jee-main

Papers (169)
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
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2017
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2016
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2015
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2014
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2013
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2012
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2011
jee-main_2011.pdf 13
2010
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2009
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2008
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2007
jee-main_2007.pdf 38
2005
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2004
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2003
jee-main_2003.pdf 9
2002
jee-main_2002.pdf 8
2025 session1_29jan_shift1

29 maths questions

Q1 Measures of Location and Spread View
Let $x_1, x_2, \ldots, x_{10}$ be ten observations such that $\sum_{i=1}^{10}(x_i - 2) = 30$, $\sum_{i=1}^{10}(x_i - \beta)^2 = 98$, $\beta > 2$, and their variance is $\frac{4}{5}$. If $\mu$ and $\sigma^2$ are respectively the mean and the variance of $2(x_1 - 1) + 4\beta$, $2(x_2 - 1) + 4\beta, \ldots, 2(x_{10} - 1) + 4\beta$, then $\frac{\beta\mu}{\sigma^2}$ is equal to:
(1) 100
(2) 120
(3) 110
(4) 90
Q2 Arithmetic Sequences and Series Find Specific Term from Given Conditions View
Consider an A.P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its $11^{\text{th}}$ term is:
(1) 90
(2) 84
(3) 122
(4) 108
Q3 Solving quadratics and applications Solving an equation via substitution to reduce to quadratic form View
The number of solutions of the equation $\left(\frac{9}{x} - \frac{9}{\sqrt{x}} + 2\right)\left(\frac{2}{x} - \frac{7}{\sqrt{x}} + 3\right) = 0$ is:
(1) 2
(2) 3
(3) 1
(4) 4
Q4 Probability Definitions Set Operations View
Define a relation R on the interval $\left[0, \frac{\pi}{2}\right)$ by $x\mathrm{R}y$ if and only if $\sec^2 x - \tan^2 y = 1$. Then R is:
(1) both reflexive and transitive but not symmetric
(2) an equivalence relation
(3) reflexive but neither symmetric nor transitive
(4) both reflexive and symmetric but not transitive
Q5 Circles Intersection of Circles or Circle with Conic View
Two parabolas have the same focus $(4,3)$ and their directrices are the $x$-axis and the $y$-axis, respectively. If these parabolas intersect at the points $A$ and $B$, then $(AB)^2$ is equal to:
(1) 392
(2) 384
(3) 192
(4) 96
Q6 Permutations & Arrangements Forming Numbers with Digit Constraints View
Let P be the set of seven digit numbers with sum of their digits equal to 11. If the numbers in P are formed by using the digits 1, 2 and 3 only, then the number of elements in the set $P$ is:
(1) 173
(2) 164
(3) 158
(4) 161
Q7 Vectors 3D & Lines Line Through Intersection Parallel to Given Direction View
Let $\overrightarrow{\mathrm{a}} = \hat{i} + 2\hat{j} + \hat{k}$ and $\overrightarrow{\mathrm{b}} = 2\hat{i} + 7\hat{j} + 3\hat{k}$. Let $\mathrm{L}_1: \overrightarrow{\mathrm{r}} = (-\hat{i} + 2\hat{j} + \hat{k}) + \lambda\overrightarrow{\mathrm{a}},\ \lambda \in \mathbf{R}$ and $\mathrm{L}_2: \overrightarrow{\mathrm{r}} = (\hat{j} + \hat{k}) + \mu\overrightarrow{\mathrm{b}},\ \mu \in \mathbf{R}$ be two lines. If the line $\mathrm{L}_3$ passes through the point of intersection of $\mathrm{L}_1$ and $\mathrm{L}_2$, and is parallel to $\vec{a} + \vec{b}$, then $\mathrm{L}_3$ passes through the point:
(1) $(5, 17, 4)$
(2) $(2, 8, 5)$
(3) $(8, 26, 12)$
(4) $(-1, -1, 1)$
Q8 Vector Product and Surfaces View
Let $\overrightarrow{\mathrm{a}} = 2\hat{i} - \hat{j} + 3\hat{k}$, $\overrightarrow{\mathrm{b}} = 3\hat{i} - 5\hat{j} + \hat{k}$ and $\overrightarrow{\mathrm{c}}$ be a vector such that $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{c}} = \overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{b}}$ and $(\vec{a} + \vec{c}) \cdot (\vec{b} + \vec{c}) = 168$. Then the maximum value of $|\vec{c}|^2$ is:
(1) 462
(2) 77
(3) 154
(4) 308
Q9 Integration by Substitution Substitution to Evaluate a Definite Integral (Numerical Answer) View
The integral $80\int_0^{\frac{\pi}{4}} \left(\frac{\sin\theta + \cos\theta}{9 + 16\sin 2\theta}\right)d\theta$ is equal to:
(1) $3\log_e 4$
(2) $4\log_e 3$
(3) $6\log_e 4$
(4) $2\log_e 3$
Q10 Conic sections Triangle or Quadrilateral Area and Perimeter with Foci View
Let the ellipse $\mathrm{E}_1: \frac{x^2}{\mathrm{a}^2} + \frac{y^2}{\mathrm{b}^2} = 1,\ \mathrm{a} > \mathrm{b}$ and $\mathrm{E}_2: \frac{x^2}{\mathrm{A}^2} + \frac{y^2}{\mathrm{B}^2} = 1,\ \mathrm{A} < \mathrm{B}$ have same eccentricity $\frac{1}{\sqrt{3}}$. Let the product of their lengths of latus rectums be $\frac{32}{\sqrt{3}}$, and the distance between the foci of $E_1$ be 4. If $E_1$ and $E_2$ meet at $A, B, C$ and $D$, then the area of the quadrilateral $ABCD$ equals:
(1) $\frac{12\sqrt{6}}{5}$
(2) $6\sqrt{6}$
(3) $\frac{18\sqrt{6}}{5}$
(4) $\frac{24\sqrt{6}}{5}$
Q11 Matrices Determinant and Rank Computation View
Let $\mathrm{A} = [\mathrm{a}_{ij}] = \begin{bmatrix} \log_5 128 & \log_4 5 \\ \log_5 8 & \log_4 25 \end{bmatrix}$. If $\mathrm{A}_{ij}$ is the cofactor of $\mathrm{a}_{ij}$, $\mathrm{C}_{ij} = \sum_{\mathrm{k}=1}^{2} \mathrm{a}_{i\mathrm{k}} \mathrm{A}_{j\mathrm{k}}$, $1 \leq i, j \leq 2$, and $\mathrm{C} = [\mathrm{C}_{ij}]$, then $8|\mathrm{C}|$ is equal to:
(1) 288
(2) 222
(3) 242
(4) 262
Q12 Complex Numbers Argand & Loci Distance and Region Optimization on Loci View
Let $|z_1 - 8 - 2i| \leq 1$ and $|z_2 - 2 + 6i| \leq 2$, $z_1, z_2 \in \mathbf{C}$. Then the minimum value of $|z_1 - z_2|$ is:
(1) 13
(2) 10
(3) 3
(4) 7
Q13 Vectors 3D & Lines Shortest Distance Between Two Lines View
Let $\mathrm{L}_1: \frac{x-1}{1} = \frac{y-2}{-1} = \frac{z-1}{2}$ and $\mathrm{L}_2: \frac{x+1}{-1} = \frac{y-2}{2} = \frac{z}{1}$ be two lines. Let $L_3$ be a line passing through the point $(\alpha, \beta, \gamma)$ and be perpendicular to both $L_1$ and $L_2$. If $L_3$ intersects $\mathrm{L}_1$, then $|5\alpha - 11\beta - 8\gamma|$ equals:
(1) 20
(2) 18
(3) 25
(4) 16
Q14 Matrices Determinant and Rank Computation View
Let M and m respectively be the maximum and the minimum values of $$f(x) = \begin{vmatrix} 1+\sin^2 x & \cos^2 x & 4\sin 4x \\ \sin^2 x & 1+\cos^2 x & 4\sin 4x \\ \sin^2 x & \cos^2 x & 1+4\sin 4x \end{vmatrix}, \quad x \in \mathbf{R}.$$ Then $M^4 - m^4$ is equal to:
(1) 1280
(2) 1295
(3) 1215
(4) 1040
Q15 Straight Lines & Coordinate Geometry Reflection and Image in a Line View
Let $ABC$ be a triangle formed by the lines $7x - 6y + 3 = 0$, $x + 2y - 31 = 0$ and $9x - 2y - 19 = 0$. Let the point $(h, k)$ be the image of the centroid of $\triangle ABC$ in the line $3x + 6y - 53 = 0$. Then $h^2 + k^2 + hk$ is equal to:
(1) 47
(2) 37
(3) 36
(4) 40
Q16 Sequences and series, recurrence and convergence Series convergence and power series analysis View
The value of $\lim_{n \rightarrow \infty}\left(\sum_{k=1}^{n} \frac{k^3 + 6k^2 + 11k + 5}{(k+3)!}\right)$ is:
(1) $4/3$
(2) $2$
(3) $7/3$
(4) $5/3$
Q17 Binomial Theorem (positive integer n) Count Integral or Rational Terms in a Binomial Expansion View
The least value of $n$ for which the number of integral terms in the Binomial expansion of $(\sqrt[3]{7} + \sqrt[12]{11})^n$ is 183, is:
(1) 2184
(2) 2196
(3) 2148
(4) 2172
Q18 First order differential equations (integrating factor) View
Let $y = y(x)$ be the solution of the differential equation $\cos x\left(\log_e(\cos x)\right)^2 \mathrm{d}y + \left(\sin x - 3y\sin x\log_e(\cos x)\right)\mathrm{d}x = 0$, $x \in \left(0, \frac{\pi}{2}\right)$. If $y\left(\frac{\pi}{4}\right) = \frac{-1}{\log_e 2}$, then $y\left(\frac{\pi}{6}\right)$ is equal to:
(1) $\frac{1}{\log_e(3) - \log_e(4)}$
(2) $\frac{2}{\log_e(3) - \log_e(4)}$
(3) $\frac{1}{\log_e(4) - \log_e(3)}$
(4) $-\frac{1}{\log_e(4)}$
Q19 Circles Area and Geometric Measurement Involving Circles View
Let the line $x + y = 1$ meet the circle $x^2 + y^2 = 4$ at the points A and B. If the line perpendicular to $AB$ and passing through the mid point of the chord $AB$ intersects the circle at $C$ and $D$, then the area of the quadrilateral ADBC is equal to:
(1) $\sqrt{14}$
(2) $3\sqrt{7}$
(3) $2\sqrt{14}$
(4) $5\sqrt{7}$
Q20 Areas Between Curves Area Between Curves with Parametric or Implicit Region Definition View
Let the area of the region $\{(x, y): 2y \leq x^2 + 3,\ y + |x| \leq 3,\ y \geq |x-1|\}$ be A. Then $6A$ is equal to:
(1) 16
(2) 12
(3) 14
(4) 18
Q21 Reciprocal Trig & Identities View
Let $\mathrm{S} = \{x: \cos^{-1}x = \pi + \sin^{-1}x + \sin^{-1}(2x+1)\}$. Then $\sum_{x \in \mathrm{S}}(2x-1)^2$ is equal to \_\_\_\_ .
Q22 Indefinite & Definite Integrals Finding a Function from an Integral Equation View
Let $f:(0,\infty) \rightarrow \mathbf{R}$ be a twice differentiable function. If for some $\mathrm{a} \neq 0$, $\int_0^1 f(\lambda x)\,\mathrm{d}\lambda = \mathrm{a}f(x)$, $f(1) = 1$ and $f(16) = \frac{1}{8}$, then $16 - f'\left(\frac{1}{16}\right)$ is equal to \_\_\_\_ .
Q23 Permutations & Arrangements Word Permutations with Repeated Letters View
The number of 6-letter words, with or without meaning, that can be formed using the letters of the word MATHS such that any letter that appears in the word must appear at least twice, is \_\_\_\_ .
Q24 Matrices Matrix Power Computation and Application View
Let $\mathrm{S} = \{\mathrm{m} \in \mathbf{Z}: \mathrm{A}^{\mathrm{m}^2} + \mathrm{A}^{\mathrm{m}} = 3\mathrm{I} - \mathrm{A}^{-6}\}$, where $\mathrm{A} = \begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix}$. Then $\mathrm{n}(\mathrm{S})$ is equal to \_\_\_\_ .
Q25 Sign Change & Interval Methods View
Let $[t]$ be the greatest integer less than or equal to $t$. Then the least value of $p \in \mathbf{N}$ for which $$\lim_{x \rightarrow 0^+}\left(x\left(\left[\frac{1}{x}\right] + \left[\frac{2}{x}\right] + \ldots + \left[\frac{\mathrm{p}}{x}\right]\right) - x^2\left(\left[\frac{1}{x^2}\right] + \left[\frac{2^2}{x^2}\right] + \ldots + \left[\frac{9^2}{x^2}\right]\right)\right) \geq 1$$ is equal to \_\_\_\_ .
Q31 Projectiles Range and Complementary Angle Relationships View
Two projectiles are fired with same initial speed from same point on ground at angles of $(45^\circ - \alpha)$ and $(45^\circ + \alpha)$, respectively, with the horizontal direction. The ratio of their maximum heights attained is:
(1) $\frac{1 - \tan\alpha}{1 + \tan\alpha}$
(2) $\frac{1 - \sin 2\alpha}{1 + \sin 2\alpha}$
(3) $\frac{1 + \sin 2\alpha}{1 - \sin 2\alpha}$
(4) $\frac{1 + \sin\alpha}{1 - \sin\alpha}$
Q45 Work done and energy Conservation of energy on frictionless tracks and pendulums View
A body of mass $m$ connected to a massless and unstretchable string goes in a vertical circle of radius $R$ under gravity $g$. The other end of the string is fixed at the center of circle. If velocity at top of circular path is $n\sqrt{gR}$, where $n \geq 1$, then ratio of kinetic energy of the body at bottom to that at top of the circle is:
(1) $\frac{n^2}{n^2 + 4}$
(2) $\frac{n^2 + 4}{n^2}$
(3) $\frac{n+4}{n}$
(4) $\frac{n}{n+4}$
Q47 Moments View
The coordinates of a particle with respect to origin in a given reference frame is $(1,1,1)$ meters. If a force of $\overrightarrow{\mathrm{F}} = \hat{i} - \hat{j} + \hat{k}$ acts on the particle, then the magnitude of torque (with respect to origin) in the $z$-direction is \_\_\_\_ .
Q50 Projectiles Projectile from a Non-Inertial or Moving Frame View
The maximum speed of a boat in still water is $27\,\mathrm{km/h}$. Now this boat is moving downstream in a river flowing at $9\,\mathrm{km/h}$. A man in the boat throws a ball vertically upwards with speed of $10\,\mathrm{m/s}$. Range of the ball as observed by an observer at rest on the river bank, is \_\_\_\_ cm. (Take $g = 10\,\mathrm{m/s^2}$)