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
2017 02apr

28 maths questions

Q61 Solving quadratics and applications Optimization or extremal value of an expression via completing the square View
If, for a positive integer $n$, the quadratic equation,
$$x(x + 1) + (x + 1)(x + 2) + \ldots + (x + \overline{n-1})(x + n) = 10n$$
has two consecutive integral solutions, then $n$ is equal to:
(1) 12
(2) 9
(3) 10
(4) 11
Q62 Complex Numbers Arithmetic Roots of Unity and Cyclotomic Expressions View
Let $\omega$ be a complex number such that $2\omega + 1 = z$ where $z = \sqrt{-3}$. If
$$\begin{vmatrix} 1 & 1 & 1 \\ 1 & -\omega^2 - 1 & \omega^2 \\ 1 & \omega^2 & \omega^7 \end{vmatrix} = 3k$$
Then $k$ can be equal to:
(1) $-z$
(2) $\frac{1}{z}$
(3) $-1$
(4) $1$
Q63 Combinations & Selection Selection with Group/Category Constraints View
A man $X$ has 7 friends, 4 of them are ladies and 3 are men. His wife $Y$ also has 7 friends, 3 of them are ladies and 4 are men. Assume $X$ and $Y$ have no common friends. Then the total number of ways in which $X$ and $Y$ together can throw a party inviting 3 ladies and 3 men, so that 3 friends of each of $X$ and $Y$ are in this party is:
(1) 485
(2) 468
(3) 469
(4) 484
Q64 Arithmetic Sequences and Series Properties of AP Terms under Transformation View
For any three positive real numbers $a, b$ and $c$. If $9(25a^2 + b^2) + 25(c^2 - 3ac) = 15b(3a + c)$. Then
(1) $b,\ c$ and $a$ are in G.P.
(2) $b,\ c$ and $a$ are in A.P.
(3) $a,\ b$ and $c$ are in A.P.
(4) $a,\ b$ and $c$ are in G.P.
Q65 Binomial Theorem (positive integer n) Evaluate a Summation Involving Binomial Coefficients View
The value of ${}^{21}C_1 - {}^{10}C_1 + {}^{21}C_2 - {}^{10}C_2 + {}^{21}C_3 - {}^{10}C_3 + {}^{21}C_4 - {}^{10}C_4 + \ldots + {}^{21}C_{10} - {}^{10}C_{10}$ is
(1) $2^{21} - 2^{11}$
(2) $2^{21} - 2^{10}$
(3) $2^{20} - 2^{9}$
(4) $2^{20} - 2^{10}$
Q66 Standard trigonometric equations Evaluate trigonometric expression given a constraint View
If $5\tan^2 x - \cos^2 x = 2\cos 2x + 9$, then the value of $\cos 4x$ is
(1) $-\dfrac{3}{5}$
(2) $\dfrac{1}{3}$
(3) $\dfrac{2}{9}$
(4) $-\dfrac{7}{9}$
Q67 Straight Lines & Coordinate Geometry Triangle Properties and Special Points View
Let $k$ be an integer such that the triangle with vertices $(k, -3)$, $(5, k)$ and $(-k, 2)$ has area 28 sq. units. Then the orthocenter of this triangle is at the point:
(1) $\left(2, -\dfrac{1}{2}\right)$
(2) $\left(1, \dfrac{3}{4}\right)$
(3) $\left(1, -\dfrac{3}{4}\right)$
(4) $\left(2, \dfrac{1}{2}\right)$
Q68 Circles Optimization on a Circle View
The radius of a circle, having minimum area, which touches the curve $y = 4 - x^2$ and the lines $y = |x|$ is:
(1) $2(\sqrt{2} + 1)$
(2) $2(\sqrt{2} - 1)$
(3) $4(\sqrt{2} - 1)$
(4) $4(\sqrt{2} + 1)$
Q69 Conic sections Tangent and Normal Line Problems View
The eccentricity of an ellipse whose centre is at the origin is $\dfrac{1}{2}$. If one of its directrices is $x = -4$, then the equation of the normal to it at $\left(1, \dfrac{3}{2}\right)$ is:
(1) $2y - x = 2$
(2) $4x - 2y = 1$
(3) $4x + 2y = 7$
(4) $x + 2y = 4$
Q70 Conic sections Tangent and Normal Line Problems View
A hyperbola passes through the point $P(\sqrt{2}, \sqrt{3})$ and has foci at $(\pm 2, 0)$. Then the tangent to this hyperbola at $P$ also passes through the point
(1) $(3\sqrt{2}, 2\sqrt{3})$
(2) $(2\sqrt{2}, 3\sqrt{3})$
(3) $(\sqrt{3}, \sqrt{2})$
(4) $(-\sqrt{2}, -\sqrt{3})$
Q71 Chain Rule Limit Evaluation Involving Composition or Substitution View
$\lim_{x \to \frac{\pi}{2}} \dfrac{\cot x - \cos x}{(\pi - 2x)^3}$ equals
(1) $\dfrac{1}{24}$
(2) $\dfrac{1}{16}$
(3) $\dfrac{1}{8}$
(4) $\dfrac{1}{4}$
Q73 Binomial Distribution Compute Expectation, Variance, or Standard Deviation View
A box contains 15 green and 10 yellow balls. If 10 balls are randomly drawn, one-by-one, with replacement, then the variance of the number of green balls drawn is:
(1) $\dfrac{12}{5}$
(2) $6$
(3) $4$
(4) $\dfrac{6}{25}$
Q75 Matrices Matrix Algebra and Product Properties View
If $A = \begin{pmatrix} 2 & -3 \\ -4 & 1 \end{pmatrix}$, then $\text{Adj}(3A^2 + 12A)$ is equal to:
(1) $\begin{pmatrix} 72 & -84 \\ -63 & 51 \end{pmatrix}$
(2) $\begin{pmatrix} 51 & 63 \\ 84 & 72 \end{pmatrix}$
(3) $\begin{pmatrix} 51 & 84 \\ 63 & 72 \end{pmatrix}$
(4) $\begin{pmatrix} 72 & -63 \\ -84 & 51 \end{pmatrix}$
Q76 Simultaneous equations View
If $S$ is the set of distinct values of $b$ for which the following system of linear equations
$$\begin{aligned} & x + y + z = 1 \\ & x + ay + z = 1 \\ & ax + by + z = 0 \end{aligned}$$
has no solution, then $S$ is:
(1) An empty set
(2) An infinite set
(3) A finite set containing two or more elements
(4) A singleton
Q77 Composite & Inverse Functions Injectivity, Surjectivity, or Bijectivity Classification View
The function $f : \mathbb{R} \rightarrow \left(-\dfrac{1}{2}, \dfrac{1}{2}\right)$ defined as $f(x) = \dfrac{x}{1 + x^2}$, is:
(1) Invertible
(2) Injective but not surjective
(3) Surjective but not injective
(4) Neither injective nor surjective
Q78 Arithmetic Sequences and Series Summation of Derived Sequence from AP View
Let $a, b, c \in \mathbb{R}$. If $f(x) = ax^2 + bx + c$ is such that $a + b + c = 3$ and $f(x + y) = f(x) + f(y) + xy,\ \forall x, y \in \mathbb{R}$, then $\displaystyle\sum_{n=1}^{10} f(n)$ is equal to:
(1) 330
(2) 165
(3) 190
(4) 255
Q79 Differentiating Transcendental Functions Compute derivative of transcendental function View
If for $x \in \left(0, \dfrac{1}{4}\right)$, the derivative of $\tan^{-1}\left(\dfrac{6x\sqrt{x}}{1 - 9x^3}\right)$ is $\sqrt{x} \cdot g(x)$, then $g(x)$ equals:
(1) $\dfrac{9}{1 + 9x^3}$
(2) $\dfrac{3x\sqrt{x}}{1 - 9x^3}$
(3) $\dfrac{3x}{1 - 9x^3}$
(4) $\dfrac{3}{1 + 9x^3}$
Q80 Stationary points and optimisation Geometric or applied optimisation problem View
Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, is:
(1) 12.5
(2) 10
(3) 25
(4) 30
Q81 Tangents, normals and gradients Normal or perpendicular line problems View
The normal to the curve $y(x - 2)(x - 3) = x + 6$ at the point where the curve intersects the $y$-axis passes through the point:
(1) $\left(-\dfrac{1}{2}, -\dfrac{1}{2}\right)$
(2) $\left(\dfrac{1}{2}, \dfrac{1}{2}\right)$
(3) $\left(\dfrac{1}{2}, -\dfrac{1}{3}\right)$
(4) $\left(\dfrac{1}{2}, \dfrac{1}{3}\right)$
Q82 Standard Integrals and Reverse Chain Rule Verify or Prove an Antiderivative/Integral Identity View
Let $I_n = \int \tan^n x\, dx$ $(n > 1)$. If $I_4 + I_6 = a\tan^5 x + bx^5 + c$, then the ordered pair $(a, b)$ is equal to
(1) $\left(-\dfrac{1}{5}, 1\right)$
(2) $\left(\dfrac{1}{5}, 0\right)$
(3) $\left(\dfrac{1}{5}, -1\right)$
(4) $\left(-\dfrac{1}{5}, 0\right)$
Q83 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
The integral $\displaystyle\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} \frac{dx}{1 + \cos x}$ is equal to
(1) $-2$
(2) $2$
(3) $4$
(4) $-1$
Q84 Areas by integration View
The area (in sq. units) of the region $\{(x, y) : x \geq 0,\ x + y \leq 3,\ x^2 \leq 4y$ and $y \leq 1 + \sqrt{x}\}$ is
(1) $\dfrac{59}{12}$ sq. units
(2) $\dfrac{3}{2}$ sq. units
(3) $\dfrac{7}{3}$ sq. units
(4) $\dfrac{5}{2}$ sq. units
Q85 Differential equations Solving Separable DEs with Initial Conditions View
If $(2 + \sin x)\dfrac{dy}{dx} + (y + 1)\cos x = 0$ and $y(0) = 1$, then $y\left(\dfrac{\pi}{2}\right)$ is equal to
(1) $\dfrac{1}{3}$
(2) $-\dfrac{2}{3}$
(3) $-\dfrac{1}{3}$
(4) $\dfrac{4}{3}$
Q86 Vectors 3D & Lines Vector Algebra and Triple Product Computation View
Given, $\vec{a} = 2\hat{i} + \hat{j} - 2\hat{k}$ 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 $\vec{a} \times \vec{b}$ be $30^\circ$. Then $\vec{a} \cdot \vec{c}$ is equal to:
(1) $\dfrac{25}{8}$
(2) $2$
(3) $5$
(4) $\dfrac{1}{8}$
Q87 Vectors 3D & Lines Line-Plane Intersection View
If the image of the point $P(1, -2, 3)$ in the plane $2x + 3y - 4z + 22 = 0$ measured parallel to the line $\dfrac{x}{1} = \dfrac{y}{4} = \dfrac{z}{5}$ is $Q$, then $PQ$ is equal to:
(1) $3\sqrt{5}$
(2) $2\sqrt{42}$
(3) $\sqrt{42}$
(4) $6\sqrt{5}$
Q88 Vectors 3D & Lines Distance from a Point to a Line (Show/Compute) View
The distance of the point $(1, 3, -7)$ from the plane passing through the point $(1, -1, -1)$, having normal perpendicular to both the lines $\dfrac{x-1}{1} = \dfrac{y+2}{-2} = \dfrac{z-4}{3}$ and $\dfrac{x-2}{2} = \dfrac{y+1}{-1} = \dfrac{z+7}{-1}$, is:
(1) $\dfrac{20}{\sqrt{74}}$
(2) $\dfrac{10}{\sqrt{83}}$
(3) $\dfrac{5}{\sqrt{83}}$
(4) $\dfrac{10}{\sqrt{74}}$
Q89 Probability Definitions Probability Using Set/Event Algebra View
For three events $A$, $B$ and $C$, $P(\text{Exactly one of } A \text{ or } B \text{ occurs}) = P(\text{Exactly one of } B \text{ or } C \text{ occurs}) = P(\text{Exactly one of } C \text{ or } A \text{ occurs}) = \dfrac{1}{4}$ and $P(\text{All the three events occur simultaneously}) = \dfrac{1}{16}$. Then the probability that at least one of the events occurs, is:
(1) $\dfrac{7}{32}$
(2) $\dfrac{7}{16}$
(3) $\dfrac{1}{64}$
(4) $\dfrac{3}{16}$
Q90 Probability Definitions Probability Involving Algebraic or Number-Theoretic Conditions View
If two different numbers are taken from the set $\{0, 1, 2, 3, \ldots, 10\}$; then the probability that their sum as well as absolute difference are both multiples of 4, is:
(1) $\dfrac{6}{55}$
(2) $\dfrac{12}{55}$
(3) $\dfrac{14}{45}$
(4) $\dfrac{7}{55}$