jee-main

Papers (169)
2025
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2024
session1_01feb_shift1 4 session1_01feb_shift2 22 session1_27jan_shift1 28 session1_27jan_shift2 30 session1_29jan_shift1 30 session1_29jan_shift2 23 session1_30jan_shift1 17 session1_30jan_shift2 30 session1_31jan_shift1 16 session1_31jan_shift2 15 session2_04apr_shift1 4 session2_04apr_shift2 30 session2_05apr_shift1 4 session2_05apr_shift2 30 session2_06apr_shift1 22 session2_06apr_shift2 30 session2_08apr_shift1 30 session2_08apr_shift2 30 session2_09apr_shift1 5 session2_09apr_shift2 30
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
session1_24jun_shift1 20 session1_24jun_shift2 25 session1_25jun_shift1 14 session1_25jun_shift2 17 session1_26jun_shift1 26 session1_26jun_shift2 23 session1_27jun_shift1 4 session1_27jun_shift2 29 session1_28jun_shift1 13 session1_29jun_shift1 20 session1_29jun_shift2 5 session2_25jul_shift1 29 session2_25jul_shift2 22 session2_26jul_shift1 29 session2_26jul_shift2 24 session2_27jul_shift1 26 session2_27jul_shift2 29 session2_28jul_shift1 12 session2_28jul_shift2 29 session2_29jul_shift1 18 session2_29jul_shift2 17
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
session1_07jan_shift1 26 session1_07jan_shift2 17 session1_08jan_shift1 5 session1_08jan_shift2 12 session1_09jan_shift1 22 session1_09jan_shift2 18 session2_02sep_shift1 19 session2_02sep_shift2 17 session2_03sep_shift1 21 session2_03sep_shift2 9 session2_04sep_shift1 10 session2_04sep_shift2 24 session2_05sep_shift1 23 session2_05sep_shift2 27 session2_06sep_shift1 13 session2_06sep_shift2 10
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
06apr 28 09apr 28 11apr 4 12apr 5 19apr 29
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
2023 session2_10apr_shift1

29 maths questions

Q61 Complex Numbers Arithmetic Systems of Equations via Real and Imaginary Part Matching View
Let the complex number $z = x + iy$ be such that $\frac{2z - 3i}{2z + i}$ is purely imaginary. If $x + y^2 = 0$, then $y^4 + y^2 - y$ is equal to
(1) $\frac{2}{3}$
(2) $\frac{3}{2}$
(3) $\frac{3}{4}$
(4) $\frac{1}{3}$
Q62 Geometric Sequences and Series Finite Geometric Sum and Term Relationships View
Let the first term $a$ and the common ratio $r$ of a geometric progression be positive integers. If the sum of squares of its first three terms is 33033, then the sum of these three terms is equal to
(1) 241
(2) 231
(3) 210
(4) 220
Q63 Binomial Theorem (positive integer n) Determine Parameters from Conditions on Coefficients or Terms View
If the coefficient of $x^7$ in $\left(ax - \frac{1}{bx^2}\right)^{13}$ and the coefficient of $x^{-5}$ in $\left(ax + \frac{1}{bx^2}\right)^{13}$ are equal, then $a^4 b^4$ is equal to:
(1) 11
(2) 44
(3) 22
(4) 33
Q64 Trig Proofs Trigonometric Identity Simplification View
$96 \cos\frac{\pi}{33} \cos\frac{2\pi}{33} \cos\frac{4\pi}{33} \cos\frac{8\pi}{33} \cos\frac{16\pi}{33}$ is equal to
(1) 3
(2) 1
(3) 4
(4) 2
Q65 Circles Circle-Related Locus Problems View
A line segment $AB$ of length $\lambda$ moves such that the points $A$ and $B$ remain on the periphery of a circle of radius $\lambda$. Then the locus of the point, that divides the line segment $AB$ in the ratio $2:3$, is a circle of radius
(1) $\frac{3}{5}\lambda$
(2) $\frac{2}{3}\lambda$
(3) $\frac{\sqrt{19}}{5}\lambda$
(4) $\frac{\sqrt{19}}{7}\lambda$
Q66 Circles Area and Geometric Measurement Involving Circles View
Let the ellipse $E$: $x^2 + 9y^2 = 9$ intersect the positive $x$- and $y$-axes at the points $A$ and $B$ respectively. Let the major axis of $E$ be a diameter of the circle $C$. Let the line passing through $A$ and $B$ meet the circle $C$ at the point $P$. If the area of the triangle with vertices $A$, $P$ and the origin $O$ is $\frac{m}{n}$, where $m$ and $n$ are coprime, then $m - n$ is equal to
(1) 16
(2) 15
(3) 17
(4) 18
Q68 Matrices Determinant and Rank Computation View
If $A$ is a $3 \times 3$ matrix and $|A| = 2$, then $|3\, \text{adj}(3A)| \cdot |A^2|$ is equal to
(1) $3^{12} \cdot 6^{11}$
(2) $3^{12} \cdot 6^{10}$
(3) $3^{10} \cdot 6^{11}$
(4) $3^{11} \cdot 6^{10}$
Q69 Simultaneous equations View
For the system of linear equations $$2x - y + 3z = 5$$ $$3x + 2y - z = 7$$ $$4x + 5y + \alpha z = \beta$$ which of the following is NOT correct?
(1) The system has infinitely many solutions for $\alpha = -5$ and $\beta = 9$
(2) The system has infinitely many solutions for $\alpha = -6$ and $\beta = 9$
(3) The system is inconsistent for $\alpha = -5$ and $\beta = 8$
(4) The system has a unique solution for $\alpha \neq -5$ and $\beta = 8$
Q70 Curve Sketching Range and Image Set Determination View
If $f(x) = \frac{\tan^{-1}x + \log_e 123}{x \log_e 1234 - \tan^{-1}x}$, $x > 0$, then the least value of $f(f(x)) + f\!\left(f\!\left(\frac{4}{x}\right)\right)$ is
(1) 0
(2) 8
(3) 2
(4) 4
Q71 Stationary points and optimisation Geometric or applied optimisation problem View
A square piece of tin of side 30 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. If the volume of the box is maximum, then its surface area (in $\mathrm{cm}^2$) is equal to
(1) 800
(2) 675
(3) 1025
(4) 900
Q72 Integration by Substitution Substitution to Compute an Indefinite Integral with Initial Condition View
If $I(x) = \int e^{\sin^2 x} \cos x (\sin 2x - \sin x)\, dx$ and $I(0) = 1$, then $I\!\left(\frac{\pi}{3}\right)$ is equal to
(1) $-\frac{1}{2}e^{\frac{3}{4}}$
(2) $\frac{1}{2}e^{\frac{3}{4}}$
(3) $-e^{\frac{3}{4}}$
(4) $e^{\frac{3}{4}}$
Q73 Differential equations Integral Equations Reducible to DEs View
Let $f$ be a differentiable function such that $x^2 f(x) - x = 4\int_0^x t\, f(t)\, dt$, $f(1) = \frac{2}{3}$. Then $18\, f(3)$ is equal to
(1) 210
(2) 160
(3) 150
(4) 180
Q74 Differential equations Solving Separable DEs with Initial Conditions View
The slope of tangent at any point $(x, y)$ on a curve $y = y(x)$ is $\frac{x^2 + y^2}{2xy}$, $x > 0$. If $y(2) = 0$, then a value of $y(8)$ is
(1) $-4\sqrt{2}$
(2) $2\sqrt{3}$
(3) $-2\sqrt{3}$
(4) $4\sqrt{3}$
Q75 Vectors Introduction & 2D Expressing a Vector as a Linear Combination View
An arc $PQ$ of a circle subtends a right angle at its centre $O$. The mid point of the arc $PQ$ is $R$. If $\overrightarrow{OP} = \vec{u}$, $\overrightarrow{OR} = \vec{v}$ and $\overrightarrow{OQ} = \alpha\vec{u} + \beta\vec{v}$, then $\alpha$, $\beta^2$ are the roots of the equation
(1) $x^2 + x - 2 = 0$
(2) $x^2 - x - 2 = 0$
(3) $3x^2 - 2x - 1 = 0$
(4) $3x^2 + 2x - 1 = 0$
Q76 Vectors Introduction & 2D Dot Product Computation View
Let $O$ be the origin and the position vector of the point $P$ be $-\hat{i} - 2\hat{j} + 3\hat{k}$. If the position vectors of the points $A$, $B$ and $C$ are $-2\hat{i} + \hat{j} - 3\hat{k}$, $2\hat{i} + 4\hat{j} - 2\hat{k}$ and $-4\hat{i} + 2\hat{j} - \hat{k}$ respectively, then the projection of the vector $\overrightarrow{OP}$ on a vector perpendicular to the vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$ is
(1) 3
(2) $\frac{8}{3}$
(3) $\frac{7}{3}$
(4) $\frac{10}{3}$
Q77 Vectors 3D & Lines Section Division and Coordinate Computation View
Let two vertices of a triangle $ABC$ be $(2, 4, 6)$ and $(0, -2, -5)$, and its centroid be $(2, 1, -1)$. If the image of the third vertex in the plane $x + 2y + 4z = 11$ is $(\alpha, \beta, \gamma)$, then $\alpha\beta + \beta\gamma + \gamma\alpha$ is equal to
(1) 70
(2) 76
(3) 74
(4) 72
Q78 Vectors 3D & Lines Shortest Distance Between Two Lines View
The shortest distance between the lines $\frac{x+2}{1} = \frac{y}{-2} = \frac{z-5}{2}$ and $\frac{x-4}{1} = \frac{y-1}{2} = \frac{z+3}{0}$ is
(1) 8
(2) 6
(3) 7
(4) 9
Q79 Vectors 3D & Lines Line-Plane Intersection View
Let $P$ be the point of intersection of the line $\frac{x+3}{3} = \frac{y+2}{1} = \frac{1-z}{2}$ and the plane $x + y + z = 2$. If the distance of the point $P$ from the plane $3x - 4y + 12z = 32$ is $q$, then $q$ and $2q$ are the roots of the equation
(1) $x^2 - 18x - 72 = 0$
(2) $x^2 - 18x + 72 = 0$
(3) $x^2 + 18x + 72 = 0$
(4) $x^2 + 18x - 72 = 0$
Q80 Discrete Probability Distributions Probability Computation for Compound or Multi-Stage Random Experiments View
Let $N$ denote the sum of the numbers obtained when two dice are rolled. If the probability that $2^N < N!$ is $\frac{m}{n}$ where $m$ and $n$ are coprime, then $4m - 3n$ is equal to
(1) 6
(2) 12
(3) 10
(4) 8
Q81 Laws of Logarithms Solve a Logarithmic Equation View
Let $a, b, c$ be three distinct positive real numbers such that $2a^{\log_e a} = bc^{\log_e b}$ and $b^{\log_e 2} = a^{\log_e c}$. Then $6a + 5bc$ is equal to $\_\_\_\_$.
Q82 Permutations & Arrangements Linear Arrangement with Constraints View
The number of permutations, of the digits $1, 2, 3, \ldots, 7$ without repetition, which neither contain the string 153 nor the string 2467, is $\_\_\_\_$.
Q83 Arithmetic Sequences and Series Counting or Combinatorial Problems on APs View
Some couples participated in a mixed doubles badminton tournament. If the number of matches played, so that no couple played in a match, is 840, then the total number of persons who participated in the tournament is $\_\_\_\_$.
Q84 Arithmetic Sequences and Series Compute Partial Sum of an Arithmetic Sequence View
The sum of all those terms, of the arithmetic progression $3, 8, 13, \ldots, 373$, which are not divisible by 3, is equal to $\_\_\_\_$.
Q85 Binomial Theorem (positive integer n) Find a Specific Coefficient in a Product of Binomial/Polynomial Expressions View
The coefficient of $x^7$ in $(1 - x + 2x^3)^{10}$ is $\_\_\_\_$.
Q86 Circles Tangent Lines and Tangent Lengths View
Let a common tangent to the curves $y^2 = 4x$ and $(x-4)^2 + y^2 = 16$ touch the curves at the points $P$ and $Q$. Then $PQ^2$ is equal to $\_\_\_\_$.
Q87 Measures of Location and Spread View
If the mean of the frequency distribution
Class :$0-10$$10-20$$20-30$$30-40$$40-50$
Frequency :23$x$5

is 28, then its variance is $\_\_\_\_$.
Q88 Sign Change & Interval Methods View
The number of elements in the set $\{n \in \mathbb{Z} : n^2 - 10n + 19 < 6\}$ is $\_\_\_\_$.
Q89 Differentiation from First Principles View
Let $f: (-2, 2) \rightarrow \mathbb{R}$ be defined by $f(x) = \begin{cases} x\lfloor x\rfloor, & 0 \leq x < 2 \\ (x-1)\lfloor x\rfloor, & -2 < x < 0 \end{cases}$ where $\lfloor x \rfloor$ denotes the greatest integer function. If $m$ and $n$ respectively are the number of points in $(-2, 2)$ at which $y = f(x)$ is not continuous and not differentiable, then $m + n$ is equal to $\_\_\_\_$.
Q90 Areas by integration View
Let $y = p(x)$ be the parabola passing through the points $(-1, 0)$, $(0, 1)$ and $(1, 0)$. If the area of the region $\{(x, y) : (x+1)^2 + (y-1)^2 \leq 1,\; y \leq p(x)\}$ is $A$, then $12\pi - 4A$ is equal to $\_\_\_\_$.