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
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2020
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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
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2016
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2015
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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 session4_27aug_shift2

28 maths questions

Q61 Discriminant and conditions for roots Parameter range for specific root conditions (location/count) View
The set of all values of $k > - 1$, for which the equation $\left( 3 x ^ { 2 } + 4 x + 3 \right) ^ { 2 } - ( k + 1 ) \left( 3 x ^ { 2 } + 4 x + 3 \right) \left( 3 x ^ { 2 } + 4 x + 2 \right) + k \left( 3 x ^ { 2 } + 4 x + 2 \right) ^ { 2 } = 0$ has real roots, is: (1) $\left[ - \frac { 1 } { 2 } , 1 \right)$ (2) $\left( 1 , \frac { 5 } { 2 } \right]$ (3) $\left( \frac { 1 } { 2 } , \frac { 3 } { 2 } \right] - \{ 1 \}$ (4) $[ 2,3 )$
Q62 Sequences and Series Evaluation of a Finite or Infinite Sum View
If $0 < x < 1$ and $y = \frac { 1 } { 2 } x ^ { 2 } + \frac { 2 } { 3 } x ^ { 3 } + \frac { 3 } { 4 } x ^ { 4 } + \ldots \ldots$, then the value of $e ^ { 1 + y }$ at $x = \frac { 1 } { 2 }$ is: (1) $\frac { 1 } { 2 } e ^ { 2 }$ (2) $2 e$ (3) $2 e ^ { 2 }$ (4) $\frac { 1 } { 2 } \sqrt { \mathrm { e } }$
Q63 Straight Lines & Coordinate Geometry Area Computation in Coordinate Geometry View
Let $A ( a , 0 ) , B ( b , 2 b + 1 )$ and $C ( 0 , b ) , b \neq 0 , | b | \neq 1$, be points such that the area of triangle $A B C$ is 1 sq. unit, then the sum of all possible values of $a$ is: (1) $\frac { - 2 b } { b + 1 }$ (2) $\frac { 2 b ^ { 2 } } { b + 1 }$ (3) $\frac { - 2 b ^ { 2 } } { b + 1 }$ (4) $\frac { 2 b } { b + 1 }$
Q64 Circles Circle-Related Locus Problems View
If two tangents drawn from a point $P$ to the parabola $y ^ { 2 } = 16 ( x - 3 )$ are at right angles, then the locus of point $P$ is: (1) $x + 4 = 0$ (2) $x + 2 = 0$ (3) $x + 3 = 0$ (4) $x + 1 = 0$
Q65 Curve Sketching Limit Computation from Algebraic Expressions View
If $\lim _ { x \rightarrow \infty } \left( \sqrt { x ^ { 2 } - x + 1 } - a x \right) = b$, then the ordered pair $( a , b )$ is: (1) $\left( 1 , - \frac { 1 } { 2 } \right)$ (2) $\left( - 1 , \frac { 1 } { 2 } \right)$ (3) $\left( - 1 , - \frac { 1 } { 2 } \right)$ (4) $\left( 1 , \frac { 1 } { 2 } \right)$
Q67 Standard trigonometric equations Evaluate trigonometric expression given a constraint View
Two poles $A B$ of length $a$ metres and $C D$ of length $a + b ( b \neq a )$ metres are erected at the same horizontal level with bases at $B$ and $D$. If $B D = x$ and $\tan \angle A C B = \frac { 1 } { 2 }$, then: (1) $x ^ { 2 } + 2 ( a + 2 b ) x - b ( a + b ) = 0$ (2) $x ^ { 2 } + 2 ( a + 2 b ) x + a ( a + b ) = 0$ (3) $x ^ { 2 } - 2 a x + b ( a + b ) = 0$ (4) $x ^ { 2 } - 2 a x + a ( a + b ) = 0$
Q68 Combinations & Selection Subset Counting with Set-Theoretic Conditions View
Let $Z$ be the set of all integers, $A = \left\{ ( x , y ) \in Z \times Z : ( x - 2 ) ^ { 2 } + y ^ { 2 } \leq 4 \right\}$ $B = \left\{ ( x , y ) \in Z \times Z : x ^ { 2 } + y ^ { 2 } \leq 4 \right\}$ and $C = \left\{ ( x , y ) \in Z \times Z : ( x - 2 ) ^ { 2 } + ( y - 2 ) ^ { 2 } \leq 4 \right\}$ If the total number of relations from $A \cap B$ to $A \cap C$ is $2 ^ { p }$, then the value of $p$ is: (1) 25 (2) 9 (3) 16 (4) 49
Q69 Simultaneous equations View
Let $[ \lambda ]$ be the greatest integer less than or equal to $\lambda$. The set of all values of $\lambda$ for which the system of linear equations $x + y + z = 4,3 x + 2 y + 5 z = 3,9 x + 4 y + ( 28 + [ \lambda ] ) z = [ \lambda ]$ has a solution is: (1) $R$ (2) $( - \infty , - 9 ) \cup [ - 8 , \infty )$ (3) $( - \infty , - 9 ) \cup ( - 9 , \infty )$ (4) $[ - 9 , - 8 )$
Q70 3x3 Matrices Determinant of Parametric or Structured Matrix View
Let $A = \left[ \begin{array} { c c c } { [ x + 1 ] } & { [ x + 2 ] } & { [ x + 3 ] } \\ { [ x ] } & { [ x + 3 ] } & { [ x + 3 ] } \\ { [ x ] } & { [ x + 2 ] } & { [ x + 4 ] } \end{array} \right]$, where $[ x ]$ denotes the greatest integer less than or equal to $x$. If $\operatorname { det } ( A ) = 192$, then the set of values of $x$ is in the interval: (1) $[ 62,63 )$ (2) $[ 65,66 )$ (3) $[ 60,61 )$ (4) $[ 68,69 )$
Q71 Differentiating Transcendental Functions Evaluate derivative at a point or find tangent slope View
If $y ( x ) = \cot ^ { - 1 } \left( \frac { \sqrt { 1 + \sin x } + \sqrt { 1 - \sin x } } { \sqrt { 1 + \sin x } - \sqrt { 1 - \sin x } } \right) , x \in \left( \frac { \pi } { 2 } , \pi \right)$, then $\frac { d y } { d x }$ at $x = \frac { 5 \pi } { 6 }$ is: (1) 0 (2) - 1 (3) $\frac { - 1 } { 2 }$ (4) $\frac { 1 } { 2 }$
Q72 Stationary points and optimisation Geometric or applied optimisation problem View
A box open from top is made from a rectangular sheet of dimension $a \times b$ by cutting squares each of side $x$ from each of the four corners and folding up the flaps. If the volume of the box is maximum, then $x$ is equal to: (1) $\frac { a + b + \sqrt { a ^ { 2 } + b ^ { 2 } - a b } } { 6 }$ (2) $\frac { a + b - \sqrt { a ^ { 2 } + b ^ { 2 } - a b } } { 12 }$ (3) $\frac { a + b - \sqrt { a ^ { 2 } + b ^ { 2 } - a b } } { 6 }$ (4) $\frac { a + b - \sqrt { a ^ { 2 } + b ^ { 2 } + a b } } { 6 }$
Q73 Stationary points and optimisation Find absolute extrema on a closed interval or domain View
Let $M$ and $m$ respectively be the maximum and minimum values of the function $f ( x ) = \tan ^ { - 1 } ( \sin x + \cos x )$ in $\left[ 0 , \frac { \pi } { 2 } \right]$. Then the value of $\tan ( M - m )$ is equal to: (1) $2 - \sqrt { 3 }$ (2) $3 - 2 \sqrt { 2 }$ (3) $3 + 2 \sqrt { 2 }$ (4) $2 + \sqrt { 3 }$
Q74 Integration by Substitution Substitution to Evaluate a Definite Integral (Numerical Answer) View
The value of the integral $\int _ { 0 } ^ { 1 } \frac { \sqrt { x } d x } { ( 1 + x ) ( 1 + 3 x ) ( 3 + x ) }$ is: (1) $\frac { \pi } { 4 } \left( 1 - \frac { \sqrt { 3 } } { 2 } \right)$ (2) $\frac { \pi } { 8 } \left( 1 - \frac { \sqrt { 3 } } { 6 } \right)$ (3) $\frac { \pi } { 8 } \left( 1 - \frac { \sqrt { 3 } } { 2 } \right)$ (4) $\frac { \pi } { 4 } \left( 1 - \frac { \sqrt { 3 } } { 6 } \right)$
Q75 Areas by integration View
The area of the region bounded by the parabola $( y - 2 ) ^ { 2 } = ( x - 1 )$, the tangent to it at the point whose ordinate is 3 and the $x$-axis, is: (1) 4 (2) 6 (3) 9 (4) 10
Q76 Differential equations First-Order Linear DE: General Solution View
If the solution curve of the differential equation $\left( 2 x - 10 y ^ { 3 } \right) d y + y d x = 0$, passes through the points $( 0,1 )$ and $( 2 , \beta )$, then $\beta$ is a root of the equation? (1) $y ^ { 5 } - 2 y - 2 = 0$ (2) $y ^ { 5 } - y ^ { 2 } - 1 = 0$ (3) $2 y ^ { 5 } - y ^ { 2 } - 2 = 0$ (4) $2 y ^ { 5 } - 2 y - 1 = 0$
Q77 Differential equations Higher-Order and Special DEs (Proof/Theory) View
A differential equation representing the family of parabolas with axis parallel to $y$-axis and whose length of latus rectum is the distance of the point $( 2 , - 3 )$ from the line $3 x + 4 y = 5$, is given by: (1) $11 \frac { d ^ { 2 } x } { d y ^ { 2 } } = 10$ (2) $11 \frac { d ^ { 2 } y } { d x ^ { 2 } } = 10$ (3) $10 \frac { d ^ { 2 } y } { d x ^ { 2 } } = 11$ (4) $10 \frac { d ^ { 2 } x } { d y ^ { 2 } } = 11$
Q78 Vectors: Lines & Planes Find Cartesian Equation of a Plane View
The equation of the plane passing through the line of intersection of the planes $\vec { r } \cdot ( \hat { i } + \hat { j } + \widehat { k } ) = 1$ and $\vec { r } \cdot ( 2 \hat { i } + 3 \hat { j } - \hat { k } ) + 4 = 0$ and parallel to the $x$-axis, is (1) $\vec { r } \cdot ( \hat { i } + 3 \widehat { k } ) + 6 = 0$ (2) $\vec { r } \cdot ( \hat { i } - 3 \widehat { k } ) + 6 = 0$ (3) $\vec { r } \cdot ( \hat { j } - 3 \widehat { k } ) - 6 = 0$ (4) $\vec { r } \cdot ( \hat { j } - 3 \widehat { k } ) + 6 = 0$
Q79 Vectors 3D & Lines MCQ: Relationship Between Two Lines View
The angle between the straight lines, whose direction cosines $l , m , n$ are given by the equations $2 l + 2 m - n = 0$ and $m n + n l + \operatorname { lm } = 0$, is: (1) $\frac { \pi } { 3 }$ (2) $\frac { \pi } { 2 }$ (3) $\cos ^ { - 1 } \left( \frac { 8 } { 9 } \right)$ (4) $\pi - \cos ^ { - 1 } \left( \frac { 4 } { 9 } \right)$
Q80 Binomial Distribution Compute Exact Binomial Probability View
Each of the persons $A$ and $B$ independently tosses three fair coins. The probability that both of them get the same number of heads is: (1) $\frac { 5 } { 8 }$ (2) $\frac { 1 } { 8 }$ (3) $\frac { 5 } { 16 }$ (4) 1
Q81 Complex Numbers Argand & Loci Locus Identification from Modulus/Argument Equation View
Let $z _ { 1 }$ and $z _ { 2 }$ be two complex numbers such that $\arg \left( z _ { 1 } - z _ { 2 } \right) = \frac { \pi } { 4 }$ and $z _ { 1 } , z _ { 2 }$ satisfy the equation $| z - 3 | = \operatorname { Re } ( z )$. Then the imaginary part $z _ { 1 } + z _ { 2 }$ is equal to
Q82 Combinations & Selection Selection with Arithmetic or Divisibility Conditions View
Let $S = \{ 1,2,3,4,5,6,9 \}$. Then the number of elements in the set $T = \{ A \subseteq S : A \neq \phi$ and the sum of all the elements of $A$ is not a multiple of $3 \}$ is
Q84 Standard trigonometric equations Solve trigonometric equation for solutions in an interval View
Let $S$ be the sum of all solutions (in radians) of the equation $\sin ^ { 4 } \theta + \cos ^ { 4 } \theta - \sin \theta \cos \theta = 0$ in $[ 0,4 \pi ]$ then $\frac { 8 S } { \pi }$ is equal to
Q85 Circles Circles Tangent to Each Other or to Axes View
Two circles each of radius 5 units touch each other at the point $( 1,2 )$. If the equation of their common tangent is $4 x + 3 y = 10$, and $C _ { 1 } ( \alpha , \beta )$ and $C _ { 2 } ( \gamma , \delta ) , C _ { 1 } \neq C _ { 2 }$ are their centres, then $| ( \alpha + \beta ) ( \gamma + \delta ) |$ is equal to
Q86 Conic sections Tangent and Normal Line Problems View
Let $P ( a \sec \theta , b \tan \theta )$ and $Q ( a \sec \phi , b \tan \phi )$ where $\theta + \phi = \frac { \pi } { 2 }$, be two points on the hyperbola $\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$. If the ordinate of the point of intersection of normals at $P$ and $Q$ is $- k \left( \frac { a ^ { 2 } + b ^ { 2 } } { 2 b } \right)$, then $k$ is equal to
Q87 Measures of Location and Spread View
An online exam is attempted by 50 candidates out of which 20 are boys. The average marks obtained by boys is 12 with a variance 2 . The variance of marks obtained by 30 girls is also 2 . The average marks of all 50 candidates is 15 . If $\mu$ is the average marks of girls and $\sigma ^ { 2 }$ is the variance of marks of 50 candidates, then $\mu + \sigma ^ { 2 }$ is equal to
Q88 Integration by Substitution Substitution to Compute an Indefinite Integral with Initial Condition View
$\int \frac { 2 e ^ { x } + 3 e ^ { - x } } { 4 e ^ { x } + 7 e ^ { - x } } d x = \frac { 1 } { 14 } \left( u x + v \log _ { e } \left( 4 e ^ { x } + 7 e ^ { - x } \right) \right) + C$, where $C$ is a constant of integration, then $u + v$ is equal to
Q89 Vectors 3D & Lines Distance from a Point to a Line (Show/Compute) View
Let $S$ be the mirror image of the point $Q ( 1,3,4 )$ with respect to the plane $2 x - y + z + 3 = 0$ and let $R ( 3,5 , \gamma )$ be a point of this plane. Then the square of the length of the line segment $S R$ is
Q90 Discrete Probability Distributions Probability Distribution Table Completion and Expectation Calculation View
The probability distribution of random variable $X$ is given by:
$X$12345
$P ( X )$$K$$2 K$$2 K$$3 K$$K$

Let $p = P ( 1 < X < 4 \mid X < 3 )$. If $5 p = \lambda K$, then $\lambda$ is equal to