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_27jul_shift1

26 maths questions

Q61 Complex Numbers Argand & Loci Distance and Region Optimization on Loci View
Let the minimum value $v _ { 0 }$ of $v = | z | ^ { 2 } + | z - 3 | ^ { 2 } + | z - 6 i | ^ { 2 } , z \in \mathbb { C }$ is attained at $z = z _ { 0 }$. Then $\left| 2 z _ { 0 } ^ { 2 } - \bar { z } _ { 0 } ^ { 3 } + 3 \right| ^ { 2 } + v _ { 0 } ^ { 2 }$ is equal to
(1) 1000
(2) 1024
(3) 1105
(4) 1196
Q62 Arithmetic Sequences and Series Compute Partial Sum of an Arithmetic Sequence View
Suppose $a _ { 1 } , a _ { 2 } , \ldots , a _ { \mathrm { n } } , \ldots$ be an arithmetic progression of natural numbers. If the ratio of the sum of the first five terms to the sum of first nine terms of the progression is 5 : 17 and $110 < a _ { 15 } < 120$, then the sum of the first ten terms of the progression is equal to
(1) 290
(2) 380
(3) 460
(4) 510
Q64 Straight Lines & Coordinate Geometry Area Computation in Coordinate Geometry View
Let $A ( 1,1 ) , B ( - 4,3 ) , C ( - 2 , - 5 )$ be vertices of a triangle $A B C , P$ be a point on side $B C$, and $\Delta _ { 1 }$ and $\Delta _ { 2 }$ be the areas of triangle $A P B$ and $A B C$ respectively. If $\Delta _ { 1 } : \Delta _ { 2 } = 4 : 7$, then the area enclosed by the lines $A P , A C$ and the $x$-axis is
(1) $\frac { 1 } { 4 }$
(2) $\frac { 3 } { 4 }$
(3) $\frac { 1 } { 2 }$
(4) 1
Q65 Circles Chord Length and Chord Properties View
If the circle $x ^ { 2 } + y ^ { 2 } - 2 g x + 6 y - 19 c = 0 , g , c \in \mathbb { R }$ passes through the point $( 6,1 )$ and its centre lies on the line $x - 2 c y = 8$, then the length of intercept made by the circle on $x$-axis is
(1) $\sqrt { 11 }$
(2) 4
(3) 3
(4) $2 \sqrt { 23 }$
Q66 Circles Tangent Lines and Tangent Lengths View
Let $P ( a , b )$ be a point on the parabola $y ^ { 2 } = 8 x$ such that the tangent at $P$ passes through the centre of the circle $x ^ { 2 } + y ^ { 2 } - 10 x - 14 y + 65 = 0$. Let $A$ be the product of all possible values of $a$ and $B$ be the product of all possible values of $b$. Then the value of $A + B$ is equal to
(1) 0
(2) 25
(3) 40
(4) 65
Q67 Indefinite & Definite Integrals Piecewise/Periodic Function Integration View
Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a function defined as $f ( x ) = a \sin \left( \frac { \pi [ x ] } { 2 } \right) + [ 2 - x ] , a \in \mathbb { R }$, where $[ t ]$ is the greatest integer less than or equal to $t$. If $\lim _ { x \rightarrow - 1 } f ( x )$ exists, then the value of $\int _ { 0 } ^ { 4 } f ( x ) d x$ is equal to
(1) - 1
(2) - 2
(3) 1
(4) 2
Q69 Standard trigonometric equations Evaluate trigonometric expression given a constraint View
Let a vertical tower $A B$ of height $2 h$ stands on a horizontal ground. Let from a point $P$ on the ground a man can see upto height $h$ of the tower with an angle of elevation $2 \alpha$. When from $P$, he moves a distance $d$ in the direction of $\overrightarrow { A P }$, he can see the top $B$ of the tower with an angle of elevation $\alpha$. If $d = \sqrt { 7 } h$, then $\tan \alpha$ is equal to
(1) $\sqrt { 5 } - 2$
(2) $\sqrt { 3 } - 1$
(3) $\sqrt { 7 } - 2$
(4) $\sqrt { 7 } - \sqrt { 3 }$
Q71 Matrices Matrix Algebra and Product Properties View
Let $A = \left( \begin{array} { c c } 1 & 2 \\ - 2 & - 5 \end{array} \right)$. Let $\alpha , \beta \in \mathbb { R }$ be such that $\alpha A ^ { 2 } + \beta A = 2 I$. Then $\alpha + \beta$ is equal to
(1) - 10
(2) - 6
(3) 6
(4) 10
Q73 Stationary points and optimisation Composite or piecewise function extremum analysis View
Let a function $f : \mathbb { R } \rightarrow \mathbb { R }$ be defined as: $f ( x ) = \begin{cases} \int _ { 0 } ^ { x } ( 5 - | t - 3 | ) d t , & x > 4 \\ x ^ { 2 } + b x , & x \leq 4 \end{cases}$ where $b \in \mathbb { R }$. If $f$ is continuous at $x = 4$, then which of the following statements is NOT true?
(1) $f$ is not differentiable at $x = 4$
(2) $f ^ { \prime } ( 3 ) + f ^ { \prime } ( 5 ) = \frac { 35 } { 4 }$
(3) $f$ is increasing in $\left( - \infty , \frac { 1 } { 8 } \right) \cup ( 8 , \infty )$
(4) $f$ has a local minima at $x = \frac { 1 } { 8 }$
Q74 Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences View
$I = \int _ { \frac { \pi } { 4 } } ^ { \frac { \pi } { 3 } } \left( \frac { 8 \sin x - \sin 2 x } { x } \right) d x$. Then
(1) $\frac { \pi } { 2 } < I < \frac { 3 \pi } { 4 }$
(2) $\frac { \pi } { 5 } < I < \frac { 5 \pi } { 12 }$
(3) $\frac { 5 \pi } { 12 } < I < \frac { \sqrt { 2 } } { 3 } \pi$
(4) $\frac { 3 \pi } { 4 } < I < \pi$
Q75 Areas by integration View
The area of the smaller region enclosed by the curves $y ^ { 2 } = 8 x + 4$ and $x ^ { 2 } + y ^ { 2 } + 4 \sqrt { 3 } x - 4 = 0$ is equal to
(1) $\frac { 1 } { 3 } ( 2 - 12 \sqrt { 3 } + 8 \pi )$
(2) $\frac { 1 } { 3 } ( 2 - 12 \sqrt { 3 } + 6 \pi )$
(3) $\frac { 1 } { 3 } ( 4 - 12 \sqrt { 3 } + 8 \pi )$
(4) $\frac { 1 } { 3 } ( 4 - 12 \sqrt { 3 } + 6 \pi )$
Q76 Differential equations Qualitative Analysis of DE Solutions View
Let $y = y _ { 1 } ( x )$ and $y = y _ { 2 } ( x )$ be two distinct solutions of the differential equation $\frac { d y } { d x } = x + y$, with $y _ { 1 } ( 0 ) = 0$ and $y _ { 2 } ( 0 ) = 1$ respectively. Then, the number of points of intersection of $y = y _ { 1 } ( x )$ and $y = y _ { 2 } ( x )$ is
(1) 0
(2) 1
(3) 2
(4) 3
Q77 Vectors Introduction & 2D Dot Product Computation View
Let $\vec { a } = \alpha \hat { i } + \hat { j } + \beta \hat { k }$ and $\vec { b } = 3 \hat { i } - 5 \hat { j } + 4 \hat { k }$ be two vectors, such that $\vec { a } \times \vec { b } = - \hat { i } + 9 \hat { i } + 12 \widehat { k }$. Then the projection of $\vec { b } - 2 \vec { a }$ on $\vec { b } + \vec { a }$ is equal to
(1) 2
(2) $\frac { 39 } { 5 }$
(3) 9
(4) $\frac { 46 } { 5 }$
Q78 Vectors Introduction & 2D Magnitude of Vector Expression View
Let $\vec { a } = 2 \hat { i } - \hat { j } + 5 \hat { k }$ and $\vec { b } = \alpha \hat { i } + \beta \hat { j } + 2 \widehat { k }$. If $( ( \vec { a } \times \vec { b } ) \times \hat { i } ) \cdot \widehat { k } = \frac { 23 } { 2 }$, then $| \vec { b } \times 2 \hat { j } |$ is equal to
(1) 4
(2) 5
(3) $\sqrt { 21 }$
(4) $\sqrt { 17 }$
Q79 Vectors: Lines & Planes Find Cartesian Equation of a Plane View
If the plane $P$ passes through the intersection of two mutually perpendicular planes $2 x + k y - 5 z = 1$ and $3 k x - k y + z = 5 , k < 3$ and intercepts a unit length on positive $x$-axis, then the intercept made by the plane $P$ on the $y$-axis is
(1) $\frac { 1 } { 11 }$
(2) $\frac { 5 } { 11 }$
(3) 6
(4) 7
Q80 Probability Definitions Probability Involving Algebraic or Number-Theoretic Conditions View
Let $S$ be the sample space of all five digit numbers. If $p$ is the probability that a randomly selected number from $S$, is a multiple of 7 but not divisible by 5 , then $9 p$ is equal to
(1) 1.0146
(2) 1.2085
(3) 1.0285
(4) 1.1521
Q81 Complex Numbers Arithmetic Solving Equations for Unknown Complex Numbers View
Let $S = \left\{ z \in \mathbb { C } : z ^ { 2 } + \bar { z } = 0 \right\}$. Then $\sum _ { z \in S } ( \operatorname { Re } ( z ) + \operatorname { Im } ( z ) )$ is equal to $\_\_\_\_$ .
Q82 Solving quadratics and applications Counting solutions or configurations satisfying a quadratic system View
Let $f ( x ) = 2 x ^ { 2 } - x - 1$ and $S = \{ n \in \mathbb { Z } : | f ( n ) | \leq 800 \}$. Then, the value of $\sum _ { n \in S } f ( n )$ is equal to $\_\_\_\_$ .
Q83 Circles Circle Equation Derivation View
If the length of the latus rectum of the ellipse $x ^ { 2 } + 4 y ^ { 2 } + 2 x + 8 y - \lambda = 0$ is 4 , and $l$ is the length of its major axis, then $\lambda + l$ is equal to $\_\_\_\_$ .
Q84 Conic sections Equation Determination from Geometric Conditions View
An ellipse $E : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ passes through the vertices of the hyperbola $H : \frac { x ^ { 2 } } { 49 } - \frac { y ^ { 2 } } { 64 } = - 1$. Let the major and minor axes of the ellipse $E$ coincide with the transverse and conjugate axes of the hyperbola $H$. Let the product of the eccentricities of $E$ and $H$ be $\frac { 1 } { 2 }$. If $l$ is the length of the latus rectum of the ellipse $E$, then the value of $113 l$ is equal to $\_\_\_\_$ .
Q85 Measures of Location and Spread View
The mean and variance of 10 observations were calculated as 15 and 15 respectively by a student who took by mistake 25 instead of 15 for one observation. Then, the correct standard deviation is $\_\_\_\_$.
Q86 Matrices Determinant and Rank Computation View
Let $S$ be the set containing all $3 \times 3$ matrices with entries from $\{ - 1,0,1 \}$. The total number of matrices $A \in S$ such that the sum of all the diagonal elements of $A ^ { T } A$ is 6 is $\_\_\_\_$ .
Q87 Standard trigonometric equations Inverse trigonometric equation View
For $k \in \mathbb { R }$, let the solutions of the equation $\cos \left( \sin ^ { - 1 } \left( x \cot \left( \tan ^ { - 1 } \left( \cos \left( \sin ^ { - 1 } x \right) \right) \right) \right) \right) = k , 0 < | x | < \frac { 1 } { \sqrt { 2 } }$ be $\alpha$ and $\beta$, where the inverse trigonometric functions take only principal values. If the solutions of the equation $x ^ { 2 } - b x - 5 = 0$ are $\frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } }$ and $\frac { \alpha } { \beta }$, then $\frac { b } { k ^ { 2 } }$ is equal to $\_\_\_\_$ .
Q88 Tangents, normals and gradients Find tangent line with a specified slope or from an external point View
Let $M$ and $N$ be the number of points on the curve $y ^ { 5 } - 9 x y + 2 x = 0$, where the tangents to the curve are parallel to $x$-axis and $y$-axis, respectively. Then the value of $M + N$ equals $\_\_\_\_$ .
Q89 First order differential equations (integrating factor) View
Let $y = y ( x )$ be the solution curve of the differential equation $\sin \left( 2 x ^ { 2 } \right) \log _ { e } \left( \tan x ^ { 2 } \right) d y + \left( 4 x y - 4 \sqrt { 2 } x \sin \left( x ^ { 2 } - \frac { \pi } { 4 } \right) \right) d x = 0,0 < x < \sqrt { \frac { \pi } { 2 } }$, which passes through the point $\left( \sqrt { \frac { \pi } { 6 } } , 1 \right)$. Then $\left| y \left( \sqrt { \frac { \pi } { 3 } } \right) \right|$ is equal to $\_\_\_\_$ .
Q90 Vectors: Lines & Planes Find Intersection of a Line and a Plane View
Let the line $\frac { x - 3 } { 7 } = \frac { y - 2 } { - 1 } = \frac { z - 3 } { - 4 }$ intersect the plane containing the lines $\frac { x - 4 } { 1 } = \frac { y + 1 } { - 2 } = \frac { z } { 1 }$ and $4 a x - y + 5 z - 7 a = 0 = 2 x - 5 y - z - 3 , a \in \mathbb { R }$ at the point $P ( \alpha , \beta , \gamma )$. Then the value of $\alpha + \beta + \gamma$ equals $\_\_\_\_$ .