jee-main

Papers (169)
2025
session1_22jan_shift1 25 session1_22jan_shift2 25 session1_23jan_shift1 25 session1_23jan_shift2 25 session1_24jan_shift1 25 session1_24jan_shift2 25 session1_28jan_shift1 25 session1_28jan_shift2 25 session1_29jan_shift1 29 session1_29jan_shift2 25
2024
session1_01feb_shift1 4 session1_01feb_shift2 22 session1_27jan_shift1 28 session1_27jan_shift2 30 session1_29jan_shift1 30 session1_29jan_shift2 23 session1_30jan_shift1 17 session1_30jan_shift2 30 session1_31jan_shift1 16 session1_31jan_shift2 15 session2_04apr_shift1 4 session2_04apr_shift2 30 session2_05apr_shift1 4 session2_05apr_shift2 30 session2_06apr_shift1 22 session2_06apr_shift2 30 session2_08apr_shift1 30 session2_08apr_shift2 30 session2_09apr_shift1 5 session2_09apr_shift2 30
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
2021 session2_16mar_shift1

29 maths questions

Q61 Laws of Logarithms Solve a Logarithmic Equation View
If for $x \in \left( 0 , \frac { \pi } { 2 } \right) , \log _ { 10 } \sin x + \log _ { 10 } \cos x = - 1$ and $\log _ { 10 } ( \sin x + \cos x ) = \frac { 1 } { 2 } \left( \log _ { 10 } n - 1 \right) , n > 0$, then the value of $n$ is equal to :
(1) 20
(2) 12
(3) 9
(4) 16
Q62 Inequalities Solve Polynomial/Rational Inequality for Solution Set View
Let a complex number $z , | z | \neq 1$, satisfy $\log _ { \frac { 1 } { \sqrt { 2 } } } \left( \frac { | z | + 11 } { ( | z | - 1 ) ^ { 2 } } \right) \leq 2$. Then, the largest value of $| z |$ is equal to
(1) 8
(2) 7
(3) 6
(4) 5
Q63 Binomial Theorem (positive integer n) Count Integral or Rational Terms in a Binomial Expansion View
If $n$ is the number of irrational terms in the expansion of $\left( 3 ^ { 1 / 4 } + 5 ^ { 1 / 8 } \right) ^ { 60 }$, then $( n - 1 )$ is divisible by :
(1) 26
(2) 30
(3) 8
(4) 7
Q64 Binomial Theorem (positive integer n) Evaluate a Summation Involving Binomial Coefficients View
Let $[ x ]$ denote greatest integer less than or equal to $x$. If for $n \in N , \left( 1 - x + x ^ { 3 } \right) ^ { n } = \sum _ { j = 0 } ^ { 3 n } a _ { j } x ^ { j }$, then $\sum _ { j = 0 } ^ { \left[ \frac { 3 n } { 2 } \right] } a _ { 2 j } + 4 \sum _ { j = 0 } ^ { \left[ \frac { 3 n - 1 } { 2 } \right] } a _ { 2 j + 1 }$ is equal to :
(1) 2
(2) $2 ^ { n - 1 }$
(3) 1
(4) $n$
Q65 Standard trigonometric equations Solve trigonometric equation for solutions in an interval View
The number of roots of the equation, $( 81 ) ^ { \sin ^ { 2 } x } + ( 81 ) ^ { \cos ^ { 2 } x } = 30$ in the interval $[ 0 , \pi ]$ is equal to :
(1) 3
(2) 4
(3) 8
(4) 2
Q66 Circles Circle Equation Derivation View
If the three normals drawn to the parabola, $y ^ { 2 } = 2 x$ pass through the point $( a , 0 ) , a \neq 0$, then $a$ must be greater than :
(1) $\frac { 1 } { 2 }$
(2) $- \frac { 1 } { 2 }$
(3) - 1
(4) 1
Q67 Circles Circle-Related Locus Problems View
The locus of the midpoints of the chord of the circle, $x ^ { 2 } + y ^ { 2 } = 25$ which is tangent to the hyperbola, $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 16 } = 1$ is :
(1) $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } - 16 x ^ { 2 } + 9 y ^ { 2 } = 0$
(2) $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } - 9 x ^ { 2 } + 144 y ^ { 2 } = 0$
(3) $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } - 9 x ^ { 2 } - 16 y ^ { 2 } = 0$
(4) $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } - 9 x ^ { 2 } + 16 y ^ { 2 } = 0$
Q69 Measures of Location and Spread View
Consider three observations $a , b$ and $c$ such that $b = a + c$. If the standard deviation of $a + 2 , b + 2 , c + 2$ is $d$, then which of the following is true?
(1) $b ^ { 2 } = 3 \left( a ^ { 2 } + c ^ { 2 } \right) + 9 d ^ { 2 }$
(2) $b ^ { 2 } = a ^ { 2 } + c ^ { 2 } + 3 d ^ { 2 }$
(3) $b ^ { 2 } = 3 \left( a ^ { 2 } + c ^ { 2 } + d ^ { 2 } \right)$
(4) $b ^ { 2 } = 3 \left( a ^ { 2 } + c ^ { 2 } \right) - 9 d ^ { 2 }$
Q70 Matrices Matrix Power Computation and Application View
Let $A = \left[ \begin{array} { c c } i & - i \\ - i & i \end{array} \right] , i = \sqrt { - 1 }$. Then, the system of linear equations $A ^ { 8 } \left[ \begin{array} { l } x \\ y \end{array} \right] = \left[ \begin{array} { c } 8 \\ 64 \end{array} \right]$ has :
(1) A unique solution
(2) Infinitely many solutions
(3) No solution
(4) Exactly two solutions
Q71 Standard trigonometric equations Inverse trigonometric equation View
Let $S _ { k } = \sum _ { r = 1 } ^ { k } \tan ^ { - 1 } \left( \frac { 6 ^ { r } } { 2 ^ { 2 r + 1 } + 3 ^ { 2 r + 1 } } \right)$, then $\lim _ { k \rightarrow \infty } S _ { k }$ is equal to :
(1) $\tan ^ { - 1 } \left( \frac { 3 } { 2 } \right)$
(2) $\frac { \pi } { 2 }$
(3) $\cot ^ { - 1 } \left( \frac { 3 } { 2 } \right)$
(4) $\tan ^ { - 1 } ( 3 )$
Q72 Modulus function Counting solutions satisfying modulus conditions View
The number of elements in the set $\{ x \in R : ( | x | - 3 ) | x + 4 | = 6 \}$ is equal to
(1) 3
(2) 2
(3) 4
(4) 1
Q73 Composite & Inverse Functions Determine Domain or Range of a Composite Function View
Let the functions $f : R \rightarrow R$ and $g : R \rightarrow R$ be defined as : $f ( x ) = \left\{ \begin{array} { l l } x + 2 , & x < 0 \\ x ^ { 2 } , & x \geq 0 \end{array} \right.$ and $g ( x ) = \begin{cases} x ^ { 3 } , & x < 1 \\ 3 x - 2 , & x \geq 1 \end{cases}$ Then, the number of points in $R$ where $( f \circ g ) ( x )$ is NOT differentiable is equal to :
(1) 3
(2) 1
(3) 0
(4) 2
Q74 Stationary points and optimisation Determine parameters from given extremum conditions View
The range of $a \in R$ for which the function $f ( x ) = ( 4 a - 3 ) \left( x + \log _ { e } 5 \right) + 2 ( a - 7 ) \cot \left( \frac { x } { 2 } \right) \sin ^ { 2 } \left( \frac { x } { 2 } \right) , x \neq 2 n \pi , n \in N$, has critical points, is :
(1) $( - 3,1 )$
(2) $\left[ - \frac { 4 } { 3 } , 2 \right]$
(3) $[ 1 , \infty )$
(4) $( - \infty , - 1 ]$
Q75 First order differential equations (integrating factor) View
If $y = y ( x )$ is the solution of the differential equation, $\frac { d y } { d x } + 2 y \tan x = \sin x , y \left( \frac { \pi } { 3 } \right) = 0$, then the maximum value of the function $y ( x )$ over $R$ is equal to :
(1) 8
(2) $\frac { 1 } { 2 }$
(3) $- \frac { 15 } { 4 }$
(4) $\frac { 1 } { 8 }$
Q76 Vectors Introduction & 2D Area Computation Using Vectors View
Let a vector $\alpha \hat { \mathrm { i } } + \beta \hat { \mathrm { j } }$ be obtained by rotating the vector $\sqrt { 3 } \hat { \mathrm { i } } + \hat { \mathrm { j } }$ by an angle $45 ^ { \circ }$ about the origin in counterclockwise direction in the first quadrant. Then the area (in sq. units) of triangle having vertices $( \alpha , \beta ) , ( 0 , \beta )$ and $( 0,0 )$ is equal to
(1) $\frac { 1 } { 2 }$
(2) 1
(3) $\frac { 1 } { \sqrt { 2 } }$
(4) $2 \sqrt { 2 }$
Q77 Vectors 3D & Lines Distance from a Point to a Line (Show/Compute) View
If for $a > 0$, the feet of perpendiculars from the points $A ( a , - 2 a , 3 )$ and $B ( 0,4,5 )$ on the plane $l x + m y + n z = 0$ are points $C ( 0 , - a , - 1 )$ and $D$ respectively, then the length of line segment $C D$ is equal to :
(1) $\sqrt { 31 }$
(2) $\sqrt { 41 }$
(3) $\sqrt { 55 }$
(4) $\sqrt { 66 }$
Q78 Vectors 3D & Lines Shortest Distance Between Two Lines View
Let the position vectors of two points $P$ and $Q$ be $3 \hat { \mathrm { i } } - \hat { \mathrm { j } } + 2 \widehat { \mathrm { k } }$ and $\hat { \mathrm { i } } + 2 \hat { \mathrm { j } } - 4 \widehat { \mathrm { k } }$, respectively. Let $R$ and $S$ be two points such that the direction ratios of lines $P R$ and $Q S$ are $( 4 , - 1,2 )$ and $( - 2,1 , - 2 )$, respectively. Let lines $P R$ and $Q S$ intersect at $T$. If the vector $\overrightarrow { T A }$ is perpendicular to both $\overrightarrow { P R }$ and $\overrightarrow { Q S }$ and the length of vector $\overrightarrow { T A }$ is $\sqrt { 5 }$ units, then the modulus of a position vector of $A$ is :
(1) $\sqrt { 482 }$
(2) $\sqrt { 171 }$
(3) $\sqrt { 5 }$
(4) $\sqrt { 227 }$
Q79 Vectors 3D & Lines Line-Plane Intersection View
Let $P$ be a plane $l x + m y + n z = 0$ containing the line, $\frac { 1 - x } { 1 } = \frac { y + 4 } { 2 } = \frac { z + 2 } { 3 }$. If plane $P$ divides the line segment $A B$ joining points $A ( - 3 , - 6,1 )$ and $B ( 2,4 , - 3 )$ in ratio $k : 1$ then the value of $k$ is equal to :
(1) 1.5
(2) 3
(3) 2
(4) 4
Q80 Probability Definitions Conditional Probability and Bayes' Theorem View
A pack of cards has one card missing. Two cards are drawn randomly and are found to be spades. The probability that the missing card is not a spade, is :
(1) $\frac { 3 } { 4 }$
(2) $\frac { 52 } { 867 }$
(3) $\frac { 39 } { 50 }$
(4) $\frac { 22 } { 425 }$
Q81 Complex Numbers Arithmetic Solving Equations for Unknown Complex Numbers View
Let $z$ and $w$ be two complex numbers such that $w = z \bar { z } - 2 z + 2 , \left| \frac { z + i } { z - 3 i } \right| = 1$ and $\operatorname { Re } ( w )$ has minimum value. Then, the minimum value of $n \in N$ for which $w ^ { n }$ is real, is equal to $\_\_\_\_$.
Q82 Sequences and series, recurrence and convergence Summation of sequence terms View
Consider an arithmetic series and a geometric series having four initial terms from the set $\{ 11,8,21,16,26,32,4 \}$. If the last terms of these series are the maximum possible four digit numbers, then the number of common terms in these two series is equal to $\_\_\_\_$.
Q83 Circles Tangent Lines and Tangent Lengths View
Let $A B C D$ be a square of side of unit length. Let a circle $C _ { 1 }$ centered at $A$ with unit radius is drawn. Another circle $C _ { 2 }$ which touches $C _ { 1 }$ and the lines $A D$ and $A B$ are tangent to it, is also drawn. Let a tangent line from the point $C$ to the circle $C _ { 2 }$ meet the side $A B$ at $E$. If the length of $E B$ is $\alpha + \sqrt { 3 } \beta$, where $\alpha , \beta$ are integers, then $\alpha + \beta$ is equal to $\_\_\_\_$.
Q84 Chain Rule Limit Evaluation Involving Composition or Substitution View
If $\lim _ { x \rightarrow 0 } \frac { a e ^ { x } - b \cos x + c e ^ { - x } } { x \sin x } = 2$, then $a + b + c$ is equal to $\_\_\_\_$.
Q85 3x3 Matrices Matrix Algebraic Properties and Abstract Reasoning View
Let $P = \left[ \begin{array} { c c c } - 30 & 20 & 56 \\ 90 & 140 & 112 \\ 120 & 60 & 14 \end{array} \right]$ and $A = \left[ \begin{array} { c c c } 2 & 7 & \omega ^ { 2 } \\ - 1 & - \omega & 1 \\ 0 & - \omega & - \omega + 1 \end{array} \right]$ where $\omega = \frac { - 1 + i \sqrt { 3 } } { 2 }$, and $I _ { 3 }$ be the identity matrix of order 3 . If the determinant of the matrix $\left( P ^ { - 1 } A P - I _ { 3 } \right) ^ { 2 }$ is $\alpha \omega ^ { 2 }$, then the value of $\alpha$ is equal to $\_\_\_\_$.
Q86 Matrices Determinant and Rank Computation View
The total number of $3 \times 3$ matrices $A$ having entries from the set $\{ 0,1,2,3 \}$ such that the sum of all the diagonal entries of $A A ^ { T }$ is 9 , is equal to $\_\_\_\_$.
Q87 Indefinite & Definite Integrals Definite Integral as a Limit of Riemann Sums View
Let $f : ( 0,2 ) \rightarrow R$ be defined as $f ( x ) = \log _ { 2 } \left( 1 + \tan \left( \frac { \pi x } { 4 } \right) \right)$. Then, $\lim _ { n \rightarrow \infty } \frac { 2 } { n } \left( f \left( \frac { 1 } { n } \right) + f \left( \frac { 2 } { n } \right) + \ldots + f ( 1 ) \right)$ is equal to $\_\_\_\_$.
Q88 Applied differentiation Tangent line computation and geometric consequences View
If the normal to the curve $y ( x ) = \int _ { 0 } ^ { x } \left( 2 t ^ { 2 } - 15 t + 10 \right) d t$ at a point $( a , b )$ is parallel to the line $x + 3 y = - 5 , a > 1$, then the value of $| a + 6 b |$ is equal to $\_\_\_\_$.
Q89 Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution View
Let $f : R \rightarrow R$ be a continuous function such that $f ( x ) + f ( x + 1 ) = 2$ for all $x \in R$. If $I _ { 1 } = \int _ { 0 } ^ { 8 } f ( x ) d x$ and $I _ { 2 } = \int _ { - 1 } ^ { 3 } f ( x ) d x$, then the value of $I _ { 1 } + 2 I _ { 2 }$ is equal to $\_\_\_\_$.
Q90 Indefinite & Definite Integrals Antiderivative Verification and Construction View
Let the curve $y = y ( x )$ be the solution of the differential equation, $\frac { d y } { d x } = 2 ( x + 1 )$. If the numerical value of area bounded by the curve $y = y ( x )$ and $x$-axis is $\frac { 4 \sqrt { 8 } } { 3 }$, then the value of $y ( 1 )$ is equal to $\_\_\_\_$.