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
2024 session2_05apr_shift2

30 maths questions

Q61 Complex Numbers Argand & Loci Intersection of Loci and Simultaneous Geometric Conditions View
Let $S _ { 1 } = \{ z \in C : | z | \leq 5 \} , S _ { 2 } = \left\{ z \in C : \operatorname { Im } \left( \frac { z + 1 - \sqrt { 3 } i } { 1 - \sqrt { 3 } i } \right) \geq 0 \right\}$ and $S _ { 3 } = \{ z \in C : \operatorname { Re } ( z ) \geq 0 \}$. Then the area of the region $S _ { 1 } \cap S _ { 2 } \cap S _ { 3 }$ is :
(1) $\frac { 125 \pi } { 12 }$
(2) $\frac { 125 \pi } { 4 }$
(3) $\frac { 125 \pi } { 24 }$
(4) $\frac { 125 \pi } { 6 }$
Q62 Permutations & Arrangements Dictionary Order / Rank of a Permutation View
60 words can be made using all the letters of the word BHBJO, with or without meaning. If these words are written as in a dictionary, then the $50 ^ { \text {th} }$ word is :
(1) JBBOH
(2) OBBJH
(3) OBBHJ
(4) HBBJO
Q63 Arithmetic Sequences and Series Find Specific Term from Given Conditions View
For $x \geqslant 0$, the least value of K , for which $4 ^ { 1 + x } + 4 ^ { 1 - x } , \frac { \mathrm {~K} } { 2 } , 16 ^ { x } + 16 ^ { - x }$ are three consecutive terms of an A.P., is equal to :
(1) 8
(2) 4
(3) 10
(4) 16
Q64 Binomial Theorem (positive integer n) Find a Specific Coefficient in a Single Binomial Expansion View
If the constant term in the expansion of $\left( \frac { \sqrt [ 5 ] { 3 } } { x } + \frac { 2 x } { \sqrt [ 3 ] { 5 } } \right) ^ { 12 } , x \neq 0$, is $\alpha \times 2 ^ { 8 } \times \sqrt [ 5 ] { 3 }$, then $25 \alpha$ is equal to :
(1) 724
(2) 742
(3) 639
(4) 693
Q65 Straight Lines & Coordinate Geometry Locus Determination View
Let $A ( - 1,1 )$ and $B ( 2,3 )$ be two points and $P$ be a variable point above the line $A B$ such that the area of $\triangle \mathrm { PAB }$ is 10 . If the locus of P is $\mathrm { a } x + \mathrm { b } y = 15$, then $5 \mathrm { a } + 2 \mathrm {~b}$ is :
(1) 6
(2) $- \frac { 6 } { 5 }$
(3) 4
(4) $- \frac { 12 } { 5 }$
Q66 Circles Circle Equation Derivation View
Let $A B C D$ and $A E F G$ be squares of side 4 and 2 units, respectively. The point $E$ is on the line segment AB and the point F is on the diagonal AC . Then the radius r of the circle passing through the point F and touching the line segments BC and CD satisfies:
(1) $r = 0$
(2) $2 r ^ { 2 } - 4 r + 1 = 0$
(3) $2 r ^ { 2 } - 8 r + 7 = 0$
(4) $r ^ { 2 } - 8 r + 8 = 0$
Q67 Circles Chord Length and Chord Properties View
Let the circle $C _ { 1 } : x ^ { 2 } + y ^ { 2 } - 2 ( x + y ) + 1 = 0$ and $C _ { 2 }$ be a circle having centre at $( - 1,0 )$ and radius 2 . If the line of the common chord of $\mathrm { C } _ { 1 }$ and $\mathrm { C } _ { 2 }$ intersects the $y$-axis at the point P , then the square of the distance of P from the centre of $\mathrm { C } _ { 1 }$ is :
(1) 2
(2) 1
(3) 4
(4) 6
Q68 Combinations & Selection Partitioning into Teams or Groups View
Let the set $S = \{ 2,4,8,16 , \ldots , 512 \}$ be partitioned into 3 sets $A , B , C$ with equal number of elements such that $\mathrm { A } \cup \mathrm { B } \cup \mathrm { C } = \mathrm { S }$ and $\mathrm { A } \cap \mathrm { B } = \mathrm { B } \cap \mathrm { C } = \mathrm { A } \cap \mathrm { C } = \phi$. The maximum number of such possible partitions of $S$ is equal to:
(1) 1680
(2) 1640
(3) 1520
(4) 1710
Q69 3x3 Matrices Determinant with Cofactor or Expansion Relationship View
Let $\alpha \beta \neq 0$ and $A = \left[ \begin{array} { r r r } \beta & \alpha & 3 \\ \alpha & \alpha & \beta \\ - \beta & \alpha & 2 \alpha \end{array} \right]$. If $B = \left[ \begin{array} { r r r } 3 \alpha & - 9 & 3 \alpha \\ - \alpha & 7 & - 2 \alpha \\ - 2 \alpha & 5 & - 2 \beta \end{array} \right]$ is the matrix of cofactors of the elements of $A$, then $\operatorname { det } ( A B )$ is equal to :
(1) 64
(2) 216
(3) 343
(4) 125
Q70 Simultaneous equations View
The values of $m , n$, for which the system of equations \begin{align*} x + y + z &= 4, 2x + 5y + 5z &= 17, x + 2y + \mathrm{m}z &= \mathrm{n} \end{align*} has infinitely many solutions, satisfy the equation:
(1) $m ^ { 2 } + n ^ { 2 } - m n = 39$
(2) $m ^ { 2 } + n ^ { 2 } - m - n = 46$
(3) $m ^ { 2 } + n ^ { 2 } + m + n = 64$
(4) $m ^ { 2 } + n ^ { 2 } + m n = 68$
Q71 Composite & Inverse Functions Injectivity, Surjectivity, or Bijectivity Classification View
Let $f , g : \mathbf { R } \rightarrow \mathbf { R }$ be defined as : $f ( x ) = | x - 1 |$ and $g ( x ) = \begin{cases} \mathrm { e } ^ { x } , & x \geq 0 \\ x + 1 , & x \leq 0 \end{cases}$ Then the function $f ( g ( x ) )$ is
(1) neither one-one nor onto.
(2) one-one but not onto.
(3) onto but not one-one.
(4) both one-one and onto.
Q72 Curve Sketching Continuity and Discontinuity Analysis of Piecewise Functions View
Let $f : [ - 1,2 ] \rightarrow \mathbf { R }$ be given by $f ( x ) = 2 x ^ { 2 } + x + \left[ x ^ { 2 } \right] - [ x ]$, where $[ t ]$ denotes the greatest integer less than or equal to $t$. The number of points, where $f$ is not continuous, is :
(1) 5
(2) 6
(3) 3
(4) 4
Q73 Differentiating Transcendental Functions Higher-order or nth derivative computation View
If $y ( \theta ) = \frac { 2 \cos \theta + \cos 2 \theta } { \cos 3 \theta + 4 \cos 2 \theta + 5 \cos \theta + 2 }$, then at $\theta = \frac { \pi } { 2 } , y ^ { \prime \prime } + y ^ { \prime } + y$ is equal to :
(1) $\frac { 1 } { 2 }$
(2) 1
(3) 2
(4) $\frac { 3 } { 2 }$
Q74 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
Let $\beta ( \mathrm { m } , \mathrm { n } ) = \int _ { 0 } ^ { 1 } x ^ { \mathrm { m } - 1 } ( 1 - x ) ^ { \mathrm { n } - 1 } \mathrm {~d} x , \mathrm {~m} , \mathrm { n } > 0$. If $\int _ { 0 } ^ { 1 } \left( 1 - x ^ { 10 } \right) ^ { 20 } \mathrm {~d} x = \mathrm { a } \times \beta ( \mathrm { b } , \mathrm { c } )$, then $100 ( \mathrm { a } + \mathrm { b } + \mathrm { c } )$ equals
(1) 1021
(2) 2120
(3) 2012
(4) 1120
Q75 Areas Between Curves Area Involving Piecewise or Composite Functions View
The area enclosed between the curves $y = x | x |$ and $y = x - | x |$ is :
(1) $\frac { 4 } { 3 }$
(2) 1
(3) $\frac { 2 } { 3 }$
(4) $\frac { 8 } { 3 }$
Q76 Differential equations Higher-Order and Special DEs (Proof/Theory) View
The differential equation of the family of circles passing through the origin and having centre at the line $y = x$ is :
(1) $\left( x ^ { 2 } - y ^ { 2 } + 2 x y \right) \mathrm { d } x = \left( x ^ { 2 } - y ^ { 2 } - 2 x y \right) \mathrm { d } y$
(2) $\left( x ^ { 2 } + y ^ { 2 } + 2 x y \right) \mathrm { d } x = \left( x ^ { 2 } + y ^ { 2 } - 2 x y \right) \mathrm { d } y$
(3) $\left( x ^ { 2 } + y ^ { 2 } - 2 x y \right) \mathrm { d } x = \left( x ^ { 2 } + y ^ { 2 } + 2 x y \right) \mathrm { d } y$
(4) $\left( x ^ { 2 } - y ^ { 2 } + 2 x y \right) \mathrm { d } x = \left( x ^ { 2 } - y ^ { 2 } + 2 x y \right) \mathrm { d } y$
Q77 Vectors Introduction & 2D Optimization of a Vector Expression View
Consider three vectors $\vec { a } , \vec { b } , \vec { c }$. Let $| \vec { a } | = 2 , | \vec { b } | = 3$ and $\vec { a } = \vec { b } \times \vec { c }$. If $\alpha \in \left[ 0 , \frac { \pi } { 3 } \right]$ is the angle between the vectors $\vec { b }$ and $\vec { c }$, then the minimum value of $27 | \vec { c } - \vec { a } | ^ { 2 }$ is equal to:
(1) 110
(2) 124
(3) 121
(4) 105
Q78 Vectors Introduction & 2D Dot Product Computation View
Let $\overrightarrow { \mathrm { a } } = 2 \hat { i } + 5 \hat { j } - \hat { k } , \overrightarrow { \mathrm {~b} } = 2 \hat { i } - 2 \hat { j } + 2 \hat { k }$ and $\overrightarrow { \mathrm { c } }$ be three vectors such that $( \vec { c } + \hat { i } ) \times ( \vec { a } + \vec { b } + \hat { i } ) = \vec { a } \times ( \vec { c } + \hat { i } )$. If $\vec { a } \cdot \vec { c } = - 29$, then $\vec { c } \cdot ( - 2 \hat { i } + \hat { j } + \hat { k } )$ is equal to:
(1) 15
(2) 12
(3) 10
(4) 5
Q79 Vectors: Lines & Planes Perpendicular/Orthogonal Projection onto a Plane View
Let $( \alpha , \beta , \gamma )$ be the image of the point $( 8,5,7 )$ in the line $\frac { x - 1 } { 2 } = \frac { y + 1 } { 3 } = \frac { z - 2 } { 5 }$. Then $\alpha + \beta + \gamma$ is equal to :
(1) 16
(2) 20
(3) 14
(4) 18
Q80 Hypothesis test of binomial distributions View
The coefficients $a , b , c$ in the quadratic equation $a x ^ { 2 } + b x + c = 0$ are from the set $\{ 1,2,3,4,5,6 \}$. If the probability of this equation having one real root bigger than the other is $p$, then 216 p equals :
(1) 57
(2) 76
(3) 38
(4) 19
Q81 Sign Change & Interval Methods View
The number of real solutions of the equation $x | x + 5 | + 2 | x + 7 | - 2 = 0$ is $\_\_\_\_$
Q82 Sequences and series, recurrence and convergence Series convergence and power series analysis View
If $1 + \frac { \sqrt { 3 } - \sqrt { 2 } } { 2 \sqrt { 3 } } + \frac { 5 - 2 \sqrt { 6 } } { 18 } + \frac { 9 \sqrt { 3 } - 11 \sqrt { 2 } } { 36 \sqrt { 3 } } + \frac { 49 - 20 \sqrt { 6 } } { 180 } + \ldots$ upto $\infty = 2 + \left( \sqrt { \frac { b } { a } } + 1 \right) \log _ { e } \left( \frac { a } { b } \right)$, where a and b are integers with $\operatorname { gcd } ( \mathrm { a } , \mathrm { b } ) = 1$, then $11 \mathrm { a } + 18 \mathrm {~b}$ is equal to $\_\_\_\_$
Q83 Standard trigonometric equations Solve trigonometric equation for solutions in an interval View
The number of solutions of $\sin ^ { 2 } x + \left( 2 + 2 x - x ^ { 2 } \right) \sin x - 3 ( x - 1 ) ^ { 2 } = 0$, where $- \pi \leq x \leq \pi$, is $\_\_\_\_$
Q84 Circles Distance from Center to Line View
Let a line perpendicular to the line $2 x - y = 10$ touch the parabola $y ^ { 2 } = 4 ( x - 9 )$ at the point $P$. The distance of the point $P$ from the centre of the circle $x ^ { 2 } + y ^ { 2 } - 14 x - 8 y + 56 = 0$ is $\_\_\_\_$
Q85 Chain Rule Limit Evaluation Involving Composition or Substitution View
Let $\mathrm { a } > 0$ be a root of the equation $2 x ^ { 2 } + x - 2 = 0$. If $\lim _ { x \rightarrow \frac { 1 } { \mathrm { a } } } \frac { 16 \left( 1 - \cos \left( 2 + x - 2 x ^ { 2 } \right) \right) } { ( 1 - \mathrm { a } x ) ^ { 2 } } = \alpha + \beta \sqrt { 17 }$, where $\alpha , \beta \in Z$, then $\alpha + \beta$ is equal to $\_\_\_\_$
Q86 Measures of Location and Spread View
Let the mean and the standard deviation of the probability distribution
X$\alpha$10- 3
$\mathrm { P } ( \mathrm { X } )$$\frac { 1 } { 3 }$K$\frac { 1 } { 6 }$$\frac { 1 } { 4 }$

be $\mu$ and $\sigma$, respectively. If $\sigma - \mu = 2$, then $\sigma + \mu$ is equal to $\_\_\_\_$
Q87 Stationary points and optimisation Find absolute extrema on a closed interval or domain View
Let the maximum and minimum values of $\left( \sqrt { 8 x - x ^ { 2 } - 12 } - 4 \right) ^ { 2 } + ( x - 7 ) ^ { 2 } , x \in \mathbf { R }$ be M and m , respectively. Then $\mathrm { M } ^ { 2 } - \mathrm { m } ^ { 2 }$ is equal to $\_\_\_\_$
Q88 Standard Integrals and Reverse Chain Rule Definite Integral Evaluation via Substitution or Standard Forms View
If $f ( t ) = \int _ { 0 } ^ { \pi } \frac { 2 x \mathrm {~d} x } { 1 - \cos ^ { 2 } \mathrm { t } \sin ^ { 2 } x } , 0 < \mathrm { t } < \pi$, then the value of $\int _ { 0 } ^ { \frac { \pi } { 2 } } \frac { \pi ^ { 2 } \mathrm { dt } } { f ( \mathrm { t } ) }$ equals $\_\_\_\_$
Q89 Areas by integration View
Let $y = y ( x )$ be the solution of the differential equation $\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 2 x } { \left( 1 + x ^ { 2 } \right) ^ { 2 } } y = x \mathrm { e } ^ { \frac { 1 } { \left( 1 + x ^ { 2 } \right) } } ; y ( 0 ) = 0$. Then the area enclosed by the curve $f ( x ) = y ( x ) \mathrm { e } ^ { - \frac { 1 } { \left( 1 + x ^ { 2 } \right) } }$ and the line $y - x = 4$ is $\_\_\_\_$
Q90 Vectors: Lines & Planes Distance Computation (Point-to-Plane or Line-to-Line) View
Let the point $( - 1 , \alpha , \beta )$ lie on the line of the shortest distance between the lines $\frac { x + 2 } { - 3 } = \frac { y - 2 } { 4 } = \frac { z - 5 } { 2 }$ and $\frac { x + 2 } { - 1 } = \frac { y + 6 } { 2 } = \frac { z - 1 } { 0 }$. Then $( \alpha - \beta ) ^ { 2 }$ is equal to $\_\_\_\_$