jee-main

Papers (169)
2025
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2024
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2023
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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
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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
2023 session1_25jan_shift1

28 maths questions

Q61 Solving quadratics and applications Optimization or extremal value of an expression via completing the square View
Let $S = \left\{ \alpha : \log _ { 2 } \left( 9 ^ { 2 \alpha - 4 } + 13 \right) - \log _ { 2 } \left( \frac { 5 } { 2 } \cdot 3 ^ { 2 \alpha - 4 } + 1 \right) = 2 \right\}$. Then the maximum value of $\beta$ for which the equation $\mathrm { x } ^ { 2 } - 2 \left( \sum _ { \alpha \in s } \alpha \right) ^ { 2 } \mathrm { x } + \sum _ { a \in s } ( \alpha + 1 ) ^ { 2 } \beta = 0$ has real roots, is $\_\_\_\_$ .
Q62 Complex Numbers Arithmetic Geometric Interpretation and Triangle/Shape Properties View
Let $z _ { 1 } = 2 + 3 i$ and $z _ { 2 } = 3 + 4 i$. The set $\mathrm { S } = \left\{ \mathrm { z } \in \mathrm { C } : \left| \mathrm { z } - \mathrm { z } _ { 1 } \right| ^ { 2 } - \left| \mathrm { z } - \mathrm { z } _ { 2 } \right| ^ { 2 } = \left| \mathrm { z } _ { 1 } - \mathrm { z } _ { 2 } \right| ^ { 2 } \right\}$ represents a
(1) straight line with sum of its intercepts on the coordinate axes equals 14
(2) hyperbola with the length of the transverse axis 7
(3) straight line with the sum of its intercepts on the coordinate axes equals $-18$
(4) hyperbola with eccentricity 2
Q63 Principle of Inclusion/Exclusion View
Let $x$ and $y$ be distinct integers where $1 \leq x \leq 25$ and $1 \leq y \leq 25$. Then, the number of ways of choosing $x$ and $y$, such that $x + y$ is divisible by 5 , is $\_\_\_\_$ .
Q65 Arithmetic Sequences and Series Compute Partial Sum of an Arithmetic Sequence View
Let $A _ { 1 } , A _ { 2 } , A _ { 3 }$ be the three A.P. with the same common difference $d$ and having their first terms as $A , A + 1 , A + 2$, respectively. Let $a , b , c$ be the $7 ^ { \text {th} } , 9 ^ { \text {th} } , 17 ^ { \text {th} }$ terms of $A _ { 1 } , A _ { 2 } , A _ { 3 }$, respectively such that $\left| \begin{array} { l l l } a & 7 & 1 \\ 2 b & 17 & 1 \\ c & 17 & 1 \end{array} \right| + 70 = 0$. If $a = 29$, then the sum of first 20 terms of an AP whose first term is $c - a - b$ and common difference is $\frac { d } { 12 }$, is equal to $\_\_\_\_$.
Q66 Binomial Theorem (positive integer n) Evaluate a Summation Involving Binomial Coefficients View
If $a _ { r }$ is the coefficient of $x ^ { 10 - r }$ in the Binomial expansion of $( 1 + x ) ^ { 10 }$, then $\sum _ { r = 1 } ^ { 10 } r ^ { 3 } \left( \frac { a _ { r } } { a _ { r - 1 } } \right) ^ { 2 }$ is equal to
(1) 4895
(2) 1210
(3) 5445
(4) 3025
Q67 Binomial Theorem (positive integer n) Find a Specific Coefficient in a Single Binomial Expansion View
The constant term in the expansion of $\left( 2 x + \frac { 1 } { x ^ { 7 } } + 3 x ^ { 2 } \right) ^ { 5 }$ is $\_\_\_\_$.
Q68 Circles Circle Equation Derivation View
The points of intersection of the line $a x + b y = 0 , ( \mathrm { a } \neq \mathrm { b } )$ and the circle $\mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } - 2 \mathrm { x } = 0$ are $A ( \alpha , 0 )$ and $B ( 1 , \beta )$. The image of the circle with $A B$ as a diameter in the line $\mathrm { x } + \mathrm { y } + 2 = 0$ is:
(1) $x ^ { 2 } + y ^ { 2 } + 5 x + 5 y + 12 = 0$
(2) $x ^ { 2 } + y ^ { 2 } + 3 x + 5 y + 8 = 0$
(3) $x ^ { 2 } + y ^ { 2 } + 3 x + 3 y + 4 = 0$
(4) $x ^ { 2 } + y ^ { 2 } - 5 x - 5 y + 12 = 0$
Q69 Straight Lines & Coordinate Geometry Point-to-Line Distance Computation View
The distance of the point $( 6 , - 2 \sqrt { 2 } )$ from the common tangent $y = m x + c , m > 0$, of the curves $x = 2 y ^ { 2 }$ and $x = 1 + y ^ { 2 }$ is
(1) $\frac { 1 } { 3 }$
(2) 5
(3) $\frac { 14 } { 3 }$
(4) $5 \sqrt { 3 }$
Q70 Conic sections Tangent and Normal Line Problems View
The vertices of a hyperbola H are $( \pm 6,0 )$ and its eccentricity is $\frac { \sqrt { 5 } } { 2 }$. Let N be the normal to H at a point in the first quadrant and parallel to the line $\sqrt { 2 } x + y = 2 \sqrt { 2 }$. If $d$ is the length of the line segment of N between H and the $y$-axis then $d ^ { 2 }$ is equal to $\_\_\_\_$ .
Q71 Sequences and series, recurrence and convergence Convergence proof and limit determination View
The value of $\lim _ { n \rightarrow \infty } \frac { 1 + 2 - 3 + 4 + 5 - 6 + \ldots + ( 3 n - 2 ) + ( 3 n - 1 ) - 3 n } { \sqrt { 2 n ^ { 4 } + 4 n + 3 } - \sqrt { n ^ { 4 } + 5 n + 4 } }$ is
(1) $\frac { \sqrt { 2 } + 1 } { 2 }$
(2) $3 ( \sqrt { 2 } + 1 )$
(3) $\frac { 3 } { 2 } ( \sqrt { 2 } + 1 )$
(4) $\frac { 3 } { 2 \sqrt { 2 } }$
Q73 Measures of Location and Spread View
The mean and variance of the marks obtained by the students in a test are 10 and 4 respectively. Later, the marks of one of the students is increased from 8 to 12 . If the new mean of the marks is 10.2 , then their new variance is equal to:
(1) 4.04
(2) 4.08
(3) 3.96
(4) 3.92
Q74 Matrices Determinant and Rank Computation View
Let $x , y , z > 1$ and $A = \left[ \begin{array} { l l l } 1 & \log _ { x } y & \log _ { x } z \\ \log _ { y } x & 2 & \log _ { y } z \\ \log _ { z } x & \log _ { z } y & 3 \end{array} \right]$. Then $\left| \operatorname { adj } \left( \operatorname { adj } \mathrm { A } ^ { 2 } \right) \right|$ is equal to
(1) $6 ^ { 4 }$
(2) $2 ^ { 8 }$
(3) $4 ^ { 8 }$
(4) $2 ^ { 4 }$
Q75 Simultaneous equations View
Let $S _ { 1 }$ and $S _ { 2 }$ be respectively the sets of all $a \in R - \{ 0 \}$ for which the system of linear equations $a x + 2 a y - 3 a z = 1$ $( 2 a + 1 ) x + ( 2 a + 3 ) y + ( a + 1 ) z = 2$ $( 3 a + 5 ) x + ( a + 5 ) y + ( a + 2 ) z = 3$ has unique solution and infinitely many solutions. Then
(1) $\mathrm { n } \left( S _ { 1 } \right) = 2$ and $S _ { 2 }$ is an infinite set
(2) $S _ { 1 }$ is an infinite set and $n \left( S _ { 2 } \right) = 2$
(3) $S _ { 1 } = \phi$ and $S _ { 2 } = \mathbb { R } - \{ 0 \}$
(4) $S _ { 1 } = \mathbb { R } - \{ 0 \}$ and $S _ { 2 } = \phi$
Q76 Standard trigonometric equations Inverse trigonometric equation View
If the sum of all the solutions of $\tan ^ { - 1 } \left( \frac { 2 x } { 1 - x ^ { 2 } } \right) + \cot ^ { - 1 } \left( \frac { 1 - x ^ { 2 } } { 2 x } \right) = \frac { \pi } { 3 } , - 1 < x < 1 , x \neq 0$, is $\alpha - \frac { 4 } { \sqrt { 3 } }$, then $\alpha$ is equal to $\_\_\_\_$ .
Q77 Stationary points and optimisation Determine intervals of increase/decrease or monotonicity conditions View
Let $f : ( 0,1 ) \rightarrow \mathbb { R }$ be a function defined by $f ( x ) = \frac { 1 } { 1 - e ^ { - x } }$, and $g ( x ) = ( f ( - x ) - f ( x ) )$. Consider two statements (I) $g$ is an increasing function in $( 0,1 )$ (II) $g$ is one-one in $( 0,1 )$ Then,
(1) Only (I) is true
(2) Only (II) is true
(3) Neither (I) nor (II) is true
(4) Both (I) and (II) are true
Q78 Composite & Inverse Functions Evaluate Composition from Algebraic Definitions View
For some $a , b , c \in \mathbb { N }$, let $f ( x ) = a x - 3$ and $g ( x ) = x ^ { b } + c , x \in \mathbb { R }$. If $( f \circ g ) ^ { - 1 } ( x ) = \left( \frac { x - 7 } { 2 } \right) ^ { \frac { 1 } { 3 } }$, then $( f \circ g ) ( a c ) + ( g \circ f ) ( b )$ is equal to $\_\_\_\_$ .
Q79 Chain Rule Higher-Order Derivatives of Products/Compositions View
Let $y ( x ) = ( 1 + x ) \left( 1 + x ^ { 2 } \right) \left( 1 + x ^ { 4 } \right) \left( 1 + x ^ { 8 } \right) \left( 1 + x ^ { 16 } \right)$. Then $y ^ { \prime } - y ^ { \prime \prime }$ at $x = - 1$ is equal to
(1) 976
(2) 464
(3) 496
(4) 944
Q80 Stationary points and optimisation Find critical points and classify extrema of a given function View
Let $x = 2$ be a local minima of the function $f ( x ) = 2 x ^ { 4 } - 18 x ^ { 2 } + 8 x + 12 , x \in ( - 4,4 )$. If $M$ is local maximum value of the function $f$ in $( - 4,4 )$, then $M =$
(1) $12 \sqrt { 6 } - \frac { 33 } { 2 }$
(2) $12 \sqrt { 6 } - \frac { 31 } { 2 }$
(3) $18 \sqrt { 6 } - \frac { 33 } { 2 }$
(4) $18 \sqrt { 6 } - \frac { 31 } { 2 }$
Q81 Integration by Substitution Substitution to Compute an Indefinite Integral with Initial Condition View
Let $f ( x ) = \int \frac { 2 x } { \left( x ^ { 2 } + 1 \right) \left( x ^ { 2 } + 3 \right) } d x$. If $f ( 3 ) = \frac { 1 } { 2 } \left( \log _ { e } 5 - \log _ { e } 6 \right)$, then $f ( 4 )$ is equal to
(1) $\frac { 1 } { 2 } \left( \log _ { e } 17 - \log _ { e } 19 \right)$
(2) $\log _ { \mathrm { e } } 17 - \log _ { \mathrm { e } } 18$
(3) $\frac { 1 } { 2 } \left( \log _ { e } 19 - \log _ { e } 17 \right)$
(4) $\log _ { e } 19 - \log _ { e } 20$
Q82 Indefinite & Definite Integrals Maximizing or Optimizing a Definite Integral View
The minimum value of the function $f ( x ) = \int _ { 0 } ^ { 2 } e ^ { | x - t | } d t$ is
(1) $2 ( e - 1 )$
(2) $2 e - 1$
(3) 2
(4) $e ( e - 1 )$
Q83 Areas by integration View
If the area enclosed by the parabolas $P _ { 1 } : 2 y = 5 x ^ { 2 }$ and $P _ { 2 } : x ^ { 2 } - y + 6 = 0$ is equal to the area enclosed by $P _ { 1 }$ and $y = \alpha x , \alpha > 0$, then $\alpha ^ { 3 }$ is equal to $\_\_\_\_$.
Q84 First order differential equations (integrating factor) View
Let $\mathrm { y } = \mathrm { y } ( \mathrm { x } )$ be the solution curve of the differential equation $\frac { d y } { d x } = \frac { y } { x } \left( 1 + x ^ { 2 } \left( 1 + \log _ { e } x \right) \right) , \mathrm { x } > 0 , \mathrm { y } ( 1 ) = 3$. Then $\frac { \mathrm { y } ^ { 2 } ( \mathrm { x } ) } { 9 }$ is equal to:
(1) $\frac { x ^ { 2 } } { 5 - 2 x ^ { 3 } \left( 2 + \log _ { e } x ^ { 3 } \right) }$
(2) $\frac { x ^ { 2 } } { 2 x ^ { 3 } \left( 2 + \log _ { e } x ^ { 3 } \right) - 3 }$
(3) $\frac { x ^ { 2 } } { 3 x ^ { 3 } \left( 1 + \log _ { e } x ^ { 2 } \right) - 2 }$
(4) $\frac { x ^ { 2 } } { 7 - 3 x ^ { 3 } \left( 2 + \log _ { e } x ^ { 2 } \right) }$
Q85 Vectors Introduction & 2D Dot Product Computation View
Let $\vec { a } , \vec { b }$ and $\vec { c }$ be three non zero vectors such that $\vec { b } \cdot \vec { c } = 0$ and $\vec { a } \times ( \vec { b } \times \vec { c } ) = \frac { \vec { b } - \vec { c } } { 2 }$. If $\vec { d }$ be a vector such that $\overrightarrow { \mathrm { b } } \cdot \overrightarrow { \mathrm { d } } = \overrightarrow { \mathrm { a } } \cdot \overrightarrow { \mathrm { b } }$, then $( \overrightarrow { \mathrm { a } } \times \overrightarrow { \mathrm { b } } ) \cdot ( \overrightarrow { \mathrm { c } } \times \overrightarrow { \mathrm { d } } )$ is equal to
(1) $\frac { 3 } { 4 }$
(2) $\frac { 1 } { 2 }$
(3) $- \frac { 1 } { 4 }$
(4) $\frac { 1 } { 4 }$
Q86 Vectors Introduction & 2D Dot Product Computation View
The vector $\vec { a } = - \hat { i } + 2 \hat { j } + \hat { k }$ is rotated through a right angle, passing through the $y$-axis in its way and the resulting vector is $\vec { b }$. Then the projection of $3 \vec { a } + \sqrt { 2 } \vec { b }$ on $\vec { c } = 5 \hat { i } + 4 \hat { j } + 3 \hat { k }$ is
(1) $3 \sqrt { 2 }$
(2) 1
(3) $\sqrt { 6 }$
(4) $2 \sqrt { 3 }$
Q87 Vectors 3D & Lines Distance from a Point to a Line (Show/Compute) View
The distance of the point $P ( 4,6 , - 2 )$ from the line passing through the point $( - 3,2,3 )$ and parallel to a line with direction ratios $3,3 , - 1$ is equal to:
(1) 3
(2) $\sqrt { 6 }$
(3) $2 \sqrt { 3 }$
(4) $\sqrt { 14 }$
Q88 Vectors 3D & Lines Shortest Distance Between Two Lines View
Consider the lines $L _ { 1 }$ and $L _ { 2 }$ given by $L _ { 1 } : \frac { x - 1 } { 2 } = \frac { y - 3 } { 1 } = \frac { z - 2 } { 2 }$ $L _ { 2 } : \frac { x - 2 } { 1 } = \frac { y - 2 } { 2 } = \frac { z - 3 } { 3 }$
A line $L _ { 3 }$ having direction ratios $1 , - 1 , - 2$, intersects $L _ { 1 }$ and $L _ { 2 }$ at the points $P$ and $Q$ respectively. Then the length of line segment $P Q$ is
(1) $2 \sqrt { 6 }$
(2) $3 \sqrt { 2 }$
(3) $4 \sqrt { 3 }$
(4) 4
Q89 Vectors: Lines & Planes Find Cartesian Equation of a Plane View
Let the equation of the plane passing through the line $x - 2 y - z - 5 = 0 = x + y + 3 z - 5$ and parallel to the line $x + y + 2 z - 7 = 0 = 2 x + 3 y + z - 2$ be $a x + b y + c z = 65$. Then the distance of the point $( a , b , c )$ from the plane $2 x + 2 y - z + 16 = 0$ is $\_\_\_\_$.
Q90 Probability Definitions Probability Involving Algebraic or Number-Theoretic Conditions View
Let M be the maximum value of the product of two positive integers when their sum is 66 . Let the sample space $S = \left\{ x \in Z : x ( 66 - x ) \geq \frac { 5 } { 9 } M \right\}$ and the event $\mathrm { A } = \{ \mathrm { x } \in \mathrm { S } : \mathrm { x }$ is a multiple of $3 \}$. Then $\mathrm { P } ( \mathrm { A } )$ is equal to
(1) $\frac { 15 } { 44 }$
(2) $\frac { 1 } { 3 }$
(3) $\frac { 1 } { 5 }$
(4) $\frac { 7 } { 22 }$