jee-main

Papers (169)
2025
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2024
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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
2020 session1_07jan_shift1

26 maths questions

Q21 Projectiles Kinetic Energy at a Point in Flight View
A particle ($\mathrm { m } = 1 \mathrm {~kg}$) slides down a frictionless track (AOC) starting from rest at a point $A$ (height 2 m). After reaching $C$, the particle continues to move freely in air as a projectile. When it reaching its highest point P (height 1 m), the kinetic energy of the particle (in J) is: (Figure drawn is schematic and not to scale; take $g = 10 \mathrm {~ms} ^ { - 2 }$) $\_\_\_\_$.
Q51 Quadratic trigonometric equations View
Let $\alpha$ and $\beta$ be two real roots of the equation $(k + 1) \tan ^ { 2 } x - \sqrt { 2 } \cdot \lambda \tan x = (1 - k)$, where $k (\neq -1)$ and $\lambda$ are real numbers. If $\tan ^ { 2 } (\alpha + \beta) = 50$, then a value of $\lambda$ is
(1) $10 \sqrt { 2 }$
(2) 10
(3) 5
(4) $5 \sqrt { 2 }$
Q52 Complex Numbers Argand & Loci Circle Equation and Properties via Complex Number Manipulation View
If $\operatorname { Re } \left( \frac { z - 1 } { 2 z + i } \right) = 1$, where $z = x + i y$, then the point $(x, y)$ lies on a
(1) circle whose centre is at $\left( - \frac { 1 } { 2 } , - \frac { 3 } { 2 } \right)$
(2) straight line whose slope is $- \frac { 2 } { 3 }$
(3) straight line whose slope is $\frac { 3 } { 2 }$
(4) circle whose diameter is $\frac { \sqrt { 5 } } { 2 }$
Q53 Permutations & Arrangements Word Permutations with Repeated Letters View
Total number of 6-digit numbers in which only and all the five digits $1, 3, 5, 7$ and 9 appears, is
(1) $\frac { 1 } { 2 } (6!)$
(2) $6!$
(3) $5 ^ { 6 }$
(4) $\frac { 5 } { 2 } (6!)$
Q54 Arithmetic Sequences and Series Find Specific Term from Given Conditions View
Five numbers are in A.P., whose sum is 25 and product is 2520. If one of these five numbers is $- \frac { 1 } { 2 }$, then the greatest number amongst them is
(1) 27
(2) 7
(3) $\frac { 21 } { 2 }$
(4) 16
Q55 Number Theory Divisibility and Divisor Analysis View
The greatest positive integer $k$, for which $49 ^ { k } + 1$ is a factor of the sum $49 ^ { 125 } + 49 ^ { 124 } + \ldots + 49 ^ { 2 } + 49 + 1$, is
(1) 32
(2) 63
(3) 60
(4) 35
Q56 Conic sections Tangent and Normal Line Problems View
If $y = m x + 4$ is a tangent to both the parabolas, $y ^ { 2 } = 4 x$ and $x ^ { 2 } = 2 b y$, then $b$ is equal to
(1) $-32$
(2) $-64$
(3) $-128$
(4) 128
Q57 Conic sections Eccentricity or Asymptote Computation View
If the distance between the foci of an ellipse is 6 and the distance between its directrix is 12, then the length of its latus rectum is
(1) $\sqrt { 3 }$
(2) $3 \sqrt { 2 }$
(3) $\frac { 3 } { \sqrt { 2 } }$
(4) $2 \sqrt { 3 }$
Q58 Proof Direct Proof of a Stated Identity or Equality View
For two statements $p$ and $q$, the logical statement $(p \rightarrow q) \wedge (q \rightarrow \sim p)$ is equivalent to
(1) $p$
(2) $q$
(3) $\sim p$
(4) $\sim q$
Q59 Matrices Matrix Power Computation and Application View
Let $\alpha$ be a root of the equation $x ^ { 2 } + x + 1 = 0$ and the matrix $A = \frac { 1 } { \sqrt { 3 } } \left[ \begin{array} { c c c } 1 & 1 & 1 \\ 1 & \alpha & \alpha ^ { 2 } \\ 1 & \alpha ^ { 2 } & \alpha ^ { 4 } \end{array} \right]$, then the matrix $A ^ { 31 }$ is equal to
(1) $A ^ { 3 }$
(2) $I _ { 3 }$
(3) $A ^ { 2 }$
(4) $A$
Q60 3x3 Matrices Linear System Existence and Uniqueness via Determinant View
If the system of linear equations $$\begin{aligned} & 2x + 2ay + az = 0 \\ & 2x + 3by + bz = 0 \\ & 2x + 4cy + cz = 0 \end{aligned}$$ where $a, b, c \in R$ are non-zero and distinct; has a non-zero solution, then
(1) $\frac { 1 } { a } , \frac { 1 } { b } , \frac { 1 } { c }$ are in $A.P$.
(2) $a, b, c$ are in $G.P$.
(3) $a + b + c = 0$
(4) $a, b, c$ are in $A.P$.
Q61 Composite & Inverse Functions Recover a Function from a Composition or Functional Equation View
If $g(x) = x ^ { 2 } + x - 1$ and $(g \circ f)(x) = 4x ^ { 2 } - 10x + 5$, then $f \left( \frac { 5 } { 4 } \right)$ is equal to
(1) $\frac { 3 } { 2 }$
(2) $- \frac { 1 } { 2 }$
(3) $\frac { 1 } { 2 }$
(4) $- \frac { 3 } { 2 }$
Q62 Differentiating Transcendental Functions Evaluate derivative at a point or find tangent slope View
If $y(\alpha) = \sqrt { 2 \left( \frac { \tan \alpha + \cot \alpha } { 1 + \tan ^ { 2 } \alpha } \right) + \frac { 1 } { \sin ^ { 2 } \alpha } }$, $\alpha \in \left( \frac { 3 \pi } { 4 } , \pi \right)$, then $\frac { d y } { d \alpha }$ at $\alpha = \frac { 5 \pi } { 6 }$ is
(1) 4
(2) $\frac { 4 } { 3 }$
(3) $-4$
(4) $- \frac { 1 } { 4 }$
Q63 Implicit equations and differentiation Eliminate parameter from implicit family and derive ODE View
Let $x ^ { k } + y ^ { k } = a ^ { k }$, $(a, k > 0)$ and $\frac { d y } { d x } + \left( \frac { y } { x } \right) ^ { \frac { 1 } { 3 } } = 0$, then $k$ is
(1) $\frac { 3 } { 2 }$
(2) $\frac { 4 } { 3 }$
(3) $\frac { 2 } { 3 }$
(4) $\frac { 1 } { 3 }$
Q64 Applied differentiation Properties of differentiable functions (abstract/theoretical) View
Let the function, $f : [-7, 0] \rightarrow R$ be continuous on $[-7, 0]$ and differentiable on $(-7, 0)$. If $f(-7) = -3$ and $f ^ { \prime } (x) \leq 2$ for all $x \in (-7, 0)$, then for all such functions $f$, $f(-1) + f(0)$ lies in the interval
(1) $(-\infty, 20]$
(2) $[-3, 11]$
(3) $(-\infty, 11]$
(4) $[-6, 20]$
Q65 Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution View
If $f(a + b + 1 - x) = f(x)$, for all $x$, where $a$ and $b$ are fixed positive real numbers, then $\frac { 1 } { a + b } \int _ { a } ^ { b } x (f(x) + f(x + 1)) d x$ is equal to
(1) $\int _ { a - 1 } ^ { b - 1 } f(x + 1) d x$
(2) $\int _ { a - 1 } ^ { b - 1 } f(x) d x$
(3) $\int _ { a + 1 } ^ { b + 1 } f(x) d x$
(4) $\int _ { a + 1 } ^ { b + 1 } f(x + 1) d x$
Q66 Areas Between Curves Area Involving Conic Sections or Circles View
The area of the region (in sq. units), enclosed by the circle $x ^ { 2 } + y ^ { 2 } = 2$ which is not common to the region bounded by the parabola $y ^ { 2 } = x$ and the straight line $y = x$, is
(1) $\frac { 1 } { 6 } (24 \pi - 1)$
(2) $\frac { 1 } { 3 } (6 \pi - 1)$
(3) $\frac { 1 } { 3 } (12 \pi - 1)$
(4) $\frac { 1 } { 6 } (12 \pi - 1)$
Q67 Differential equations Solving Separable DEs with Initial Conditions View
If $y = y(x)$ is the solution of the differential equation, $e ^ { y } \left( \frac { d y } { d x } - 1 \right) = e ^ { x }$ such that $y(0) = 0$, then $y(1)$ is equal to
(1) $1 + \log _ { e } 2$
(2) $2 + \log _ { e } 2$
(3) $2e$
(4) $\log _ { e } 2$
Q68 Vectors 3D & Lines Vector Algebra and Triple Product Computation View
A vector $\vec { a } = \alpha \hat { i } + 2 \hat { j } + \beta \hat { k }$ $(\alpha, \beta \in R)$ lies in the plane of the vectors, $\vec { b } = \hat { i } + \hat { j }$ and $\vec { c } = \hat { i } - \hat { j } + 4 \hat { k }$. If $\vec { a }$ bisects the angle between $\vec { b }$ and $\vec { c }$, then
(1) $\vec { a } \cdot \hat { i } + 3 = 0$
(2) $\vec { a } \cdot \hat { i } + 1 = 0$
(3) $\vec { a } \cdot \widehat { k } + 2 = 0$
(4) $\vec { a } \cdot \widehat { k } + 4 = 0$
Q69 Vectors: Lines & Planes Perpendicular/Orthogonal Projection onto a Plane View
Let $P$ be a plane passing through the points $(2,1,0)$, $(4,1,1)$ and $(5,0,1)$ and $R$ be any point $(2,1,6)$. Then the image of $R$ in the plane $P$ is
(1) $(6,5,2)$
(2) $(6,5,-2)$
(3) $(4,3,2)$
(4) $(3,4,-2)$
Q70 Discrete Probability Distributions Expectation and Variance from Context-Based Random Variables View
An unbiased coin is tossed 5 times. Suppose that a variable $X$ is assigned the value $k$ when $k$ consecutive heads are obtained for $k = 3, 4, 5$, otherwise $X$ takes the value $-1$. Then the expected value of $X$, is
(1) $\frac { 3 } { 16 }$
(2) $\frac { 1 } { 8 }$
(3) $- \frac { 3 } { 16 }$
(4) $- \frac { 1 } { 8 }$
Q71 Binomial Theorem (positive integer n) Determine Parameters from Conditions on Coefficients or Terms View
If the sum of the coefficients of all even powers of $x$ in the product $\left(1 + x + x ^ { 2 } + \ldots + x ^ { 2n} \right) \left(1 - x + x ^ { 2 } - x ^ { 3 } + \ldots + x ^ { 2n } \right)$ is 61, then $n$ is equal to
Q72 Straight Lines & Coordinate Geometry Triangle Properties and Special Points View
Let $A(1,0)$, $B(6,2)$ and $C \left( \frac { 3 } { 2 } , 6 \right)$ be the vertices of a triangle $ABC$. If $P$ is a point inside the triangle $ABC$ such that the triangles $APC$, $APB$ and $BPC$ have equal areas, then the length of the line segment $PQ$, where $Q$ is the point $\left( - \frac { 7 } { 6 } , - \frac { 1 } { 3 } \right)$, is
Q73 Exponential Functions Limit Evaluation View
$\lim _ { x \rightarrow 2 } \frac { 3 ^ { x } + 3 ^ { 3 - x } - 12 } { 3 ^ { - \frac { x } { 2 } } - 3 ^ { 1 - x } }$ is equal to
Q74 Measures of Location and Spread View
If the variance of the first $n$ natural numbers is 10 and the variance of the first $m$ even natural numbers is 16, then the value of $m + n$ is equal to
Q75 Modulus function Differentiability of functions involving modulus View
Let $S$ be the set of points where the function, $f(x) = |2 - |x - 3||$, $x \in R$, is not differentiable. Then $\sum _ { x \in S } f(f(x))$ is equal to