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
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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
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
2022 session1_27jun_shift2

29 maths questions

Q61 Complex Numbers Argand & Loci Intersection of Loci and Simultaneous Geometric Conditions View
The number of points of intersection $| z - ( 4 + 3 i ) | = 2$ and $| z | + | z - 4 | = 6 , z \in C$ is
(1) 1
(2) 2
(3) 3
(4) 4
Q62 Complex Numbers Arithmetic Systems of Equations via Real and Imaginary Part Matching View
Let for some real numbers $\alpha$ and $\beta , a = \alpha - i \beta$. If the system of equations $4 i x + ( 1 + i ) y = 0$ and $8 \left( \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 } \right) x + \bar { a } y = 0$ has more than one solution then $\frac { \alpha } { \beta }$ is equal to
(1) $2 - \sqrt { 3 }$
(2) $2 + \sqrt { 3 }$
(3) $- 2 + \sqrt { 3 }$
(4) $- 2 - \sqrt { 3 }$
Q63 Geometric Sequences and Series Sum of an Infinite Geometric Series (Direct Computation) View
Let $S = 2 + \frac { 6 } { 7 } + \frac { 12 } { 7 ^ { 2 } } + \frac { 20 } { 7 ^ { 3 } } + \frac { 30 } { 7 ^ { 4 } } + \ldots$. then $4 S$ is equal to
(1) $\left( \frac { 7 } { 2 } \right) ^ { 2 }$
(2) $\left( \frac { 7 } { 3 } \right) ^ { 3 }$
(3) $\frac { 7 } { 3 }$
(4) $\left( \frac { 7 } { 3 } \right) ^ { 4 }$
Q64 Arithmetic Sequences and Series Find Specific Term from Given Conditions View
If $a _ { 1 } , a _ { 2 } , a _ { 3 } \ldots$ and $b _ { 1 } , b _ { 2 } , b _ { 3 } \ldots$ are A.P. and $a _ { 1 } = 2 , a _ { 10 } = 3 , a _ { 1 } b _ { 1 } = 1 = a _ { 10 } b _ { 10 }$ then $a _ { 4 } b _ { 4 }$ is equal to
(1) $\frac { 28 } { 27 }$
(2) $\frac { 28 } { 24 }$
(3) $\frac { 23 } { 26 }$
(4) $\frac { 22 } { 23 }$
Q65 Solving quadratics and applications Finding roots or coefficients of a quadratic using Vieta's relations View
$\alpha = \sin 36 ^ { \circ }$ is a root of which of the following equation
(1) $16 x ^ { 4 } - 20 x ^ { 2 } + 5 = 0$
(2) $16 x ^ { 4 } + 20 x ^ { 2 } + 5 = 0$
(3) $10 x ^ { 4 } - 10 x ^ { 2 } - 5 = 0$
(4) $16 x ^ { 4 } - 10 x ^ { 2 } + 5 = 0$
Q66 Circles Circle Identification and Classification View
The set of values of $k$ for which the circle $C : 4 x ^ { 2 } + 4 y ^ { 2 } - 12 x + 8 y + k = 0$ lies inside the fourth quadrant and the point $\left( 1 , - \frac { 1 } { 3 } \right)$ lies on or inside the circle $C$ is
(1) An empty set
(2) $\left( 6 , \frac { 95 } { 9 } \right]$
(3) $\left[ \frac { 80 } { 9 } , 10 \right)$
(4) $\left( 9 , \frac { 92 } { 9 } \right]$
Q67 Conic sections Equation Determination from Geometric Conditions View
If the equation of the parabola, whose vertex is at $( 5,4 )$ and the directrix is $3 x + y - 29 = 0$, is $x ^ { 2 } + a y ^ { 2 } + b x y + c x + d y + k = 0$, then $a + b + c + d + k$ is equal to
(1) 575
(2) - 575
(3) 576
(4) - 576
Q69 Measures of Location and Spread View
The mean and variance of the data $4,5,6,6,7,8 , x , y$ where $x < y$ are 6 and $\frac { 9 } { 4 }$ respectively. Then $x ^ { 4 } + y ^ { 2 }$ is equal to
(1) 320
(2) 420
(3) 162
(4) 674
Q70 3x3 Matrices Determinant of Parametric or Structured Matrix View
Let $A$ and $B$ be two $3 \times 3$ matrices such that $A B = I$ and $| A | = \frac { 1 } { 8 }$ then $| \operatorname{adj} ( B \operatorname{adj} ( 2 A ) ) |$ is equal to
(1) 128
(2) 32
(3) 64
(4) 102
Q71 Differentiation from First Principles View
$f ( x ) = \left| \begin{array} { c c c } a & - 1 & 0 \\ a x & a & - 1 \\ a x ^ { 2 } & a x & a \end{array} \right| , a \in R$. Then the sum of the squares of all the values of $a$ for $2 f ^ { \prime } ( 10 ) - f ^ { \prime } ( 5 ) + 100 = 0$ is
(1) 117
(2) 106
(3) 125
(4) 136
Q72 Standard trigonometric equations Inverse trigonometric equation View
The value of $\cot \left( \sum _ { n = 1 } ^ { 50 } \tan ^ { - 1 } \left( \frac { 1 } { 1 + n + n ^ { 2 } } \right) \right)$ is
(1) $\frac { 25 } { 26 }$
(2) $\frac { 50 } { 51 }$
(3) $\frac { 26 } { 25 }$
(4) $\frac { 52 } { 51 }$
Q73 Stationary points and optimisation Find critical points and classify extrema of a given function View
If $m$ and $n$ respectively are the number of local maximum and local minimum points of the function $f ( x ) = \int _ { 0 } ^ { x ^ { 2 } } \frac { t ^ { 2 } - 5 t + 4 } { 2 + e ^ { t } } d t$, then the ordered pair $( m , n )$ is equal to
(1) $( 2,3 )$
(2) $( 3,2 )$
(3) $( 2,2 )$
(4) $( 3,4 )$
Q74 Integration by Substitution Substitution to Compute an Indefinite Integral with Initial Condition View
Let $f$ be a differentiable function in $\left( 0 , \frac { \pi } { 2 } \right)$. If $\int _ { \cos x } ^ { 1 } t ^ { 2 } f ( t ) d t = \sin ^ { 3 } x + \cos x$, then $\frac { 1 } { \sqrt { 3 } } f ^ { \prime } \left( \frac { 1 } { \sqrt { 3 } } \right)$ is equal to
(1) $6 - 9 \sqrt { 2 }$
(2) $6 + \frac { 9 } { \sqrt { 2 } }$
(3) $6 - \frac { 9 } { \sqrt { 2 } }$
(4) $3 + \sqrt { 2 }$
Q75 Indefinite & Definite Integrals Piecewise/Periodic Function Integration View
The integral $\int _ { 0 } ^ { 1 } \frac { 1 } { 7 ^ { \left[ \frac { 1 } { x } \right] } } d x$, where $[ \cdot ]$ denotes the greatest integer function, is equal to
(1) $1 - 6 \ln \left( \frac { 6 } { 7 } \right)$
(2) $1 + 6 \ln \left( \frac { 6 } { 7 } \right)$
(3) $1 - 7 \ln \left( \frac { 6 } { 7 } \right)$
(4) $1 + 7 \ln \left( \frac { 6 } { 7 } \right)$
Q76 First order differential equations (integrating factor) View
If the solution curve of the differential equation $\left( \left( \tan ^ { - 1 } y \right) - x \right) d y = \left( 1 + y ^ { 2 } \right) d x$ passes through the point $( 1,0 )$ then the abscissa of the point on the curve whose ordinate is $\tan ( 1 )$ is
(1) 2
(2) $\frac { 2 } { e }$
(3) $\frac { 3 } { e }$
(4) $2 e$
Q77 Vectors: Cross Product & Distances View
Let $\vec { a }$ and $\vec { b }$ be the vectors along the diagonal of a parallelogram having area $2 \sqrt { 2 }$. Let the angle between $\vec { a }$ and $\vec { b }$ be acute. $| \vec { a } | = 1$ and $| \vec { a } \cdot \vec { b } | = | \vec { a } \times \vec { b } |$. If $\vec { c } = 2 \sqrt { 2 } ( \vec { a } \times \vec { b } ) - 2 \vec { b }$, then an angle between $\vec { b }$ and $\vec { c }$ is
(1) $\frac { - \pi } { 4 }$
(2) $\frac { 5 \pi } { 6 }$
(3) $\frac { \pi } { 3 }$
(4) $\frac { 3 \pi } { 4 }$
Q78 Vectors: Lines & Planes Distance Computation (Point-to-Plane or Line-to-Line) View
Let the foot of the perpendicular from the point $( 1,2,4 )$ on the line $\frac { x + 2 } { 4 } = \frac { y - 1 } { 2 } = \frac { z + 1 } { 3 }$ be $P$. Then the distance of $P$ from the plane $3 x + 4 y + 12 z + 23 = 0$ is
(1) $\frac { 50 } { 13 }$
(2) $\frac { 63 } { 13 }$
(3) $\frac { 65 } { 13 }$
(4) 4
Q79 Vectors: Lines & Planes Distance Computation (Point-to-Plane or Line-to-Line) View
The shortest distance between the lines $\frac { x - 3 } { 2 } = \frac { y - 2 } { 3 } = \frac { z - 1 } { - 1 }$ and $\frac { x + 3 } { 2 } = \frac { y - 6 } { 1 } = \frac { z - 5 } { 3 }$ is
(1) $\frac { 18 } { \sqrt { 5 } }$
(2) $\frac { 22 } { 3 \sqrt { 5 } }$
(3) $\frac { 46 } { 3 \sqrt { 5 } }$
(4) $6 \sqrt { 3 }$
Q80 Geometric Probability View
If a point $A ( x , y )$ lies in the region bounded by the $y$-axis, straight lines $2 y + x = 6$ and $5 x - 6 y = 30$, then the probability that $y < 1$ is
(1) $\frac { 1 } { 6 }$
(2) $\frac { 5 } { 6 }$
(3) $\frac { 2 } { 3 }$
(4) $\frac { 6 } { 7 }$
Q81 Roots of polynomials Determine coefficients or parameters from root conditions View
Let $\alpha , \beta$ be the roots of the equation $x ^ { 2 } - 4 \lambda x + 5 = 0$ and $\alpha , \gamma$ be the roots of the equation $x ^ { 2 } - ( 3 \sqrt { 2 } + 2 \sqrt { 3 } ) x + 7 + 3 \lambda \sqrt { 3 } = 0$. If $\beta + \gamma = 3 \sqrt { 2 }$, then $( \alpha + 2 \beta + \gamma ) ^ { 2 }$ is equal to
Q82 Binomial Theorem (positive integer n) Determine Parameters from Conditions on Coefficients or Terms View
If the sum of the coefficients of all the positive powers of $x$, in the binomial expansion of $\left( x ^ { n } + \frac { 2 } { x ^ { 5 } } \right) ^ { 7 }$ is 939, then the sum of all the possible integral values of $n$ is
Q83 Circles Distance from Center to Line View
Let a circle $C$ of radius 5 lie below the $x$-axis. The line $L _ { 1 } : 4 x + 3 y + 2 = 0$ passes through the centre $P$ of the circle $C$ and intersects the line $L _ { 2 } : 3 x - 4 y - 11 = 0$ at $Q$. The line $L _ { 2 }$ touches $C$ at the point $Q$. Then the distance of $P$ from the line $5 x - 12 y + 51 = 0$ is
Q84 Modulus function Solving equations involving modulus View
Let $[ t ]$ denote the greatest integer $\leq t$ and $\{ t \}$ denote the fractional part of $t$. Then integral value of $\alpha$ for which the left hand limit of the function $f ( x ) = [ 1 + x ] + \frac { \alpha ^ { 2 [ x ] + \{ x \} } + [ x ] - 1 } { 2 [ x ] + \{ x \} }$ at $x = 0$ is equal to $\alpha - \frac { 4 } { 3 }$ is $\_\_\_\_$
Q85 Permutations & Arrangements Distribution of Objects into Bins/Groups View
Let $A$ be a matrix of order $2 \times 2$, whose entries are from the set $\{ 0,1,2,3,4,5 \}$. If the sum of all the entries of $A$ is a prime number $p , 2 < p < 8$, then the number of such matrices $A$ is
Q86 Composite & Inverse Functions Evaluate Composition from Diagram or Mapping View
Let $S = \{ 1,2,3,4,5,6,7,8,9,10 \}$. Define $f : S \rightarrow S$ as $f ( n ) = \left\{ \begin{array} { c l } 2 n , & \text { if } n = 1,2,3,4,5 \\ 2 n - 11 & \text { if } n = 6,7,8,9,10 \end{array} \right.$ Let $g : S \rightarrow S$ be a function such that $f \circ g ( n ) = \left\{ \begin{array} { l l } n + 1 & , \text { if } n \text { is odd } \\ n - 1 & , \text { if } n \text { is even } \end{array} \right.$, then $g ( 10 ) ( g ( 1 ) + g ( 2 ) + g ( 3 ) + g ( 4 ) + g ( 5 ) )$ is equal to
Q87 Differentiating Transcendental Functions Higher-order or nth derivative computation View
If $y ( x ) = \left( x ^ { x } \right) ^ { x } , x > 0$ then $\frac { d ^ { 2 } x } { d y ^ { 2 } } + 20$ at $x = 1$ is equal to
Q88 Areas by integration View
If the area of the region $\left\{ ( x , y ) : x ^ { \frac { 2 } { 3 } } + y ^ { \frac { 2 } { 3 } } \leq 1 , x + y \geq 0 , y \geq 0 \right\}$ is $A$, then $\frac { 256 A } { \pi }$ is
Q89 First order differential equations (integrating factor) View
Let $y = y ( x )$ be the solution of the differential equation $\left( 1 - x ^ { 2 } \right) d y = \left( x y + \left( x ^ { 3 } + 2 \right) \sqrt { 1 - x ^ { 2 } } \right) d x , - 1 < x < 1$ and $y ( 0 ) = 0$. If $\int _ { - \frac { 1 } { 2 } } ^ { \frac { 1 } { 2 } } \sqrt { 1 - x ^ { 2 } } y ( x ) d x = k$ then $k ^ { - 1 }$ is equal to
Q90 Probability Definitions Combinatorial Counting (Non-Probability) View
Let $S = \left\{ E _ { 1 } , E _ { 2 } \ldots E _ { 8 } \right\}$ be a sample space of a random experiment such that $P \left( E _ { n } \right) = \frac { n } { 36 }$ for every $n = 1,2 \ldots 8$. Then the number of elements in the set $\left\{ A \subset S : P ( A ) \geq \frac { 4 } { 5 } \right\}$ is $\_\_\_\_$.