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
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 session2_08apr_shift2

30 maths questions

Q61 Complex Numbers Arithmetic Identifying Real/Imaginary Parts or Components View
The sum of all possible values of $\theta \in [ - \pi , 2 \pi ]$, for which $\frac { 1 + i \cos \theta } { 1 - 2 i \cos \theta }$ is purely imaginary, is equal (1) $3 \pi$ (2) $2 \pi$ (3) $5 \pi$ (4) $4 \pi$
Q62 Combinations & Selection Selection with Group/Category Constraints View
The number of ways five alphabets can be chosen from the alphabets of the word MATHEMATICS, where the chosen alphabets are not necessarily distinct, is equal to : (1) 179 (2) 177 (3) 181 (4) 175
Q63 Geometric Sequences and Series Finite Geometric Sum and Term Relationships View
In an increasing geometric progression of positive terms, the sum of the second and sixth terms is $\frac { 70 } { 3 }$ and the product of the third and fifth terms is 49 . Then the sum of the $4 ^ { \text {th} } , 6 ^ { \text {th} }$ and $8 ^ { \text {th} }$ terms is equal to : (1) 96 (2) 91 (3) 84 (4) 78
Q64 Binomial Theorem (positive integer n) Determine Parameters from Conditions on Coefficients or Terms View
If the term independent of $x$ in the expansion of $\left( \sqrt { \mathrm { a } } x ^ { 2 } + \frac { 1 } { 2 x ^ { 3 } } \right) ^ { 10 }$ is 105 , then $\mathrm { a } ^ { 2 }$ is equal to : (1) 2 (2) 4 (3) 6 (4) 9
Q65 Reciprocal Trig & Identities View
If the value of $\frac { 3 \cos 36 ^ { \circ } + 5 \sin 18 ^ { \circ } } { 5 \cos 36 ^ { \circ } - 3 \sin 18 ^ { \circ } }$ is $\frac { a \sqrt { 5 } - b } { c }$, where $a , b , c$ are natural numbers and $\operatorname { gcd } ( a , c ) = 1$, then $a + b + c$ is equal to : (1) 40 (2) 52 (3) 50 (4) 54
Q66 Circles Circle Equation Derivation View
If the image of the point $( - 4,5 )$ in the line $x + 2 y = 2$ lies on the circle $( x + 4 ) ^ { 2 } + ( y - 3 ) ^ { 2 } = r ^ { 2 }$, then r is equal to: (1) 2 (2) 3 (3) 1 (4) 4
Q67 Straight Lines & Coordinate Geometry Slope and Angle Between Lines View
If the line segment joining the points $( 5,2 )$ and $( 2 , a )$ subtends an angle $\frac { \pi } { 4 }$ at the origin, then the absolute value of the product of all possible values of $a$ is : (1) 6 (2) 8 (3) 2 (4) - 4
Q68 Probability Definitions Finite Equally-Likely Probability Computation View
Let $A = \{ 2,3,6,8,9,11 \}$ and $B = \{ 1,4,5,10,15 \}$. Let $R$ be a relation on $A \times B$ defined by ( $a , b$ ) $R ( c , d )$ if and only if $3 a d - 7 b c$ is an even integer. Then the relation $R$ is (1) an equivalence relation. (2) reflexive and symmetric but not transitive. (3) transitive but not symmetric. (4) reflexive but not symmetric.
Q69 Matrices Determinant and Rank Computation View
If $\alpha \neq \mathrm { a } , \beta \neq \mathrm { b } , \gamma \neq \mathrm { c }$ and $\left| \begin{array} { c c c } \alpha & \mathrm { b } & \mathrm { c } \\ \mathrm { a } & \beta & \mathrm { c } \\ \mathrm { a } & \mathrm { b } & \gamma \end{array} \right| = 0$, then $\frac { \mathrm { a } } { \alpha - \mathrm { a } } + \frac { \mathrm { b } } { \beta - \mathrm { b } } + \frac { \gamma } { \gamma - \mathrm { c } }$ is equal to: (1) 3 (2) 0 (3) 1 (4) 2
Q70 Simultaneous equations View
If the system of equations $x + 4 y - z = \lambda , 7 x + 9 y + \mu z = - 3,5 x + y + 2 z = - 1$ has infinitely many solutions, then $( 2 \mu + 3 \lambda )$ is equal to : (1) 3 (2) - 3 (3) - 2 (4) 2
Q71 Composite & Inverse Functions Injectivity, Surjectivity, or Bijectivity Classification View
Let $f ( x ) = \left\{ \begin{array} { c c c } - \mathrm { a } & \text { if } & - \mathrm { a } \leq x \leq 0 \\ x + \mathrm { a } & \text { if } & 0 < x \leq \mathrm { a } \end{array} \right.$ where $\mathrm { a } > 0$ and $\mathrm { g } ( x ) = ( f ( x \mid ) - | f ( x ) | ) / 2$. Then the function $g : [ - a , a ] \rightarrow [ - a , a ]$ is (1) neither one-one nor onto. (2) onto. (3) both one-one and onto. (4) one-one.
Q72 Curve Sketching Finding Parameters for Continuity View
For $\mathrm { a } , \mathrm { b } > 0$, let $f ( x ) = \left\{ \begin{array} { c l } \frac { \tan ( ( \mathrm { a } + 1 ) x ) + \mathrm { b } \tan x } { x } , & x < 0 \\ 3 , & x = 0 \\ \frac { \sqrt { \mathrm { a } x + \mathrm { b } ^ { 2 } x ^ { 2 } } - \sqrt { \mathrm { a } x } } { \mathrm {~b} \sqrt { \mathrm { a } } \sqrt { x } } , & x > 0 \end{array} \right.$ be a continous function at $x = 0$. Then $\frac { \mathrm { b } } { \mathrm { a } }$ is equal to : (1) 6 (2) 4 (3) 5 (4) 8
Q73 Stationary points and optimisation Determine parameters from given extremum conditions View
If the function $f ( x ) = 2 x ^ { 3 } - 9 x ^ { 2 } + 12 \mathrm { a } ^ { 2 } x + 1 , \mathrm { a } > 0$ has a local maximum at $x = \alpha$ and a local minimum at $x = \alpha ^ { 2 }$, then $\alpha$ and $\alpha ^ { 2 }$ are the roots of the equation : (1) $x ^ { 2 } - 6 x + 8 = 0$ (2) $x ^ { 2 } + 6 x + 8 = 0$ (3) $8 x ^ { 2 } + 6 x - 1 = 0$ (4) $8 x ^ { 2 } - 6 x + 1 = 0$
Q74 Solving quadratics and applications Finding roots or coefficients of a quadratic using Vieta's relations View
Let $\int _ { \alpha } ^ { \log _ { e } 4 } \frac { \mathrm {~d} x } { \sqrt { \mathrm { e } ^ { x } - 1 } } = \frac { \pi } { 6 }$. Then $\mathrm { e } ^ { \alpha }$ and $\mathrm { e } ^ { - \alpha }$ are the roots of the equation : (1) $x ^ { 2 } + 2 x - 8 = 0$ (2) $x ^ { 2 } - 2 x - 8 = 0$ (3) $2 x ^ { 2 } - 5 x + 2 = 0$ (4) $2 x ^ { 2 } - 5 x - 2 = 0$
Q75 Areas by integration View
The area of the region in the first quadrant inside the circle $x ^ { 2 } + y ^ { 2 } = 8$ and outside the parabola $y ^ { 2 } = 2 x$ is equal to : (1) $\frac { \pi } { 2 } - \frac { 1 } { 3 }$ (2) $\pi - \frac { 1 } { 3 }$ (3) $\frac { \pi } { 2 } - \frac { 2 } { 3 }$ (4) $\pi - \frac { 2 } { 3 }$
Q76 First order differential equations (integrating factor) View
Let $y = y ( x )$ be the solution curve of the differential equation $\sec y \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 x \sin y = x ^ { 3 } \cos y , y ( 1 ) = 0$. Then $y ( \sqrt { 3 } )$ is equal to : (1) $\frac { \pi } { 3 }$ (2) $\frac { \pi } { 6 }$ (3) $\frac { \pi } { 12 }$ (4) $\frac { \pi } { 4 }$
Q77 Vectors: Cross Product & Distances View
Let $\vec { a } = 4 \hat { i } - \hat { j } + \hat { k } , \vec { b } = 11 \hat { i } - \hat { j } + \hat { k }$ and $\vec { c }$ be a vector such that $( \vec { a } + \vec { b } ) \times \vec { c } = \vec { c } \times ( - 2 \vec { a } + 3 \vec { b } )$. If $( 2 \vec { a } + 3 \vec { b } ) \cdot \vec { c } = 1670$, then $| \vec { c } | ^ { 2 }$ is equal to : (1) 1609 (2) 1618 (3) 1600 (4) 1627
Q78 Vectors 3D & Lines Vector Algebra and Triple Product Computation View
Let $\overrightarrow { \mathrm { a } } = \hat { i } + 2 \hat { j } + 3 \hat { k } , \overrightarrow { \mathrm {~b} } = 2 \hat { i } + 3 \hat { j } - 5 \hat { k }$ and $\overrightarrow { \mathrm { c } } = 3 \hat { i } - \hat { j } + \lambda \hat { k }$ be three vectors. Let $\overrightarrow { \mathrm { r } }$ be a unit vector along $\vec { b } + \vec { c }$. If $\vec { r } \cdot \vec { a } = 3$, then $3 \lambda$ is equal to: (1) 21 (2) 30 (3) 25 (4) 27
Q79 Vectors: Lines & Planes Distance Computation (Point-to-Plane or Line-to-Line) View
If the shortest distance between the lines $\frac { x - \lambda } { 2 } = \frac { y - 4 } { 3 } = \frac { z - 3 } { 4 }$ and $\frac { x - 2 } { 4 } = \frac { y - 4 } { 6 } = \frac { z - 7 } { 8 }$ is $\frac { 13 } { \sqrt { 29 } }$, then a value of $\lambda$ is : (1) - 1 (2) $- \frac { 13 } { 25 }$ (3) $\frac { 13 } { 25 }$ (4) 1
Q80 Conditional Probability Bayes' Theorem with Production/Source Identification View
There are three bags $X , Y$ and $Z$. Bag $X$ contains 5 one-rupee coins and 4 five-rupee coins; Bag $Y$ contains 4 one-rupee coins and 5 five-rupee coins and Bag Z contains 3 one-rupee coins and 6 five-rupee coins. A bag is selected at random and a coin drawn from it at random is found to be a one-rupee coin. Then the probability, that it came from bag Y , is : (1) $\frac { 1 } { 4 }$ (2) $\frac { 1 } { 2 }$ (3) $\frac { 5 } { 12 }$ (4) $\frac { 1 } { 3 }$
Q81 Modulus function Counting solutions satisfying modulus conditions View
The number of distinct real roots of the equation $| x + 1 | | x + 3 | - 4 | x + 2 | + 5 = 0$, is
Q82 Arithmetic Sequences and Series Compute Partial Sum of an Arithmetic Sequence View
An arithmetic progression is written in the following way
2
11514817
20232629

The sum of all the terms of the $10 ^ { \text {th} }$ row is $\_\_\_\_$
Q83 Straight Lines & Coordinate Geometry Reflection and Image in a Line View
Let a ray of light passing through the point $( 3,10 )$ reflects on the line $2 x + y = 6$ and the reflected ray passes through the point $( 7,2 )$. If the equation of the incident ray is $a x + b y + 1 = 0$, then $a ^ { 2 } + b ^ { 2 } + 3 a b$ is equal to $\_\_\_\_$
Q84 Circles Area and Geometric Measurement Involving Circles View
Let S be the focus of the hyperbola $\frac { x ^ { 2 } } { 3 } - \frac { y ^ { 2 } } { 5 } = 1$, on the positive $x$-axis. Let C be the circle with its centre at $A ( \sqrt { 6 } , \sqrt { 5 } )$ and passing through the point $S$. If $O$ is the origin and $S A B$ is a diameter of $C$, then the square of the area of the triangle OSB is equal to $\_\_\_\_$
Q85 Exponential Functions Limit Evaluation View
If $\alpha = \lim _ { x \rightarrow 0 ^ { + } } \left( \frac { \mathrm { e } ^ { \sqrt { \tan x } } - \mathrm { e } ^ { \sqrt { x } } } { \sqrt { \tan x } - \sqrt { x } } \right)$ and $\beta = \lim _ { x \rightarrow 0 } ( 1 + \sin x ) ^ { \frac { 1 } { 2 } \cot x }$ are the roots of the quadratic equation $a x ^ { 2 } + b x - \sqrt { \mathrm { e } } = 0$, then $12 \log _ { \mathrm { e } } ( \mathrm { a } + \mathrm { b } )$ is equal to $\_\_\_\_$
Q86 Measures of Location and Spread View
Let $\mathrm { a } , \mathrm { b } , \mathrm { c } \in \mathrm { N }$ and $\mathrm { a } < \mathrm { b } < \mathrm { c }$. Let the mean, the mean deviation about the mean and the variance of the 5 observations $9,25 , \mathrm { a } , \mathrm { b } , \mathrm { c }$ be 18,4 and $\frac { 136 } { 5 }$, respectively. Then $2 \mathrm { a } + \mathrm { b } - \mathrm { c }$ is equal to $\_\_\_\_$
Q87 Stationary points and optimisation Geometric or applied optimisation problem View
Let A be the region enclosed by the parabola $y ^ { 2 } = 2 x$ and the line $x = 24$. Then the maximum area of the rectangle inscribed in the region A is $\_\_\_\_$
Q88 Integration by Substitution Substitution to Transform Integral Form (Show Transformed Expression) View
If $\int \frac { 1 } { \sqrt [ 5 ] { ( x - 1 ) ^ { 4 } ( x + 3 ) ^ { 6 } } } \mathrm {~d} x = \mathrm { A } \left( \frac { \alpha x - 1 } { \beta x + 3 } \right) ^ { B } + \mathrm { C }$, where C is the constant of integration, then the value of $\alpha + \beta + 20 \mathrm { AB }$ is $\_\_\_\_$
Q89 Differential equations Solving Separable DEs with Initial Conditions View
Let $\alpha | x | = | y | \mathrm { e } ^ { x y - \beta } , \alpha , \beta \in \mathbf { N }$ be the solution of the differential equation $x \mathrm {~d} y - y \mathrm {~d} x + x y ( x \mathrm {~d} y + y \mathrm {~d} x ) = 0$, $y ( 1 ) = 2$. Then $\alpha + \beta$ is equal to $\_\_\_\_$
Q90 Vectors: Lines & Planes Perpendicular/Orthogonal Projection onto a Plane View
Let $\mathrm { P } ( \alpha , \beta , \gamma )$ be the image of the point $\mathrm { Q } ( 1,6,4 )$ in the line $\frac { x } { 1 } = \frac { y - 1 } { 2 } = \frac { z - 2 } { 3 }$. Then $2 \alpha + \beta + \gamma$ is equal to $\_\_\_\_$