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
2016 09apr

30 maths questions

Q61 Arithmetic Sequences and Series Arithmetic-Geometric Hybrid Problem View
If the 2nd, 5th and 9th terms of a non-constant A.P. are in G.P., then the common ratio of this G.P. is:
(1) $\frac{8}{5}$
(2) $\frac{4}{3}$
(3) $1$
(4) $\frac{7}{4}$
Q62 Binomial Theorem (positive integer n) Determine Parameters from Conditions on Coefficients or Terms View
If the number of terms in the expansion of $\left(1 - \frac{2}{x} + \frac{4}{x^2}\right)^n$, $x \neq 0$, is 28, then the sum of the coefficients of all the terms in this expansion, is:
(1) 64
(2) 2187
(3) 243
(4) 729
Q63 Permutations & Arrangements Dictionary Order / Rank of a Permutation View
If all the words (with or without meaning) having five letters, formed using the letters of the word SMALL and arranged as in a dictionary; then the position of the word SMALL is:
(1) 46th
(2) 59th
(3) 52nd
(4) 58th
Q64 Binomial Theorem (positive integer n) Evaluate a Summation Involving Binomial Coefficients View
The sum $\sum_{r=1}^{9} \frac{10!}{r!(10-r)!}$ is equal to:
(1) $2^{10} - 2$
(2) $2^{10} - 1$
(3) $2^9$
(4) $2^{10}$
Q65 Matrices Determinant and Rank Computation View
If $A = \begin{bmatrix} 5a & -b \\ 3 & 2 \end{bmatrix}$ and $A$ adj $A = A A^T$, then $5a + b$ is equal to:
(1) $-1$
(2) $5$
(3) $4$
(4) $13$
Q66 Matrices Linear System and Inverse Existence View
The system of linear equations \begin{align*} x + \lambda y - z &= 0 \lambda x - y - z &= 0 x + y - \lambda z &= 0 \end{align*} has a non-trivial solution for:
(1) infinitely many values of $\lambda$
(2) exactly one value of $\lambda$
(3) exactly two values of $\lambda$
(4) exactly three values of $\lambda$
Q67 Matrices Determinant and Rank Computation View
If $A = \begin{bmatrix} 2 & -3 \\ -4 & 1 \end{bmatrix}$, then adj$(3A^2 + 12A)$ is equal to:
(1) $\begin{bmatrix} 72 & -84 \\ -63 & 51 \end{bmatrix}$
(2) $\begin{bmatrix} 51 & 63 \\ 84 & 72 \end{bmatrix}$
(3) $\begin{bmatrix} 51 & 84 \\ 63 & 72 \end{bmatrix}$
(4) $\begin{bmatrix} 72 & -63 \\ -84 & 51 \end{bmatrix}$
Q68 Proof Direct Proof of a Stated Identity or Equality View
The Boolean expression $(p \wedge \sim q) \vee q \vee (\sim p \wedge q)$ is equivalent to:
(1) $p \wedge q$
(2) $p \vee q$
(3) $p \vee \sim q$
(4) $\sim p \wedge q$
Q69 Proof Direct Proof of a Stated Identity or Equality View
The contrapositive of the following statement, "If the side of a square doubles, then its area increases four times", is:
(1) If the area of a square increases four times, then its side is not doubled.
(2) If the area of a square does not increase four times, then its side is not doubled.
(3) If the area of a square does not increase four times, then its side is doubled.
(4) If the side of a square is not doubled, then its area does not increase four times.
Q70 Conic sections Tangent and Normal Line Problems View
The eccentricity of an ellipse whose centre is at the origin is $\frac{1}{2}$. If one of its directrices is $x = -4$, then the equation of the normal to it at $\left(1, \frac{3}{2}\right)$ is:
(1) $4x - 2y = 1$
(2) $4x + 2y = 7$
(3) $x + 2y = 4$
(4) $2y - x = 2$
Q71 Areas Between Curves Area Involving Conic Sections or Circles View
The area (in sq. units) of the region $\{(x, y) : y^2 \geq 2x$ and $x^2 + y^2 \leq 4x, x \geq 0, y \geq 0\}$ is:
(1) $\pi - \frac{4\sqrt{2}}{3}$
(2) $\pi - \frac{8}{3}$
(3) $\pi - \frac{4}{3}$
(4) $\frac{\pi}{2} - \frac{2\sqrt{2}}{3}$
Q72 Straight Lines & Coordinate Geometry Geometric Figure on Coordinate Plane View
Two sides of a rhombus are along the lines, $x - y + 1 = 0$ and $7x - y - 5 = 0$. If its diagonals intersect at $(-1, -2)$, then which one of the following is a vertex of this rhombus?
(1) $(-3, -9)$
(2) $(-3, -8)$
(3) $\left(\frac{1}{3}, -\frac{8}{3}\right)$
(4) $\left(-\frac{1}{3}, -\frac{8}{3}\right)$
Q73 Conic sections Locus and Trajectory Derivation View
The centres of those circles which touch the circle, $x^2 + y^2 - 8x - 8y - 4 = 0$, externally and also touch the $x$-axis, lie on:
(1) a circle
(2) an ellipse which is not a circle
(3) a hyperbola
(4) a parabola
Q74 Circles Circle Equation Derivation View
If one of the diameters of the circle, given by the equation, $x^2 + y^2 - 4x + 6y - 12 = 0$, is a chord of a circle $S$, whose centre is at $(-3, 2)$, then the radius of $S$ is:
(1) $5\sqrt{2}$
(2) $5\sqrt{3}$
(3) $5$
(4) $10$
Q75 Circles Circle Equation Derivation View
Let $P$ be the point on the parabola, $y^2 = 8x$, which is at a minimum distance from the centre $C$ of the circle, $x^2 + (y+6)^2 = 1$. Then the equation of the circle, passing through $C$ and having its centre at $P$ is:
(1) $x^2 + y^2 - 4x + 8y + 12 = 0$
(2) $x^2 + y^2 - x + 4y - 12 = 0$
(3) $x^2 + y^2 - \frac{x}{4} + 2y - 24 = 0$
(4) $x^2 + y^2 - 4x + 9y + 18 = 0$
Q76 Conic sections Optimization on Conics View
If the tangent at a point on the ellipse $\frac{x^2}{27} + \frac{y^2}{3} = 1$ meets the coordinate axes at $A$ and $B$, and $O$ is the origin, then the minimum area (in sq. units) of the triangle $OAB$ is:
(1) $\frac{9}{2}$
(2) $9$
(3) $9\sqrt{3}$
(4) $\frac{\sqrt{3}}{2}$
Q77 Vectors: Lines & Planes Distance Computation (Point-to-Plane or Line-to-Line) View
The distance of the point $(1, -5, 9)$ from the plane $x - y + z = 5$ measured along the line $x = y = z$ is:
(1) $3\sqrt{10}$
(2) $10\sqrt{3}$
(3) $\frac{10}{\sqrt{3}}$
(4) $\frac{20}{3}$
Q78 Vectors: Lines & Planes Parallelism Between Line and Plane or Constraint on Parameters View
If the line $\frac{x-3}{2} = \frac{y+2}{-1} = \frac{z+4}{3}$ lies in the plane $lx + my - z = 9$, then $l^2 + m^2$ is equal to:
(1) $26$
(2) $18$
(3) $1$
(4) $2$
Q79 Vectors Introduction & 2D Magnitude of Vector Expression View
If $\vec{a}, \vec{b}$ and $\vec{c}$ are unit vectors satisfying $|\vec{a} - \vec{b}|^2 + |\vec{b} - \vec{c}|^2 + |\vec{c} - \vec{a}|^2 = 9$, then $|2\vec{a} + 5\vec{b} + 5\vec{c}|$ is:
(1) $3$
(2) $\sqrt{10}$
(3) $2$
(4) $\sqrt{5}$
Q80 Vectors: Cross Product & Distances View
Let $\vec{a} = 2\hat{i} + \hat{j} - 2\hat{k}$ and $\vec{b} = \hat{i} + \hat{j}$. Let $\vec{c}$ be a vector such that $|\vec{c} - \vec{a}| = 3$, $|(\vec{a} \times \vec{b}) \times \vec{c}| = 3$ and the angle between $\vec{c}$ and $\vec{a} \times \vec{b}$ is $30^\circ$. Then $\vec{a} \cdot \vec{c}$ is equal to:
(1) $\frac{1}{8}$
(2) $25$
(3) $2$
(4) $5$
Q81 Small angle approximation View
$\lim_{x \to \pi/2} \frac{\cot x - \cos x}{(\pi - 2x)^3}$ equals:
(1) $\frac{1}{24}$
(2) $\frac{1}{16}$
(3) $\frac{1}{8}$
(4) $\frac{1}{4}$
Q82 Differentiating Transcendental Functions Compute derivative of transcendental function View
For $x \in \mathbb{R}$, $f(x) = |\log 2 - \sin x|$ and $g(x) = f(f(x))$, then:
(1) $g$ is not differentiable at $x = 0$
(2) $g'(0) = \cos(\log 2)$
(3) $g'(0) = -\cos(\log 2)$
(4) $g$ is differentiable at $x = 0$ and $g'(0) = -\sin(\log 2)$
Q83 Tangents, normals and gradients Normal or perpendicular line problems View
Consider $f(x) = \tan^{-1}\left(\sqrt{\frac{1+\sin x}{1-\sin x}}\right)$, $x \in \left(0, \frac{\pi}{2}\right)$. A normal to $y = f(x)$ at $x = \frac{\pi}{6}$ also passes through the point:
(1) $(0, 0)$
(2) $\left(0, \frac{2\pi}{3}\right)$
(3) $\left(\frac{\pi}{6}, 0\right)$
(4) $\left(\frac{\pi}{4}, 0\right)$
Q84 Stationary points and optimisation Geometric or applied optimisation problem View
A wire of length 2 units is cut into two parts which are bent respectively to form a square of side $= x$ units and a circle of radius $= r$ units. If the sum of the areas of the square and the circle so formed is minimum, then:
(1) $2x = (\pi + 4)r$
(2) $(4 - \pi)x = \pi r$
(3) $x = 2r$
(4) $2x = r$
Q85 Integration by Substitution Substitution to Compute an Indefinite Integral with Initial Condition View
The integral $\int \frac{2x^{12} + 5x^9}{(x^5 + x^3 + 1)^3} dx$ is equal to:
(1) $\frac{-x^{10}}{2(x^5 + x^3 + 1)^2} + C$
(2) $\frac{x^{10}}{2(x^5 + x^3 + 1)^2} + C$
(3) $\frac{-x^5}{(x^5 + x^3 + 1)^2} + C$
(4) $\frac{x^5}{2(x^5 + x^3 + 1)^2} + C$
Q86 Integration by Substitution Substitution to Evaluate a Definite Integral (Numerical Answer) View
The integral $\int_0^{\pi/4} \frac{\sin x + \cos x}{9 + 16\sin 2x} dx$ is equal to:
(1) $\frac{1}{20} \log 3$
(2) $\log 3$
(3) $\frac{1}{20} \log 9$
(4) $\frac{1}{10} \log 3$
Q87 First order differential equations (integrating factor) View
If a curve $y = f(x)$ passes through the point $(1, -1)$ and satisfies the differential equation, $y(1 + xy) dx = x\, dy$, then $f\left(-\frac{1}{2}\right)$ is equal to:
(1) $-\frac{2}{5}$
(2) $-\frac{4}{5}$
(3) $\frac{2}{5}$
(4) $\frac{4}{5}$
Q88 Integration by Substitution Substitution to Compute an Indefinite Integral with Initial Condition View
If $m$ is a non-zero number and $\int \frac{x^{5m-1} + 2x^{4m-1}}{(x^{2m} + x^m + 1)^3} dx = f(x) + C$, then $f(x)$ is:
(1) $\frac{x^{5m}}{2m(x^{2m} + x^m + 1)^2}$
(2) $\frac{x^{4m}}{2m(x^{2m} + x^m + 1)^2}$
(3) $\frac{(2x^{2m} + x^m)}{(x^{2m} + x^m + 1)^2}$
(4) $\frac{(x^{5m} + x^{4m})}{2m(x^{2m} + x^m + 1)^2}$
Q89 First order differential equations (integrating factor) View
The solution of the differential equation $\frac{dy}{dx} + \frac{y}{2}\sec^2 x = \frac{\tan x \sec^2 x}{2y}$, where $y(0) = 1$, is given by:
(1) $y^2 = 1 + \frac{\tan x}{x}$
(2) $y^2 = 1 + \tan x$
(3) $y = 1 - \tan x$
(4) $y^2 = 1 - \tan x$
Q90 Measures of Location and Spread View
If the standard deviation of the numbers $2, 3, a$ and $11$ is $3.5$, then which of the following is true?
(1) $3a^2 - 26a + 55 = 0$
(2) $3a^2 - 32a + 84 = 0$
(3) $3a^2 - 34a + 91 = 0$
(4) $3a^2 - 23a + 44 = 0$