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
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
2024 session1_29jan_shift1

30 maths questions

Q61 Complex Numbers Arithmetic Systems of Equations via Real and Imaginary Part Matching View
If $z = \frac { 1 } { 2 } - 2 i$, is such that $| z + 1 | = \alpha z + \beta ( 1 + i ) , i = \sqrt { - 1 }$ and $\alpha , \beta \in \mathrm { R }$, then $\alpha + \beta$ is equal to
(1) - 4
(2) 3
(3) 2
(4) - 1
Q62 Arithmetic Sequences and Series Optimization Involving an Arithmetic Sequence View
In an A.P., the sixth term $\mathbf { a } _ { 6 } = 2$. If the $\mathbf { a } _ { 1 } \mathbf { a } _ { 4 } \mathbf { a } _ { 5 }$ is the greatest, then the common difference of the A.P., is equal to
(1) $\frac { 3 } { 2 }$
(2) $\frac { 8 } { 5 }$
(3) $\frac { 2 } { 3 }$
(4) $\frac { 5 } { 8 }$
Q63 Geometric Sequences and Series Finite Geometric Sum and Term Relationships View
If in a G.P. of 64 terms, the sum of all the terms is 7 times the sum of the odd terms of the G.P, then the common ratio of the G.P. is equal to
(1) 7
(2) 4
(3) 5
(4) 6
Q64 Standard trigonometric equations Solve trigonometric equation for solutions in an interval View
If $\alpha , - \frac { \pi } { 2 } < \alpha < \frac { \pi } { 2 }$ is the solution of $4 \cos \theta + 5 \sin \theta = 1$, then the value of $\tan \alpha$ is
(1) $\frac { 10 - \sqrt { 10 } } { 6 }$
(2) $\frac { 10 - \sqrt { 10 } } { 12 }$
(3) $\frac { \sqrt { 10 } - 10 } { 12 }$
(4) $\frac { \sqrt { 10 } - 10 } { 6 }$
Q65 Circles Inscribed/Circumscribed Circle Computations View
Let $\left( 5 , \frac { a } { 4 } \right)$, be the circumcenter of a triangle with vertices $A ( a , - 2 ) , B ( a , 6 )$ and $C \left( \frac { a } { 4 } , - 2 \right)$. Let $\alpha$ denote the circumradius, $\beta$ denote the area and $\gamma$ denote the perimeter of the triangle. Then $\alpha + \beta + \gamma$ is
(1) 60
(2) 53
(3) 62
(4) 30
Q66 Straight Lines & Coordinate Geometry Slope and Angle Between Lines View
In a $\triangle \mathrm { ABC }$, suppose $\mathrm { y } = \mathrm { x }$ is the equation of the bisector of the angle $B$ and the equation of the side $A C$ is $2 x - y = 2$. If $2 A B = B C$ and the point $A$ and $B$ are respectively $( 4,6 )$ and $( \alpha , \beta )$, then $\alpha + 2 \beta$ is equal to
(1) - 4
(2) 42
(3) 2
(4) - 1
Q67 Indefinite & Definite Integrals Accumulation Function Analysis View
$\lim _ { x \rightarrow \frac { \pi } { 2 } } \left( \frac { 1 } { \left( x - \frac { \pi } { 2 } \right) ^ { 2 } } \int _ { x ^ { 3 } } ^ { \left( \frac { \pi } { 2 } \right) ^ { 3 } } \cos \left( \frac { 1 } { t ^ { 3 } } \right) d t \right)$ is equal to
(1) $\frac { 3 \pi } { 8 }$
(2) $\frac { 3 \pi ^ { 2 } } { 4 }$
(3) $\frac { 3 \pi ^ { 2 } } { 8 }$
(4) $\frac { 3 \pi } { 4 }$
Q68 Probability Definitions Finite Equally-Likely Probability Computation View
Let $R$ be a relation on $Z \times Z$ defined by $( a , b ) R ( c , d )$ if and only if $a d - b c$ is divisible by 5 . Then R is
(1) Reflexive and symmetric but not transitive
(2) Reflexive but neither symmetric not transitive
(3) Reflexive, symmetric and transitive
(4) Reflexive and transitive but not symmetric
Q69 Matrices Determinant and Rank Computation View
Let $A = \left[ \begin{array} { l l l } 1 & 0 & 0 \\ 0 & \alpha & \beta \\ 0 & \beta & \alpha \end{array} \right]$ and $| 2 A | ^ { 3 } = 2 ^ { 21 }$ where $\alpha , \beta \in Z$, Then a value of $\alpha$ is
(1) 3
(2) 5
(3) 17
(4) 9
Q70 Matrices Matrix Algebra and Product Properties View
Let A be a square matrix such that $\mathrm { AA } ^ { \mathrm { T } } = \mathrm { I }$. Then $\frac { 1 } { 2 } \mathrm { ~A} \left[ \left( \mathrm { ~A} + \mathrm { A } ^ { \mathrm { T } } \right) ^ { 2 } + \left( \mathrm { A } - \mathrm { A } ^ { \mathrm { T } } \right) ^ { 2 } \right]$ is equal to
(1) $A ^ { 2 } + I$
(2) $A ^ { 3 } + I$
(3) $A ^ { 2 } + A ^ { T }$
(4) $\mathrm { A } ^ { 3 } + \mathrm { A } ^ { \mathrm { T } }$
Q71 Composite & Inverse Functions Determine Domain or Range of a Composite Function View
If $f ( x ) = \left\{ \begin{array} { l } 2 + 2 x , - 1 \leq x < 0 \\ 1 - \frac { x } { 3 } , 0 \leq x \leq 3 \end{array} ; g ( x ) = \left\{ \begin{array} { l } - x , - 3 \leq x \leq 0 \\ x , 0 < x \leq 1 \end{array} \right. \right.$, then range of $( f \circ g ( x ) )$ is
(1) $( 0,1 ]$
(2) $[ 0,3 )$
(3) $[ 0,1 ]$
(4) $[ 0,1 )$
Q72 Curve Sketching Number of Solutions / Roots via Curve Analysis View
Consider the function $f : \left[ \frac { 1 } { 2 } , 1 \right] \rightarrow \mathrm { R }$ defined by $f ( x ) = 4 \sqrt { 2 } x ^ { 3 } - 3 \sqrt { 2 } x - 1$. Consider the statements (I) The curve $y = f ( x )$ intersects the $x$-axis exactly at one point (II) The curve $y = f ( x )$ intersects the $x$-axis at $x = \cos \frac { \pi } { 12 }$ Then
(1) Only (II) is correct
(2) Both (I) and (II) are incorrect
(3) Only (I) is correct
(4) Both (I) and (II) are correct
Q73 Chain Rule Chain Rule with Composition of Explicit Functions View
Suppose $f ( x ) = \frac { \left( 2 ^ { x } + 2 ^ { - x } \right) \tan x \sqrt { \tan ^ { - 1 } \left( x ^ { 2 } - x + 1 \right) } } { \left( 7 x ^ { 2 } + 3 x + 1 \right) ^ { 3 } }$. Then the value of $f ^ { \prime } ( 0 )$ is equal to
(1) $\pi$
(2) 0
(3) $\sqrt { \pi }$
(4) $\frac { \pi } { 2 }$
Q74 Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution View
If the value of the integral $\int _ { - \frac { \pi } { 2 } } ^ { \frac { \pi } { 2 } } \left( \frac { x ^ { 2 } \cos x } { 1 + \pi ^ { x } } + \frac { 1 + \sin ^ { 2 } x } { 1 + e ^ { ( \sin x ) ^ { 2023 } } } \right) d x = \frac { \pi } { 4 } ( \pi + a ) - 2$, then the value of $a$ is
(1) 3
(2) $- \frac { 3 } { 2 }$
(3) 2
(4) $\frac { 3 } { 2 }$
Q75 Integration by Substitution Substitution to Compute an Indefinite Integral with Initial Condition View
For $x \in \left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$, if $y ( x ) = \int \frac { \operatorname { cosec } x + \sin x } { \operatorname { cosec } x \sec x + \tan x \sin ^ { 2 } x } d x$ and $\lim _ { x \rightarrow \left( \frac { \pi } { 2 } \right) ^ { - } } y ( x ) = 0$ then $y \left( \frac { \pi } { 4 } \right)$ is equal to
(1) $\tan ^ { - 1 } \left( \frac { 1 } { \sqrt { 2 } } \right)$
(2) $\frac { 1 } { 2 } \tan ^ { - 1 } \left( \frac { 1 } { \sqrt { 2 } } \right)$
(3) $- \frac { 1 } { \sqrt { 2 } } \tan ^ { - 1 } \left( \frac { 1 } { \sqrt { 2 } } \right)$
(4) $\frac { 1 } { \sqrt { 2 } } \tan ^ { - 1 } \left( - \frac { 1 } { 2 } \right)$
Q76 Differential equations First-Order Linear DE: General Solution View
A function $y = f ( x )$ satisfies $f ( x ) \sin 2 x + \sin x - \left( 1 + \cos ^ { 2 } x \right) f ^ { \prime } ( x ) = 0$ with condition $f ( 0 ) = 0$. Then $f \left( \frac { \pi } { 2 } \right)$ is equal to
(1) 1
(2) 0
(3) - 1
(4) 2
Q77 Vectors Introduction & 2D Expressing a Vector as a Linear Combination View
Let $\vec { a } , \vec { b }$ and $\vec { c }$ be three non-zero vectors such that $\vec { b }$ and $\vec { c }$ are non-collinear if $\vec { a } + 5 \vec { b }$ is collinear with $\overrightarrow { c , b } + 6 \overrightarrow { c c }$ is collinear with $\vec { a }$ and $\vec { a } + \alpha \vec { b } + \beta \vec { c } = \overrightarrow { 0 }$, then $\alpha + \beta$ is equal to
(1) 35
(2) 30
(3) - 30
(4) - 25
Q78 Vectors Introduction & 2D Section Ratios and Intersection via Vectors View
Let $O$ be the origin and the position vector of $A$ and $B$ be $2 \hat { i } + 2 \hat { j } + \widehat { k }$ and $2 \hat { i } + 4 \hat { j } + 4 \widehat { k }$ respectively. If the internal bisector of $\angle A O B$ meets the line $A B$ at $C$, then the length of $O C$ is
(1) $\frac { 2 } { 3 } \sqrt { 31 }$
(2) $\frac { 2 } { 3 } \sqrt { 34 }$
(3) $\frac { 3 } { 4 } \sqrt { 34 }$
(4) $\frac { 3 } { 2 } \sqrt { 31 }$
Q79 Vectors: Lines & Planes Find Intersection of a Line and a Plane View
Let $P Q R$ be a triangle with $R ( - 1,4,2 )$. Suppose $M ( 2,1,2 )$ is the mid point of $P Q$. The distance of the centroid of $\triangle P Q R$ from the point of intersection of the line $\frac { x - 2 } { 0 } = \frac { y } { 2 } = \frac { z + 3 } { - 1 }$ and $\frac { x - 1 } { 1 } = \frac { y + 3 } { - 3 } = \frac { z + 1 } { 1 }$ is
(1) 69
(2) 9
(3) $\sqrt { 69 }$
(4) $\sqrt { 99 }$
Q80 Discrete Probability Distributions Properties of Named Discrete Distributions (Non-Binomial) View
A fair die is thrown until 2 appears. Then the probability, that 2 appears in even number of throws, is
(1) $\frac { 5 } { 6 }$
(2) $\frac { 1 } { 6 }$
(3) $\frac { 5 } { 11 }$
(4) $\frac { 6 } { 11 }$
Q81 Complex Numbers Arithmetic Powers of i or Complex Number Integer Powers View
Let $\alpha , \beta$ be the roots of the equation $x ^ { 2 } - x + 2 = 0$ with $\operatorname { Im } ( \alpha ) > \operatorname { Im } ( \beta )$. Then $\alpha ^ { 6 } + \alpha ^ { 4 } + \beta ^ { 4 } - 5 \alpha ^ { 2 }$ is equal to
Q82 Permutations & Arrangements Dictionary Order / Rank of a Permutation View
All the letters of the word $G T W E N T Y$ are written in all possible ways with or without meaning and these words are written as in a dictionary. The serial number of the word $G T W E N T Y$ IS
Q83 Binomial Theorem (positive integer n) Evaluate a Summation Involving Binomial Coefficients View
If $\frac { { } ^ { 11 } C _ { 1 } } { 2 } + \frac { { } ^ { 11 } C _ { 2 } } { 3 } + \ldots . . + \frac { { } ^ { 11 } C _ { 9 } } { 10 } = \frac { n } { m }$ with $\operatorname { gcd } ( n , m ) = 1$, then $n + m$ is equal to
Q84 Circles Tangent Lines and Tangent Lengths View
Equations of two diameters of a circle are $2 x - 3 y = 5$ and $3 x - 4 y = 7$. The line joining the points $\left( - \frac { 22 } { 7 } , - 4 \right)$ and $\left( - \frac { 1 } { 7 } , 3 \right)$ intersects the circle at only one point $P ( \alpha , \beta )$. Then $17 \beta - \alpha$ is equal to
Q85 Circles Intersection of Circles or Circle with Conic View
If the points of intersection of two distinct conics $x ^ { 2 } + y ^ { 2 } = 4 b$ and $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ lie on the curve $y ^ { 2 } = 3 x ^ { 2 }$, then $3 \sqrt { 3 }$ times the area of the rectangle formed by the intersection points is $\_\_\_\_$ .
Q86 Measures of Location and Spread View
If the mean and variance of the data $65,68,58,44,48,45,60 , \alpha , \beta , 60$ where $\alpha > \beta$ are 56 and 66.2 respectively, then $\alpha ^ { 2 } + \beta ^ { 2 }$ is equal to
Q87 Sign Change & Interval Methods View
Let $\mathrm { f } ( \mathrm { x } ) = 2 ^ { \mathrm { x } } - \mathrm { x } ^ { 2 } , \mathrm { x } \in \mathrm { R }$. If m and n are respectively the number of points at which the curves $\mathrm { y } = \mathrm { f } ( \mathrm { x } )$ and $\mathrm { y } = \mathrm { f } ^ { \prime } ( \mathrm { x } )$ intersects the x-axis, then the value of $\mathrm { m } + \mathrm { n }$ is
Q88 Circles Area and Geometric Measurement Involving Circles View
The area (in sq. units) of the part of circle $x ^ { 2 } + y ^ { 2 } = 169$ which is below the line $5 x - y = 13$ is $\frac { \pi \alpha } { 2 \beta } - \frac { 65 } { 2 } + \frac { \alpha } { \beta } \sin ^ { - 1 } \left( \frac { 12 } { 13 } \right)$ where $\alpha , \beta$ are coprime numbers. Then $\alpha + \beta$ is equal to
Q89 First order differential equations (integrating factor) View
If the solution curve $y = y ( x )$ of the differential equation $\left( 1 + y ^ { 2 } \right) \left( 1 + \log _ { e } x \right) d x + x d y = 0 , x > 0$ passes through the point $( 1,1 )$ and $y ( e ) = \frac { \alpha - \tan \left( \frac { 3 } { 2 } \right) } { \beta + \tan \left( \frac { 3 } { 2 } \right) }$, then $\alpha + 2 \beta$ is
Q90 Vectors 3D & Lines Distance from a Point to a Line (Show/Compute) View
A line with direction ratio $2,1,2$ meets the lines $\mathrm { x } = \mathrm { y } + 2 = \mathrm { z }$ and $\mathrm { x } + 2 = 2 \mathrm { y } = 2 \mathrm { z }$ respectively at the point P and Q . if the length of the perpendicular from the point $( 1,2,12 )$ to the line PQ is $l$, then $l ^ { 2 }$ is