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
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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
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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
2022 session1_24jun_shift2

25 maths questions

Q61 Exponential Equations & Modelling Solve Exponential Equation for Unknown Variable View
The sum of all real roots of equation $\left( e ^ { 2 x } - 4 \right) \left( 6 e ^ { 2 x } - 5 e ^ { x } + 1 \right) = 0$ is
(1) $\ln 4$
(2) $- \ln 3$
(3) $\ln 3$
(4) $\ln 5$
Q62 Stationary points and optimisation Geometric or applied optimisation problem View
Let $x , y > 0$. If $x ^ { 3 } y ^ { 2 } = 2 ^ { 15 }$, then the least value of $3 x + 2 y$ is
(1) 30
(2) 32
(3) 36
(4) 40
Q63 Standard trigonometric equations Solve trigonometric equation for solutions in an interval View
The number of solutions of the equation $\cos \left( x + \frac { \pi } { 3 } \right) \cos \left( \frac { \pi } { 3 } - x \right) = \frac { 1 } { 4 } \cos ^ { 2 } 2 x , x \in [ - 3 \pi , 3 \pi ]$ is:
(1) 8
(2) 5
(3) 6
(4) 7
Q64 Straight Lines & Coordinate Geometry Collinearity and Concurrency View
Let the area of the triangle with vertices $A ( 1 , \alpha ) , B ( \alpha , 0 )$ and $C ( 0 , \alpha )$ be 4 sq. units. If the points $( \alpha , - \alpha ) , ( - \alpha , \alpha )$ and $\left( \alpha ^ { 2 } , \beta \right)$ are collinear, then $\beta$ is equal to
(1) 64
(2) - 8
(3) - 64
(4) 512
Q65 Circles Circle-Related Locus Problems View
A particle is moving in the $x y$-plane along a curve $C$ passing through the point $( 3,3 )$. The tangent to the curve $C$ at the point $P$ meets the $x$-axis at $Q$. If the $y$-axis bisects the segment $P Q$, then $C$ is a parabola with
(1) length of latus rectum 3
(2) length of latus rectum 6
(3) focus $\left( \frac { 4 } { 3 } , 0 \right)$
(4) focus $\left( 0 , \frac { 3 } { 3 } \right)$
Q66 Conic sections Eccentricity or Asymptote Computation View
Let the maximum area of the triangle that can be inscribed in the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { 4 } = 1 , a > 2$, having one of its vertices at one end of the major axis of the ellipse and one of its sides parallel to the $y$-axis, be $6 \sqrt { 3 }$. Then the eccentricity of the ellipse is:
(1) $\frac { \sqrt { 3 } } { 2 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 1 } { \sqrt { 2 } }$
(4) $\frac { \sqrt { 3 } } { 4 }$
Q68 Matrices Linear System and Inverse Existence View
Let the system of linear equations $x + y + a z = 2$ $3 x + y + z = 4$ $x + 2 z = 1$ have a unique solution $\left( x ^ { * } , y ^ { * } , z ^ { * } \right)$. If $\left( \left( a , x ^ { * } \right) , \left( y ^ { * } , \alpha \right) \right.$ and $\left( x ^ { * } , - y ^ { * } \right)$ are collinear points, then the sum of absolute values of all possible values of $\alpha$ is:
(1) 4
(2) 3
(3) 2
(4) 1
Q69 Standard trigonometric equations Inverse trigonometric equation View
Let $x \times y = x ^ { 2 } + y ^ { 3 }$ and $( x \times 1 ) \times 1 = x \times ( 1 \times 1 )$. Then a value of $2 \sin ^ { - 1 } \left( \frac { x ^ { 4 } + x ^ { 2 } - 2 } { x ^ { 4 } + x ^ { 2 } + 2 } \right)$ is
(1) $\frac { \pi } { 4 }$
(2) $\frac { \pi } { 3 }$
(3) $\frac { \pi } { 6 }$
(4) $\pi$
Q70 Curve Sketching Continuity and Discontinuity Analysis of Piecewise Functions View
Let $f ( x ) = \begin{cases} \frac { \sin ( x - [ x ] ) } { x - [ x ] } , & x \in ( - 2 , - 1 ) \\ \max ( 2 x , 3 [ | x | ] ) , & | x | < 1 \\ 1 , & \text { otherwise } \end{cases}$ where $[ t ]$ denotes greatest integer $\leq t$. If $m$ is the number of points where $f$ is not continuous and $n$ is the number of points where $f$ is not differentiable, the ordered pair $( m , n )$ is:
(1) $( 3,3 )$
(2) $( 2,4 )$
(3) $( 2,3 )$
(4) $( 3,4 )$
Q71 Differentiating Transcendental Functions Higher-order or nth derivative computation View
If $y = \tan ^ { - 1 } \left( \sec x ^ { 3 } - \tan x ^ { 3 } \right) , \frac { \pi } { 2 } < x ^ { 3 } < \frac { 3 \pi } { 2 }$, then
(1) $x y ^ { \prime \prime } + 2 y ^ { \prime } = 0$
(2) $x ^ { 2 } y ^ { \prime \prime } - 6 y + \frac { 3 \pi } { 2 } = 0$
(3) $x ^ { 2 } y ^ { \prime \prime } - 6 y + 3 \pi = 0$
(4) $x y ^ { \prime \prime } - 4 y ^ { \prime } = 0$
Q72 Curve Sketching Number of Solutions / Roots via Curve Analysis View
The number of distinct real roots of the equation $x ^ { 7 } - 7 x - 2 = 0$ is
(1) 5
(2) 7
(3) 1
(4) 3
Q73 Stationary points and optimisation Determine intervals of increase/decrease or monotonicity conditions View
Let $\lambda ^ { * }$ be the largest value of $\lambda$ for which the function $f _ { \lambda } ( x ) = 4 \lambda x ^ { 3 } - 36 \lambda x ^ { 2 } + 36 x + 48$ is increasing for all $x \in \mathbb { R }$. Then $f _ { \lambda ^ { * } } ( 1 ) + f _ { \lambda ^ { * } } ( - 1 )$ is equal to:
(1) 36
(2) 48
(3) 64
(4) 72
Q74 Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution View
The value of the integral $\int _ { - \frac { \pi } { 2 } } ^ { \frac { \pi } { 2 } } \frac { d x } { \left( 1 + e ^ { x } \right) \left( \sin ^ { 6 } x + \cos ^ { 6 } x \right) }$ is equal to
(1) $2 \pi$
(2) 0
(3) $\pi$
(4) $\frac { \pi } { 2 }$
Q75 Indefinite & Definite Integrals Definite Integral as a Limit of Riemann Sums View
$\lim _ { n \rightarrow \infty } \left( \frac { n ^ { 2 } } { \left( n ^ { 2 } + 1 \right) ( n + 1 ) } + \frac { n ^ { 2 } } { \left( n ^ { 2 } + 4 \right) ( n + 2 ) } + \frac { n ^ { 2 } } { \left( n ^ { 2 } + 9 \right) ( n + 3 ) } + \ldots + \frac { n ^ { 2 } } { \left( n ^ { 2 } + n ^ { 2 } \right) ( n + n ) } \right)$ is equal to
(1) $\frac { \pi } { 8 } + \frac { 1 } { 4 } \ln 2$
(2) $\frac { \pi } { 4 } + \frac { 1 } { 8 } \ln 2$
(3) $\frac { \pi } { 4 } - \frac { 1 } { 8 } \ln 2$
(4) $\frac { \pi } { 8 } + \ln \sqrt { 2 }$
Q76 Differential equations Solving Separable DEs with Initial Conditions View
The slope of normal at any point $( x , y ) , x > 0 , y > 0$ on the curve $y = y ( x )$ is given by $\frac { x ^ { 2 } } { x y - x ^ { 2 } y ^ { 2 } - 1 }$. If the curve passes through the point $( 1,1 )$, then $e \cdot y ( e )$ is equal to
(1) $\frac { 1 - \tan ( 1 ) } { 1 + \tan ( 1 ) }$
(2) $\tan ( 1 )$
(3) 1
(4) $\frac { 1 + \tan ( 1 ) } { 1 - \tan ( 1 ) }$
Q77 Vectors Introduction & 2D True/False or Multiple-Statement Verification View
Let $a$ and $b$ be two unit vectors such that $| ( a + b ) + 2 ( a \times b ) | = 2$. If $\theta \in ( 0 , \pi )$ is the angle between $\hat { \mathrm { a } }$ and $\widehat { \mathrm { b } }$, then among the statements: $( S 1 ) : 2 | \widehat { a } \times \hat { b } | = | \widehat { a } - \hat { b } |$ $( S 2 )$ : The projection of $\widehat { a }$ on $( \widehat { a } + \widehat { b } )$ is $\frac { 1 } { 2 }$
(1) Only $( S 1 )$ is true.
(2) Only $( S 2 )$ is true.
(3) Both $( S 1 )$ and $( S 2 )$ are true.
(4) Both $( S 1 )$ and $( S 2 )$ are false.
Q78 Vectors 3D & Lines Shortest Distance Between Two Lines View
If the shortest distance between the lines $\frac { x - 1 } { 2 } = \frac { y - 2 } { 3 } = \frac { z - 3 } { \lambda }$ and $\frac { x - 2 } { 1 } = \frac { y - 4 } { 4 } = \frac { z - 5 } { 5 }$ is $\frac { 1 } { \sqrt { 3 } }$, then the sum of all possible values of $\lambda$ is:
(1) 16
(2) 6
(3) 12
(4) 15
Q79 Vectors 3D & Lines Dihedral Angle Computation View
Let the points on the plane $P$ be equidistant from the points $( - 4,2,1 )$ and $( 2 , - 2,3 )$. Then the acute angle between the plane $P$ and the plane $2 x + y + 3 z = 1$ is
(1) $\frac { \pi } { 6 }$
(2) $\frac { \pi } { 4 }$
(3) $\frac { \pi } { 3 }$
(4) $\frac { 5 \pi } { 12 }$
Q80 Conditional Probability Conditional Probability with Discrete Random Variable View
A random variable $X$ has the following probability distribution:
$X$01234
$P ( X )$$k$$2 k$$4 k$$6 k$$8 k$

The value of $P \left( \frac { 1 < x < 4 } { x \leq 2 } \right)$ is equal to
(1) $\frac { 4 } { 7 }$
(2) $\frac { 2 } { 3 }$
(3) $\frac { 3 } { 7 }$
(4) $\frac { 4 } { 5 }$
Q81 Complex Numbers Argand & Loci Distance and Region Optimization on Loci View
Let $S = \{ z \in \mathbb { C } : | z - 3 | \leq 1$ and $z ( 4 + 3 i ) + \bar { z } ( 4 - 3 i ) \leq 24 \}$. If $\alpha + i \beta$ is the point in $S$ which is closest to $4 i$, then $25 ( \alpha + \beta )$ is equal to $\_\_\_\_$.
Q82 Permutations & Arrangements Forming Numbers with Digit Constraints View
The number of 7-digit numbers which are multiples of 11 and are formed using all the digits $1,2,3,4,5,7$ and 9 is $\_\_\_\_$.
Q83 Arithmetic Sequences and Series Compute Partial Sum of an Arithmetic Sequence View
The remainder on dividing $1 + 3 + 3 ^ { 2 } + 3 ^ { 3 } + \ldots + 3 ^ { 2021 }$ by 50 is $\_\_\_\_$.
Q84 Circles Chord Length and Chord Properties View
Let a circle $C : ( x - h ) ^ { 2 } + ( y - k ) ^ { 2 } = r ^ { 2 } , k > 0$, touch the $x$-axis at $( 1,0 )$. If the line $x + y = 0$ intersects the circle $C$ at $P$ and $Q$ such that the length of the chord $P Q$ is 2, then the value of $h + k + r$ is equal to $\_\_\_\_$.
Q85 Circles Circle Equation Derivation View
Let $P _ { 1 }$ be a parabola with vertex $( 3,2 )$ and focus $( 4,4 )$ and $P _ { 2 }$ be its mirror image with respect to the line $x + 2 y = 6$. Then the directrix of $P _ { 2 }$ is $x + 2 y =$ $\_\_\_\_$.
Q86 Conic sections Eccentricity or Asymptote Computation View
Let the hyperbola $H : \frac { x ^ { 2 } } { a ^ { 2 } } - y ^ { 2 } = 1$ and the ellipse $E : 3 x ^ { 2 } + 4 y ^ { 2 } = 12$ be such that the length of latus rectum of $H$ is equal to the length of latus rectum of $E$. If $e _ { H }$ and $e _ { E }$ are the eccentricities of $H$ and $E$ respectively, then the value of $12 \left( e _ { H } ^ { 2 } + e _ { E } ^ { 2 } \right)$ is equal to $\_\_\_\_$.