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
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
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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_shift1

30 maths questions

Q61 Laws of Logarithms Solve a Logarithmic Equation View
The sum of all the solutions of the equation $( 8 ) ^ { 2 x } - 16 \cdot ( 8 ) ^ { x } + 48 = 0$ is :
(1) $1 + \log _ { 8 } ( 6 )$
(2) $1 + \log _ { 6 } ( 8 )$
(3) $\log _ { 8 } ( 6 )$
(4) $\log _ { 8 } ( 4 )$
Q62 Complex Numbers Argand & Loci Circle Equation and Properties via Complex Number Manipulation View
Let $z$ be a complex number such that $| z + 2 | = 1$ and $\operatorname { Im } \left( \frac { z + 1 } { z + 2 } \right) = \frac { 1 } { 5 }$. Then the value of $| \operatorname { Re } ( \overline { z + 2 } ) |$ is
(1) $\frac { 2 \sqrt { 6 } } { 5 }$
(2) $\frac { 24 } { 5 }$
(3) $\frac { 1 + \sqrt { 6 } } { 5 }$
(4) $\frac { \sqrt { 6 } } { 5 }$
Q63 Complex Numbers Arithmetic Powers of i or Complex Number Integer Powers View
If the set $R = \{ ( a , b ) : a + 5 b = 42 , a , b \in \mathbb { N } \}$ has $m$ elements and $\sum _ { n = 1 } ^ { m } \left( 1 - i ^ { n ! } \right) = x + i y$, where $i = \sqrt { - 1 }$, then the value of $m + x + y$ is
(1) 12
(2) 4
(3) 8
(4) 5
Q64 Trig Proofs Trigonometric Equation Constraint Deduction View
If $\sin x = - \frac { 3 } { 5 }$, where $\pi < x < \frac { 3 \pi } { 2 }$, then $80 \left( \tan ^ { 2 } x - \cos x \right)$ is equal to
(1) 108
(2) 109
(3) 18
(4) 19
Q65 Straight Lines & Coordinate Geometry Line Equation and Parametric Representation View
The equations of two sides AB and AC of a triangle ABC are $4 x + y = 14$ and $3 x - 2 y = 5$, respectively. The point $\left( 2 , - \frac { 4 } { 3 } \right)$ divides the third side BC internally in the ratio $2 : 1$. the equation of the side BC is
(1) $x + 3 y + 2 = 0$
(2) $x - 6 y - 10 = 0$
(3) $x - 3 y - 6 = 0$
(4) $x + 6 y + 6 = 0$
Q66 Circles Circles Tangent to Each Other or to Axes View
Let the circles $C _ { 1 } : ( x - \alpha ) ^ { 2 } + ( y - \beta ) ^ { 2 } = r _ { 1 } ^ { 2 }$ and $C _ { 2 } : ( x - 8 ) ^ { 2 } + \left( y - \frac { 15 } { 2 } \right) ^ { 2 } = r _ { 2 } ^ { 2 }$ touch each other externally at the point $( 6,6 )$. If the point $( 6,6 )$ divides the line segment joining the centres of the circles $C _ { 1 }$ and $C _ { 2 }$ internally in the ratio $2 : 1$, then $( \alpha + \beta ) + 4 \left( r _ { 1 } ^ { 2 } + r _ { 2 } ^ { 2 } \right)$ equals
(1) 125
(2) 130
(3) 110
(4) 145
Q67 Conic sections Focal Distance and Point-on-Conic Metric Computation View
Let $H : \frac { - x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ be the hyperbola, whose eccentricity is $\sqrt { 3 }$ and the length of the latus rectum is $4 \sqrt { 3 }$. Suppose the point $( \alpha , 6 ) , \alpha > 0$ lies on $H$. If $\beta$ is the product of the focal distances of the point $( \alpha , 6 )$, then $\alpha ^ { 2 } + \beta$ is equal to
(1) 172
(2) 171
(3) 169
(4) 170
Q68 3x3 Matrices Matrix Algebraic Properties and Abstract Reasoning View
Let $A = \left[ \begin{array} { l l l } 2 & a & 0 \\ 1 & 3 & 1 \\ 0 & 5 & b \end{array} \right]$. If $A ^ { 3 } = 4 A ^ { 2 } - A - 21 I$, where $I$ is the identity matrix of order $3 \times 3$, then $2 a + 3 b$ is equal to
(1) $- 9$
(2) $- 13$
(3) $- 10$
(4) $- 12$
Q69 Permutations & Arrangements Counting Functions with Constraints View
Let $[ t ]$ be the greatest integer less than or equal to $t$. Let $A$ be the set of all prime factors of 2310 and $f : A \rightarrow \mathbb { Z }$ be the function $f ( x ) = \left[ \log _ { 2 } \left( x ^ { 2 } + \left[ \frac { x ^ { 3 } } { 5 } \right] \right) \right]$. The number of one-to-one functions from $A$ to the range of $f$ is
(1) 25
(2) 24
(3) 20
(4) 120
Q70 Stationary points and optimisation Determine intervals of increase/decrease or monotonicity conditions View
For the function $f ( x ) = ( \cos x ) - x + 1 , x \in \mathbb { R }$, between the following two statements (S1) $f ( x ) = 0$ for only one value of $x$ in $[ 0 , \pi ]$. (S2) $f ( x )$ is decreasing in $\left[ 0 , \frac { \pi } { 2 } \right]$ and increasing in $\left[ \frac { \pi } { 2 } , \pi \right]$.
(1) Both (S1) and (S2) are correct.
(2) Both (S1) and (S2) are incorrect.
(3) Only (S2) is correct.
(4) Only (S1) is correct.
Q71 Stationary points and optimisation Find critical points and classify extrema of a given function View
Let $f ( x ) = 4 \cos ^ { 3 } x + 3 \sqrt { 3 } \cos ^ { 2 } x - 10$. The number of points of local maxima of $f$ in interval $( 0,2 \pi )$ is
(1) 3
(2) 4
(3) 1
(4) 2
Q72 Stationary points and optimisation Find critical points and classify extrema of a given function View
The number of critical points of the function $f ( x ) = ( x - 2 ) ^ { 2 / 3 } ( 2 x + 1 )$ is
(1) 1
(2) 2
(3) 0
(4) 3
Q73 Standard Integrals and Reverse Chain Rule Antiderivative with Initial Condition View
Let $I ( x ) = \int \frac { 6 } { \sin ^ { 2 } x ( 1 - \cot x ) ^ { 2 } } d x$. If $I ( 0 ) = 3$, then $I \left( \frac { \pi } { 12 } \right)$ is equal to
(1) $2 \sqrt { 3 }$
(2) $\sqrt { 3 }$
(3) $3 \sqrt { 3 }$
(4) $6 \sqrt { 3 }$
Q74 Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences View
The value of $k \in \mathrm {~N}$ for which the integral $I _ { n } = \int _ { 0 } ^ { 1 } \left( 1 - x ^ { k } \right) ^ { n } d x , n \in \mathbb { N }$, satisfies $147 I _ { 20 } = 148 I _ { 21 }$ is
(1) 14
(2) 8
(3) 10
(4) 7
Q75 Differential equations Higher-Order and Special DEs (Proof/Theory) View
Let $f ( x )$ be a positive function such that the area bounded by $y = f ( x ) , y = 0$ from $x = 0$ to $x = a > 0$ is $e ^ { - a } + 4 a ^ { 2 } + a - 1$. Then the differential equation, whose general solution is $y = c _ { 1 } f ( x ) + c _ { 2 }$, where $c _ { 1 }$ and $c _ { 2 }$ are arbitrary constants, is
(1) $\left( 8 e ^ { x } - 1 \right) \frac { d ^ { 2 } y } { d x ^ { 2 } } + \frac { d y } { d x } = 0$
(2) $\left( 8 e ^ { x } - 1 \right) \frac { d ^ { 2 } y } { d x ^ { 2 } } - \frac { d y } { d x } = 0$
(3) $\left( 8 e ^ { x } + 1 \right) \frac { d ^ { 2 } y } { d x ^ { 2 } } - \frac { d y } { d x } = 0$
(4) $\left( 8 e ^ { x } + 1 \right) \frac { d ^ { 2 } y } { d x ^ { 2 } } + \frac { d y } { d x } = 0$
Q76 First order differential equations (integrating factor) View
Let $y = y ( x )$ be the solution of the differential equation $\left( 1 + y ^ { 2 } \right) e ^ { \tan x } d x + \cos ^ { 2 } x \left( 1 + e ^ { 2 \tan x } \right) d y = 0 , y ( 0 ) = 1$. Then $y \left( \frac { \pi } { 4 } \right)$ is equal to
(1) $\frac { 2 } { e }$
(2) $\frac { 2 } { e ^ { 2 } }$
(3) $\frac { 1 } { e }$
(4) $\frac { 1 } { e ^ { 2 } }$
Q77 Vectors Introduction & 2D Angle or Cosine Between Vectors View
The set of all $\alpha$, for which the vectors $\vec { a } = \alpha t \hat { i } + 6 \hat { j } - 3 \hat { k }$ and $\vec { b } = t \hat { i } - 2 \hat { j } - 2 \alpha t \hat { k }$ are inclined at an obtuse angle for all $t \in \mathbb { R }$, is
(1) $\left( - \frac { 4 } { 3 } , 1 \right)$
(2) $[ 0,1 )$
(3) $\left( - \frac { 4 } { 3 } , 0 \right]$
(4) $( - 2,0 ]$
Q78 Vectors: Lines & Planes Distance Computation (Point-to-Plane or Line-to-Line) View
If the shortest distance between the lines $\begin{aligned} & L _ { 1 } : \vec { r } = ( 2 + \lambda ) \hat { i } + ( 1 - 3 \lambda ) \hat { j } + ( 3 + 4 \lambda ) \hat { k } , \quad \lambda \in \mathbb { R } \\ & L _ { 2 } : \vec { r } = 2 ( 1 + \mu ) \hat { i } + 3 ( 1 + \mu ) \hat { j } + ( 5 + \mu ) \hat { k } , \quad \mu \in \mathbb { R } \end{aligned}$ is $\frac { m } { \sqrt { n } }$, where $\operatorname { gcd } ( m , n ) = 1$, then the value of $m + n$ equals
(1) 390
(2) 384
(3) 377
(4) 387
Q79 Vectors Introduction & 2D Angle or Cosine Between Vectors View
Let $P ( x , y , z )$ be a point in the first octant, whose projection in the $x y$-plane is the point $Q$. Let $O P = \gamma$; the angle between $O Q$ and the positive $x$-axis be $\theta$; and the angle between $O P$ and the positive $z$-axis be $\phi$, where $O$ is the origin. Then the distance of $P$ from the $x$-axis is
(1) $\gamma \sqrt { 1 - \sin ^ { 2 } \phi \cos ^ { 2 } \theta }$
(2) $\gamma \sqrt { 1 - \sin ^ { 2 } \theta \cos ^ { 2 } \phi }$
(3) $\gamma \sqrt { 1 + \cos ^ { 2 } \phi \sin ^ { 2 } \theta }$
(4) $\gamma \sqrt { 1 + \cos ^ { 2 } \theta \sin ^ { 2 } \phi }$
Q80 Probability Definitions Probability Involving Algebraic or Number-Theoretic Conditions View
Let the sum of two positive integers be 24. If the probability, that their product is not less than $\frac { 3 } { 4 }$ times their greatest possible product, is $\frac { m } { n }$, where $\operatorname { gcd } ( m , n ) = 1$, then $n - m$ equals
(1) 10
(2) 9
(3) 11
(4) 8
Q81 Permutations & Arrangements Forming Numbers with Digit Constraints View
The number of 3-digit numbers, formed using the digits $2,3,4,5$ and 7, when the repetition of digits is not allowed, and which are not divisible by 3, is equal to $\_\_\_\_$
Q82 Arithmetic Sequences and Series Counting or Combinatorial Problems on APs View
Let the positive integers be written in the form : If the $k ^ { \text {th} }$ row contains exactly $k$ numbers for every natural number $k$, then the row in which the number 5310 will be, is $\_\_\_\_$
Q83 Binomial Theorem (positive integer n) Evaluate a Summation Involving Binomial Coefficients View
Let $\alpha = \sum _ { r = 0 } ^ { n } \left( 4 r ^ { 2 } + 2 r + 1 \right) ^ { n } C _ { r }$ and $\beta = \left( \sum _ { r = 0 } ^ { n } \frac { { } ^ { n } C _ { r } } { r + 1 } \right) + \frac { 1 } { n + 1 }$. If $140 < \frac { 2 \alpha } { \beta } < 281$, then the value of $n$ is $\_\_\_\_$
Q84 Straight Lines & Coordinate Geometry Triangle Properties and Special Points View
If the orthocentre of the triangle formed by the lines $2 x + 3 y - 1 = 0 , x + 2 y - 1 = 0$ and $a x + b y - 1 = 0$, is the centroid of another triangle, whose circumcentre and orthocentre respectively are $( 3,4 )$ and $( - 6 , - 8 )$, then the value of $| a - b |$ is $\_\_\_\_$
Q85 Chain Rule Limit Evaluation Involving Composition or Substitution View
The value of $\lim _ { x \rightarrow 0 } 2 \left( \frac { 1 - \cos x \sqrt { \cos 2 x } \sqrt [ 3 ] { \cos 3 x } \ldots \ldots \sqrt [ 10 ] { \cos 10 x } } { x ^ { 2 } } \right)$ is $\_\_\_\_$
Q86 Matrices Matrix Power Computation and Application View
Let $A = \left[ \begin{array} { c c } 2 & - 1 \\ 1 & 1 \end{array} \right]$. If the sum of the diagonal elements of $A ^ { 13 }$ is $3 ^ { n }$, then $n$ is equal to $\_\_\_\_$
Q87 Geometric Sequences and Series Sum of an Infinite Geometric Series (Direct Computation) View
If the range of $f ( \theta ) = \frac { \sin ^ { 4 } \theta + 3 \cos ^ { 2 } \theta } { \sin ^ { 4 } \theta + \cos ^ { 2 } \theta } , \theta \in \mathbb { R }$ is $[ \alpha , \beta ]$, then the sum of the infinite G.P., whose first term is 64 and the common ratio is $\frac { \alpha } { \beta }$, is equal to $\_\_\_\_$
Q88 Areas by integration View
Let the area of the region enclosed by the curve $y = \min \{ \sin x , \cos x \}$ and the $x$ axis between $x = - \pi$ to $x = \pi$ be $A$. Then $A ^ { 2 }$ is equal to $\_\_\_\_$
Q89 Vectors Introduction & 2D Magnitude of Vector Expression View
Let $\vec { a } = 9 \hat { i } - 13 \hat { j } + 25 \hat { k } , \vec { b } = 3 \hat { i } + 7 \hat { j } - 13 \hat { k }$ and $\vec { c } = 17 \hat { i } - 2 \hat { j } + \hat { k }$ be three given vectors. If $\vec { r }$ is a vector such that $\vec { r } \times \vec { a } = ( \vec { b } + \vec { c } ) \times \vec { a }$ and $\vec { r } \cdot ( \vec { b } - \vec { c } ) = 0$, then $\frac { | 593 \vec { r } + 67 \vec { a } | ^ { 2 } } { ( 593 ) ^ { 2 } }$ is equal to $\_\_\_\_$
Q90 Discrete Probability Distributions Expectation and Variance from Context-Based Random Variables View
Three balls are drawn at random from a bag containing 5 blue and 4 yellow balls. Let the random variables $X$ and $Y$ respectively denote the number of blue and yellow balls. If $\bar { X }$ and $\bar { Y }$ are the means of $X$ and $Y$ respectively, then $7 \bar { X } + 4 \bar { Y }$ is equal to $\_\_\_\_$