jee-main

Papers (191)
2026
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2025
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2024
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2023
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2022
session1_24jun_shift1 19 session1_24jun_shift2 25 session1_25jun_shift1 14 session1_25jun_shift2 14 session1_26jun_shift1 29 session1_26jun_shift2 24 session1_27jun_shift1 4 session1_27jun_shift2 29 session1_28jun_shift1 13 session1_29jun_shift1 20 session1_29jun_shift2 4 session2_25jul_shift1 29 session2_25jul_shift2 20 session2_26jul_shift1 29 session2_26jul_shift2 23 session2_27jul_shift1 28 session2_27jul_shift2 29 session2_28jul_shift1 11 session2_28jul_shift2 29 session2_29jul_shift1 17 session2_29jul_shift2 18
2021
session1_24feb_shift1 9 session1_24feb_shift2 4 session1_25feb_shift1 29 session1_25feb_shift2 29 session1_26feb_shift2 15 session2_16mar_shift1 29 session2_16mar_shift2 18 session2_17mar_shift1 21 session2_17mar_shift2 27 session2_18mar_shift1 18 session2_18mar_shift2 9 session3_20jul_shift1 29 session3_20jul_shift2 29 session3_22jul_shift1 9 session3_25jul_shift1 8 session3_25jul_shift2 14 session3_27jul_shift1 4 session3_27jul_shift2 7 session4_01sep_shift2 14 session4_26aug_shift1 5 session4_26aug_shift2 2 session4_27aug_shift1 3 session4_27aug_shift2 29 session4_31aug_shift1 28 session4_31aug_shift2 4
2020
session1_07jan_shift1 28 session1_07jan_shift2 20 session1_08jan_shift1 5 session1_08jan_shift2 11 session1_09jan_shift1 26 session1_09jan_shift2 16 session2_02sep_shift1 18 session2_02sep_shift2 16 session2_03sep_shift1 23 session2_03sep_shift2 8 session2_04sep_shift1 14 session2_04sep_shift2 27 session2_05sep_shift1 22 session2_05sep_shift2 29 session2_06sep_shift1 11 session2_06sep_shift2 10
2019
session1_09jan_shift1 6 session1_09jan_shift2 29 session1_10jan_shift1 29 session1_10jan_shift2 14 session1_11jan_shift1 6 session1_11jan_shift2 5 session1_12jan_shift1 10 session1_12jan_shift2 29 session2_08apr_shift1 29 session2_08apr_shift2 29 session2_09apr_shift1 29 session2_09apr_shift2 29 session2_10apr_shift1 2 session2_10apr_shift2 5 session2_12apr_shift1 3 session2_12apr_shift2 9
2018
08apr 30 15apr 28 15apr_shift1 28 15apr_shift2 6 16apr 19
2017
02apr 30 08apr 30 09apr 34
2016
03apr 28 09apr 29 10apr 30
2015
04apr 29 10apr 29 11apr 8
2014
06apr 28 09apr 28 11apr 4 12apr 5 19apr 29
2013
07apr 29 09apr 12 22apr 5 23apr 14 25apr 13
2012
07may 17 12may 21 19may 14 26may 17 offline 30
2011
jee-main_2011.pdf 18
2010
jee-main_2010.pdf 6
2009
jee-main_2009.pdf 2
2008
jee-main_2008.pdf 4
2007
jee-main_2007.pdf 38
2006
jee-main_2006.pdf 15
2005
jee-main_2005.pdf 25
2004
jee-main_2004.pdf 22
2003
jee-main_2003.pdf 8
2002
jee-main_2002.pdf 12
2023 session2_12apr_shift1

26 maths questions

Q61 Roots of polynomials Vieta's formulas: compute symmetric functions of roots View
Let $\alpha , \beta$ be the roots of the quadratic equation $x ^ { 2 } + \sqrt { 6 } x + 3 = 0$. Then $\frac { \alpha ^ { 23 } + \beta ^ { 23 } + \alpha ^ { 14 } + \beta ^ { 14 } } { \alpha ^ { 15 } + \beta ^ { 15 } + \alpha ^ { 10 } + \beta ^ { 10 } }$ is equal to
(1) 81
(2) 9
(3) 72
(4) 729
Q62 Complex Numbers Argand & Loci Distance and Region Optimization on Loci View
Let $C$ be the circle in the complex plane with centre $z _ { 0 } = \frac { 1 } { 2 } ( 1 + 3 i )$ and radius $r = 1$. Let $z _ { 1 } = 1 + i$ and the complex number $z _ { 2 }$ be outside circle $C$ such that $\left| z _ { 1 } - z _ { 0 } \right| \left| z _ { 2 } - z _ { 0 } \right| = 1$. If $z _ { 0 } , z _ { 1 }$ and $z _ { 2 }$ are collinear, then the smaller value of $\left| z _ { 2 } \right| ^ { 2 }$ is equal to
(1) $\frac { 5 } { 2 }$
(2) $\frac { 7 } { 2 }$
(3) $\frac { 13 } { 2 }$
(4) $\frac { 3 } { 2 }$
Q63 Permutations & Arrangements Forming Numbers with Digit Constraints View
The number of five-digit numbers, greater than 40000 and divisible by 5 , which can be formed using the digits $0,1,3,5,7$ and 9 without repetition, is equal to
(1) 132
(2) 120
(3) 72
(4) 96
Q64 Arithmetic Sequences and Series Counting or Combinatorial Problems on APs View
Let the digits $a , b , c$ be in A.P. Nine-digit numbers are to be formed using each of these three digits thrice such that three consecutive digits are in A.P. at least once. How many such numbers can be formed?
Q65 Sequences and series, recurrence and convergence Summation of sequence terms View
Let $\left\langle a _ { n } \right\rangle$ be a sequence such that $a _ { 1 } + a _ { 2 } + \ldots + a _ { n } = \frac { n ^ { 2 } + 3 n } { ( n + 1 ) ( n + 2 ) }$. If $28 \sum _ { k = 1 } ^ { 10 } \frac { 1 } { a _ { k } } = p _ { 1 } p _ { 2 } p _ { 3 } \ldots p _ { m }$, where $p _ { 1 } , p _ { 2 } , \ldots p _ { m }$ are the first $m$ prime numbers, then $m$ is equal to
(1) 5
(2) 8
(3) 6
(4) 7
Q66 Binomial Theorem (positive integer n) Evaluate a Summation Involving Binomial Coefficients View
If $\frac { 1 } { n + 1 } { } ^ { n } C _ { n } + \frac { 1 } { n } { } ^ { n } C _ { n - 1 } + \ldots + \frac { 1 } { 2 } { } ^ { n } C _ { 1 } + { } ^ { n } C _ { 0 } = \frac { 1023 } { 10 }$ then $n$ is equal to
(1) 9
(2) 8
(3) 7
(4) 6
Q67 Binomial Theorem (positive integer n) Evaluate a Summation Involving Binomial Coefficients View
The sum, of the coefficients of the first 50 terms in the binomial expansion of $( 1 - x ) ^ { 100 }$, is equal to
(1) ${ } ^ { 101 } C _ { 50 }$
(2) ${ } ^ { 99 } C _ { 49 }$
(3) $- { } ^ { 101 } C _ { 50 }$
(4) $- { } ^ { 99 } C _ { 49 }$
Q68 Circles Locus Determination View
If the point $\left( \alpha , \frac { 7 \sqrt { 3 } } { 3 } \right)$ lies on the curve traced by the mid-points of the line segments of the lines $x \cos \theta + y \sin \theta = 7 , \theta \in \left( 0 , \frac { \pi } { 2 } \right)$ between the co-ordinates axes, then $\alpha$ is equal to
(1) - 7
(2) $- 7 \sqrt { 3 }$
(3) $7 \sqrt { 3 }$
(4) 7
Q69 Sine and Cosine Rules Determine an angle or side from a trigonometric identity/equation View
In a triangle $A B C$, if $\cos A + 2 \cos B + \cos C = 2$ and the lengths of the sides opposite to the angles $A$ and $C$ are 3 and 7 respectively, then $\cos A - \cos C$ is equal to
(1) $\frac { 9 } { 7 }$
(2) $\frac { 10 } { 7 }$
(3) $\frac { 5 } { 7 }$
(4) $\frac { 3 } { 7 }$
Q70 Circles Circles Tangent to Each Other or to Axes View
Two circles in the first quadrant of radii $r _ { 1 }$ and $r _ { 2 }$ touch the coordinate axes. Each of them cuts off an intercept of 2 units with the line $x + y = 2$. Then $r _ { 1 } { } ^ { 2 } + r _ { 2 } { } ^ { 2 } - r _ { 1 } r _ { 2 }$ is equal to $\_\_\_\_$ .
Q71 Conic sections Chord Properties and Midpoint Problems View
Let $P \left( \frac { 2 \sqrt { 3 } } { \sqrt { 7 } } , \frac { 6 } { \sqrt { 7 } } \right) , Q , R$ and $S$ be four points on the ellipse $9 x ^ { 2 } + 4 y ^ { 2 } = 36$. Let $P Q$ and $R S$ be mutually perpendicular and pass through the origin. If $\frac { 1 } { ( P Q ) ^ { 2 } } + \frac { 1 } { ( R S ) ^ { 2 } } = \frac { p } { q }$, where $p$ and $q$ are coprime, then $p + q$ is equal to
(1) 147
(2) 143
(3) 137
(4) 157
Q73 Geometric Sequences and Series Finite Geometric Sum and Term Relationships View
Let the positive numbers $a _ { 1 } , a _ { 2 } , a _ { 3 } , a _ { 4 }$ and $a _ { 5 }$ be in a G.P. Let their mean and variance be $\frac { 31 } { 10 }$ and $\frac { m } { n }$ respectively, where $m$ and $n$ are co-prime. If the mean of their reciprocals is $\frac { 31 } { 10 }$ and $a _ { 3 } + a _ { 4 } + a _ { 5 } = 14$, then $m + n$ is equal to $\_\_\_\_$ .
Q75 Matrices Matrix Power Computation and Application View
Let $A = \left[ \begin{array} { c c } 1 & \frac { 1 } { 51 } \\ 0 & 1 \end{array} \right]$. If $B = \left[ \begin{array} { c c } 1 & 2 \\ - 1 & - 1 \end{array} \right] A \left[ \begin{array} { c c } - 1 & - 2 \\ 1 & 1 \end{array} \right]$, then the sum of all the elements of the matrix $\sum _ { n = 1 } ^ { 50 } B ^ { n }$ is equal to
(1) 75
(2) 125
(3) 50
(4) 100
Q77 Function Transformations Determine Domain or Range of a Composite Function View
Let $D$ be the domain of the function $f ( x ) = \sin ^ { - 1 } \left( \log _ { 3 x } \left( \frac { 6 + 2 \log _ { 3 } x } { - 5 x } \right) \right)$. If the range of the function $g : D \rightarrow \mathbb { R }$ defined by $g ( x ) = x - [ x ]$, ([x] is the greatest integer function), is $( \alpha , \beta )$, then $\alpha ^ { 2 } + \frac { 5 } { \beta }$ is equal to
(1) 135
(2) 45
(3) 46
(4) 136
Q78 Curve Sketching Continuity and Discontinuity Analysis of Piecewise Functions View
Let $[ x ]$ be the greatest integer $\leq x$. Then the number of points in the interval $( - 2,1 )$ where the function $f ( x ) = | [ x ] | + \sqrt { x - [ x ] }$ is discontinuous, is $\_\_\_\_$ .
Q79 Exponential Functions Full function study with transcendental functions View
If the total maximum value of the function $f ( x ) = \left( \frac { \sqrt { 3 e } } { 2 \sin x } \right) ^ { \sin ^ { 2 } x } , x \in \left( 0 , \frac { \pi } { 2 } \right)$, is $\frac { k } { e }$, then $\left( \frac { k } { e } \right) ^ { 8 } + \frac { k ^ { 8 } } { e ^ { 5 } } + k ^ { 8 }$ is equal to
(1) $e ^ { 3 } + e ^ { 6 } + e ^ { 11 }$
(2) $e ^ { 5 } + e ^ { 6 } + e ^ { 11 }$
(3) $e ^ { 3 } + e ^ { 6 } + e ^ { 10 }$
(4) $e ^ { 3 } + e ^ { 5 } + e ^ { 11 }$
Q80 Integration by Substitution Substitution to Compute an Indefinite Integral with Initial Condition View
Let $I ( x ) = \int \sqrt { \frac { x + 7 } { x } } d x$ and $I ( 9 ) = 12 + 7 \log _ { e } 7$. If $I ( 1 ) = \alpha + 7 \log _ { e } ( 1 + 2 \sqrt { 2 } )$, then $\alpha ^ { 4 }$ is equal to $\_\_\_\_$ .
Q81 Indefinite & Definite Integrals Piecewise/Periodic Function Integration View
If $\int _ { - 0.15 } ^ { 0.15 } \left| 100 x ^ { 2 } - 1 \right| d x = \frac { k } { 3000 }$, then $k$ is equal to $\_\_\_\_$ .
Q82 Areas Between Curves Compute Area Directly (Numerical Answer) View
The area of the region enclosed by the curve $y = x ^ { 3 }$ and its tangent at the point $( - 1 , - 1 )$ is
(1) $\frac { 19 } { 4 }$
(2) $\frac { 23 } { 4 }$
(3) $\frac { 31 } { 4 }$
(4) $\frac { 27 } { 4 }$
Q83 First order differential equations (integrating factor) Solving Separable DEs with Initial Conditions View
Let $y = y ( x ) , y > 0$, be a solution curve of the differential equation $\left( 1 + x ^ { 2 } \right) d y = y ( x - y ) d x$. If $y ( 0 ) = 1$ and $y ( 2 \sqrt { 2 } ) = \beta$, then
(1) $e ^ { 3 \beta - 1 } = e ( 3 + 2 \sqrt { 2 } )$
(2) $e ^ { 3 \beta - 1 } = e ( 5 + \sqrt { 2 } )$
(3) $e ^ { \beta - 1 } = e ^ { - 2 } ( 3 + 2 \sqrt { 2 } )$
(4) $e ^ { \beta - 1 } = e ^ { - 2 } ( 5 + \sqrt { 2 } )$
Q84 Vectors 3D & Lines Geometric Property Identification via Vectors View
Let $a , b , c$ be three distinct real numbers, none equal to one. If the vectors $a \hat { i } + \hat { j } + \widehat { k } , \hat { i } + b \hat { j } + \widehat { k }$ and $\hat { i } + \hat { j } + c \hat { k }$ are coplanar, then $\frac { 1 } { 1 - a } + \frac { 1 } { 1 - b } + \frac { 1 } { 1 - c }$ is equal to
(1) 2
(2) - 1
(3) - 2
(4) 1
Q85 Vectors 3D & Lines Magnitude of Vector Expression View
Let $\lambda \in \mathbb { Z } , \vec { a } = \lambda \hat { i } + \hat { j } - \widehat { k }$ and $\vec { b } = 3 \hat { i } - \hat { j } + 2 \widehat { k }$. Let $\vec { c }$ be a vector such that $( \vec { a } + \vec { b } + \vec { c } ) \times \vec { c } = \overrightarrow { 0 } , \vec { a } \cdot \vec { c } = - 17$ and $\vec { b } \cdot \vec { c } = - 20$. Then $| \vec { c } \times ( \lambda \hat { i } + \hat { j } + \hat { k } ) | ^ { 2 }$ is equal to
(1) 46
(2) 53
(3) 62
(4) 49
Q86 Vectors 3D & Lines Perpendicular/Orthogonal Projection onto a Plane View
Let the plane $x + 3 y - 2 z + 6 = 0$ meet the co-ordinate axes at the points $A , B , C$. If the orthocenter of the triangle $A B C$ is $\left( \alpha , \beta , \frac { 6 } { 7 } \right)$, then $98 ( \alpha + \beta ) ^ { 2 }$ is equal to $\_\_\_\_$ .
Q87 Vectors 3D & Lines Coplanarity and Relative Position of Planes View
Let the lines $L _ { 1 } : \frac { x + 5 } { 3 } = \frac { y + 4 } { 1 } = \frac { z - \alpha } { - 2 }$ and $L _ { 2 } : 3 x + 2 y + z - 2 = 0 = x - 3 y + 2 z - 13$ be coplanar. If the point $P ( a , b , c )$ on $L _ { 1 }$ is nearest to the point $Q ( - 4 , - 3,2 )$, then $| a | + | b | + | c |$ is equal to
(1) 12
(2) 14
(3) 8
(4) 10
Q88 Vectors 3D & Lines Find Cartesian Equation of a Plane View
Let the plane $P : 4 x - y + z = 10$ be rotated by an angle $\frac { \pi } { 2 }$ about its line of intersection with the plane $x + y - z = 4$. If $\alpha$ is the distance of the point $( 2,3 , - 4 )$ from the new position of the plane $P$, then $35 \alpha$ is equal to
(1) 85
(2) 105
(3) 126
(4) 90
Q89 Measures of Location and Spread View
Two dice $A$ and $B$ are rolled. Let the numbers obtained on $A$ and $B$ be $\alpha$ and $\beta$ respectively. If the variance of $\alpha - \beta$ is $\frac { p } { q }$, where $p$ and $q$ are co-prime, then the sum of the positive divisors of $p$ is equal to
(1) 72
(2) 36
(3) 48
(4) 31