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
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
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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
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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
2024 session1_31jan_shift1

16 maths questions

Q61 Inequalities Quadratic Inequality Holding for All x (or a Restricted Domain) View
Let $S$ be the set of positive integral values of $a$ for which $\frac { a x ^ { 2 } + 2 a + 1 x + 9 a + 4 } { x ^ { 2 } - 8 x + 32 } < 0 , \quad \forall x \in \mathbb { R }$. Then, the number of elements in $S$ is:
(1) 1
(2) 0
(3) $\infty$
(4) 3
Q62 Solving quadratics and applications Finding a ratio or relationship between variables from an equation View
For $0 < c < b < a$, let $( a + b - 2 c ) x ^ { 2 } + ( b + c - 2 a ) x + ( c + a - 2 b ) = 0$ and $\alpha \neq 1$ be one of its root. Then, among the two statements (I) If $\alpha \in ( -1, 0)$, then $b$ cannot be the geometric mean of $a$ and $c$. (II) If $\alpha \in (0, 1)$, then $b$ may be the geometric mean of $a$ and $c$.
(1) Both (I) and (II) are true
(2) Neither (I) nor (II) is true
(3) Only (II) is true
(4) Only (I) is true
Q63 Sequences and Series Evaluation of a Finite or Infinite Sum View
The sum of the series $\frac { 1 } { 1 - 3 \cdot 1 ^ { 2 } + 1 ^ { 4 } } + \frac { 2 } { 1 - 3 \cdot 2 ^ { 2 } + 2 ^ { 4 } } + \frac { 3 } { 1 - 3 \cdot 3 ^ { 2 } + 3 ^ { 4 } } +\ldots$ up to 10 terms is
(1) $\frac { 45 } { 109 }$
(2) $- \frac { 45 } { 109 }$
(3) $\frac { 55 } { 109 }$
(4) $- \frac { 55 } { 109 }$
Q64 Straight Lines & Coordinate Geometry Geometric Figure on Coordinate Plane View
Let $\alpha , \quad \beta , \quad \gamma , \quad \delta \in Z$ and let $A(\alpha , \beta)$, $B(1, 0)$, $C(\gamma , \delta)$ and $D(1, 2)$ be the vertices of a parallelogram $ABCD$. If $AB = \sqrt { 10 }$ and the points $A$ and $C$ lie on the line $3 y = 2 x + 1$, then $2 \alpha + \beta + \gamma + \delta$ is equal to
(1) 10
(2) 5
(3) 12
(4) 8
Q65 Circles Circle Equation Derivation View
If one of the diameters of the circle $x ^ { 2 } + y ^ { 2 } - 10 x + 4 y + 13 = 0$ is a chord of another circle $C$, whose center is the point of intersection of the lines $2 x + 3 y = 12$ and $3 x - 2 y = 5$, then the radius of the circle $C$ is
(1) $\sqrt { 20 }$
(2) 4
(3) 6
(4) $3 \sqrt { 2 }$
Q66 Conic sections Focal Distance and Point-on-Conic Metric Computation View
If the foci of a hyperbola are same as that of the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 25 } = 1$ and the eccentricity of the hyperbola is $\frac { 15 } { 8 }$ times the eccentricity of the ellipse, then the smaller focal distance of the point $\left(\sqrt { 2 } , \frac { 14 } { 3 } \sqrt { \frac { 2 } { 5 } }\right)$ on the hyperbola is equal to
(1) $7 \sqrt { \frac { 2 } { 5 } } - \frac { 8 } { 3 }$
(2) $14 \sqrt { \frac { 2 } { 5 } } - \frac { 4 } { 3 }$
(3) $14 \sqrt { \frac { 2 } { 5 } } - \frac { 16 } { 3 }$
(4) $7 \sqrt { \frac { 2 } { 5 } } + \frac { 8 } { 3 }$
Q67 Chain Rule Limit Evaluation Involving Composition or Substitution View
$\lim _ { x \rightarrow 0 } \frac { e ^ { 2 \sin x } - 2 \sin x - 1 } { x ^ { 2 } }$
(1) is equal to $-1$
(2) does not exist
(3) is equal to 1
(4) is equal to 2
Q68 Binomial Theorem (positive integer n) Determine Parameters from Conditions on Coefficients or Terms View
Let $a$ be the sum of all coefficients in the expansion of $\left( 1 - 2 x + 2 x ^ { 2 } \right) ^ { 2023 } \left( 3 - 4 x ^ { 2 } + 2 x ^ { 3 } \right) ^ { 2024 }$ and $b = \lim _ { x \rightarrow 0 } \frac { \int _ { 0 } ^ { x } \frac { \log ( 1 + t ) } { t ^ { 2024 } + 1 } dt } { x ^ { 2 } }$. If the equations $c x ^ { 2 } + d x + e = 0$ and $2 b x ^ { 2 } + a x + 4 = 0$ have a common root, where $c , d , e \in R$, then $d : c : e$ equals
(1) $2 : 1 : 4$
(2) $4 : 1 : 4$
(3) $1 : 2 : 4$
(4) $1 : 1 : 4$
Q69 Matrices Determinant and Rank Computation View
If $f(x) = \begin{vmatrix} x^3 & 2x^2+1 & 1+3x \\ 3x^2+2 & 2x & x^3+6 \\ x^3-x & 4 & x^2-2 \end{vmatrix}$ for all $x \in \mathbb{R}$, then $2f(0) + f'(0)$ is equal to
(1) 48
(2) 24
(3) 42
(4) 18
Q70 3x3 Matrices Linear System with Parameter — Infinite Solutions View
If the system of linear equations $$\begin{aligned} & x - 2 y + z = - 4 \\ & 2 x + \alpha y + 3 z = 5 \\ & 3 x - y + \beta z = 3 \end{aligned}$$ has infinitely many solutions, then $12 \alpha + 13 \beta$ is equal to
(1) 60
(2) 64
(3) 54
(4) 58
Q71 Reciprocal Trig & Identities View
For $\alpha , \beta , \gamma \neq 0$. If $\sin ^ { - 1 } \alpha + \sin ^ { - 1 } \beta + \sin ^ { - 1 } \gamma = \pi$ and $(\alpha + \beta + \gamma)(\alpha - \gamma + \beta) = 3 \alpha \beta$, then $\gamma$ equals
(1) $\frac { \sqrt { 3 } } { 2 }$
(2) $\frac { 1 } { \sqrt { 2 } }$
(3) $\frac { \sqrt { 3 } - 1 } { 2 \sqrt { 2 } }$
(4) $\sqrt { 3 }$
Q72 Composite & Inverse Functions Evaluate Composition from Algebraic Definitions View
If $f(x) = \frac { 4 x + 3 } { 6 x - 4 } , \quad x \neq \frac { 2 } { 3 }$ and $(f \circ f)(x) = g(x)$, where $g : \mathbb{R} - \left\{\frac { 2 } { 3 }\right\} \rightarrow \mathbb{R} - \left\{\frac { 2 } { 3 }\right\}$, then $(g \circ g \circ g)(4)$ is equal to
(1) $- \frac { 19 } { 20 }$
(2) $\frac { 19 } { 20 }$
(3) $-4$
(4) 4
Q73 Chain Rule Continuity Conditions via Composition View
Let $g(x)$ be a linear function and $f(x) = \begin{cases} g(x), & x \leq 0 \\ \frac { 1 + x } { 2 + x } , & x > 0 \end{cases}$, is continuous at $x = 0$. If $f'(1) = f(-1)$, then the value of $g(3)$ is
(1) $\frac { 1 } { 3 } \log _ { e } \frac { 4 } { e^{1/3} }$
(2) $\frac { 1 } { 3 } \log _ { e } \frac { 4 } { 9 } + 1$
(3) $\log _ { e } \frac { 4 } { 9 } - 1$
(4) $\log _ { e } \frac { 4 } { 9 e ^ { 1/3 } }$
Q74 Areas Between Curves Area Between Curves with Parametric or Implicit Region Definition View
The area of the region $\left\{(x, y) : y ^ { 2 } \leq 4 x , x < 4 , \frac { x y (x - 1)(x - 2) } { (x - 3)(x - 4) } > 0 , x \neq 3 \right\}$ is
(1) $\frac { 16 } { 3 }$
(2) $\frac { 64 } { 3 }$
(3) $\frac { 8 } { 3 }$
(4) $\frac { 32 } { 3 }$
Q75 Differential equations Solving Separable DEs with Initial Conditions View
The solution curve of the differential equation $y \frac { d x } { d y } = x \left( \log _ { e } x - \log _ { e } y + 1 \right) , \quad x > 0 , \quad y > 0$ passing through the point $(e, 1)$ is
(1) $\log _ { e } \frac { y } { x } = x$
(2) $\log _ { e } \frac { y } { x } = y ^ { 2 }$
(3) $\log _ { e } \frac { x } { y } = y$
(4) $2 \log _ { e } \frac { x } { y } = y + 1$
Q76 First order differential equations (integrating factor) View
Let $y = y(x)$ be the solution of the differential equation $\frac { d y } { d x } = \frac { \tan x + y } { \sin x \sec x - \sin x \tan x } , \quad x \in \left(0, \frac { \pi } { 2 }\right)$ satisfying the condition $y\left(\frac { \pi } { 4 }\right) = 2$. Then, $y\left(\frac { \pi } { 3 }\right)$ is
(1) $\sqrt { 32 } + \log _ { e } \sqrt { 3 }$
(2) $\frac { \sqrt { 3 } } { 2} \left(2 + \log _ { e } 3\right)$
(3) $\sqrt { 3 } \left(1 + 2 \log _ { e } 3\right)$
(4) $\sqrt { 3 } \left(2 + \log _ { e } 3\right)$