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
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2018
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2017
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2016
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2015
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2014
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2013
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2012
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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 session3_20jul_shift2

29 maths questions

Q61 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
If the real part of the complex number $( 1 - \cos \theta + 2i \sin \theta ) ^ { - 1 }$ is $\frac { 1 } { 5 }$ for $\theta \in ( 0 , \pi )$, then the value of the integral $\int _ { 0 } ^ { \theta } \sin x \mathrm {~d} x$ is equal to:
(1) 1
(2) 2
(3) - 1
(4) 0
Q62 Laws of Logarithms Solve a Logarithmic Equation View
If sum of the first 21 terms of the series $\log _ { 9^{1/2} } x + \log _ { 9^{1/3} } x + \log _ { 9^{1/4} } x + \ldots$ where $x > 0$ is 504, then $x$ is equal to
(1) 243
(2) 9
(3) 7
(4) 81
Q63 Binomial Theorem (positive integer n) Determine Parameters from Conditions on Coefficients or Terms View
For the natural numbers $m , n$, if $( 1 - y ) ^ { m } ( 1 + y ) ^ { n } = 1 + a _ { 1 } y + a _ { 2 } y ^ { 2 } + \ldots + a _ { m + n } y ^ { m + n }$ and $a _ { 1 } = a _ { 2 } = 10$, then the value of $m + n$, is equal to:
(1) 88
(2) 64
(3) 100
(4) 80
Q64 Circles Optimization on a Circle View
Let $r _ { 1 }$ and $r _ { 2 }$ be the radii of the largest and smallest circles, respectively, which pass through the point $( - 4,1 )$ and having their centres on the circumference of the circle $x ^ { 2 } + y ^ { 2 } + 2 x + 4 y - 4 = 0$. If $\frac { r _ { 1 } } { r _ { 2 } } = a + b \sqrt { 2 }$, then $a + b$ is equal to:
(1) 3
(2) 11
(3) 5
(4) 7
Q65 Curve Sketching Sketching a Curve from Analytical Properties View
Let $P$ be a variable point on the parabola $y = 4 x ^ { 2 } + 1$. Then, the locus of the mid-point of the point $P$ and the foot of the perpendicular drawn from the point $P$ to the line $y = x$ is:
(1) $( 3 x - y ) ^ { 2 } + ( x - 3 y ) + 2 = 0$
(2) $2 ( 3 x - y ) ^ { 2 } + ( x - 3 y ) + 2 = 0$
(3) $( 3 x - y ) ^ { 2 } + 2 ( x - 3 y ) + 2 = 0$
(4) $2 ( x - 3 y ) ^ { 2 } + ( 3 x - y ) + 2 = 0$
Q67 Measures of Location and Spread View
If the mean and variance of six observations $7,10,11,15 , a , b$ are 10 and $\frac { 20 } { 3 }$, respectively, then the value of $| a - b |$ is equal to:
(1) 9
(2) 11
(3) 7
(4) 1
Q68 Standard trigonometric equations Evaluate trigonometric expression given a constraint View
Let in a right angled triangle, the smallest angle be $\theta$. If a triangle formed by taking the reciprocal of its sides is also a right angled triangle, then $\sin \theta$ is equal to:
(1) $\frac { \sqrt { 5 } + 1 } { 4 }$
(2) $\frac { \sqrt { 5 } - 1 } { 2 }$
(3) $\frac { \sqrt { 2 } - 1 } { 2 }$
(4) $\frac { \sqrt { 5 } - 1 } { 4 }$
Q69 Simultaneous equations View
The value of $k \in R$, for which the following system of linear equations $3 x - y + 4 z = 3$ $x + 2 y - 3 z = - 2$ $6 x + 5 y + k z = - 3$ has infinitely many solutions, is:
(1) 3
(2) - 5
(3) 5
(4) - 3
Q70 Chi-squared test of independence View
The value of $\tan \left( 2 \tan ^ { - 1 } \left( \frac { 3 } { 5 } \right) + \sin ^ { - 1 } \left( \frac { 5 } { 13 } \right) \right)$ is equal to:
(1) $\frac { - 181 } { 69 }$
(2) $\frac { 220 } { 21 }$
(3) $\frac { - 291 } { 76 }$
(4) $\frac { 151 } { 63 }$
Q71 Composite & Inverse Functions Recover a Function from a Composition or Functional Equation View
Let $f : R - \left\{ \frac { \alpha } { 6 } \right\} \rightarrow R$ be defined by $f ( x ) = \left( \frac { 5 x + 3 } { 6 x - \alpha } \right)$. Then the value of $\alpha$ for which $( f \circ f ) ( x ) = x$, for all $x \in R - \left\{ \frac { \alpha } { 6 } \right\}$, is
(1) No such $\alpha$ exists
(2) 5
(3) 8
(4) 6
Q72 Stationary points and optimisation Find critical points and classify extrema of a given function View
The sum of all the local minimum values of the twice differentiable function $f : R \rightarrow R$ defined by $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - \frac { 3 f ^ { \prime \prime } ( 2 ) } { 2 } x + f ^ { \prime \prime } ( 1 )$ is:
(1) - 22
(2) 5
(3) - 27
(4) 0
Q73 Indefinite & Definite Integrals Piecewise/Periodic Function Integration View
If $[ x ]$ denotes the greatest integer less than or equal to $x$, then the value of the integral $\int _ { - \pi / 2 } ^ { \pi / 2 } [ [ x ] - \sin x ] \, d x$ is equal to:
(1) $- \pi$
(2) $\pi$
(3) 0
(4) 1
Q74 Indefinite & Definite Integrals Definite Integral as a Limit of Riemann Sums View
If $f : R \rightarrow R$ is given by $f ( x ) = x + 1$, then the value of $\lim _ { n \rightarrow \infty } \frac { 1 } { n } \left[ f ( 0 ) + f \left( \frac { 5 } { n } \right) + f \left( \frac { 10 } { n } \right) + \ldots + f \left( \frac { 5 ( n - 1 ) } { n } \right) \right]$ is:
(1) $\frac { 3 } { 2 }$
(2) $\frac { 5 } { 2 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 7 } { 2 }$
Q75 Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution View
Let $g ( t ) = \int _ { - \pi / 2 } ^ { \pi / 2 } \left( \cos \frac { \pi } { 4 } t + f ( x ) \right) d x$, where $f ( x ) = \log _ { e } \left( x + \sqrt { x ^ { 2 } + 1 } \right) , x \in R$. Then which one of the following is correct?
(1) $g ( 1 ) = g ( 0 )$
(2) $\sqrt { 2 } g ( 1 ) = g ( 0 )$
(3) $g ( 1 ) = \sqrt { 2 } g ( 0 )$
(4) $g ( 1 ) + g ( 0 ) = 0$
Q76 First order differential equations (integrating factor) View
Let $y = y ( x )$ satisfies the equation $\frac { d y } { d x } - | A | = 0$, for all $x > 0$, where $A = \left[ \begin{array} { c c c } y & \sin x & 1 \\ 0 & - 1 & 1 \\ 2 & 0 & \frac { 1 } { x } \end{array} \right]$. If $y ( \pi ) = \pi + 2$, then the value of $y \left( \frac { \pi } { 2 } \right)$ is:
(1) $\frac { \pi } { 2 } + \frac { 4 } { \pi }$
(2) $\frac { \pi } { 2 } - \frac { 1 } { \pi }$
(3) $\frac { 3 \pi } { 2 } - \frac { 1 } { \pi }$
(4) $\frac { \pi } { 2 } - \frac { 4 } { \pi }$
Q77 Vectors Introduction & 2D Dot Product Computation View
In a triangle $ABC$, if $| \overrightarrow { BC } | = 3 , | \overrightarrow { CA } | = 5$ and $| \overrightarrow { BA } | = 7$, then the projection of the vector $\overrightarrow { BA }$ on $\overrightarrow { BC }$ is equal to
(1) $\frac { 19 } { 2 }$
(2) $\frac { 13 } { 2 }$
(3) $\frac { 11 } { 2 }$
(4) $\frac { 15 } { 2 }$
Q78 Vectors: Lines & Planes Coplanarity and Relative Position of Planes View
The lines $x = ay - 1 = z - 2$ and $x = 3y - 2 = bz - 2 , ( ab \neq 0 )$ are coplanar, if:
(1) $b = 1 , a \in R - \{ 0 \}$
(2) $a = 1 , b \in R - \{ 0 \}$
(3) $a = 2 , b = 2$
(4) $a = 2 , b = 3$
Q79 Vectors: Lines & Planes Perpendicular/Orthogonal Projection onto a Plane View
Consider the line $L$ given by the equation $\frac { x - 3 } { 2 } = \frac { y - 1 } { 1 } = \frac { z - 2 } { 1 }$. Let $Q$ be the mirror image of the point $( 2,3 , - 1 )$ with respect to $L$. Let a plane $P$ be such that it passes through $Q$, and the line $L$ is perpendicular to $P$. Then which of the following points is on the plane $P$?
(1) $( - 1,1,2 )$
(2) $( 1,1,1 )$
(3) $( 1,1,2 )$
(4) $( 1,2,2 )$
Q80 Probability Definitions Probability Using Set/Event Algebra View
Let $A , B$ and $C$ be three events such that the probability that exactly one of $A$ and $B$ occurs is $( 1 - k )$, the probability that exactly one of $B$ and $C$ occurs is $( 1 - 2k )$, the probability that exactly one of $C$ and $A$ occurs is $( 1 - k )$ and the probability of all $A , B$ and $C$ occur simultaneously is $k ^ { 2 }$, where $0 < k < 1$. Then the probability that at least one of $A , B$ and $C$ occur is:
(1) greater than $\frac { 1 } { 8 }$ but less than $\frac { 1 } { 4 }$
(2) greater than $\frac { 1 } { 2 }$
(3) greater than $\frac { 1 } { 4 }$ but less than $\frac { 1 } { 2 }$
(4) exactly equal to $\frac { 1 } { 2 }$
Q81 Laws of Logarithms Solve a Logarithmic Equation View
The number of solutions of the equation $\log _ { ( x + 1 ) } \left( 2 x ^ { 2 } + 7 x + 5 \right) + \log _ { ( 2 x + 5 ) } ( x + 1 ) ^ { 2 } - 4 = 0 , x > 0$, is
Q82 Sequences and series, recurrence and convergence Summation of sequence terms View
Let $\left\{ a _ { n } \right\} _ { n = 1 } ^ { \infty }$ be a sequence such that $a _ { 1 } = 1 , a _ { 2 } = 1$ and $a _ { n + 2 } = 2 a _ { n + 1 } + a _ { n }$ for all $n \geq 1$. Then the value of $47 \sum _ { n = 1 } ^ { \infty } \left( \frac { a _ { n } } { 2 ^ { 3 n } } \right)$ is equal to $\underline{\hspace{1cm}}$.
Q83 Partial Fractions View
For $k \in N$, let $\frac { 1 } { \alpha ( \alpha + 1 ) ( \alpha + 2 ) \ldots ( \alpha + 20 ) } = \sum _ { K = 0 } ^ { 20 } \frac { A _ { k } } { \alpha + k }$, where $\alpha > 0$. Then the value of $100 \left( \frac { A _ { 14 } + A _ { 15 } } { A _ { 13 } } \right) ^ { 2 }$ is equal to $\underline{\hspace{1cm}}$.
Q84 Circles Inscribed/Circumscribed Circle Computations View
Consider a triangle having vertices $A ( - 2,3 ) , B ( 1,9 )$ and $C ( 3,8 )$. If a line $L$ passing through the circumcentre of triangle $ABC$, bisects line $BC$, and intersects $y$-axis at point $\left( 0 , \frac { \alpha } { 2 } \right)$, then the value of real number $\alpha$ is $\underline{\hspace{1cm}}$.
Q85 Stationary points and optimisation Geometric or applied optimisation problem View
If the point on the curve $y ^ { 2 } = 6 x$, nearest to the point $\left( 3 , \frac { 3 } { 2 } \right)$ is $( \alpha , \beta )$, then $2 ( \alpha + \beta )$ is equal to $\underline{\hspace{1cm}}$.
Q86 Chain Rule Limit Evaluation Involving Composition or Substitution View
If $\lim _ { x \rightarrow 0 } \left[ \frac { \alpha x e ^ { x } - \beta \log _ { e } ( 1 + x ) + \gamma x ^ { 2 } e ^ { - x } } { x \sin ^ { 2 } x } \right] = 10 , \alpha , \beta , \gamma \in R$, then the value of $\alpha + \beta + \gamma$ is $\underline{\hspace{1cm}}$.
Q87 3x3 Matrices Determinant of Parametric or Structured Matrix View
Let $A = \left\{ a _ { i j } \right\}$ be a $3 \times 3$ matrix, where $a _ { i j } = \left\{ \begin{array} { l l } ( - 1 ) ^ { j - i } & \text { if } i < j \\ 2 & \text { if } i = j \\ ( - 1 ) ^ { i + j } & \text { if } i > j \end{array} \right.$ then $\det \left( 3 \operatorname{Adj} \left( 2 A ^ { - 1 } \right) \right)$ is equal to $\underline{\hspace{1cm}}$.
Q88 Curve Sketching Continuity and Discontinuity Analysis of Piecewise Functions View
Let a function $g : [ 0,4 ] \rightarrow R$ be defined as $g ( x ) = \left\{ \begin{array} { c c } \max \left\{ t ^ { 3 } - 6 t ^ { 2 } + 9 t - 3 \right\} , & 0 \leq x \leq 3 \\ 0 \leq t \leq x & \\ 4 - x, & 3 < x \leq 4 \end{array} \right.$ then the number of points in the interval $( 0,4 )$ where $g ( x )$ is NOT differentiable, is $\underline{\hspace{1cm}}$.
Q89 Differential equations Solving Separable DEs with Initial Conditions View
Let a curve $y = y ( x )$ be given by the solution of the differential equation $\cos \left( \frac { 1 } { 2 } \cos ^ { - 1 } \left( e ^ { - x } \right) \right) dx = \left( \sqrt { e ^ { 2 x } - 1 } \right) dy$. If it intersects $y$-axis at $y = - 1$, and the intersection point of the curve with $x$-axis is $( \alpha , 0 )$, then $e ^ { \alpha }$ is equal to $\underline{\hspace{1cm}}$.
Q90 Vectors Introduction & 2D Angle or Cosine Between Vectors View
For $p > 0$, a vector $\vec { v } _ { 2 } = 2 \hat { i } + ( p + 1 ) \hat { j }$ is obtained by rotating the vector $\vec { v } _ { 1 } = \sqrt { 3 } p \hat { i } + \hat { j }$ by an angle $\theta$ about origin in counter clockwise direction. If $\tan \theta = \frac { ( \alpha \sqrt { 3 } - 2 ) } { ( 4 \sqrt { 3 } + 3 ) }$, then the value of $\alpha$ is equal to $\underline{\hspace{1cm}}$.