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
2022 session2_26jul_shift1

29 maths questions

Q61 Complex Numbers Argand & Loci Geometric Properties of Triangles/Polygons from Affixes View
Let $O$ be the origin and $A$ be the point $z _ { 1 } = 1 + 2i$. If $B$ is the point $z _ { 2 } , \operatorname { Re } \left( z _ { 2 } \right) < 0$, such that $OAB$ is a right angled isosceles triangle with $OB$ as hypotenuse, then which of the following is NOT true?
(1) $\arg z _ { 2 } = \pi - \tan ^ { - 1 } 3$
(2) $\arg \left( z _ { 1 } - 2 z _ { 2 } \right) = - \tan ^ { - 1 } \frac { 4 } { 3 }$
(3) $\left| z _ { 2 } \right| = \sqrt { 10 }$
(4) $\left| 2 z _ { 1 } - z _ { 2 } \right| = 5$
Q62 Geometric Sequences and Series Finite Geometric Sum and Term Relationships View
Consider two G.Ps. $2,2 ^ { 2 } , 2 ^ { 3 } , \ldots$ and $4,4 ^ { 2 } , 4 ^ { 3 } , \ldots$ of 60 and $n$ terms respectively. If the geometric mean of all the $60 + n$ terms is $( 2 ) ^ { \frac { 225 } { 8 } }$, then $\sum _ { k = 1 } ^ { n } k ( n - k )$ is equal to:
(1) 560
(2) 1540
(3) 1330
(4) 2600
Q63 Standard trigonometric equations Solve trigonometric equation for solutions in an interval View
Let $S = \left\{ \theta \in [ 0,2 \pi ] : 8 ^ { 2 \sin ^ { 2 } \theta } + 8 ^ { 2 \cos ^ { 2 } \theta } = 16 \right\}$. Then $n ( S ) + \sum _ { \theta \in \mathrm { S } } \left( \sec \left( \frac { \pi } { 4 } + 2 \theta \right) \operatorname { cosec } \left( \frac { \pi } { 4 } + 2 \theta \right) \right)$ is equal to:
(1) 0
(2) $- 2$
(3) $- 4$
(4) 12
Q64 Circles Area and Geometric Measurement Involving Circles View
A point $P$ moves so that the sum of squares of its distances from the points $( 1,2 )$ and $( - 2,1 )$ is 14. Let $f ( x , y ) = 0$ be the locus of $P$, which intersects the $x$-axis at the points $A , B$ and the $y$-axis at the point $C , D$. Then the area of the quadrilateral $ACBD$ is equal to
(1) $\frac { 9 } { 2 }$
(2) $\frac { 3 \sqrt { 17 } } { 2 }$
(3) $\frac { 3 \sqrt { 17 } } { 4 }$
(4) 9
Q65 Conic sections Tangent and Normal Line Problems View
Let the tangent drawn to the parabola $y ^ { 2 } = 24 x$ at the point $( \alpha , \beta )$ is perpendicular to the line $2 x + 2 y = 5$. Then the normal to the hyperbola $\frac { x ^ { 2 } } { \alpha ^ { 2 } } - \frac { y ^ { 2 } } { \beta ^ { 2 } } = 1$ at the point $( \alpha + 4 , \beta + 4 )$ does NOT pass through the point:
(1) $( 25,10 )$
(2) $( 20,12 )$
(3) $( 30,8 )$
(4) $( 15,13 )$
Q67 Matrices Determinant and Rank Computation View
Let $A$ be a $2 \times 2$ matrix with $\operatorname { det } ( A ) = - 1$ and $\operatorname { det } ( ( A + I ) ( \operatorname { Adj } ( A ) + I ) ) = 4$. Then the sum of the diagonal elements of $A$ can be:
(1) $- 1$
(2) 2
(3) 1
(4) $- \sqrt { 2 }$
Q68 Simultaneous equations View
If the system of linear equations $8 x + y + 4 z = - 2$ $x + y + z = 0$ $\lambda x - 3 y = \mu$ has infinitely many solutions, then the distance of the point $\left( \lambda , \mu , - \frac { 1 } { 2 } \right)$ from the plane $8 x + y + 4 z + 2 = 0$ is:
(1) $3 \sqrt { 5 }$
(2) 4
(3) $\frac { 26 } { 9 }$
(4) $\frac { 10 } { 3 }$
Q69 Addition & Double Angle Formulae Multi-Step Composite Problem Using Identities View
$\tan \left( 2 \tan ^ { - 1 } \frac { 1 } { 5 } + \sec ^ { - 1 } \frac { \sqrt { 5 } } { 2 } + 2 \tan ^ { - 1 } \frac { 1 } { 8 } \right)$ is equal to:
(1) 1
(2) 2
(3) $\frac { 1 } { 4 }$
(4) $\frac { 5 } { 4 }$
Q70 Differential equations Finding a DE from a Limit or Implicit Condition View
Let $f : R \rightarrow R$ be a continuous function such that $f ( 3 x ) - f ( x ) = x$. If $f ( 8 ) = 7$, then $f ( 14 )$ is equal to:
(1) 4
(2) 10
(3) 11
(4) 16
Q71 Differentiating Transcendental Functions Determine parameters from function or curve conditions View
If the function $f ( x ) = \left\{ \begin{array} { l l } \frac { \log _ { e } \left( 1 - x + x ^ { 2 } \right) + \log _ { e } \left( 1 + x + x ^ { 2 } \right) } { \sec x - \cos x } , & x \in \left( \frac { - \pi } { 2 } , \frac { \pi } { 2 } \right) - \{ 0 \} \\ k & , x = 0 \end{array} \right.$ is continuous at $x = 0$, then $k$ is equal to:
(1) 1
(2) $- 1$
(3) $e$
(4) 0
Q72 Composite & Inverse Functions Evaluate Composition from Algebraic Definitions View
If $f ( x ) = \left\{ \begin{array} { l l } x + a , & x \leq 0 \\ | x - 4 | , & x > 0 \end{array} \right.$ and $g ( x ) = \left\{ \begin{array} { l l } x + 1 , & x < 0 \\ ( x - 4 ) ^ { 2 } + b , & x \geq 0 \end{array} \right.$ are continuous on $R$, then $( g \circ f ) ( 2 ) + ( f \circ g ) ( - 2 )$ is equal to:
(1) $- 10$
(2) 10
(3) 8
(4) $- 8$
Q73 Stationary points and optimisation Composite or piecewise function extremum analysis View
Let $f ( x ) = \left\{ \begin{array} { c c } x ^ { 3 } - x ^ { 2 } + 10 x - 7 , & x \leq 1 \\ - 2 x + \log _ { 2 } \left( b ^ { 2 } - 4 \right) , & x > 1 \end{array} \right.$ Then the set of all values of $b$, for which $f ( x )$ has maximum value at $x = 1$, is:
(1) $( - 6 , - 2 )$
(2) $( 2,6 )$
(3) $[ - 6 , - 2 ) \cup ( 2,6 ]$
(4) $[ - \sqrt { 6 } , - 2 ) \cup ( 2 , \sqrt { 6 } ]$
Q74 Differentiating Transcendental Functions Evaluate derivative at a point or find tangent slope View
If $a = \lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \frac { 2 n } { n ^ { 2 } + k ^ { 2 } }$ and $f ( x ) = \sqrt { \frac { 1 - \cos x } { 1 + \cos x } } , x \in ( 0,1 )$, then:
(1) $2 \sqrt { 2 } f \left( \frac { a } { 2 } \right) = f ^ { \prime } \left( \frac { a } { 2 } \right)$
(2) $f \left( \frac { a } { 2 } \right) f ^ { \prime } \left( \frac { a } { 2 } \right) = \sqrt { 2 }$
(3) $\sqrt { 2 } f \left( \frac { a } { 2 } \right) = f ^ { \prime } \left( \frac { a } { 2 } \right)$
(4) $f \left( \frac { a } { 2 } \right) = \sqrt { 2 } f ^ { \prime } \left( \frac { a } { 2 } \right)$
Q75 Areas by integration View
The odd natural number $a$, such that the area of the region bounded by $y = 1 , y = 3 , x = 0 , x = y ^ { a }$ is $\frac { 364 } { 3 }$, equal to:
(1) 3
(2) 5
(3) 7
(4) 9
Q76 First order differential equations (integrating factor) View
If $\frac { d y } { d x } + 2 y \tan x = \sin x , 0 < x < \frac { \pi } { 2 }$ and $y \left( \frac { \pi } { 3 } \right) = 0$, then the maximum value of $y ( x )$ is
(1) $\frac { 1 } { 8 }$
(2) $\frac { 3 } { 4 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 3 } { 8 }$
Q77 Vectors Introduction & 2D Dot Product Computation View
Let $\vec { a } = \alpha \hat { i } + \hat { j } - \hat { k }$ and $\vec { b } = 2 \hat { i } + \hat { j } - \alpha \hat { k } , \alpha > 0$. If the projection of $\vec { a } \times \vec { b }$ on the vector $- \hat { i } + 2 \hat { j } - 2 \hat { k }$ is 30, then $\alpha$ is equal to
(1) $\frac { 15 } { 2 }$
(2) 8
(3) $\frac { 13 } { 2 }$
(4) 7
Q78 Vectors 3D & Lines Distance from a Point to a Line (Show/Compute) View
The length of the perpendicular from the point $( 1 , - 2,5 )$ on the line passing through $( 1,2,4 )$ and parallel to the line $x + y - z = 0 = x - 2 y + 3 z - 5$ is:
(1) $\sqrt { \frac { 21 } { 2 } }$
(2) $\sqrt { \frac { 9 } { 2 } }$
(3) $\sqrt { \frac { 73 } { 2 } }$
(4) 1
Q79 Binomial Distribution Find Parameters from Moment Conditions View
The mean and variance of a binomial distribution are $\alpha$ and $\frac { \alpha } { 3 }$ respectively. If $P ( X = 1 ) = \frac { 4 } { 243 }$, then $P ( X = 4$ or $5 )$ is equal to:
(1) $\frac { 5 } { 9 }$
(2) $\frac { 64 } { 81 }$
(3) $\frac { 16 } { 27 }$
(4) $\frac { 145 } { 243 }$
Q80 Probability Definitions Probability Using Set/Event Algebra View
Let $E _ { 1 } , E _ { 2 } , E _ { 3 }$ be three mutually exclusive events such that $P \left( E _ { 1 } \right) = \frac { 2 + 3 p } { 6 } , P \left( E _ { 2 } \right) = \frac { 2 - p } { 8 }$ and $P \left( E _ { 3 } \right) = \frac { 1 - p } { 2 }$. If the maximum and minimum values of $p$ are $p _ { 1 }$ and $p _ { 2 }$ then $\left( p _ { 1 } + p _ { 2 } \right)$ is equal to:
(1) $\frac { 2 } { 3 }$
(2) $\frac { 5 } { 3 }$
(3) $\frac { 5 } { 4 }$
(4) 1
Q81 Solving quadratics and applications Finding roots or coefficients of a quadratic using Vieta's relations View
If for some $p , q , r \in R$, all have positive sign, one of the roots of the equation $\left( p ^ { 2 } + q ^ { 2 } \right) x ^ { 2 } - 2 q ( p + r ) x + q ^ { 2 } + r ^ { 2 } = 0$ is also a root of the equation $x ^ { 2 } + 2 x - 8 = 0$, then $\frac { q ^ { 2 } + r ^ { 2 } } { p ^ { 2 } }$ is equal to $\_\_\_\_$.
Q82 Permutations & Arrangements Forming Numbers with Digit Constraints View
The number of 5-digit natural numbers, such that the product of their digits is 36, is $\_\_\_\_$.
Q83 Arithmetic Sequences and Series Compute Partial Sum of an Arithmetic Sequence View
The series of positive multiples of 3 is divided into sets: $\{ 3 \} , \{ 6,9,12 \} , \{ 15,18,21,24,27 \} , \ldots$ Then the sum of the elements in the $11 ^ { \text {th} }$ set is equal to $\_\_\_\_$.
Q84 Binomial Theorem (positive integer n) Find a Specific Coefficient in a Product of Binomial/Polynomial Expressions View
If the coefficients of $x$ and $x ^ { 2 }$ in the expansion of $( 1 + x ) ^ { p } ( 1 - x ) ^ { q } , p , q \leq 15$, are $-3$ and $-5$ respectively, then the coefficient of $x ^ { 3 }$ is equal to $\_\_\_\_$.
Q85 Straight Lines & Coordinate Geometry Triangle Properties and Special Points View
The equations of the sides $AB , BC$ and $CA$ of a triangle $ABC$ are $2 x + y = 0 , x + p y = 15 a$ and $x - y = 3$ respectively. If its orthocentre is $( 2 , a ) , - \frac { 1 } { 2 } < a < 2$, then $p$ is equal to $\_\_\_\_$.
Q86 Sign Change & Interval Methods View
The number of distinct real roots of the equation $x ^ { 5 } \left( x ^ { 3 } - x ^ { 2 } - x + 1 \right) + x \left( 3 x ^ { 3 } - 4 x ^ { 2 } - 2 x + 4 \right) - 1 = 0$ is $\_\_\_\_$.
Q87 Tangents, normals and gradients Normal or perpendicular line problems View
Let the function $f ( x ) = 2 x ^ { 2 } - \log _ { e } x , x > 0$, be decreasing in $( 0 , a )$ and increasing in $( a , 4 )$. A tangent to the parabola $y ^ { 2 } = 4 a x$ at a point $P$ on it passes through the point $( 8 a , 8 a - 1 )$ but does not pass through the point $\left( - \frac { 1 } { a } , 0 \right)$. If the equation of the normal at $P$ is $\frac { x } { \alpha } + \frac { y } { \beta } = 1$, then $\alpha + \beta$ is equal to $\_\_\_\_$.
Q88 Indefinite & Definite Integrals Maximizing or Optimizing a Definite Integral View
If $n ( 2 n + 1 ) \int _ { 0 } ^ { 1 } \left( 1 - x ^ { n } \right) ^ { 2 n } d x = 1177 \int _ { 0 } ^ { 1 } \left( 1 - x ^ { n } \right) ^ { 2 n + 1 } d x$, then $n \in N$ is equal to $\_\_\_\_$.
Q89 Differential equations Finding a DE from a Limit or Implicit Condition View
Let a curve $y = y ( x )$ pass through the point $( 3,3 )$ and the area of the region under this curve, above the $x$-axis and between the abscissae 3 and $x ( > 3 )$ be $\left( \frac { y } { x } \right) ^ { 3 }$. If this curve also passes through the point $( \alpha , 6 \sqrt { 10 } )$ in the first quadrant, then $\alpha$ is equal to $\_\_\_\_$.
Q90 Vectors 3D & Lines Distance from a Point to a Line (Show/Compute) View
Let $Q$ and $R$ be two points on the line $\frac { x + 1 } { 2 } = \frac { y + 2 } { 3 } = \frac { z - 1 } { 2 }$ at a distance $\sqrt { 26 }$ from the point $P ( 4,2,7 )$. Then the square of the area of the triangle $PQR$ is $\_\_\_\_$.