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
session1_24jun_shift1 19 session1_24jun_shift2 25 session1_25jun_shift1 14 session1_25jun_shift2 14 session1_26jun_shift1 29 session1_26jun_shift2 24 session1_27jun_shift1 4 session1_27jun_shift2 29 session1_28jun_shift1 13 session1_29jun_shift1 20 session1_29jun_shift2 4 session2_25jul_shift1 29 session2_25jul_shift2 20 session2_26jul_shift1 29 session2_26jul_shift2 23 session2_27jul_shift1 28 session2_27jul_shift2 29 session2_28jul_shift1 11 session2_28jul_shift2 29 session2_29jul_shift1 17 session2_29jul_shift2 18
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
2022 session2_27jul_shift1

28 maths questions

Q61 Complex Numbers Arithmetic Distance and Region Optimization on Loci View
Let the minimum value $v _ { 0 }$ of $v = | z | ^ { 2 } + | z - 3 | ^ { 2 } + | z - 6 i | ^ { 2 } , z \in \mathbb { C }$ is attained at $z = z _ { 0 }$. Then $\left| 2 z _ { 0 } ^ { 2 } - \bar { z } _ { 0 } ^ { 3 } + 3 \right| ^ { 2 } + v _ { 0 } ^ { 2 }$ is equal to
(1) 1000
(2) 1024
(3) 1105
(4) 1196
Q62 Arithmetic Sequences and Series Compute Partial Sum of an Arithmetic Sequence View
Suppose $a _ { 1 } , a _ { 2 } , \ldots , a _ { \mathrm { n } } , \ldots$ be an arithmetic progression of natural numbers. If the ratio of the sum of the first five terms to the sum of first nine terms of the progression is 5 : 17 and $110 < a _ { 15 } < 120$, then the sum of the first ten terms of the progression is equal to
(1) 290
(2) 380
(3) 460
(4) 510
Q63 Number Theory Modular Arithmetic Computation View
The remainder when $( 2021 ) ^ { 2022 } + ( 2022 ) ^ { 2021 }$ is divided by 7 is
(1) 0
(2) 1
(3) 2
(4) 6
Q64 Areas by integration Area Computation in Coordinate Geometry View
Let $A ( 1,1 ) , B ( - 4,3 ) , C ( - 2 , - 5 )$ be vertices of a triangle $A B C , P$ be a point on side $B C$, and $\Delta _ { 1 }$ and $\Delta _ { 2 }$ be the areas of triangle $A P B$ and $A B C$ respectively. If $\Delta _ { 1 } : \Delta _ { 2 } = 4 : 7$, then the area enclosed by the lines $A P , A C$ and the $x$-axis is
(1) $\frac { 1 } { 4 }$
(2) $\frac { 3 } { 4 }$
(3) $\frac { 1 } { 2 }$
(4) 1
Q65 Circles Chord Length and Chord Properties View
If the circle $x ^ { 2 } + y ^ { 2 } - 2 g x + 6 y - 19 c = 0 , g , c \in \mathbb { R }$ passes through the point $( 6,1 )$ and its centre lies on the line $x - 2 c y = 8$, then the length of intercept made by the circle on $x$-axis is
(1) $\sqrt { 11 }$
(2) 4
(3) 3
(4) $2 \sqrt { 23 }$
Q66 Circles Tangent Lines and Tangent Lengths View
Let $P ( a , b )$ be a point on the parabola $y ^ { 2 } = 8 x$ such that the tangent at $P$ passes through the centre of the circle $x ^ { 2 } + y ^ { 2 } - 10 x - 14 y + 65 = 0$. Let $A$ be the product of all possible values of $a$ and $B$ be the product of all possible values of $b$. Then the value of $A + B$ is equal to
(1) 0
(2) 25
(3) 40
(4) 65
Q67 Indefinite & Definite Integrals Piecewise/Periodic Function Integration View
Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a function defined as $f ( x ) = a \sin \left( \frac { \pi [ x ] } { 2 } \right) + [ 2 - x ] , a \in \mathbb { R }$, where $[ t ]$ is the greatest integer less than or equal to $t$. If $\lim _ { x \rightarrow - 1 } f ( x )$ exists, then the value of $\int _ { 0 } ^ { 4 } f ( x ) d x$ is equal to
(1) - 1
(2) - 2
(3) 1
(4) 2
Q69 Sine and Cosine Rules Evaluate trigonometric expression given a constraint View
Let a vertical tower $A B$ of height $2 h$ stands on a horizontal ground. Let from a point $P$ on the ground a man can see upto height $h$ of the tower with an angle of elevation $2 \alpha$. When from $P$, he moves a distance $d$ in the direction of $\overrightarrow { A P }$, he can see the top $B$ of the tower with an angle of elevation $\alpha$. If $d = \sqrt { 7 } h$, then $\tan \alpha$ is equal to
(1) $\sqrt { 5 } - 2$
(2) $\sqrt { 3 } - 1$
(3) $\sqrt { 7 } - 2$
(4) $\sqrt { 7 } - \sqrt { 3 }$
Q71 Matrices Matrix Algebra and Product Properties View
Let $A = \left( \begin{array} { c c } 1 & 2 \\ - 2 & - 5 \end{array} \right)$. Let $\alpha , \beta \in \mathbb { R }$ be such that $\alpha A ^ { 2 } + \beta A = 2 I$. Then $\alpha + \beta$ is equal to
(1) - 10
(2) - 6
(3) 6
(4) 10
Q72 Number Theory Arithmetic Functions and Multiplicative Number Theory View
Let $f , g : \mathbb { N } - \{ 1 \} \rightarrow \mathbb { N }$ be functions defined by $f ( \mathrm { a } ) = \alpha$, where $\alpha$ is the maximum of the powers of those primes $p$ such that $p ^ { \alpha }$ divides $a$, and $g ( a ) = a + 1$, for all $a \in \mathbb { N } - \{ 1 \}$. Then, the function $f + g$ is
(1) one-one but not onto
(2) onto but not one-one
(3) both one-one and onto
(4) neither one-one nor onto
Q73 Stationary points and optimisation Composite or piecewise function extremum analysis View
Let a function $f : \mathbb { R } \rightarrow \mathbb { R }$ be defined as: $f ( x ) = \begin{cases} \int _ { 0 } ^ { x } ( 5 - | t - 3 | ) d t , & x > 4 \\ x ^ { 2 } + b x , & x \leq 4 \end{cases}$ where $b \in \mathbb { R }$. If $f$ is continuous at $x = 4$, then which of the following statements is NOT true?
(1) $f$ is not differentiable at $x = 4$
(2) $f ^ { \prime } ( 3 ) + f ^ { \prime } ( 5 ) = \frac { 35 } { 4 }$
(3) $f$ is increasing in $\left( - \infty , \frac { 1 } { 8 } \right) \cup ( 8 , \infty )$
(4) $f$ has a local minima at $x = \frac { 1 } { 8 }$
Q74 Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences View
$I = \int _ { \frac { \pi } { 4 } } ^ { \frac { \pi } { 3 } } \left( \frac { 8 \sin x - \sin 2 x } { x } \right) d x$. Then
(1) $\frac { \pi } { 2 } < I < \frac { 3 \pi } { 4 }$
(2) $\frac { \pi } { 5 } < I < \frac { 5 \pi } { 12 }$
(3) $\frac { 5 \pi } { 12 } < I < \frac { \sqrt { 2 } } { 3 } \pi$
(4) $\frac { 3 \pi } { 4 } < I < \pi$
Q75 Areas by integration View
The area of the smaller region enclosed by the curves $y ^ { 2 } = 8 x + 4$ and $x ^ { 2 } + y ^ { 2 } + 4 \sqrt { 3 } x - 4 = 0$ is equal to
(1) $\frac { 1 } { 3 } ( 2 - 12 \sqrt { 3 } + 8 \pi )$
(2) $\frac { 1 } { 3 } ( 2 - 12 \sqrt { 3 } + 6 \pi )$
(3) $\frac { 1 } { 3 } ( 4 - 12 \sqrt { 3 } + 8 \pi )$
(4) $\frac { 1 } { 3 } ( 4 - 12 \sqrt { 3 } + 6 \pi )$
Q76 First order differential equations (integrating factor) Qualitative Analysis of DE Solutions View
Let $y = y _ { 1 } ( x )$ and $y = y _ { 2 } ( x )$ be two distinct solutions of the differential equation $\frac { d y } { d x } = x + y$, with $y _ { 1 } ( 0 ) = 0$ and $y _ { 2 } ( 0 ) = 1$ respectively. Then, the number of points of intersection of $y = y _ { 1 } ( x )$ and $y = y _ { 2 } ( x )$ is
(1) 0
(2) 1
(3) 2
(4) 3
Q77 Vectors 3D & Lines Dot Product Computation View
Let $\vec { a } = \alpha \hat { i } + \hat { j } + \beta \hat { k }$ and $\vec { b } = 3 \hat { i } - 5 \hat { j } + 4 \hat { k }$ be two vectors, such that $\vec { a } \times \vec { b } = - \hat { i } + 9 \hat { i } + 12 \widehat { k }$. Then the projection of $\vec { b } - 2 \vec { a }$ on $\vec { b } + \vec { a }$ is equal to
(1) 2
(2) $\frac { 39 } { 5 }$
(3) 9
(4) $\frac { 46 } { 5 }$
Q78 Vectors 3D & Lines Magnitude of Vector Expression View
Let $\vec { a } = 2 \hat { i } - \hat { j } + 5 \hat { k }$ and $\vec { b } = \alpha \hat { i } + \beta \hat { j } + 2 \widehat { k }$. If $( ( \vec { a } \times \vec { b } ) \times \hat { i } ) \cdot \widehat { k } = \frac { 23 } { 2 }$, then $| \vec { b } \times 2 \hat { j } |$ is equal to
(1) 4
(2) 5
(3) $\sqrt { 21 }$
(4) $\sqrt { 17 }$
Q79 Vectors: Lines & Planes Find Cartesian Equation of a Plane View
If the plane $P$ passes through the intersection of two mutually perpendicular planes $2 x + k y - 5 z = 1$ and $3 k x - k y + z = 5 , k < 3$ and intercepts a unit length on positive $x$-axis, then the intercept made by the plane $P$ on the $y$-axis is
(1) $\frac { 1 } { 11 }$
(2) $\frac { 5 } { 11 }$
(3) 6
(4) 7
Q80 Probability Definitions Probability Involving Algebraic or Number-Theoretic Conditions View
Let $S$ be the sample space of all five digit numbers. If $p$ is the probability that a randomly selected number from $S$, is a multiple of 7 but not divisible by 5 , then $9 p$ is equal to
(1) 1.0146
(2) 1.2085
(3) 1.0285
(4) 1.1521
Q81 Complex Numbers Arithmetic Solving Equations for Unknown Complex Numbers View
Let $S = \left\{ z \in \mathbb { C } : z ^ { 2 } + \bar { z } = 0 \right\}$. Then $\sum _ { z \in S } ( \operatorname { Re } ( z ) + \operatorname { Im } ( z ) )$ is equal to $\_\_\_\_$ .
Q82 Proof Counting solutions or configurations satisfying a quadratic system View
Let $f ( x ) = 2 x ^ { 2 } - x - 1$ and $S = \{ n \in \mathbb { Z } : | f ( n ) | \leq 800 \}$. Then, the value of $\sum _ { n \in S } f ( n )$ is equal to $\_\_\_\_$ .
Q83 Circles Circle Equation Derivation View
If the length of the latus rectum of the ellipse $x ^ { 2 } + 4 y ^ { 2 } + 2 x + 8 y - \lambda = 0$ is 4 , and $l$ is the length of its major axis, then $\lambda + l$ is equal to $\_\_\_\_$ .
Q84 Circles Equation Determination from Geometric Conditions View
An ellipse $E : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ passes through the vertices of the hyperbola $H : \frac { x ^ { 2 } } { 49 } - \frac { y ^ { 2 } } { 64 } = - 1$. Let the major and minor axes of the ellipse $E$ coincide with the transverse and conjugate axes of the hyperbola $H$. Let the product of the eccentricities of $E$ and $H$ be $\frac { 1 } { 2 }$. If $l$ is the length of the latus rectum of the ellipse $E$, then the value of $113 l$ is equal to $\_\_\_\_$ .
Q85 Measures of Location and Spread View
The mean and variance of 10 observations were calculated as 15 and 15 respectively by a student who took by mistake 25 instead of 15 for one observation. Then, the correct standard deviation is $\_\_\_\_$.
Q86 Matrices Determinant and Rank Computation View
Let $S$ be the set containing all $3 \times 3$ matrices with entries from $\{ - 1,0,1 \}$. The total number of matrices $A \in S$ such that the sum of all the diagonal elements of $A ^ { T } A$ is 6 is $\_\_\_\_$ .
Q87 Complex Numbers Argand & Loci Inverse trigonometric equation View
For $k \in \mathbb { R }$, let the solutions of the equation $\cos \left( \sin ^ { - 1 } \left( x \cot \left( \tan ^ { - 1 } \left( \cos \left( \sin ^ { - 1 } x \right) \right) \right) \right) \right) = k , 0 < | x | < \frac { 1 } { \sqrt { 2 } }$ be $\alpha$ and $\beta$, where the inverse trigonometric functions take only principal values. If the solutions of the equation $x ^ { 2 } - b x - 5 = 0$ are $\frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } }$ and $\frac { \alpha } { \beta }$, then $\frac { b } { k ^ { 2 } }$ is equal to $\_\_\_\_$ .
Q88 Tangents, normals and gradients Find tangent line with a specified slope or from an external point View
Let $M$ and $N$ be the number of points on the curve $y ^ { 5 } - 9 x y + 2 x = 0$, where the tangents to the curve are parallel to $x$-axis and $y$-axis, respectively. Then the value of $M + N$ equals $\_\_\_\_$ .
Q89 Differential equations First-Order Linear DE: General Solution View
Let $y = y ( x )$ be the solution curve of the differential equation $\sin \left( 2 x ^ { 2 } \right) \log _ { e } \left( \tan x ^ { 2 } \right) d y + \left( 4 x y - 4 \sqrt { 2 } x \sin \left( x ^ { 2 } - \frac { \pi } { 4 } \right) \right) d x = 0,0 < x < \sqrt { \frac { \pi } { 2 } }$, which passes through the point $\left( \sqrt { \frac { \pi } { 6 } } , 1 \right)$. Then $\left| y \left( \sqrt { \frac { \pi } { 3 } } \right) \right|$ is equal to $\_\_\_\_$ .
Q90 Vectors: Lines & Planes Find Intersection of a Line and a Plane View
Let the line $\frac { x - 3 } { 7 } = \frac { y - 2 } { - 1 } = \frac { z - 3 } { - 4 }$ intersect the plane containing the lines $\frac { x - 4 } { 1 } = \frac { y + 1 } { - 2 } = \frac { z } { 1 }$ and $4 a x - y + 5 z - 7 a = 0 = 2 x - 5 y - z - 3 , a \in \mathbb { R }$ at the point $P ( \alpha , \beta , \gamma )$. Then the value of $\alpha + \beta + \gamma$ equals $\_\_\_\_$ .