jee-main

Papers (169)
2025
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2024
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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 session1_31jan_shift1

28 maths questions

Q61 Inequalities Solve Polynomial/Rational Inequality for Solution Set View
The number of real roots of the equation $\sqrt{x^2 - 4x + 3} + \sqrt{x^2 - 9} = \sqrt{4x^2 - 14x + 6}$, is:
(1) 0
(2) 1
(3) 3
(4) 2
Q62 Complex Numbers Arithmetic Geometric Interpretation and Triangle/Shape Properties View
For all $z \in C$ on the curve $C_1 : |z| = 4$, let the locus of the point $z + \frac{1}{z}$ be the curve $C_2$. Then
(1) the curves $C_1$ and $C_2$ intersect at 4 points
(2) the curves $C_1$ lies inside $C_2$
(3) the curves $C_1$ and $C_2$ intersect at 2 points
(4) the curves $C_2$ lies inside $C_1$
Q63 Geometric Sequences and Series Finite Geometric Sum and Term Relationships View
If the sum and product of four positive consecutive terms of a G.P., are 126 and 1296, respectively, then the sum of common ratios of all such GPs is
(1) 7
(2) $\frac{9}{2}$
(3) 3
(4) 14
Q64 Circles Area and Geometric Measurement Involving Circles View
Let a circle $C_1$ be obtained on rolling the circle $x^2 + y^2 - 4x - 6y + 11 = 0$ upwards 4 units on the tangent $T$ to it at the point $(3,2)$. Let $C_2$ be the image of $C_1$ in $T$. Let $A$ and $B$ be the centers of circles $C_1$ and $C_2$ respectively, and $M$ and $N$ be respectively the feet of perpendiculars drawn from $A$ and $B$ on the $x$-axis. Then the area of the trapezium AMNB is:
(1) $22 + \sqrt{2}$
(2) $41 + \sqrt{2}$
(3) $3 + 2\sqrt{2}$
(4) $21 + \sqrt{2}$
Q65 Conic sections Eccentricity or Asymptote Computation View
If the maximum distance of normal to the ellipse $\frac{x^2}{4} + \frac{y^2}{b^2} = 1$, $b < 2$, from the origin is 1, then the eccentricity of the ellipse is:
(1) $\frac{1}{\sqrt{2}}$
(2) $\frac{\sqrt{3}}{2}$
(3) $\frac{1}{2}$
(4) $\frac{\sqrt{3}}{4}$
Q68 Matrices Matrix Power Computation and Application View
Let $A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 4 & -1 \\ 0 & 12 & -3 \end{pmatrix}$. Then the sum of the diagonal elements of the matrix $(A+I)^{11}$ is equal to:
(1) 6144
(2) 4094
(3) 4097
(4) 2050
Q69 Simultaneous equations View
For the system of linear equations $x + y + z = 6$ $\alpha x + \beta y + 7z = 3$ $x + 2y + 3z = 14$ which of the following is NOT true?
(1) If $\alpha = \beta = 7$, then the system has no solution
(2) If $\alpha = \beta$ and $\alpha \neq 7$ then the system has a unique solution.
(3) There is a unique point $(\alpha, \beta)$ on the line $x + 2y + 18 = 0$ for which the system has infinitely many solutions
(4) For every point $(\alpha, \beta) \neq (7,7)$ on the line $x - 2y + 7 = 0$, the system has infinitely many solutions.
Q70 Standard trigonometric equations Inverse trigonometric equation View
If $\sin^{-1}\frac{\alpha}{17} + \cos^{-1}\frac{4}{5} - \tan^{-1}\frac{77}{36} = 0$, $0 < \alpha < 13$, then $\sin^{-1}(\sin\alpha) + \cos^{-1}(\cos\alpha)$ is equal to
(1) $\pi$
(2) 16
(3) 0
(4) $16 - 5\pi$
Q71 Standard trigonometric equations Inverse trigonometric equation View
Let $y = f(x)$ represent a parabola with focus $\left(-\frac{1}{2}, 0\right)$ and directrix $y = -\frac{1}{2}$. Then $S = \left\{x \in \mathbb{R} : \tan^{-1}\sqrt{f(x)} + \sin^{-1}\sqrt{f(x)+1} = \frac{\pi}{2}\right\}$:
(1) contains exactly two elements
(2) contains exactly one element
(3) is an infinite set
(4) is an empty set
Q72 Composite & Inverse Functions Determine Domain or Range of a Composite Function View
If the domain of the function $f(x) = \frac{x}{1+\lfloor x \rfloor^2}$, where $\lfloor x \rfloor$ is greatest integer $\leq x$, is $[2,6)$, then its range is
(1) $\left\{\frac{5}{26}, \frac{2}{5}\right\} \cup \left\{\frac{9}{29}, \frac{27}{109}, \frac{18}{89}, \frac{9}{53}\right\}$
(2) $\left[\frac{5}{26}, \frac{2}{5}\right]$
(3) $\left\{\frac{5}{37}, \frac{2}{5}\right\} \cup \left\{\frac{9}{29}, \frac{27}{109}, \frac{18}{89}, \frac{9}{53}\right\}$
(4) $\left[\frac{5}{37}, \frac{2}{5}\right]$
Q73 Chain Rule Chain Rule with Composition of Explicit Functions View
Let $y = f(x) = \sin^3\left(\frac{\pi}{3}\cos\left(\frac{\pi}{3\sqrt{2}}\left(-4x^3 + 5x^2 + 1\right)^{3/2}\right)\right)$. Then, at $x = 1$,
(1) $2y' + \sqrt{3}\pi^2 y = 0$
(2) $2y' + 3\pi^2 y = 0$
(3) $\sqrt{2}y' - 3\pi^2 y = 0$
(4) $y' + 3\pi^2 y = 0$
Q74 Stationary points and optimisation Geometric or applied optimisation problem View
A wire of length 20 m is to be cut into two pieces. A piece of length $\ell_1$ is bent to make a square of area $A_1$ and the other piece of length $\ell_2$ is made into a circle of area $A_2$. If $2A_1 + 3A_2$ is minimum then $\pi\ell_1 : \ell_2$ is equal to:
(1) 6:1
(2) $3:1$
(3) $1:6$
(4) $4:1$
Q75 Integration by Substitution Substitution to Prove an Integral Identity or Equality View
Let $\alpha \in (0,1)$ and $\beta = \log_e(1-\alpha)$. Let $P_n(x) = x + \frac{x^2}{2} + \frac{x^3}{3} + \ldots + \frac{x^n}{n}$, $x \in (0,1)$. Then the integral $\int_0^{\alpha} \frac{t^{50}}{1-t}\,dt$ is equal to
(1) $\beta - P_{50}(\alpha)$
(2) $-\beta + P_{50}(\alpha)$
(3) $P_{50}(\alpha) - \beta$
(4) $\beta + P_{50}(\alpha)$
Q76 Integration by Substitution Substitution to Evaluate a Definite Integral (Numerical Answer) View
The value of $\int_{\pi/3}^{\pi/2} \frac{2 + 3\sin x}{\sin x(1 + \cos x)}\,dx$ is equal to
(1) $\frac{7}{2} - \sqrt{3} - \log_e\sqrt{3}$
(2) $-2 + 3\sqrt{3} + \log_e\sqrt{3}$
(3) $\frac{10}{3} - \sqrt{3} + \log_e\sqrt{3}$
(4) $\frac{10}{3} - \sqrt{3} - \log_e\sqrt{3}$
Q77 First order differential equations (integrating factor) View
Let a differentiable function $f$ satisfy $f(x) + \int_3^x \frac{f(t)}{t}\,dt = \sqrt{x+1}$, $x \geq 3$. Then $12f(8)$ is equal to:
(1) 34
(2) 19
(3) 17
(4) 1
Q78 Vectors 3D & Lines Vector Algebra and Triple Product Computation View
Let $\vec{a} = 2\hat{i} + \hat{j} + \hat{k}$, and $\vec{b}$ and $\vec{c}$ be two nonzero vectors such that $|\vec{a} + \vec{b} + \vec{c}| = |\vec{a} + \vec{b} - \vec{c}|$ and $\vec{b} \cdot \vec{c} = 0$. Consider the following two statements: $A$: $|\vec{a} + \lambda\vec{c}| \geq |\vec{a}|$ for all $\lambda \in \mathbb{R}$. $B$: $\vec{a}$ and $\vec{c}$ are always parallel.
(1) only (B) is correct
(2) neither (A) nor (B) is correct
(3) only (A) is correct
(4) both (A) and (B) are correct.
Q79 Vectors 3D & Lines Shortest Distance Between Two Lines View
Let the shortest distance between the lines $L: \frac{x-5}{-2} = \frac{y-\lambda}{0} = \frac{z+\lambda}{1}$, $\lambda \geq 0$ and $L_1: x+1 = y-1 = 4-z$ be $2\sqrt{6}$. If $(\alpha, \beta, \gamma)$ lies on $L$, then which of the following is NOT possible?
(1) $\alpha + 2\gamma = 24$
(2) $2\alpha + \gamma = 7$
(3) $2\alpha - \gamma = 9$
(4) $\alpha - 2\gamma = 19$
Q80 Conditional Probability Bayes' Theorem with Production/Source Identification View
A bag contains 6 balls. Two balls are drawn from it at random and both are found to be black. The probability that the bag contains at least 5 black balls is
(1) $\frac{5}{3}$
(2) $\frac{2}{7}$
(3) $\frac{3}{7}$
(4) $\frac{5}{6}$
Q81 Permutations & Arrangements Dictionary Order / Rank of a Permutation View
Let 5 digit numbers be constructed using the digits $0, 2, 3, 4, 7, 9$ with repetition allowed, and are arranged in ascending order with serial numbers. Then the serial number of the number 42923 is $\underline{\hspace{1cm}}$.
Q82 Arithmetic Sequences and Series Telescoping or Non-Standard Summation Involving an AP View
Let $a_1, a_2, \ldots, a_n$ be in A.P. If $a_5 = 2a_7$ and $a_{11} = 18$, then $12\left(\frac{1}{\sqrt{a_{10}} + \sqrt{a_{11}}} + \frac{1}{\sqrt{a_{11}} + \sqrt{a_{12}}} + \ldots + \frac{1}{\sqrt{a_{17}} + \sqrt{a_{18}}}\right)$ is equal to $\underline{\hspace{1cm}}$.
Q83 Permutations & Arrangements Forming Numbers with Digit Constraints View
Number of 4-digit numbers that are less than or equal to 2800 and either divisible by 3 or by 11, is equal to $\underline{\hspace{1cm}}$.
Q84 Binomial Theorem (positive integer n) Find a Specific Coefficient in a Single Binomial Expansion View
Let $\alpha > 0$ be the smallest number such that the expansion of $\left(x^{\frac{2}{3}} + \frac{2}{x^3}\right)^{30}$ has a term $\beta x^{-\alpha}$, $\beta \in \mathbb{N}$. Then $\alpha$ is equal to $\underline{\hspace{1cm}}$.
Q85 Number Theory Modular Arithmetic Computation View
The remainder on dividing $5^{99}$ by 11 is $\underline{\hspace{1cm}}$.
Q86 Measures of Location and Spread View
If the variance of the frequency distribution
$x_i$2345678
Frequency $f_i$3616$\alpha$956

is 3, then $\alpha$ is equal to $\underline{\hspace{1cm}}$.
Q87 Areas by integration View
Let for $x \in \mathbb{R}$, $f(x) = \frac{x + |x|}{2}$ and $g(x) = \begin{cases} x, & x < 0 \\ x^2, & x \geq 0 \end{cases}$. Then area bounded by the curve $y = f(g(x))$ and the lines $y = 0$, $2y - x = 15$ is equal to $\underline{\hspace{1cm}}$.
Q88 Vectors 3D & Lines Vector Algebra and Triple Product Computation View
Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{a}| = \sqrt{14}$, $|\vec{b}| = \sqrt{6}$ and $|\vec{a} \times \vec{b}| = \sqrt{48}$. Then $(\vec{a} \cdot \vec{b})^2$ is equal to $\underline{\hspace{1cm}}$.
Q89 Vectors 3D & Lines Distance from a Point to a Line (Show/Compute) View
Let the line $L: \frac{x-1}{2} = \frac{y+1}{-1} = \frac{z-3}{1}$ intersect the plane $2x + y + 3z = 16$ at the point $P$. Let the point $Q$ be the foot of perpendicular from the point $R(1,-1,-3)$ on the line $L$. If $\alpha$ is the area of triangle $PQR$, then $\alpha^2$ is equal to $\underline{\hspace{1cm}}$.
Q90 Vectors 3D & Lines Dihedral Angle Computation View
Let $\theta$ be the angle between the planes $P_1: \vec{r} \cdot (\hat{i} + \hat{j} + 2\hat{k}) = 9$ and $P_2: \vec{r} \cdot (2\hat{i} - \hat{j} + \hat{k}) = 15$. Let $L$ be the line that meets $P_2$ at the point $(4,-2,5)$ and makes an angle $\theta$ with the normal of $P_2$. If $\alpha$ is the angle between $L$ and $P_2$, then $\tan^2\theta \cdot \cot^2\alpha$ is equal to $\underline{\hspace{1cm}}$.