jee-main

Papers (169)
2025
session1_22jan_shift1 25 session1_22jan_shift2 25 session1_23jan_shift1 25 session1_23jan_shift2 25 session1_24jan_shift1 25 session1_24jan_shift2 25 session1_28jan_shift1 25 session1_28jan_shift2 25 session1_29jan_shift1 29 session1_29jan_shift2 25
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_30jan_shift2

29 maths questions

Q61 Solving quadratics and applications Finding roots or coefficients of a quadratic using Vieta's relations View
If the value of real number $\alpha > 0$ for which $x^{2} - 5\alpha x + 1 = 0$ and $x^{2} - \alpha x - 5 = 0$ have a common real roots is $\frac{3}{\sqrt{2\beta}}$ then $\beta$ is equal to $\_\_\_\_$
Q62 Permutations & Arrangements Forming Numbers with Digit Constraints View
The number of ways of selecting two numbers $a$ and $b$, $a \in \{2, 4, 6, \ldots, 100\}$ and $b \in \{1, 3, 5, \ldots, 99\}$ such that 2 is the remainder when $a + b$ is divided by 23 is
(1) 186
(2) 54
(3) 108
(4) 268
Q63 Permutations & Arrangements Word Permutations with Repeated Letters View
The number of seven digits odd numbers, that can be formed using all the seven digits $1, 2, 2, 2, 3, 3, 5$ is
Q64 Arithmetic Sequences and Series Compute Partial Sum of an Arithmetic Sequence View
Let $a, b, c > 1$, $a^{3}$, $b^{3}$ and $c^{3}$ be in A.P. and $\log_{a} b$, $\log_{c} a$ and $\log_{b} c$ be in G.P. If the sum of first 20 terms of an A.P., whose first term is $\frac{a + 4b + c}{3}$ and the common difference is $\frac{a - 8b + c}{10}$ is $-444$, then $abc$ is equal to
(1) 343
(2) 216
(3) $\frac{343}{8}$
(4) $\frac{125}{8}$
Q65 Arithmetic Sequences and Series Counting or Combinatorial Problems on APs View
The $8^{\text{th}}$ common term of the series $$\begin{aligned} & S_{1} = 3 + 7 + 11 + 15 + 19 + \ldots \\ & S_{2} = 1 + 6 + 11 + 16 + 21 + \ldots \end{aligned}$$ is
Q66 Proof True/False Justification View
Let $x = (8\sqrt{3} + 13)^{13}$ and $y = (7\sqrt{2} + 9)^{9}$. If $[t]$ denotes the greatest integer $\leq t$, then
(1) $[x] + [y]$ is even
(2) $[x]$ is odd but $[y]$ is even
(3) $[x]$ is even but $[y]$ is odd
(4) $[x]$ and $[y]$ are both odd
Q67 Indices and Surds Number-Theoretic Reasoning with Indices View
$50^{\text{th}}$ root of a number $x$ is 12 and $50^{\text{th}}$ root of another number $y$ is 18. Then the remainder obtained on dividing $(x + y)$ by 25 is $\_\_\_\_$.
Q68 Circles Area and Geometric Measurement Involving Circles View
Let $P(a_{1}, b_{1})$ and $Q(a_{2}, b_{2})$ be two distinct points on a circle with center $C(\sqrt{2}, \sqrt{3})$. Let $O$ be the origin and $OC$ be perpendicular to both $CP$ and $CQ$. If the area of the triangle $OCP$ is $\frac{\sqrt{35}}{2}$, then $a_{1}^{2} + a_{2}^{2} + b_{1}^{2} + b_{2}^{2}$ is equal to $\_\_\_\_$
Q69 Geometric Sequences and Series True/False or Multiple-Statement Verification View
The parabolas: $ax^{2} + 2bx + cy = 0$ and $dx^{2} + 2ex + fy = 0$ intersect on the line $y = 1$. If $a, b, c, d, e, f$ are positive real numbers and $a, b, c$ are in G.P., then
(1) $d, e, f$ are in A.P.
(2) $\frac{d}{a}, \frac{e}{b}, \frac{f}{c}$ are in G.P.
(3) $\frac{d}{a}, \frac{e}{b}, \frac{f}{c}$ are in A.P.
(4) $d, e, f$ are in G.P.
Q70 Circles Tangent Lines and Tangent Lengths View
Let $A$ be a point on the $x$-axis. Common tangents are drawn from $A$ to the curves $x^{2} + y^{2} = 8$ and $y^{2} = 16x$. If one of these tangents touches the two curves at $Q$ and $R$, then $(QR)^{2}$ is equal to
(1) 64
(2) 76
(3) 81
(4) 72
Q71 Composite & Inverse Functions Evaluate Composition from Algebraic Definitions View
Let $f$, $g$ and $h$ be the real valued functions defined on $\mathbb{R}$ as $f(x) = \left\{ \begin{array}{cc} \frac{x}{|x|}, & x \neq 0 \\ 1, & x = 0 \end{array} \right.$, $\quad g(x) = \left\{ \begin{array}{cc} \frac{\sin(x+1)}{(x+1)}, & x \neq -1 \\ 1, & x = -1 \end{array} \right.$ and $h(x) = 2[x] - f(x)$, where $[x]$ is the greatest integer $\leq x$. Then the value of $\lim_{x \rightarrow 1} g(h(x-1))$ is
(1) 1
(2) $\sin(1)$
(3) $-1$
(4) 0
Q73 Measures of Location and Spread View
Let $S$ be the set of all values of $a_{1}$ for which the mean deviation about the mean of 100 consecutive positive integers $a_{1}, a_{2}, a_{3}, \ldots, a_{100}$ is 25. Then $S$ is
(1) $\phi$
(2) $\{99\}$
(3) $\mathbb{N}$
(4) $\{9\}$
Q74 Matrices Determinant and Rank Computation View
If $P$ is a $3 \times 3$ real matrix such that $P^{T} = aP + (a-1)I$, where $a > 1$, then
(1) $P$ is a singular matrix
(2) $|\operatorname{Adj} P| > 1$
(3) $|\operatorname{Adj} P| = \frac{1}{2}$
(4) $|\operatorname{Adj} P| = 1$
Q75 Simultaneous equations View
For $\alpha, \beta \in \mathbb{R}$, suppose the system of linear equations $x - y + z = 5$ $2x + 2y + \alpha z = 8$ $3x - y + 4z = \beta$ has infinitely many solutions. Then $\alpha$ and $\beta$ are the roots of
(1) $x^{2} - 10x + 16 = 0$
(2) $x^{2} + 18x + 56 = 0$
(3) $x^{2} - 18x + 56 = 0$
(4) $x^{2} + 14x + 24 = 0$
Q76 Arithmetic Sequences and Series Telescoping or Non-Standard Summation Involving an AP View
Let $a_{1} = 1, a_{2}, a_{3}, a_{4}, \ldots$ be consecutive natural numbers. Then $\tan^{-1}\left(\frac{1}{1 + a_{1}a_{2}}\right) + \tan^{-1}\left(\frac{1}{1 + a_{2}a_{3}}\right) + \ldots + \tan^{-1}\left(\frac{1}{1 + a_{2021}a_{2022}}\right)$ is equal to
(1) $\frac{\pi}{4} - \cot^{-1}(2022)$
(2) $\cot^{-1}(2022) - \frac{\pi}{4}$
(3) $\tan^{-1}(2022) - \frac{\pi}{4}$
(4) $\frac{\pi}{4} - \tan^{-1}(2022)$
Q77 Composite & Inverse Functions Determine Domain or Range of a Composite Function View
The range of the function $f(x) = \sqrt{3 - x} + \sqrt{2 + x}$ is
(1) $[\sqrt{5}, \sqrt{10}]$
(2) $[2\sqrt{2}, \sqrt{11}]$
(3) $[\sqrt{5}, \sqrt{13}]$
(4) $[\sqrt{2}, \sqrt{7}]$
Q78 Composite & Inverse Functions Counting Functions with Composition or Mapping Constraints View
Let $A = \{1, 2, 3, 5, 8, 9\}$. Then the number of possible functions $f: A \rightarrow A$ such that $f(m \cdot n) = f(m) \cdot f(n)$ for every $m, n \in A$ with $m \cdot n \in A$ is equal to
Q79 Stationary points and optimisation Determine parameters from given extremum conditions View
If the functions $f(x) = \frac{x^{3}}{3} + 2bx + \frac{ax^{2}}{2}$ and $g(x) = \frac{x^{3}}{3} + ax + bx^{2}$, $a \neq 2b$ have a common extreme point, then $a + 2b + 7$ is equal to
(1) 4
(2) $\frac{3}{2}$
(3) 3
(4) 6
Q80 Integration by Substitution Substitution to Transform Integral Form (Show Transformed Expression) View
If $\int \sqrt{\sec 2x - 1}\, dx = \alpha \log_{e}\left|\cos 2x + \beta + \sqrt{\cos 2x\left(1 + \cos \frac{1}{\beta}x\right)}\right| + \text{constant}$, then $\beta - \alpha$ is equal to $\_\_\_\_$.
Q81 Numerical integration Riemann Sum Computation from a Given Formula View
$\lim_{n \rightarrow \infty} \frac{3}{n}\left\{4 + \left(2 + \frac{1}{n}\right)^{2} + \left(2 + \frac{2}{n}\right)^{2} + \ldots + \left(3 - \frac{1}{n}\right)^{2}\right\}$ is equal to
(1) 12
(2) $\frac{19}{3}$
(3) 0
(4) 19
Q82 Solving quadratics and applications Optimization or extremal value of an expression via completing the square View
Let $q$ be the maximum integral value of $p$ in $[0, 10]$ for which the roots of the equation $x^{2} - px + \frac{5}{4}p = 0$ are rational. Then the area of the region $\left\{(x, y): 0 \leq y \leq (x - q)^{2},\, 0 \leq x \leq q\right\}$ is
(1) 243
(2) 25
(3) $\frac{125}{3}$
(4) 164
Q83 Areas by integration View
Let $A$ be the area of the region $\left\{(x, y): y \geq x^{2},\, y \geq (1-x)^{2},\, y \leq 2x(1-x)\right\}$. Then $540A$ is equal to
Q84 Differential equations Solving Separable DEs with Initial Conditions View
The solution of the differential equation $\frac{dy}{dx} = -\left(\frac{x^{2} + 3y^{2}}{3x^{2} + y^{2}}\right)$, $y(1) = 0$ is
(1) $\log_{e}|x + y| - \frac{xy}{(x+y)^{2}} = 0$
(2) $\log_{e}|x + y| + \frac{xy}{(x+y)^{2}} = 0$
(3) $\log_{e}|x + y| + \frac{2xy}{(x+y)^{2}} = 0$
(4) $\log_{e}|x + y| - \frac{2xy}{(x+y)^{2}} = 0$
Q85 Vectors 3D & Lines Vector Algebra and Triple Product Computation View
Let $\lambda \in \mathbb{R}$, $\vec{a} = \lambda\hat{i} + 2\hat{j} - 3\hat{k}$, $\vec{b} = \hat{i} - \lambda\hat{j} + 2\hat{k}$. If $((\vec{a} + \vec{b}) \times (\vec{a} \times \vec{b})) \times (\vec{a} - \vec{b}) = 8\hat{i} - 40\hat{j} - 24\hat{k}$, then $|\lambda(\vec{a} + \vec{b}) \times (\vec{a} - \vec{b})|^{2}$ is equal to
(1) 140
(2) 132
(3) 144
(4) 136
Q86 Vectors 3D & Lines Vector Algebra and Triple Product Computation View
Let $\vec{a}$ and $\vec{b}$ be two vectors. Let $|\vec{a}| = 1$, $|\vec{b}| = 4$ and $\vec{a} \cdot \vec{b} = 2$. If $\vec{c} = (2\vec{a} \times \vec{b}) - 3\vec{b}$, then the value of $\vec{b} \cdot \vec{c}$ is
(1) $-24$
(2) $-48$
(3) $-84$
(4) $-60$
Q87 Vectors 3D & Lines Normal Vector and Plane Equation View
A vector $\vec{v}$ in the first octant is inclined to the $x$-axis at $60^{\circ}$, to the $y$-axis at $45^{\circ}$ and to the $z$-axis at an acute angle. If a plane passing through the points $(\sqrt{2}, -1, 1)$ and $(a, b, c)$, is normal to $\vec{v}$, then
(1) $\sqrt{2}a + b + c = 1$
(2) $a + b + \sqrt{2}c = 1$
(3) $a + \sqrt{2}b + c = 1$
(4) $\sqrt{2}a - b + c = 1$
Q88 Vectors 3D & Lines Normal Vector and Plane Equation View
If a plane passes through the points $(-1, k, 0)$, $(2, k, -1)$, $(1, 1, 2)$ and is parallel to the line $\frac{x-1}{1} = \frac{2y+1}{2} = \frac{z+1}{-1}$, then the value of $\frac{k^{2}+1}{(k-1)(k-2)}$ is
(1) $\frac{17}{5}$
(2) $\frac{5}{17}$
(3) $\frac{6}{13}$
(4) $\frac{13}{6}$
Q89 Vectors 3D & Lines Distance from a Point to a Line (Show/Compute) View
Let a line $L$ pass through the point $P(2, 3, 1)$ and be parallel to the line $x + 3y - 2z - 2 = 0 = x - y + 2z$. If the distance of $L$ from the point $(5, 3, 8)$ is $\alpha$, then $3\alpha^{2}$ is equal to $\_\_\_\_$
Q90 Discrete Probability Distributions Probability Computation for Compound or Multi-Stage Random Experiments View
A bag contains six balls of different colours. Two balls are drawn in succession with replacement. The probability that both the balls are of the same colour is $p$. Next four balls are drawn in succession with replacement and the probability that exactly three balls are of the same colours is $q$. If $p : q = m : n$, where $m$ and $n$ are co-prime, then $m + n$ is equal to