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
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
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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
2021 session2_17mar_shift2

24 maths questions

Q61 Complex Numbers Argand & Loci Intersection of Loci and Simultaneous Geometric Conditions View
Let $S _ { 1 } , S _ { 2 }$ and $S _ { 3 }$ be three sets defined as $S _ { 1 } = \{ z \in \mathbb { C } : | z - 1 | \leq \sqrt { 2 } \}$, $S _ { 2 } = \{ z \in \mathbb { C } : \operatorname { Re } ( ( 1 - i ) z ) \geq 1 \}$ and $S _ { 3 } = \{ z \in \mathbb { C } : \operatorname { Im } ( z ) \leq 1 \}$. Then, the set $S _ { 1 } \cap S _ { 2 } \cap S _ { 3 }$
(1) is a singleton
(2) has exactly two elements
(3) has infinitely many elements
(4) has exactly three elements
Q62 Combinations & Selection Geometric Combinatorics View
If the sides $A B , B C$ and $C A$ of a triangle $A B C$ have 3,5 and 6 interior points respectively, then the total number of triangles that can be constructed using these points as vertices, is equal to:
(1) 364
(2) 240
(3) 333
(4) 360
Q63 Binomial Theorem (positive integer n) Evaluate a Summation Involving Binomial Coefficients View
The value of $\sum _ { r = 0 } ^ { 6 } \left( { } ^ { 6 } C _ { r } \cdot { } ^ { 6 } C _ { 6 - r } \right)$ is equal to:
(1) 1124
(2) 1324
(3) 1024
(4) 924
Q64 Standard trigonometric equations Solve trigonometric equation for solutions in an interval View
The number of solutions of the equation $x + 2 \tan x = \frac { \pi } { 2 }$ in the interval $[ 0,2 \pi ]$ is
(1) 3
(2) 4
(3) 2
(4) 5
Q65 Circles Area and Geometric Measurement Involving Circles View
Two tangents are drawn from a point $P$ to the circle $x ^ { 2 } + y ^ { 2 } - 2 x - 4 y + 4 = 0$, such that the angle between these tangents is $\tan ^ { - 1 } \left( \frac { 12 } { 5 } \right)$, where $\tan ^ { - 1 } \left( \frac { 12 } { 5 } \right) \in ( 0 , \pi )$. If the centre of the circle is denoted by $C$ and these tangents touch the circle at points $A$ and $B$, then the ratio of the areas of $\triangle P A B$ and $\triangle C A B$ is:
(1) $11 : 4$
(2) $9 : 4$
(3) $3 : 1$
(4) $2 : 1$
Q66 Circles Inscribed/Circumscribed Circle Computations View
Let the tangent to the circle $x ^ { 2 } + y ^ { 2 } = 25$ at the point $R ( 3,4 )$ meet $x$-axis and $y$-axis at point $P$ and $Q$, respectively. If $r$ is the radius of the circle passing through the origin $O$ and having centre at the incentre of the triangle $O P Q$, then $r ^ { 2 }$ is equal to
(1) $\frac { 529 } { 64 }$
(2) $\frac { 125 } { 72 }$
(3) $\frac { 625 } { 72 }$
(4) $\frac { 585 } { 66 }$
Q67 Circles Tangent Lines and Tangent Lengths View
Let $L$ be a tangent line to the parabola $y ^ { 2 } = 4 x - 20$ at (6, 2). If $L$ is also a tangent to the ellipse $\frac { x ^ { 2 } } { 2 } + \frac { y ^ { 2 } } { b } = 1$, then the value of $b$ is equal to:
(1) 11
(2) 14
(3) 16
(4) 20
Q71 3x3 Matrices Determinant of Parametric or Structured Matrix View
If $x , y , z$ are in arithmetic progression with common difference $d , x \neq 3 d$, and the determinant of the matrix $\left[ \begin{array} { c c c } 3 & 4 \sqrt { 2 } & x \\ 4 & 5 \sqrt { 2 } & y \\ 5 & k & z \end{array} \right]$ is zero, then the value of $k ^ { 2 }$ is
(1) 72
(2) 12
(3) 36
(4) 6
Q72 Standard trigonometric equations Inverse trigonometric equation View
The number of solutions of the equation $\sin ^ { - 1 } \left[ x ^ { 2 } + \frac { 1 } { 3 } \right] + \cos ^ { - 1 } \left[ x ^ { 2 } - \frac { 2 } { 3 } \right] = x ^ { 2 }$ for $x \in [ - 1,1 ]$, and $[ x ]$ denotes the greatest integer less than or equal to $x$, is:
(1) 2
(2) 0
(3) 4
(4) Infinite
Q73 Curve Sketching Continuity and Differentiability of Special Functions View
Consider the function $f : R \rightarrow R$ defined by $f ( x ) = \left\{ \begin{array} { c c } \left( 2 - \sin \left( \frac { 1 } { x } \right) \right) | x | , & x \neq 0 \\ 0 , & x = 0 \end{array} \right.$. Then $f$ is:
(1) monotonic on $( - \infty , 0 ) \cup ( 0 , \infty )$
(2) not monotonic on $( - \infty , 0 )$ and $( 0 , \infty )$
(3) monotonic on $( 0 , \infty )$ only
(4) monotonic on $( - \infty , 0 )$ only
Q74 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
Let $f : R \rightarrow R$ be defined as $f ( x ) = e ^ { - x } \sin x$. If $F : [ 0,1 ] \rightarrow R$ is a differentiable function such that $F ( x ) = \int _ { 0 } ^ { x } f ( t ) d t$, then the value of $\int _ { 0 } ^ { 1 } \left( F ^ { \prime } ( x ) + f ( x ) \right) e ^ { x } d x$ lies in the interval
(1) $\left[ \frac { 327 } { 360 } , \frac { 329 } { 360 } \right]$
(2) $\left[ \frac { 330 } { 360 } , \frac { 331 } { 360 } \right]$
(3) $\left[ \frac { 331 } { 360 } , \frac { 334 } { 360 } \right]$
(4) $\left[ \frac { 335 } { 360 } , \frac { 336 } { 360 } \right]$
Q75 Indefinite & Definite Integrals Piecewise/Periodic Function Integration View
If the integral $\int _ { 0 } ^ { 10 } \frac { [ \sin 2 \pi x ] } { \mathrm { e } ^ { x - [ x ] } } d x = \alpha e ^ { - 1 } + \beta e ^ { - \frac { 1 } { 2 } } + \gamma$, where $\alpha , \beta , \gamma$ are integers and $[ x ]$ denotes the greatest integer less than or equal to $x$, then the value of $\alpha + \beta + \gamma$ is equal to:
(1) 0
(2) 20
(3) 25
(4) 10
Q76 First order differential equations (integrating factor) View
Let $y = y ( x )$ be the solution of the differential equation $\cos x ( 3 \sin x + \cos x + 3 ) d y = ( 1 + y \sin x ( 3 \sin x + \cos x + 3 ) ) d x , 0 \leq x \leq \frac { \pi } { 2 } , y ( 0 ) = 0$. Then, $y \left( \frac { \pi } { 3 } \right)$ is equal to:
(1) $2 \log _ { \mathrm { e } } \left( \frac { 2 \sqrt { 3 } + 9 } { 6 } \right)$
(2) $2 \log _ { \mathrm { e } } \left( \frac { 2 \sqrt { 3 } + 10 } { 11 } \right)$
(3) $2 \log _ { \mathrm { e } } \left( \frac { \sqrt { 3 } + 7 } { 2 } \right)$
(4) $2 \log _ { \mathrm { e } } \left( \frac { 3 \sqrt { 3 } - 8 } { 4 } \right)$
Q77 First order differential equations (integrating factor) View
If the curve $y = y ( x )$ is the solution of the differential equation $2 \left( x ^ { 2 } + x ^ { 5 / 4 } \right) d y - y \left( x + x ^ { 1 / 4 } \right) d x = 2 x ^ { 9 / 4 } d x , x > 0$ which passes through the point $\left( 1,1 - \frac { 4 } { 3 } \log _ { \mathrm { e } } 2 \right)$, then the value of $y ( 16 )$ is equal to
(1) $4 \left( \frac { 31 } { 3 } + \frac { 8 } { 3 } \log _ { e } 3 \right)$
(2) $\left( \frac { 31 } { 3 } + \frac { 8 } { 3 } \log _ { e } 3 \right)$
(3) $4 \left( \frac { 31 } { 3 } - \frac { 8 } { 3 } \log _ { \mathrm { e } } 3 \right)$
(4) $\left( \frac { 31 } { 3 } - \frac { 8 } { 3 } \log _ { e } 3 \right)$
Q78 Vectors Introduction & 2D Perpendicularity or Parallel Condition View
Let $O$ be the origin. Let $\overrightarrow { O P } = x \hat { i } + y \hat { j } - \widehat { k }$ and $\overrightarrow { O Q } = - \hat { i } + 2 \hat { j } + 3 x \hat { k } , x , y \in R , x > 0$, be such that $| \overrightarrow { P Q } | = \sqrt { 20 }$ and the vector $\overrightarrow { O P }$ is perpendicular to $\overrightarrow { O Q }$. If $\overrightarrow { O R } = 3 \hat { i } + \mathrm { z } \hat { j } - 7 \hat { k } , z \in R$, is coplanar with $\overrightarrow { O P }$ and $\overrightarrow { O Q }$, then the value of $x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$ is equal to
(1) 7
(2) 9
(3) 2
(4) 1
Q79 Vectors: Lines & Planes Find Cartesian Equation of a Plane View
If the equation of plane passing through the mirror image of a point $( 2,3,1 )$ with respect to line $\frac { x + 1 } { 2 } = \frac { y - 3 } { 1 } = \frac { z + 2 } { - 1 }$ and containing the line $\frac { x - 2 } { 3 } = \frac { 1 - y } { 2 } = \frac { z + 1 } { 1 }$ is $\alpha x + \beta y + \gamma z = 24$ then $\alpha + \beta + \gamma$ is equal to:
(1) 20
(2) 19
(3) 18
(4) 21
Q81 3x3 Matrices Direct Determinant Computation View
If $1 , \log _ { 10 } \left( 4 ^ { x } - 2 \right)$ and $\log _ { 10 } \left( 4 ^ { x } + \frac { 18 } { 5 } \right)$ are in arithmetic progression for a real number $x$ then the value of the determinant $\left| \begin{array} { c c c } 2 \left( x - \frac { 1 } { 2 } \right) & x - 1 & x ^ { 2 } \\ 1 & 0 & x \\ x & 1 & 0 \end{array} \right|$ is equal to:
Q82 Binomial Theorem (positive integer n) Determine Parameters from Conditions on Coefficients or Terms View
Let the coefficients of third, fourth and fifth terms in the expansion of $\left( x + \frac { a } { x ^ { 2 } } \right) ^ { n } , x \neq 0$, be in the ratio $12 : 8 : 3$. Then the term independent of $x$ in the expansion, is equal to $\_\_\_\_$ .
Q84 Measures of Location and Spread View
Consider a set of $3 n$ numbers having variance 4 . In this set, the mean of first $2 n$ numbers is 6 and the mean of the remaining $n$ numbers is 3 . A new set is constructed by adding 1 into each of the first $2 n$ numbers, and subtracting 1 from each of the remaining $n$ numbers. If the variance of the new set is $k$, then $9 k$ is equal to $\_\_\_\_$ .
Q85 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Let $A = \left[ \begin{array} { l l } a & b \\ c & d \end{array} \right]$ and $B = \left[ \begin{array} { l } \alpha \\ \beta \end{array} \right] \neq \left[ \begin{array} { l } 0 \\ 0 \end{array} \right]$ such that $A B = B$ and $a + d = 2021$, then the value of $a d - b c$ is equal to $\_\_\_\_$.
Q86 Stationary points and optimisation Find absolute extrema on a closed interval or domain View
Let $f : [ - 1,1 ] \rightarrow R$ be defined as $f ( x ) = a x ^ { 2 } + b x + c$ for all $x \in [ - 1,1 ]$, where $a , b , c \in R$ such that $f ( - 1 ) = 2 , f ^ { \prime } ( - 1 ) = 1$ and for $x \in ( - 1,1 )$ the maximum value of $f ^ { \prime \prime } ( x )$ is $\frac { 1 } { 2 }$. If $f ( x ) \leq \alpha , x \in [ - 1,1 ]$, then the least value of $\alpha$ is equal to
Q87 Reduction Formulae Derive a Reduction/Recurrence Formula via Integration by Parts View
Let $I _ { n } = \int _ { 1 } ^ { e } x ^ { 19 } ( \log | x | ) ^ { n } d x$, where $n \in N$. If (20) $I _ { 10 } = \alpha I _ { 9 } + \beta I _ { 8 }$, for natural numbers $\alpha$ and $\beta$, then $\alpha - \beta$ equal to $\_\_\_\_$ .
Q88 Areas by integration View
Let $f : [ - 3,1 ] \rightarrow R$ be given as $f ( x ) = \left\{ \begin{array} { l l } \min \left\{ ( x + 6 ) , x ^ { 2 } \right\} , & - 3 \leq x \leq 0 \\ \max \left\{ \sqrt { x } , x ^ { 2 } \right\} , & 0 \leq x \leq 1 \end{array} \right.$. If the area bounded by $y = f ( x )$ and $x$-axis is $A$ sq units, then the value of $6 A$ is equal to
Q89 Vectors Introduction & 2D Magnitude of Vector Expression View
Let $\vec { x }$ be a vector in the plane containing vectors $\vec { a } = 2 \hat { i } - \hat { j } + \hat { k }$ and $\vec { b } = \hat { i } + 2 \hat { j } - \hat { k }$. If the vector $\vec { x }$ is perpendicular to $( 3 \hat { i } + 2 \hat { j } - \widehat { k } )$ and its projection on $\vec { a }$ is $\frac { 17 \sqrt { 6 } } { 2 }$, then the value of $| \vec { x } | ^ { 2 }$ is equal to $\_\_\_\_$ .