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
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
2015 10apr

30 maths questions

Q61 Permutations & Arrangements Forming Numbers with Digit Constraints View
The number of integers greater than 6,000 that can be formed, using the digits 3, 5, 6, 7 and 8, without repetition, is:
(1) 216
(2) 192
(3) 120
(4) 72
Q62 Geometric Sequences and Series Arithmetic-Geometric Sequence Interplay View
If $m$ is the A.M. of two distinct real numbers $l$ and $n$ $(l, n > 1)$ and $G_1, G_2$ and $G_3$ are three geometric means between $l$ and $n$, then $G_1^4 + 2G_2^4 + G_3^4$ equals:
(1) $4l^2 m n$
(2) $4lm^2 n$
(3) $4lmn^2$
(4) $4l^2 m^2 n^2$
Q63 3x3 Matrices Linear System Existence and Uniqueness via Determinant View
The set of all values of $\lambda$ for which the system of linear equations: $2x_1 - 2x_2 + x_3 = \lambda x_1$ $2x_1 - 3x_2 + 2x_3 = \lambda x_2$ $-x_1 + 2x_2 = \lambda x_3$ has a non-trivial solution:
(1) is an empty set
(2) is a singleton
(3) contains two elements
(4) contains more than two elements
Q64 3x3 Matrices Direct Determinant Computation View
The number of distinct real roots of $\begin{vmatrix} \sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x \end{vmatrix} = 0$ in the interval $-\frac{\pi}{4} \leq x \leq \frac{\pi}{4}$ is:
(1) 0
(2) 2
(3) 1
(4) 3
Q65 3x3 Matrices Matrix Algebraic Properties and Abstract Reasoning View
If $A = \begin{pmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b \end{pmatrix}$ is a matrix satisfying the equation $AA^T = 9I$, where $I$ is $3 \times 3$ identity matrix, then the ordered pair $(a, b)$ is equal to:
(1) $(2, -1)$
(2) $(-2, 1)$
(3) $(2, 1)$
(4) $(-2, -1)$
Q66 Circles Tangent Lines and Tangent Lengths View
The number of common tangents to the circles $x^2 + y^2 - 4x - 6y - 12 = 0$ and $x^2 + y^2 + 6x + 18y + 26 = 0$, is:
(1) 1
(2) 2
(3) 3
(4) 4
Q67 Areas Between Curves Area Between Curves with Parametric or Implicit Region Definition View
The area (in sq. units) of the region described by $\{(x, y) : y^2 \leq 2x \text{ and } y \geq 4x - 1\}$ is:
(1) $\frac{7}{32}$
(2) $\frac{5}{64}$
(3) $\frac{15}{64}$
(4) $\frac{9}{32}$
Q68 First order differential equations (integrating factor) View
Let $y(x)$ be the solution of the differential equation $(x \log x) \frac{dy}{dx} + y = 2x \log x$, $(x \geq 1)$. Then $y(e)$ is equal to:
(1) $e$
(2) $0$
(3) $2$
(4) $2e$
Q69 Integration by Substitution Substitution to Compute an Indefinite Integral with Initial Condition View
The integral $\int \frac{dx}{x^2(x^4+1)^{3/4}}$ equals:
(1) $-\left(1 + \frac{1}{x^4}\right)^{1/4} + C$
(2) $-\left(\frac{x^4+1}{x^4}\right)^{1/4} + C$
(3) $\left(\frac{x^4+1}{x^4}\right)^{1/4} + C$
(4) $\left(1 + \frac{1}{x^4}\right)^{1/4} + C$
Q70 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
The integral $\int_{\pi/4}^{3\pi/4} \frac{dx}{1 + \cos x}$ is equal to:
(1) $-1$
(2) $-2$
(3) $2$
(4) $4$
Q71 Composite & Inverse Functions Evaluate Composition from Algebraic Definitions View
Let $f(x) = x^2$, $g(x) = \sin x$ for all $x \in \mathbb{R}$ and $h(x) = (gof)(x) = g(f(x))$. Statement I: $h$ is not differentiable at $x = 0$. Statement II: $(hog)(x) = \sin^2(\sin x)$. Which of the following is correct?
(1) Statement I is false, Statement II is true
(2) Statement I is true, Statement II is false
(3) Both Statement I and Statement II are true
(4) Both Statement I and Statement II are false
Q72 Stationary points and optimisation Determine parameters from given extremum conditions View
Let $f: \mathbb{R} \to \mathbb{R}$ be defined by $f(x) = \begin{cases} k - 2x, & \text{if } x \leq -1 \\ 2x + 3, & \text{if } x > -1 \end{cases}$. If $f$ has a local minimum at $x = -1$, then a possible value of $k$ is:
(1) $0$
(2) $-\frac{1}{2}$
(3) $-1$
(4) $1$
Q73 Chain Rule Piecewise Function Differentiability Analysis View
If the function $g(x) = \begin{cases} k\sqrt{x+1}, & 0 \leq x \leq 3 \\ mx + 2, & 3 < x \leq 5 \end{cases}$ is differentiable, then the value of $k + m$ is:
(1) $2$
(2) $\frac{16}{5}$
(3) $\frac{10}{3}$
(4) $4$
Q74 Implicit equations and differentiation Compute slope at a point via implicit differentiation (single-step) View
The normal to the curve $x^2 + 2xy - 3y^2 = 0$ at $(1, 1)$:
(1) does not meet the curve again
(2) meets the curve again in the second quadrant
(3) meets the curve again in the third quadrant
(4) meets the curve again in the fourth quadrant
Q75 Quadratic trigonometric equations View
If $12 \cot^2\theta - 31 \csc\theta + 32 = 0$, then the value of $\sin\theta$ is:
(1) $\frac{3}{5}$ or $1$
(2) $\frac{2}{3}$ or $-\frac{2}{3}$
(3) $\frac{4}{5}$ or $\frac{3}{4}$
(4) $\pm\frac{1}{2}$
Q76 Complex Numbers Argand & Loci Circle Equation and Properties via Complex Number Manipulation View
A complex number $z$ is said to be unimodular if $|z| = 1$. Suppose $z_1$ and $z_2$ are complex numbers such that $\frac{z_1 - 2z_2}{2 - z_1\bar{z}_2}$ is unimodular and $z_2$ is not unimodular. Then the point $z_1$ lies on a:
(1) straight line parallel to $x$-axis
(2) straight line parallel to $y$-axis
(3) circle of radius 2
(4) circle of radius $\sqrt{2}$
Q77 Roots of polynomials Vieta's formulas: compute symmetric functions of roots View
Let $\alpha$ and $\beta$ be the roots of equation $px^2 + qx + r = 0$, $p \neq 0$. If $p$, $q$, $r$ are in A.P. and $\frac{1}{\alpha} + \frac{1}{\beta} = 4$, then the value of $|\alpha - \beta|$ is:
(1) $\frac{\sqrt{61}}{9}$
(2) $\frac{2\sqrt{17}}{9}$
(3) $\frac{\sqrt{34}}{9}$
(4) $\frac{2\sqrt{13}}{9}$
Q78 Binomial Theorem (positive integer n) Count Integral or Rational Terms in a Binomial Expansion View
The sum of coefficients of integral powers of $x$ in the binomial expansion of $(1 - 2\sqrt{x})^{50}$ is:
(1) $\frac{1}{2}(3^{50} + 1)$
(2) $\frac{1}{2}(3^{50})$
(3) $\frac{1}{2}(3^{50} - 1)$
(4) $\frac{1}{2}(2^{50} + 1)$
Q79 Addition & Double Angle Formulae Multi-Step Composite Problem Using Identities View
If $\tan\left(\frac{\pi}{4} + \frac{\theta}{2}\right) = \tan^3\left(\frac{\pi}{4} + \frac{\alpha}{2}\right)$, then $\sin\theta = $:
(1) $\frac{\sin\alpha(3 + \sin^2\alpha)}{1 + 3\sin^2\alpha}$
(2) $\frac{\sin\alpha(3 - \sin^2\alpha)}{1 + 3\sin^2\alpha}$
(3) $\frac{\sin\alpha(3 + \cos^2\alpha)}{1 + 3\cos^2\alpha}$
(4) $\frac{\sin\alpha(3 - \cos^2\alpha)}{1 + 3\cos^2\alpha}$
Q80 Vectors 3D & Lines Line-Plane Intersection View
The distance of the point $(1, 0, 2)$ from the point of intersection of the line $\frac{x-2}{3} = \frac{y+1}{4} = \frac{z-2}{12}$ and the plane $x - y + z = 16$, is:
(1) $2\sqrt{14}$
(2) $8$
(3) $3\sqrt{21}$
(4) $13$
Q81 Vectors: Lines & Planes Find Cartesian Equation of a Plane View
The equation of the plane containing the line $2x - 5y + z = 3$; $x + y + 4z = 5$, and parallel to the plane $x + 3y + 6z = 1$, is:
(1) $2x + 6y + 12z = 13$
(2) $x + 3y + 6z = -7$
(3) $x + 3y + 6z = 7$
(4) $2x + 6y + 12z = -13$
Q82 Vectors: Cross Product & Distances View
Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be three non-zero vectors such that no two of them are collinear and $(\vec{a} \times \vec{b}) \times \vec{c} = \frac{1}{3}|\vec{b}||\vec{c}|\vec{a}$. If $\theta$ is the angle between vectors $\vec{b}$ and $\vec{c}$, then a value of $\sin\theta$ is:
(1) $\frac{2\sqrt{2}}{3}$
(2) $-\frac{\sqrt{2}}{3}$
(3) $\frac{2}{3}$
(4) $-\frac{2\sqrt{3}}{3}$
Q83 Measures of Location and Spread View
The mean of the data set comprising of 16 observations is 16. If one of the observation valued 16 is deleted and three new observations valued 3, 4 and 5 are added to the data, then the mean of the resultant data, is:
(1) 16.8
(2) 15.8
(3) 14.0
(4) 16.0
Q84 Binomial Theorem (positive integer n) Determine Parameters from Conditions on Coefficients or Terms View
If $A$ and $B$ are coefficients of $x^n$ in the expansions of $(1+x)^{2n}$ and $(1+x)^{2n-1}$ respectively, then $\frac{A}{B}$ equals:
(1) $1$
(2) $2$
(3) $\frac{1}{2}$
(4) $\frac{1}{n}$
Q85 Permutations & Arrangements Lattice Path / Grid Route Counting View
The number of points, having both co-ordinates as integers, that lie in the interior of the triangle with vertices $(0, 0)$, $(0, 41)$ and $(41, 0)$, is:
(1) 820
(2) 780
(3) 901
(4) 861
Q86 Circles Circle-Related Locus Problems View
Locus of the image of the point $(2, 3)$ in the line $(2x - 3y + 4) + k(x - 2y + 3) = 0$, $k \in \mathbb{R}$, is a:
(1) straight line parallel to $x$-axis
(2) straight line parallel to $y$-axis
(3) circle of radius $\sqrt{2}$
(4) circle of radius $\sqrt{3}$
Q87 Arithmetic Sequences and Series Counting or Combinatorial Problems on APs View
The number of terms in an A.P. is even; the sum of the odd terms in it is 24 and that the even terms is 30. If the last term exceeds the first term by $10\frac{1}{2}$, then the number of terms in the A.P. is:
(1) 4
(2) 8
(3) 12
(4) 16
Q88 Proof Direct Proof of a Stated Identity or Equality View
The negation of $\sim s \vee (\sim r \wedge s)$ is equivalent to:
(1) $s \wedge \sim r$
(2) $s \wedge (r \wedge \sim s)$
(3) $s \vee (r \vee \sim s)$
(4) $s \wedge r$
Q89 Measures of Location and Spread View
The variance of first 50 even natural numbers is:
(1) $833$
(2) $437$
(3) $\frac{833}{4}$
(4) $833$
Q90 Sine and Cosine Rules Heights and distances / angle of elevation problem View
If the angles of elevation of the top of a tower from three collinear points A, B and C, on a line leading to the foot of the tower, are $30^\circ$, $45^\circ$ and $60^\circ$ respectively, then the ratio $AB : BC$, is:
(1) $\sqrt{3} : 1$
(2) $\sqrt{3} : \sqrt{2}$
(3) $1 : \sqrt{3}$
(4) $2 : 3$