jee-main

Papers (191)
2026
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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
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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
07apr 29 09apr 12 22apr 5 23apr 14 25apr 13
2012
07may 17 12may 21 19may 14 26may 17 offline 30
2011
jee-main_2011.pdf 18
2010
jee-main_2010.pdf 6
2009
jee-main_2009.pdf 2
2008
jee-main_2008.pdf 4
2007
jee-main_2007.pdf 38
2006
jee-main_2006.pdf 15
2005
jee-main_2005.pdf 25
2004
jee-main_2004.pdf 22
2003
jee-main_2003.pdf 8
2002
jee-main_2002.pdf 12
2022 session2_28jul_shift2

29 maths questions

Q61 Inequalities Set Operations Using Inequality-Defined Sets View
Let $S = \left\{ x \in [ - 6,3 ] - \{ - 2,2 \} : \frac { | x + 3 | - 1 } { | x | - 2 } \geq 0 \right\}$ and $T = \left\{ x \in Z : x ^ { 2 } - 7 | x | + 9 \leq 0 \right\}$. Then the number of elements in $S \cap T$ is
(1) 7
(2) 5
(3) 4
(4) 3
Q62 Discriminant and conditions for roots Root relationships and Vieta's formulas View
Let $\alpha , \beta$ be the roots of the equation $x ^ { 2 } - \sqrt { 2 } x + \sqrt { 6 } = 0$ and $\frac { 1 } { \alpha ^ { 2 } } + 1 , \frac { 1 } { \beta ^ { 2 } } + 1$ be the roots of the equation $x ^ { 2 } + a x + b = 0$. Then the roots of the equation $x ^ { 2 } - ( a + b - 2 ) x + ( a + b + 2 ) = 0$ are :
(1) non-real complex numbers
(2) real and both negative
(3) real and both positive
(4) real and exactly one of them is positive
Q63 Circles Area and Geometric Measurement Involving Circles View
Let the tangents at two points $A$ and $B$ on the circle $x ^ { 2 } + y ^ { 2 } - 4 x + 3 = 0$ meet at origin $O ( 0,0 )$. Then the area of the triangle of $O A B$ is
(1) $\frac { 3 \sqrt { 3 } } { 2 }$
(2) $\frac { 3 \sqrt { 3 } } { 4 }$
(3) $\frac { 3 } { 2 \sqrt { 3 } }$
(4) $\frac { 3 } { 4 \sqrt { 3 } }$
Q64 Conic sections Confocal or Related Conic Construction View
Let the hyperbola $H : \frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ pass through the point $( 2 \sqrt { 2 } , - 2 \sqrt { 2 } )$. A parabola is drawn whose focus is same as the focus of $H$ with positive abscissa and the directrix of the parabola passes through the other focus of $H$. If the length of the latus rectum of the parabola is $e$ times the length of the latus rectum of $H$, where $e$ is the eccentricity of $H$, then which of the following points lies on the parabola?
(1) $( 2 \sqrt { 3 } , 3 \sqrt { 2 } )$
(2) $( 3 \sqrt { 3 } , - 6 \sqrt { 2 } )$
(3) $( \sqrt { 3 } , - \sqrt { 6 } )$
(4) $( 3 \sqrt { 6 } , 6 \sqrt { 2 } )$
Q66 SUVAT in 2D & Gravity Heights and distances / angle of elevation problem View
A horizontal park is in the shape of a triangle $O A B$ with $A B = 16$. A vertical lamp post $O P$ is erected at the point $O$ such that $\angle P A O = \angle P B O = 15 ^ { \circ }$ and $\angle P C O = 45 ^ { \circ }$, where $C$ is the midpoint of $A B$. Then $( O P ) ^ { 2 }$ is equal to
(1) $\frac { 32 } { \sqrt { 3 } } ( \sqrt { 3 } - 1 )$
(2) $\frac { 32 } { \sqrt { 3 } } ( 2 - \sqrt { 3 } )$
(3) $\frac { 16 } { \sqrt { 3 } } ( \sqrt { 3 } - 1 )$
(4) $\frac { 16 } { \sqrt { 3 } } ( 2 - \sqrt { 3 } )$
Q67 Matrices Matrix Algebra and Product Properties View
Let $A$ and $B$ be any two $3 \times 3$ symmetric and skew symmetric matrices respectively. Then which of the following is NOT true?
(1) $A ^ { 4 } - B ^ { 4 }$ is a symmetric matrix
(2) $A B - B A$ is a symmetric matrix
(3) $B ^ { 5 } - A ^ { 5 }$ is a skew-symmetric matrix
(4) $A B + B A$ is a skew-symmetric matrix
Q68 Solving quadratics and applications Determining quadratic function from given conditions View
Let $f ( x ) = a x ^ { 2 } + b x + c$ be such that $f ( 1 ) = 3 , f ( - 2 ) = \lambda$ and $f ( 3 ) = 4$. If $f ( 0 ) + f ( 1 ) + f ( - 2 ) + f ( 3 ) = 14$, then $\lambda$ is equal to
(1) $- 4$
(2) $\frac { 13 } { 2 }$
(3) $\frac { 23 } { 2 }$
(4) $4$
Q69 Composite & Inverse Functions Determine Domain or Range of a Composite Function View
The function $f : R \rightarrow R$ defined by $f ( x ) = \lim _ { n \rightarrow \infty } \frac { \cos ( 2 \pi x ) - x ^ { 2 n } \sin ( x - 1 ) } { 1 + x ^ { 2 n + 1 } - x ^ { 2 n } }$ is continuous for all $x$ in
(1) $R - \{ - 1 \}$
(2) $R - \{ - 1,1 \}$
(3) $R - \{ 1 \}$
(4) $R - \{ 0 \}$
Q70 Parametric differentiation View
Let $x ( t ) = 2 \sqrt { 2 } \cos t \sqrt { \sin 2 t }$ and $y ( t ) = 2 \sqrt { 2 } \sin t \sqrt { \sin 2 t } , t \in \left( 0 , \frac { \pi } { 2 } \right)$. Then $\frac { 1 + \left( \frac { d y } { d x } \right) ^ { 2 } } { \frac { d ^ { 2 } y } { d x ^ { 2 } } }$ at $t = \frac { \pi } { 4 }$ is equal to
(1) $\frac { - 2 \sqrt { 2 } } { 3 }$
(2) $\frac { 2 } { 3 }$
(3) $\frac { 1 } { 3 }$
(4) $\frac { - 2 } { 3 }$
Q71 Stationary points and optimisation Determine intervals of increase/decrease or monotonicity conditions View
The function $f ( x ) = x e ^ { x ( 1 - x ) } , x \in R$, is
(1) increasing in $\left( - \frac { 1 } { 2 } , 1 \right)$
(2) decreasing in $\left( \frac { 1 } { 2 } , 2 \right)$
(3) increasing in $\left( - 1 , - \frac { 1 } { 2 } \right)$
(4) decreasing in $\left( - \frac { 1 } { 2 } , \frac { 1 } { 2 } \right)$
Q72 Stationary points and optimisation Find absolute extrema on a closed interval or domain View
The sum of the absolute maximum and absolute minimum values of the function $f ( x ) = \tan ^ { - 1 } ( \sin x - \cos x )$ in the interval $[ 0 , \pi ]$ is
(1) $0$
(2) $\tan ^ { - 1 } \left( \frac { 1 } { \sqrt { 2 } } \right) - \frac { \pi } { 4 }$
(3) $\cos ^ { - 1 } \left( \frac { 1 } { \sqrt { 3 } } \right) - \frac { \pi } { 4 }$
(4) $\frac { - \pi } { 12 }$
Q73 Reduction Formulae Derive a Reduction/Recurrence Formula via Integration by Parts View
Let $I _ { n } ( x ) = \int _ { 0 } ^ { x } \frac { 1 } { \left( t ^ { 2 } + 5 \right) ^ { n } } d t , n = 1,2,3 , \ldots$. Then
(1) $50 I _ { 6 } - 9 I _ { 5 } = x I _ { 5 } ^ { \prime }$
(2) $50 I _ { 6 } - 11 I _ { 5 } = x I _ { 5 } ^ { \prime }$
(3) $50 I _ { 6 } - 9 I _ { 5 } = I _ { 5 } ^ { \prime }$
(4) $50 I _ { 6 } - 11 I _ { 5 } = I _ { 5 } ^ { \prime }$
Q74 Areas by integration View
The area enclosed by the curves $y = \log _ { e } \left( x + e ^ { 2 } \right) , x = \log _ { e } \left( \frac { 2 } { y } \right)$ and $x = \log _ { e } 2$, above the line $y = 1$ is
(1) $2 + e - \log _ { e } 2$
(2) $1 + e - \log _ { e } 2$
(3) $e - \log _ { e } 2$
(4) $1 + \log _ { e } 2$
Q75 First order differential equations (integrating factor) First-Order Linear DE: General Solution View
Let $y = y ( x )$ be the solution curve of the differential equation $\frac { d y } { d x } + \frac { 1 } { x ^ { 2 } - 1 } y = \left( \frac { x - 1 } { x + 1 } \right) ^ { \frac { 1 } { 2 } } , x > 1$ passing through the point $\left( 2 , \sqrt { \frac { 1 } { 3 } } \right)$. Then $\sqrt { 7 } y ( 8 )$ is equal to
(1) $11 + 6 \log _ { e } 3$
(2) $19$
(3) $12 - 2 \log _ { e } 3$
(4) $19 - 6 \log _ { e } 3$
Q76 Differential equations Higher-Order and Special DEs (Proof/Theory) View
The differential equation of the family of circles passing through the points $( 0,2 )$ and $( 0 , - 2 )$ is
(1) $2 x y \frac { d y } { d x } + \left( x ^ { 2 } - y ^ { 2 } + 4 \right) = 0$
(2) $2 x y \frac { d y } { d x } + \left( x ^ { 2 } + y ^ { 2 } - 4 \right) = 0$
(3) $2 x y \frac { d y } { d x } + \left( y ^ { 2 } - x ^ { 2 } + 4 \right) = 0$
(4) $2 x y \frac { d y } { d x } - \left( x ^ { 2 } - y ^ { 2 } + 4 \right) = 0$
Q77 Vectors 3D & Lines Angle or Cosine Between Vectors View
Let $S$ be the set of all $a \in R$ for which the angle between the vectors $\vec { u } = a \left( \log _ { e } b \right) \hat { i } - 6 \hat { j } + 3 \hat { k }$ and $\vec { v } = \left( \log _ { e } b \right) \hat { i } + 2 \hat { j } + 2 a \left( \log _ { e } b \right) \hat { k } , ( b > 1 )$ is acute. Then $S$ is equal to
(1) $\left( - \infty , - \frac { 4 } { 3 } \right)$
(2) $\Phi$
(3) $\left( - \frac { 4 } { 3 } , 0 \right)$
(4) $\left( \frac { 12 } { 7 } , \infty \right)$
Q78 Vectors: Lines & Planes Coplanarity and Relative Position of Planes View
Let the lines $\frac { x - 1 } { \lambda } = \frac { y - 2 } { 1 } = \frac { z - 3 } { 2 }$ and $\frac { x + 26 } { - 2 } = \frac { y + 18 } { 3 } = \frac { z + 28 } { \lambda }$ be coplanar and $P$ be the plane containing these two lines. Then which of the following points does NOT lie on $P$?
(1) $( 0 , - 2 , - 2 )$
(2) $( - 5,0 , - 1 )$
(3) $( 3 , - 1,0 )$
(4) $( 0,4,5 )$
Q79 Vectors: Lines & Planes Volume of Pyramid/Tetrahedron Using Planes and Lines View
A plane $P$ is parallel to two lines whose direction ratios are $- 2,1 , - 3$ and $- 1,2 , - 2$ and it contains the point $( 2,2 , - 2 )$. Let $P$ intersect the co-ordinate axes at the points $A , B , C$ making the intercepts $\alpha , \beta , \gamma$. If $V$ is the volume of the tetrahedron $O A B C$, where $O$ is the origin and $p = \alpha + \beta + \gamma$, then the ordered pair $( V , p )$ is equal to
(1) $( 48 , - 13 )$
(2) $( 24 , - 13 )$
(3) $( 48,11 )$
(4) $( 24 , - 5 )$
Q80 Probability Definitions Direct Conditional Probability Computation from Definitions View
Let $A$ and $B$ be two events such that $P ( B \mid A ) = \frac { 2 } { 5 } , P ( A \mid B ) = \frac { 1 } { 7 }$ and $P ( A \cap B ) = \frac { 1 } { 9 }$. Consider $( S1 )\; P \left( A ^ { \prime } \cup B \right) = \frac { 5 } { 6 }$, $( S2 )\; P \left( A ^ { \prime } \cap B ^ { \prime } \right) = \frac { 1 } { 18 }$. Then
(1) Both $(S1)$ and $(S2)$ are true
(2) Both $(S1)$ and $(S2)$ are false
(3) Only $(S1)$ is true
(4) Only $(S2)$ is true
Q81 Complex Numbers Arithmetic Solving Equations for Unknown Complex Numbers View
Let $\mathrm { z } = \mathrm { a } + i b , \mathrm { b } \neq 0$ be complex numbers satisfying $\mathrm { z } ^ { 2 } = \overline { \mathrm { z } } \cdot 2 ^ { 1 - | z | }$. Then the least value of $n \in N$, such that $z ^ { n } = ( z + 1 ) ^ { n }$, is equal to $\_\_\_\_$.
Q82 Combinations & Selection Selection with Group/Category Constraints View
A class contains $b$ boys and $g$ girls. If the number of ways of selecting 3 boys and 2 girls from the class is 168, then $b + 3g$ is equal to $\_\_\_\_$.
Q83 Binomial Theorem (positive integer n) Finite Geometric Sum and Term Relationships View
If $\frac { 6 } { 3 ^ { 12 } } + \frac { 10 } { 3 ^ { 11 } } + \frac { 20 } { 3 ^ { 10 } } + \frac { 40 } { 3 ^ { 9 } } + \ldots + \frac { 10240 } { 3 } = 2 ^ { n } \cdot m$, where $m$ is odd, then $m \cdot n$ is equal to $\_\_\_\_$.
Q84 Binomial Theorem (positive integer n) Determine Parameters from Conditions on Coefficients or Terms View
Let the coefficients of the middle terms in the expansion of $\left( \frac { 1 } { \sqrt { 6 } } + \beta x \right) ^ { 4 } , ( 1 - 3 \beta x ) ^ { 2 }$ and $\left( 1 - \frac { \beta } { 2 } x \right) ^ { 6 } , \beta > 0$, respectively form the first three terms of an A.P. If $d$ is the common difference of this A.P., then $50 - \frac { 2 d } { \beta ^ { 2 } }$ is equal to $\_\_\_\_$.
Q85 Binomial Theorem (positive integer n) Evaluate a Summation Involving Binomial Coefficients View
If $1 + \left( 2 + { } ^ { 49 } C _ { 1 } + { } ^ { 49 } C _ { 2 } + \ldots + { } ^ { 49 } C _ { 49 } \right) \left( { } ^ { 50 } C _ { 2 } + { } ^ { 50 } C _ { 4 } + \ldots + { } ^ { 50 } C _ { 50 } \right)$ is equal to $2 ^ { n } \cdot m$, where $m$ is odd, then $n + m$ is equal to $\_\_\_\_$.
Q86 Standard trigonometric equations Solve trigonometric equation for solutions in an interval View
Let $S = \left[ - \pi , \frac { \pi } { 2 } \right) - \left\{ - \frac { \pi } { 2 } , - \frac { \pi } { 4 } , - \frac { 3 \pi } { 4 } , \frac { \pi } { 4 } \right\}$. Then the number of elements in the set $A = \{ \theta \in S : \tan \theta ( 1 + \sqrt { 5 } \tan ( 2 \theta ) ) = \sqrt { 5 } - \tan ( 2 \theta ) \}$ is $\_\_\_\_$.
Q87 Circles Tangent Lines and Tangent Lengths View
Two tangent lines $l _ { 1 }$ and $l _ { 2 }$ are drawn from the point $( 2,0 )$ to the parabola $2 y ^ { 2 } = - x$. If the lines $l _ { 1 }$ and $l _ { 2 }$ are also tangent to the circle $( x - 5 ) ^ { 2 } + y ^ { 2 } = r$, then $17 r ^ { 2 }$ is equal to $\_\_\_\_$.
Q88 Circles Tangent Lines and Tangent Lengths View
Let the tangents at the points $P$ and $Q$ on the ellipse $\frac { x ^ { 2 } } { 2 } + \frac { y ^ { 2 } } { 4 } = 1$ meet at the point $R ( \sqrt { 2 } , 2 \sqrt { 2 } - 2 )$. If $S$ is the focus of the ellipse on its negative major axis, then $S P ^ { 2 } + S Q ^ { 2 }$ is equal to $\_\_\_\_$.
Q89 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
The value of the integral $\int _ { 0 } ^ { \frac { \pi } { 2 } } 60 \frac { \sin ( 6 x ) } { \sin x } d x$ is equal to $\_\_\_\_$.
Q90 Discrete Probability Distributions Expectation and Variance from Context-Based Random Variables View
A bag contains 4 white and 6 black balls. Three balls are drawn at random from the bag. Let $X$ be the number of white balls, among the drawn balls. If $\sigma ^ { 2 }$ is the variance of $X$, then $100 \sigma ^ { 2 }$ is equal to $\_\_\_\_$.