jee-main

Papers (169)
2025
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2024
session1_01feb_shift1 4 session1_01feb_shift2 22 session1_27jan_shift1 28 session1_27jan_shift2 30 session1_29jan_shift1 30 session1_29jan_shift2 23 session1_30jan_shift1 17 session1_30jan_shift2 30 session1_31jan_shift1 16 session1_31jan_shift2 15 session2_04apr_shift1 4 session2_04apr_shift2 30 session2_05apr_shift1 4 session2_05apr_shift2 30 session2_06apr_shift1 22 session2_06apr_shift2 30 session2_08apr_shift1 30 session2_08apr_shift2 30 session2_09apr_shift1 5 session2_09apr_shift2 30
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_01feb_shift1

24 maths questions

Q61 Exponential Functions Exponential Equation Solving View
Let $S = \{x : x \in \mathbb{R}$ and $\left(\sqrt{3} + \sqrt{2}\right)^{x^2 - 4} + \left(\sqrt{3} - \sqrt{2}\right)^{x^2 - 4} = 10\}$. Then $n(S)$ is equal to
(1) 2
(2) 4
(3) 6
(4) 0
Q62 Complex Numbers Argand & Loci Circle Equation and Properties via Complex Number Manipulation View
If the center and radius of the circle $\left|\frac{z - 2}{z - 3}\right| = 2$ are respectively $(\alpha, \beta)$ and $\gamma$, then $3\alpha + \beta + \gamma$ is equal to
(1) 11
(2) 9
(3) 10
(4) 12
Q63 Arithmetic Sequences and Series Telescoping or Non-Standard Summation Involving an AP View
The sum to 10 terms of the series $\frac{1}{1 + 1^2 + 1^4} + \frac{2}{1 + 2^2 + 2^4} + \frac{3}{1 + 3^2 + 3^4} + \ldots$ is:
(1) $\frac{59}{111}$
(2) $\frac{55}{111}$
(3) $\frac{56}{111}$
(4) $\frac{58}{111}$
Q64 Binomial Theorem (positive integer n) Evaluate a Summation Involving Binomial Coefficients View
The value of $\frac{1}{1! \cdot 50!} + \frac{1}{3! \cdot 48!} + \frac{1}{5! \cdot 46!} + \ldots + \frac{1}{49! \cdot 2!} + \frac{1}{51! \cdot 1!}$ is
(1) $\frac{2^{50}}{50!}$
(2) $\frac{2^{50}}{51!}$
(3) $\frac{2^{51}}{51!}$
(4) $\frac{2^{51}}{50!}$
Q65 Curve Sketching Sketching a Curve from Analytical Properties View
The combined equation of the two lines $ax + by + c = 0$ and $a'x + b'y + c' = 0$ can be written as $(ax + by + c)(a'x + b'y + c') = 0$. The equation of the angle bisectors of the lines represented by the equation $2x^2 + xy - 3y^2 = 0$ is
(1) $3x^2 + 5xy + 2y^2 = 0$
(2) $x^2 - y^2 + 10xy = 0$
(3) $3x^2 + xy - 2y^2 = 0$
(4) $x^2 - y^2 - 10xy = 0$
Q66 Straight Lines & Coordinate Geometry Triangle Properties and Special Points View
If the orthocentre of the triangle, whose vertices are $(1,2)$, $(2,3)$ and $(3,1)$ is $(\alpha, \beta)$, then the quadratic equation whose roots are $\alpha + 4\beta$ and $4\alpha + \beta$, is
(1) $x^2 - 19x + 90 = 0$
(2) $x^2 - 18x + 80 = 0$
(3) $x^2 - 22x + 120 = 0$
(4) $x^2 - 20x + 99 = 0$
Q68 Measures of Location and Spread View
The mean and variance of 5 observations are 5 and 8 respectively. If 3 observations are $1, 3, 5$, then the sum of cubes of the remaining two observations is
(1) 1072
(2) 1792
(3) 1216
(4) 1456
Q69 Sine and Cosine Rules Multi-step composite figure problem View
For a triangle $ABC$, the value of $\cos 2A + \cos 2B + \cos 2C$ is least. If its inradius is 3 and incentre is $M$, then which of the following is NOT correct?
(1) Perimeter of $\triangle ABC$ is $18\sqrt{3}$
(2) $\sin 2A + \sin 2B + \sin 2C = \sin A + \sin B + \sin C$
(3) $\overrightarrow{MA} \cdot \overrightarrow{MB} = -18$
(4) area of $\triangle ABC$ is $\frac{27\sqrt{3}}{2}$
Q71 Matrices Linear System and Inverse Existence View
Let $S$ denote the set of all real values of $\lambda$ such that the system of equations $$\lambda x + y + z = 1$$ $$x + \lambda y + z = 1$$ $$x + y + \lambda z = 1$$ is inconsistent, then $\sum_{\lambda \in S} (\lambda^2 + \lambda)$ is equal to
(1) 2
(2) 12
(3) 4
(4) 6
Q73 Differentiating Transcendental Functions Monotonicity or convexity of transcendental functions View
Let $f(x) = 2x + \tan^{-1} x$ and $g(x) = \log_e\left(\sqrt{1 + x^2} + x\right)$, $x \in [0, 3]$. Then
(1) There exists $x \in (0, 3)$ such that $f'(x) < g'(x)$
(2) $\max f(x) > \max g(x)$
(3) There exist $0 < x_1 < x_2 < 3$ such that $f(x) < g(x)$, $\forall x \in (x_1, x_2)$
(4) $\min f'(x) = 1 + \max g'(x)$
Q74 Matrices Determinant and Rank Computation View
Let $f(x) = \begin{vmatrix} 1 + \sin^2 x & \cos^2 x & \sin 2x \\ \sin^2 x & 1 + \cos^2 x & \sin 2x \\ \sin^2 x & \cos^2 x & 1 + \sin 2x \end{vmatrix}$, $x \in \left[\frac{\pi}{6}, \frac{\pi}{3}\right]$. If $\alpha$ and $\beta$ respectively are the maximum and the minimum values of $f$, then
(1) $\beta^2 - 2\sqrt{\alpha} = \frac{19}{4}$
(2) $\beta^2 + 2\sqrt{\alpha} = \frac{19}{4}$
(3) $\alpha^2 - \beta^2 = 4\sqrt{3}$
(4) $\alpha^2 + \beta^2 = \frac{9}{2}$
Q76 Differential equations Solving Separable DEs with Initial Conditions View
The area enclosed by the closed curve $C$ given by the differential equation $\frac{dy}{dx} + \frac{x + a}{y - 2} = 0$, $y(1) = 0$ is $4\pi$. Let $P$ and $Q$ be the points of intersection of the curve $C$ and the $y$-axis. If normals at $P$ and $Q$ on the curve $C$ intersect $x$-axis at points $R$ and $S$ respectively, then the length of the line segment $RS$ is
(1) $2\sqrt{3}$
(2) $\frac{2\sqrt{3}}{3}$
(3) 2
(4) $\frac{4\sqrt{3}}{3}$
Q77 First order differential equations (integrating factor) View
If $y = y(x)$ is the solution curve of the differential equation $\frac{dy}{dx} + y\tan x = x\sec x$, $0 \leq x \leq \frac{\pi}{3}$, $y(0) = 1$, then $y\left(\frac{\pi}{6}\right)$ is equal to
(1) $\frac{\pi}{12} - \frac{\sqrt{3}}{2}\log_e\frac{2}{e\sqrt{3}}$
(2) $\frac{\pi}{12} + \frac{\sqrt{3}}{2}\log_e\frac{2\sqrt{3}}{e}$
(3) $\frac{\pi}{12} - \frac{\sqrt{3}}{2}\log_e\frac{2\sqrt{3}}{e}$
(4) $\frac{\pi}{12} + \frac{\sqrt{3}}{2}\log_e\frac{2}{e\sqrt{3}}$
Q78 Vectors 3D & Lines Line-Plane Intersection View
Let the image of the point $P(2, -1, 3)$ in the plane $x + 2y - z = 0$ be $Q$. Then the distance of the plane $3x + 2y + z + 29 = 0$ from the point $Q$ is
(1) $\frac{22\sqrt{2}}{7}$
(2) $\frac{24\sqrt{2}}{7}$
(3) $2\sqrt{14}$
(4) $3\sqrt{14}$
Q79 Vectors 3D & Lines Shortest Distance Between Two Lines View
The shortest distance between the lines $\frac{x-5}{1} = \frac{y-2}{2} = \frac{z-4}{-3}$ and $\frac{x+3}{1} = \frac{y+5}{4} = \frac{z-1}{-5}$ is
(1) $7\sqrt{3}$
(2) $5\sqrt{3}$
(3) $6\sqrt{3}$
(4) $4\sqrt{3}$
Q80 Binomial Distribution Find Parameters from Moment Conditions View
In a binomial distribution $B(n, p)$, the sum and product of the mean and variance are 5 and 6 respectively, then $6(n + p - q)$ is equal to:
(1) 51
(2) 52
(3) 53
(4) 50
Q81 Permutations & Arrangements Word Permutations with Repeated Letters View
The number of words, with or without meaning, that can be formed using all the letters of the word ASSASSINATION so that the vowels occur together, is $\_\_\_\_$.
Q82 Arithmetic Sequences and Series Find Specific Term from Given Conditions View
Let $a_1 = 8, a_2, a_3, \ldots, a_n$ be an A.P. If the sum of its first four terms is 50 and the sum of its last four terms is 170, then the product of its middle two terms is $\_\_\_\_$.
Q85 Chain Rule Higher-Order Derivatives of Products/Compositions View
If $f(x) = x^2 + g'(1)x + g''(2)$ and $g(x) = f(1)x^2 + xf'(x) + f''(x)$, then the value of $f(4) - g(4)$ is equal to $\_\_\_\_$.
Q86 Integration by Substitution Substitution to Evaluate a Definite Integral (Numerical Answer) View
If $\int_0^1 (x^{21} + x^{14} + x^7)(2x^{14} + 3x^7 + 6)^{1/7}\, dx = \frac{1}{l} \cdot 11^{m/n}$ where $l, m, n \in \mathbb{N}$, $m$ and $n$ are co-prime, then $l + m + n$ is equal to $\_\_\_\_$.
Q87 Areas by integration View
Let $A$ be the area bounded by the curve $y = x(x - 3)$, the $x$-axis and the ordinates $x = -1$ and $x = 2$. Then $12A$ is equal to $\_\_\_\_$.
Q88 First order differential equations (integrating factor) View
Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function such that $f'(x) + f(x) = \int_0^2 f(t)\, dt$. If $f(0) = e^{-2}$, then $2f(0) - f(2)$ is equal to $\_\_\_\_$.
Q89 Vectors Introduction & 2D Optimization of a Vector Expression View
Let $\vec{v} = \alpha\hat{i} + 2\hat{j} - 3\hat{k}$, $\vec{w} = 2\alpha\hat{i} + \hat{j} - \hat{k}$, and $\vec{u}$ be a vector such that $|\vec{u}| = \alpha > 0$. If the minimum value of the scalar triple product $[\vec{u}\, \vec{v}\, \vec{w}]$ is $-\alpha\sqrt{3401}$, and $|\vec{u} \cdot \hat{i}|^2 = \frac{m}{n}$ where $m$ and $n$ are coprime natural numbers, then $m + n$ is equal to $\_\_\_\_$.
Q90 Vectors Introduction & 2D Area Computation Using Vectors View
$A(2, 6, 2)$, $B(-4, 0, \lambda)$, $C(2, 3, -1)$ and $D(4, 5, 0)$, $\lambda \leq 5$ are the vertices of a quadrilateral $ABCD$. If its area is 18 square units, then $5 - 6\lambda$ is equal to $\_\_\_\_$.