } = 4 A ^ { 2 } - A - 21 I$, where $I$ is the identity matrix of order $3 \times 3$, then $2 a + 3 b$ is equal to
(1) - 9
(2) - 13
(3) - 10
(4) - 12
Q69 Permutations & Arrangements Counting Functions with Constraints View
Q69. Let $[ t ]$ be the greatest integer less than or equal to $t$. Let $A$ be the set of all prime factors of 2310 and $f : A \rightarrow \mathbb { Z }$ be the function $f ( x ) = \left[ \log _ { 2 } \left( x ^ { 2 } + \left[ \frac { x ^ { 3 } } { 5 } \right] \right) \right]$. The number of one-to-one functions from $A$ to the range of $f$ is
(1) 25
(2) 24
(3) 20
(4) 120
Q70 Stationary points and optimisation Count or characterize roots using extremum values View
Q70. For the function $f ( x ) = ( \cos x ) - x + 1 , x \in \mathbb { R }$, between the following two statements (S1) $f ( x ) = 0$ for only one value of $x$ in $[ 0 , \pi ]$. (S2) $f ( x )$ is decreasing in $\left[ 0 , \frac { \pi } { 2 } \right]$ and increasing in $\left[ \frac { \pi } { 2 } , \pi \right]$.
(1) Both (S1) and (S2) are correct.
(2) Both (S1) and (S2) are incorrect.
(3) Only (S2) is correct.
(4) Only (S1) is correct.
Q71 Stationary points and optimisation Find critical points and classify extrema of a given function View
Q71. Let $f ( x ) = 4 \cos ^ { 3 } x + 3 \sqrt { 3 } \cos ^ { 2 } x - 10$. The number of points of local maxima of $f$ in interval $( 0,2 \pi )$ is
(1) 3
(2) 4
(3) 1
(4) 2
Q72 Stationary points and optimisation Find critical points and classify extrema of a given function View
Q72. The number of critical points of the function $f ( x ) = ( x - 2 ) ^ { 2 / 3 } ( 2 x + 1 )$ is
(1) 1
(2) 2
(3) 0
(4) 3
Q73 Standard Integrals and Reverse Chain Rule Antiderivative with Initial Condition View
Q73. Let $I ( x ) = \int \frac { 6 } { \sin ^ { 2 } x ( 1 - \cot x ) ^ { 2 } } d x$. If $I ( 0 ) = 3$, then $I \left( \frac { \pi } { 12 } \right)$ is equal to
(1) $2 \sqrt { 3 }$
(2) $\sqrt { 3 }$
(3) $3 \sqrt { 3 }$
(4) $6 \sqrt { 3 }$
Q74 Standard Integrals and Reverse Chain Rule Integral Equation to Determine a Function Value View
Q74. The value of $k \in \mathrm {~N}$ for which the integral $I _ { n } = \int _ { 0 } ^ { 1 } \left( 1 - x ^ { k } \right) ^ { n } d x , n \in \mathbb { N }$, satisfies $147 I _ { 20 } = 148 I _ { 21 }$ is
(1) 14
(2) 8
(3) 10
(4) 7
Q75 Second order differential equations Solving homogeneous second-order linear ODE View
Q75. Let $f ( x )$ be a positive function such that the area bounded by $y = f ( x ) , y = 0$ from $x = 0$ to $x = a > 0$ is $e ^ { - a } + 4 a ^ { 2 } + a - 1$. Then the differential equation, whose general solution is $y = c _ { 1 } f ( x ) + c _ { 2 }$, where $c _ { 1 }$ and $c _ { 2 }$ are arbitrary constants, is
(1) $\left( 8 e ^ { x } - 1 \right) \frac { d ^ { 2 } y } { d x ^ { 2 } } + \frac { d y } { d x } = 0$
(2) $\left( 8 e ^ { x } - 1 \right) \frac { d ^ { 2 } y } { d x ^ { 2 } } - \frac { d y } { d x } = 0$
(3) $\left( 8 e ^ { x } + 1 \right) \frac { d ^ { 2 } y } { d x ^ { 2 } } - \frac { d y } { d x } = 0$
(4) $\left( 8 e ^ { x } + 1 \right) \frac { d ^ { 2 } y } { d x ^ { 2 } } + \frac { d y } { d x } = 0$
Q76 First order differential equations (integrating factor) View
Q76. Let $y = y ( x )$ be the solution of the differential equation $\left( 1 + y ^ { 2 } \right) e ^ { \tan x } d x + \cos ^ { 2 } x \left( 1 + e ^ { 2 \tan x } \right) d y = 0 , y ( 0 ) = 1$. Then $y \left( \frac { \pi } { 4 } \right)$ is equal to
(1) $\frac { 2 } { e }$
(2) $\frac { 2 } { e ^ { 2 } }$
(3) $\frac { 1 } { e }$
(4) $\frac { 1 } { e ^ { 2 } }$
Q77 Vectors 3D & Lines Vector Algebra and Triple Product Computation View
Q77. The set of all $\alpha$, for which the vectors $\vec { a } = \alpha t \hat { i } + 6 \hat { j } - 3 \hat { k }$ and $\vec { b } = t \hat { i } - 2 \hat { j } - 2 \alpha t \hat { k }$ are inclined at an obtuse angle for all $t \in \mathbb { R }$, is
(1) $\left( - \frac { 4 } { 3 } , 1 \right)$
(2) $[ 0,1 )$
(3) $\left( - \frac { 4 } { 3 } , 0 \right]$
(4) $( - 2,0 ]$
Q78 Vectors: Lines & Planes Distance Computation (Point-to-Plane or Line-to-Line) View
Q78. If the shortest distance between the lines $\begin{aligned} & L _ { 1 } : \vec { r } = ( 2 + \lambda ) \hat { i } + ( 1 - 3 \lambda ) \hat { j } + ( 3 + 4 \lambda ) \hat { k } , \quad \lambda \in \mathbb { R } \\ & L _ { 2 } : \vec { r } = 2 ( 1 + \mu ) \hat { i } + 3 ( 1 + \mu ) \hat { j } + ( 5 + \mu ) \hat { k } , \quad \mu \in \mathbb { R } \end{aligned}$ is $\frac { m } { \sqrt { n } }$ , where $\operatorname { gcd } ( m , n ) = 1$, then the value of $m + n$ equals
(1) 390
(2) 384
(3) 377
(4) 387
Q79 Vectors 3D & Lines Vector Algebra and Triple Product Computation View
Q79. Let $P ( x , y , z )$ be a point in the first octant, whose projection in the $x y$-plane is the point $Q$. Let $O P = \gamma$; the angle between $O Q$ and the positive $x$-axis be $\theta$; and the angle between $O P$ and the positive $z$-axis be $\phi$, where $O$ is the origin. Then the distance of $P$ from the $x$-axis is
(1) $\gamma \sqrt { 1 - \sin ^ { 2 } \phi \cos ^ { 2 } \theta }$
(2) $\gamma \sqrt { 1 - \sin ^ { 2 } \theta \cos ^ { 2 } \phi }$
(3) $\gamma \sqrt { 1 + \cos ^ { 2 } \phi \sin ^ { 2 } \theta }$
(4) $\gamma \sqrt { 1 + \cos ^ { 2 } \theta \sin ^ { 2 } \phi }$
Q80 Probability Definitions Probability Involving Algebraic or Number-Theoretic Conditions View
Q80. Let the sum of two positive integers be 24 . If the probability, that their product is not less than $\frac { 3 } { 4 }$ times their greatest possible product, is $\frac { m } { n }$, where $\operatorname { gcd } ( m , n ) = 1$, then $n - m$ equals
(1) 10
(2) 9
(3) 11
(4) 8
Q81 Permutations & Arrangements Forming Numbers with Digit Constraints View
Q81. The number of 3-digit numbers, formed using the digits $2,3,4,5$ and 7 , when the repetition of digits is not allowed, and which are not divisible by 3 , is equal to $\_\_\_\_$
Q82 Sequences and Series Recurrence Relations and Sequence Properties View
Q82. Let the positive integers be written in the form : [Figure]
If the $k ^ { \text {th } }$ row contains exactly $k$ numbers for every natural number $k$, then the row in which the number 5310 will be, is $\_\_\_\_$
Q83 Binomial Theorem (positive integer n) Evaluate a Summation Involving Binomial Coefficients View
Q83. Let $\alpha = \sum _ { r = 0 } ^ { n } \left( 4 r ^ { 2 } + 2 r + 1 \right) ^ { n } C _ { r }$ and $\beta = \left( \sum _ { r = 0 } ^ { n } \frac { { } ^ { n } C _ { r } } { r + 1 } \right) + \frac { 1 } { n + 1 }$. If $140 < \frac { 2 \alpha } { \beta } < 281$, then the value of $n$ is $\_\_\_\_$
Q84 Straight Lines & Coordinate Geometry Triangle Properties and Special Points View
Q84. If the orthocentre of the triangle formed by the lines $2 x + 3 y - 1 = 0 , x + 2 y - 1 = 0$ and $a x + b y - 1 = 0$, is the centroid of another triangle, whose circumcentre and orthocentre respectively are $( 3,4 )$ and $( - 6 , - 8 )$, then the value of $| a - b |$ is $\_\_\_\_$
Q85 Chain Rule Limit Evaluation Involving Composition or Substitution View
Q85. The value of $\lim _ { x \rightarrow 0 } 2 \left( \frac { 1 - \cos x \sqrt { \cos 2 x } \sqrt [ 3 ] { \cos 3 x } \ldots \ldots \sqrt [ 10 ] { \cos 10 x } } { x ^ { 2 } } \right)$ is
Q86 Matrices Matrix Power Computation and Application View
Q86. Let $A = \left[ \begin{array} { c c } 2 & - 1 \\ 1 & 1 \end{array} \right]$. If the sum of the diagonal elements of $A ^ { 13 }$ is $3 ^ { n }$, then $n$ is equal to $\_\_\_\_$
Q87 Geometric Sequences and Series Sum of an Infinite Geometric Series (Direct Computation) View
Q87. If the range of $f ( \theta ) = \frac { \sin ^ { 4 } \theta + 3 \cos ^ { 2 } \theta } { \sin ^ { 4 } \theta + \cos ^ { 2 } \theta } , \theta \in \mathbb { R }$ is $[ \alpha , \beta ]$, then the sum of the infinite G.P., whose first term is 64 and the common ratio is $\frac { \alpha } { \beta }$, is equal to $\_\_\_\_$
Q88 Areas by integration View
Q88. Let the area of the region enclosed by the curve $y = \min \{ \sin x , \cos x \}$ and the $x$ axis between $x = - \pi$ to $x = \pi$ be $A$. Then $A ^ { 2 }$ is equal to $\_\_\_\_$
Q89 Vectors 3D & Lines Vector Algebra and Triple Product Computation View
Q89. Let $\vec { a } = 9 \hat { i } - 13 \hat { j } + 25 \hat { k } , \vec { b } = 3 \hat { i } + 7 \hat { j } - 13 \hat { k }$ and $\vec { c } = 17 \hat { i } - 2 \hat { j } + \hat { k }$ be three given vectors. If $\vec { r }$ is a vector such that $\vec { r } \times \vec { a } = ( \vec { b } + \vec { c } ) \times \vec { a }$ and $\vec { r } \cdot ( \vec { b } - \vec { c } ) = 0$, then $\frac { | 593 \vec { r } + 67 \vec { a } | ^ { 2 } } { ( 593 ) ^ { 2 } }$ is equal to $\_\_\_\_$
Q90 Discrete Probability Distributions Expectation and Variance from Context-Based Random Variables View
Q90. Three balls are drawn at random from a bag containing 5 blue and 4 yellow balls. Let the random variables $X$ and $Y$ respectively denote the number of blue and yellow balls. If $\bar { X }$ and $\bar { Y }$ are the means of $X$ and $Y$ respectively, then $7 \bar { X } + 4 \bar { Y }$ is equal to $\_\_\_\_$
ANSWER KEYS

\begin{tabular}{|l|l|l|l|} \hline 1. (2) & 2. (2) & 3. (3) & 4. (1) \hline 9. (2) & 10. (2) & 11. (1) & 12. (1) \hline 17. (4) & 18. (2) & 19. (2) & 20. (1) \hline 25. (12) & 26. (748) & 27. (4) & 28. (3) \hline 33. (2) & 34. (1) & 35. (1) & 36. (4) \hline