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 session3_20jul_shift1

30 maths questions

Q61 Complex Numbers Arithmetic Powers of i or Complex Number Integer Powers View
If $\alpha$ and $\beta$ are the distinct roots of the equation $x ^ { 2 } + ( 3 ) ^ { \frac { 1 } { 4 } } x + 3 ^ { \frac { 1 } { 2 } } = 0$, then the value of $\alpha ^ { 96 } \left( \alpha ^ { 12 } - 1 \right) + \beta ^ { 96 } \left( \beta ^ { 12 } - 1 \right)$ is equal to:
(1) $56 \times 3 ^ { 25 }$
(2) $56 \times 3 ^ { 24 }$
(3) $52 \times 3 ^ { 24 }$
(4) $28 \times 3 ^ { 25 }$
Q62 Discriminant and conditions for roots Probability involving discriminant conditions View
The probability of selecting integers $a \in [ - 5,30 ]$ such that $x ^ { 2 } + 2 ( a + 4 ) x - 5 a + 64 > 0$, for all $x \in R$, is:
(1) $\frac { 7 } { 36 }$
(2) $\frac { 2 } { 9 }$
(3) $\frac { 1 } { 6 }$
(4) $\frac { 1 } { 4 }$
Q63 Complex numbers 2 Modulus and Argument Computation View
If $z$ and $\omega$ are two complex numbers such that $| z \omega | = 1$ and $\arg ( z ) - \arg ( \omega ) = \frac { 3 \pi } { 2 }$, then $\arg \left( \frac { 1 - 2 \bar { z } \omega } { 1 + 3 \bar { z } \omega } \right)$ is: (Here $\arg ( z )$ denotes the principal argument of complex number $z$)
(1) $\frac { \pi } { 4 }$
(2) $- \frac { 3 \pi } { 4 }$
(3) $- \frac { \pi } { 4 }$
(4) $\frac { 3 \pi } { 4 }$
Q64 Binomial Theorem (positive integer n) Find a Specific Coefficient in a Product of Binomial/Polynomial Expressions View
The coefficient of $x ^ { 256 }$ in the expansion of $( 1 - x ) ^ { 101 } \left( x ^ { 2 } + x + 1 \right) ^ { 100 }$ is:
(1) ${ } ^ { 100 } C _ { 16 }$
(2) ${ } ^ { 100 } C _ { 15 }$
(3) ${ } ^ { - 100 } C _ { 16 }$
(4) ${ } ^ { - 100 } C _ { 15 }$
Q65 Conic sections Triangle or Quadrilateral Area and Perimeter with Foci View
Let the tangent to the parabola $S : y ^ { 2 } = 2 x$ at the point $P ( 2,2 )$ meet the $x$-axis at $Q$ and normal at it meet the parabola $S$ at the point $R$. Then the area (in sq. units) of the triangle $P Q R$ is equal to:
(1) $\frac { 25 } { 2 }$
(2) $\frac { 35 } { 2 }$
(3) $\frac { 15 } { 2 }$
(4) 25
Q66 Proof Proof of Equivalence or Logical Relationship Between Conditions View
The Boolean expression $( p \wedge \sim q ) \Rightarrow ( q \vee \sim p )$ is equivalent to:
(1) $q \Rightarrow p$
(2) $p \Rightarrow q$
(3) $\sim q \Rightarrow p$
(4) $p \Rightarrow \sim q$
Q67 Measures of Location and Spread View
The mean of 6 distinct observations is 6.5 and their variance is 10.25 . If 4 out of 6 observations are $2,4,5$ and 7, then the remaining two observations are:
(1) 10,11
(2) 3,18
(3) 8,13
(4) 1,20
Q68 Sine and Cosine Rules Circumradius or incircle radius computation View
If in a triangle $A B C , A B = 5$ units, $\angle B = \cos ^ { - 1 } \left( \frac { 3 } { 5 } \right)$ and radius of circumcircle of $\triangle A B C$ is 5 units, then the area (in sq. units) of $\triangle A B C$ is:
(1) $10 + 6 \sqrt { 2 }$
(2) $8 + 2 \sqrt { 2 }$
(3) $6 + 8 \sqrt { 3 }$
(4) $4 + 2 \sqrt { 3 }$
Q69 Matrices Determinant and Rank Computation View
Let $A = \left[ \begin{array} { l l } 2 & 3 \\ a & 0 \end{array} \right] , a \in R$ be written as $P + Q$ where $P$ is a symmetric matrix and $Q$ is skew symmetric matrix. If $\operatorname { det } ( Q ) = 9$, then the modulus of the sum of all possible values of determinant of $P$ is equal to:
(1) 36
(2) 24
(3) 45
(4) 18
Q70 Composite & Inverse Functions Find or Apply an Inverse Function Formula View
The number of real roots of the equation $\tan ^ { - 1 } \sqrt { x ( x + 1 ) } + \sin ^ { - 1 } \sqrt { x ^ { 2 } + x + 1 } = \frac { \pi } { 4 }$ is:
(1) 1
(2) 2
(3) 4
(4) 0
Q71 Composite & Inverse Functions Determine Domain or Range of a Composite Function View
Let $[ x ]$ denote the greatest integer $\leq x$, where $x \in R$. If the domain of the real valued function $f ( x ) = \sqrt { \frac { | [ x ] | - 2 } { | [ x ] | - 3 } }$ is $( - \infty , a ) \cup [ b , c ) \cup [ 4 , \infty ) , a < b < c$, then the value of $a + b + c$ is:
(1) 8
(2) 1
(3) $- 2$
(4) $- 3$
Q72 Composite & Inverse Functions Determine Domain or Range of a Composite Function View
Let a function $f : R \rightarrow R$ be defined as, $f ( x ) = \begin{cases} \sin x - e ^ { x } & \text { if } x \leq 0 \\ a + [ - x ] & \text { if } 0 < x < 1 \\ 2 x - b & \text { if } x \geq 1 \end{cases}$
Where $[ x ]$ is the greatest integer less than or equal to $x$. If $f$ is continuous on $R$, then ( $a + b$ ) is equal to:
(1) 4
(2) 3
(3) 2
(4) 5
Q73 Matrices Determinant and Rank Computation View
Let $A = \left[ a _ { i j } \right]$ be a $3 \times 3$ matrix, where $a _ { i j } = \left\{ \begin{array} { c c } 1 , & \text { if } i = j \\ - x , & \text { if } | i - j | = 1 \\ 2 x + 1 , & \text { otherwise } \end{array} \right.$ Let a function $f : R \rightarrow R$ be defined as $f ( x ) = \operatorname { det } ( A )$. Then the sum of maximum and minimum values of $f$ on $R$ is equal to:
(1) $- \frac { 20 } { 27 }$
(2) $\frac { 88 } { 27 }$
(3) $\frac { 20 } { 27 }$
(4) $- \frac { 88 } { 27 }$
Q74 Stationary points and optimisation Find critical points and classify extrema of a given function View
Let $a$ be a real number such that the function $f ( x ) = a x ^ { 2 } + 6 x - 15 , x \in R$ is increasing in $( - \infty , \frac { 3 } { 4 } )$ and decreasing in $\left( \frac { 3 } { 4 } , \infty \right)$. Then the function $g ( x ) = a x ^ { 2 } - 6 x + 15 , x \in R$ has a
(1) local maximum at $x = - \frac { 3 } { 4 }$
(2) local minimum at $x = - \frac { 3 } { 4 }$
(3) local maximum at $x = \frac { 3 } { 4 }$
(4) local minimum at $x = \frac { 3 } { 4 }$
Q75 Indefinite & Definite Integrals Piecewise/Periodic Function Integration View
Let $a$ be a positive real number such that $\int _ { 0 } ^ { a } e ^ { x - [ x ] } d x = 10 e - 9$ where, $[ x ]$ is the greatest integer less than or equal to $x$. Then, $a$ is equal to:
(1) $10 - \log _ { e } ( 1 + e )$
(2) $10 + \log _ { e } 2$
(3) $10 + \log _ { e } ( 1 + e )$
(4) $10 - \log _ { e } 2$
Q76 Combinations & Selection Combinatorial Identity or Bijection Proof View
If $\sum _ { k = 1 } ^ { 10 } K ^ { 2 } \left( { } ^ { 10 } C _ { K } \right) ^ { 2 } = 22000 L$, then $L$ is equal to
Q77 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
Q78 Vectors 3D & Lines Vector Algebra and Triple Product Computation View
Let $\vec { a } = \hat { i } + \hat { j } + \hat { k }$ and $\vec { b } = \hat { j } - \hat { k }$. If $\vec { c }$ is a vector such that $\vec { a } \times \vec { c } = \vec { b }$ and $\vec { a } \cdot \vec { c } = 3$, then $\vec { a } \cdot ( \vec { b } \times \vec { c } )$ is equal to
Q79 Indefinite & Definite Integrals Finding a Function from an Integral Equation View
Let $f ( x ) = \int _ { 0 } ^ { x } e ^ { t } f ( t ) d t + e ^ { x }$ be a differentiable function for all $x \in R$. Then $f ( x )$ equals:
Q80 Conic sections Circle-Conic Interaction with Tangency or Intersection 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:
Q81 Laws of Logarithms Solve a Logarithmic Equation View
The number of solutions of the equation $\log _ { 4 } ( x - 1 ) = \log _ { 2 } ( x - 3 )$ is
Q82 First order differential equations (integrating factor) View
If $y = y ( x )$ is the solution of the differential equation $\frac { d y } { d x } + ( \tan x ) y = \sin x , 0 \leq x \leq \frac { \pi } { 3 }$, with $y ( 0 ) = 0$, then $y \left( \frac { \pi } { 4 } \right)$ equal to
Q83 Permutations & Arrangements Forming Numbers with Digit Constraints View
The number of seven digit integers with sum of digits equal to 10 and formed by using the digits 1, 2 and 3 only is
Q84 Vectors: Lines & Planes Parallelism Between Line and Plane or Constraint on Parameters 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
Q85 Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution View
The value of $\int _ { - \pi / 2 } ^ { \pi / 2 } \left( \frac { 1 + \sin ^ { 2 } x } { 1 + \pi ^ { \sin x } } \right) d x$ is
Q86 Number Theory Modular Arithmetic Computation View
Let $A = \{ n \in N : n$ is a 3-digit number $\}$, $B = \{ 9 k + 2 : k \in N \}$ and $C = \{ 9 k + l : k \in N \}$ for some $l ( 0 < l < 9 )$. If the sum of all the elements of the set $A \cap ( B \cup C )$ is $274 \times 400$, then $l$ is equal to
Q87 Small angle approximation 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
Q88 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
Q89 Geometric Sequences and Series Find a Threshold Index (Algorithm or Calculation) View
Let $A _ { 1 } , A _ { 2 } , A _ { 3 } , \ldots$ be squares such that for each $n \geq 1$, the length of the side of $A _ { n }$ equals the length of diagonal of $A _ { n + 1 }$. If the length of $A _ { 1 }$ is 12 cm, then the smallest value of $n$ for which area of $A _ { n }$ is less than one is
Q90 Matrices Determinant and Rank Computation View
Let $\theta = \frac { \pi } { 5 }$ and $A = \begin{bmatrix} \cos \theta & \sin \theta \\ - \sin \theta & \cos \theta \end{bmatrix}$. If $B = A + A ^ { 4 }$, then $\det ( B )$