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
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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
2018 15apr_shift1

28 maths questions

Q61 Roots of polynomials Vieta's formulas: compute symmetric functions of roots View
If $\lambda \in \mathrm { R }$ is such that the sum of the cubes of the roots of the equation, $x ^ { 2 } + ( 2 - \lambda ) x + ( 10 - \lambda ) = 0$ is minimum, then the magnitude of the difference of the roots of this equation is
(1) 20
(2) $2 \sqrt { 5 }$
(3) $2 \sqrt { 7 }$
(4) $4 \sqrt { 2 }$
Q62 Complex Numbers Arithmetic Identifying Real/Imaginary Parts or Components View
The set of all $\alpha \in R$, for which $w = \frac { 1 + ( 1 - 8 \alpha ) z } { 1 - z }$ is a purely imaginary number, for all $z \in C$ satisfying $| z | = 1$ and $\operatorname { Re } z \neq 1$, is
(1) $\{ 0 \}$
(2) an empty set
(3) $\left\{ 0 , \frac { 1 } { 4 } , - \frac { 1 } { 4 } \right\}$
(4) equal to $R$
Q63 Permutations & Arrangements Forming Numbers with Digit Constraints View
$n$ - digit numbers are formed using only three digits 2,5 and 7 . The smallest value of $n$ for which 900 such distinct numbers can be formed, is
(1) 6
(2) 8
(3) 9
(4) 7
Q64 Geometric Sequences and Series Sum of an Infinite Geometric Series (Direct Computation) View
If $b$ is the first term of an infinite G.P whose sum is five, then $b$ lies in the interval.
(1) $( - \infty , - 10 )$
(2) $( 10 , \infty )$
(3) $( 0,10 )$
(4) $( - 10,0 )$
Q65 Arithmetic Sequences and Series Find Specific Term from Given Conditions View
If $x _ { 1 } , x _ { 2 } , \ldots , x _ { n }$ and $\frac { 1 } { h _ { 1 } } , \frac { 1 } { h ^ { 2 } } , \ldots \ldots \frac { 1 } { h _ { n } }$ are two A.P's such that $x _ { 3 } = h _ { 2 } = 8$ and $x _ { 8 } = h _ { 7 } = 20$, then $x _ { 5 }$. $h _ { 10 }$ equals.
(1) 2560
(2) 2650
(3) 3200
(4) 1600
Q66 Binomial Theorem (positive integer n) Find a Specific Coefficient in a Single Binomial Expansion View
If $n$ is the degree of the polynomial,
$$\left[ \frac { 1 } { \sqrt { 5 x ^ { 3 } + 1 } - \sqrt { 5 x ^ { 3 } - 1 } } \right] ^ { 8 } + \left[ \frac { 1 } { \sqrt { 5 x ^ { 3 } + 1 } + \sqrt { 5 x ^ { 3 } - 1 } } \right] ^ { 8 }$$
and $m$ is the coefficient of $x ^ { n }$ in it, then the ordered pair ( $n , m$ ) is equal to
(1) $\left( 12 , ( 20 ) ^ { 4 } \right)$
(2) $\left( 8,5 ( 10 ) ^ { 4 } \right)$
(3) $\left( 24 , ( 10 ) ^ { 8 } \right)$
(4) $\left( 12,8 ( 10 ) ^ { 4 } \right)$
Q67 Trig Proofs Trigonometric Equation Constraint Deduction View
If $\tan A$ and $\tan B$ are the roots of the quadratic equation, $3 x ^ { 2 } - 10 x - 25 = 0$ then the value of $3 \sin ^ { 2 } ( A + B ) - 10 \sin ( A + B ) \cdot \cos ( A + B ) - 25 \cos ^ { 2 } ( A + B )$ is
(1) 25
(2) - 25
(3) - 10
(4) 10
Q68 Straight Lines & Coordinate Geometry Area Computation in Coordinate Geometry View
In a triangle $A B C$, coordiantates of $A$ are $( 1,2 )$ and the equations of the medians through $B$ and $C$ are $x + y = 5$ and $x = 4$ respectively. Then area of $\triangle A B C$ (in sq. units) is
(1) 5
(2) 9
(3) 12
(4) 4
Q69 Circles Circle Equation Derivation View
A circle passes through the points $( 2,3 )$ and $( 4,5 )$. If its centre lies on the line, $y - 4 x + 3 = 0$, then its radius is equal to
(1) $\sqrt { 5 }$
(2) 1
(3) $\sqrt { 2 }$
(4) 2
Q70 Circles Tangent Lines and Tangent Lengths View
Two parabolas with a common vertex and with axes along $x$-axis and $y$-axis, respectively, intersect each other in the first quadrant. if the length of the latus rectum of each parabola is 3 , then the equation of the common tangent to the two parabolas is?
(1) $3 ( x + y ) + 4 = 0$
(2) $8 ( 2 x + y ) + 3 = 0$
(3) $4 ( x + y ) + 3 = 0$
(4) $x + 2 y + 3 = 0$
Q71 Tangents, normals and gradients Normal or perpendicular line problems View
If $\beta$ is one of the angles between the normals to the ellipse, $x ^ { 2 } + 3 y ^ { 2 } = 9$ at the points ( $3 \cos \theta , \sqrt { 3 } \sin \theta$ ) and $( - 3 \sin \theta , \sqrt { 3 } \cos \theta ) ; \in \left( 0 , \frac { \pi } { 2 } \right) ;$ then $\frac { 2 \cot \beta } { \sin 2 \theta }$ is equal to
(1) $\sqrt { 2 }$
(2) $\frac { 2 } { \sqrt { 3 } }$
(3) $\frac { 1 } { \sqrt { 3 } }$
(4) $\frac { \sqrt { 3 } } { 4 }$
Q72 Circles Circle-Related Locus Problems View
If the tangents drawn to the hyperbola $4 y ^ { 2 } = x ^ { 2 } + 1$ intersect the co-ordinate axes at the distinct points $A$ and $B$, then the locus of the mid point of $A B$ is
(1) $x ^ { 2 } - 4 y ^ { 2 } + 16 x ^ { 2 } y ^ { 2 } = 0$
(2) $4 x ^ { 2 } - y ^ { 2 } + 16 x ^ { 2 } y ^ { 2 } = 0$
(3) $4 x ^ { 2 } - y ^ { 2 } - 16 x ^ { 2 } y ^ { 2 } = 0$
(4) $x ^ { 2 } - 4 y ^ { 2 } - 16 x ^ { 2 } y ^ { 2 } = 0$
Q74 Measures of Location and Spread View
The mean of a set of 30 observations is 75 . If each other observation is multiplied by a nonzero number $\lambda$ and then each of them is decreased by 25 , their mean remains the same. The $\lambda$ is equal to
(1) $\frac { 10 } { 3 }$
(2) $\frac { 4 } { 3 }$
(3) $\frac { 1 } { 3 }$
(4) $\frac { 2 } { 3 }$
Q75 Projectiles Horizontal Launch or Dropped Object Problems View
An aeroplane flying at a constant speed, parallel to the horizontal ground, $\sqrt { 3 } \mathrm {~km}$ above it, is observed at an elevation of $60 ^ { \circ }$ from a point on the ground. If, after five seconds, its elevation from the same point, is $30 ^ { \circ }$, then the speed (in $\mathrm { km } / \mathrm { hr }$ ) of the aeroplane is
(1) 1500
(2) 750
(3) 720
(4) 1440
Q77 Matrices Matrix Algebra and Product Properties View
Let $A$ be a matrix such that $A$. $\left[ \begin{array} { l l } 1 & 2 \\ 0 & 3 \end{array} \right]$ is a scalar matrix and $| 3 A | = 108$. Then $A ^ { 2 }$ equals
(1) $\left[ \begin{array} { c c } 4 & - 32 \\ 0 & 36 \end{array} \right]$
(2) $\left[ \begin{array} { c c } 4 & 0 \\ - 32 & 36 \end{array} \right]$
(3) $\left[ \begin{array} { c c } 36 & 0 \\ - 32 & 4 \end{array} \right]$
(4) $\left[ \begin{array} { c c } 36 & - 32 \\ 0 & 4 \end{array} \right]$
Q78 Chain Rule Limit Involving Derivative Definition of Composed Functions View
$f ( x ) = \left| \begin{array} { c c c } \cos x & x & 1 \\ 2 \sin x & x ^ { 2 } & 2 x \\ \tan x & x & 1 \end{array} \right|$, then $\lim _ { x \rightarrow 0 } \frac { f ^ { \prime } ( x ) } { x }$
(1) Exists and is equal to - 2
(2) Does not exist
(3) Exist and is equal to 0
(4) Exists and is equal to 2
Q79 Simultaneous equations View
Let $S$ be the set of all real values of $k$ for which the system of linear equations
$$\begin{aligned} & x + y + z = 2 \\ & 2 x + y - z = 3 \\ & 3 x + 2 y + k z = 4 \end{aligned}$$
has a unique solution. Then $S$ is
(1) an empty set
(2) equal to $\mathrm { R } - \{ 0 \}$
(3) equal to $\{ 0 \}$
(4) equal to $R$
Q80 Differential equations Qualitative Analysis of DE Solutions View
Let $S = \left\{ ( \lambda , \mu ) \in R \times R : f ( t ) = \left( | \lambda | e ^ { t } - \mu \right) \cdot \sin ( 2 | t | ) , t \in R \right.$, is a differentiable function $\}$. Then $S$ is a subest of?
(1) $R \times [ 0 , \infty )$
(2) $( - \infty , 0 ) \times R$
(3) $[ 0 , \infty ) \times R$
(4) $R \times ( - \infty , 0 )$
Q81 Implicit equations and differentiation Second derivative via implicit differentiation View
If $x ^ { 2 } + y ^ { 2 } + \sin y = 4$, then the value of $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ at the point $( - 2,0 )$ is
(1) - 34
(2) - 32
(3) - 2
(4) 4
Q82 Stationary points and optimisation Geometric or applied optimisation problem View
If a right circular cone having maximum volume, is inscribed in a sphere of radius 3 cm , then the curved surface area ( $\mathrm { in } \mathrm { cm } ^ { 2 }$ ) of this cone is
(1) $8 \sqrt { 3 } \pi$
(2) $6 \sqrt { 2 } \pi$
(3) $6 \sqrt { 3 } \pi$
(4) $8 \sqrt { 2 } \pi$
Q83 Integration by Substitution Substitution to Transform Integral Form (Show Transformed Expression) View
If $f \left( \frac { x - 4 } { x + 2 } \right) = 2 x + 1 , ( x \in R = \{ 1 , - 2 \} )$, then $\int f ( x ) d x$ is equal to (where $C$ is a constant of integration)
(1) $12 \log _ { e } | 1 - x | - 3 x + c$
(2) $- 12 \log _ { e } | 1 - x | - 3 x + c$
(3) $- 12 \log _ { e } | 1 - x | + 3 x + c$
(4) $12 \log _ { e } | 1 - x | + 3 x + c$
Q84 Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution View
The value of the integral
$$\int _ { - \frac { \pi } { 2 } } ^ { \frac { \pi } { 2 } } \sin ^ { 4 } x \left( 1 + \log \left( \frac { 2 + \sin x } { 2 - \sin x } \right) \right) d x$$
is
(1) $\frac { 3 } { 16 } \pi$
(2) 0
(3) $\frac { 3 } { 8 } \pi$
(4) $\frac { 3 } { 4 }$
Q85 Areas by integration View
The area (in sq. units) of the region $\{ x \in R : x \geq 0 , y \geq 0 , y \geq x - 2$ and $y \leq \sqrt { x } \}$, is
(1) $\frac { 13 } { 3 }$
(2) $\frac { 10 } { 3 }$
(3) $\frac { 5 } { 3 }$
(4) $\frac { 8 } { 3 }$
Q86 First order differential equations (integrating factor) View
Let $y = y ( x )$ be the solution of the differential equation $\frac { d y } { d x } + 2 y = f ( x )$, where
$$f ( x ) = \left\{ \begin{array} { l c } 1 , & x \in [ 0,1 ] \\ 0 , & \text { otherwise } \end{array} \right.$$
If $y ( 0 ) = 0$, then $y \left( \frac { 3 } { 2 } \right)$ is
(1) $\frac { e ^ { 2 } - 1 } { 2 e ^ { 3 } }$
(2) $\frac { e ^ { 2 } - 1 } { e ^ { 3 } }$
(3) $\frac { 1 } { 2 e }$
(4) $\frac { e ^ { 2 } + 1 } { 2 e ^ { 4 } }$
Q87 Vectors Introduction & 2D Dot Product Computation View
If $\vec { a } , \vec { b }$, and $\overrightarrow { \mathrm { c } }$ are unit vectors such that $\vec { a } + 2 \vec { b } + 2 \overrightarrow { \mathbf { c } } = \overrightarrow { 0 }$, then $| \vec { a } \times \overrightarrow { \mathbf { c } } |$ is equal to
(1) $\frac { 1 } { 4 }$
(2) $\frac { \sqrt { 15 } } { 4 }$
(3) $\frac { 15 } { 16 }$
(4) $\frac { \sqrt { 15 } } { 16 }$
Q88 Vectors: Lines & Planes Coplanarity and Relative Position of Planes View
A variable plane passes through a fixed point ( $3,2,1$ ) and meets $x , y$ and $z$ axes at $A , B$ and $C$ respectively. A plane is drawn parallel to $y z$ - plane through $A$, a second plane is drawn parallel $z x$ plane through $B$ and a third plane is drawn parallel to $x y$ - plane through $C$. Then the locus of the point of intersection of these three planes, is
(1) $( x + y + z = 6 )$
(2) $\frac { x } { 3 } + \frac { y } { 2 } + \frac { z } { 1 } = 1$
(3) $\frac { 3 } { x } + \frac { 2 } { y } + \frac { 1 } { z } = 1$
(4) $\frac { 1 } { x } + \frac { 1 } { y } + \frac { 1 } { z } = \frac { 11 } { 6 }$
Q89 Vectors: Lines & Planes Dihedral Angle or Angle Between Planes/Lines View
An angle between the plane, $x + y + z = 5$ and the line of intersection of the planes, $3 x + 4 y + z - 1 = 0$ and $5 x + 8 y + 2 z + 14 = 0$, is
(1) $\cos ^ { - 1 } \left( \frac { 3 } { \sqrt { 17 } } \right)$
(2) $\cos ^ { - 1 } \left( \sqrt { \frac { 3 } { 17 } } \right)$
(3) $\sin ^ { - 1 } \left( \frac { 3 } { \sqrt { 17 } } \right)$
(4) $\sin ^ { - 1 } \left( \sqrt { \frac { 3 } { 17 } } \right)$
Q90 Conditional Probability Bayes' Theorem with Production/Source Identification View
A box ' $A$ ' contains 2 white, 3 red and 2 black balls. Another box ' $B ^ { \prime }$ contains 4 white, 2 red and 3 black balls. If two balls are drawn at random, without replacement, from a randomly selected box and one ball turns out to be white while the other ball turns out to be red, then the probability that both balls are drawn from box ' $B ^ { \prime }$ is
(1) $\frac { 7 } { 16 }$
(2) $\frac { 9 } { 32 }$
(3) $\frac { 7 } { 8 }$
(4) $\frac { 9 } { 16 }$