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
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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
2025 session1_23jan_shift2

25 maths questions

Q1 Vectors: Lines & Planes Distance Computation (Point-to-Plane or Line-to-Line) View
The distance of the line $\frac { x - 2 } { 2 } = \frac { y - 6 } { 3 } = \frac { z - 3 } { 4 }$ from the point $( 1,4,0 )$ along the line $\frac { x } { 1 } = \frac { y - 2 } { 2 } = \frac { z + 3 } { 3 }$ is :
(1) $\sqrt { 17 }$
(2) $\sqrt { 15 }$
(3) $\sqrt { 14 }$
(4) $\sqrt { 13 }$
Q2 Inequalities Set Operations Using Inequality-Defined Sets View
Let $\mathrm { A } = \{ ( x , y ) \in \mathbf { R } \times \mathbf { R } : | x + y | \geqslant 3 \}$ and $\mathrm { B } = \{ ( x , y ) \in \mathbf { R } \times \mathbf { R } : | x | + | y | \leq 3 \}$. If $\mathrm { C } = \{ ( x , y ) \in \mathrm { A } \cap \mathrm { B } : x = 0$ or $y = 0 \}$, then $\sum _ { ( x , y ) \in \mathrm { C } } | x + y |$ is :
(1) 15
(2) 24
(3) 18
(4) 12
Q3 Proof True/False Justification View
Let $X = \mathbf { R } \times \mathbf { R }$. Define a relation $R$ on $X$ as : $\left( a _ { 1 } , b _ { 1 } \right) R \left( a _ { 2 } , b _ { 2 } \right) \Leftrightarrow b _ { 1 } = b _ { 2 }$
Statement I : $\quad \mathrm { R }$ is an equivalence relation.
Statement II : For some $( a , b ) \in X$, the set $S = \{ ( x , y ) \in X : ( x , y ) R ( a , b ) \}$ represents a line parallel to $y = x$.
In the light of the above statements, choose the correct answer from the options given below :
(1) Both Statement I and Statement II are false
(2) Statement I is true but Statement II is false
(3) Both Statement I and Statement II are true
(4) Statement I is false but Statement II is true
Q4 Integration by Parts Definite Integral Evaluation by Parts View
Let $\int x ^ { 3 } \sin x \mathrm {~d} x = g ( x ) + C$, where $C$ is the constant of integration. If $8 \left( g \left( \frac { \pi } { 2 } \right) + g ^ { \prime } \left( \frac { \pi } { 2 } \right) \right) = \alpha \pi ^ { 3 } + \beta \pi ^ { 2 } + \gamma , \alpha , \beta , \gamma \in Z$, then $\alpha + \beta - \gamma$ equals :
(1) 48
(2) 55
(3) 62
(4) 47
Q5 Parametric curves and Cartesian conversion View
A rod of length eight units moves such that its ends $A$ and $B$ always lie on the lines $x - y + 2 = 0$ and $y + 2 = 0$, respectively. If the locus of the point $P$, that divides the rod $AB$ internally in the ratio $2 : 1$ is $9 \left( x ^ { 2 } + \alpha y ^ { 2 } + \beta x y + \gamma x + 28 y \right) - 76 = 0$, then $\alpha - \beta - \gamma$ is equal to :
(1) 22
(2) 21
(3) 23
(4) 24
Q6 Vectors 3D & Lines Shortest Distance Between Two Lines View
If the square of the shortest distance between the lines $\frac { x - 2 } { 1 } = \frac { y - 1 } { 2 } = \frac { z + 3 } { - 3 }$ and $\frac { x + 1 } { 2 } = \frac { y + 3 } { 4 } = \frac { z + 5 } { - 5 }$ is $\frac { \mathrm { m } } { \mathrm { n } }$, where $\mathrm { m } , \mathrm { n }$ are coprime numbers, then $\mathrm { m } + \mathrm { n }$ is equal to :
(1) 21
(2) 9
(3) 14
(4) 6
Q7 Sign Change & Interval Methods View
$\lim _ { x \rightarrow \infty } \frac { \left( 2 x ^ { 2 } - 3 x + 5 \right) ( 3 x - 1 ) ^ { \frac { x } { 2 } } } { \left( 3 x ^ { 2 } + 5 x + 4 \right) \sqrt { ( 3 x + 2 ) ^ { x } } }$ is equal to :
(1) $\frac { 2 } { \sqrt { 3 \mathrm { e } } }$
(2) $\frac { 2 \mathrm { e } } { \sqrt { 3 } }$
(3) $\frac { 2 } { 3 \sqrt { e } }$
(4) $\frac { 2 e } { 3 }$
Q8 Vectors Introduction & 2D Section Ratios and Intersection via Vectors View
Let the point A divide the line segment joining the points $P ( - 1 , - 1,2 )$ and $Q ( 5,5,10 )$ internally in the ratio $\mathrm { r } : 1 ( \mathrm { r } > 0 )$. If O is the origin and $( \overrightarrow { \mathrm { OQ } } \cdot \overrightarrow { \mathrm { OA } } ) - \frac { 1 } { 5 } | \overrightarrow { \mathrm { OP } } \times \overrightarrow { \mathrm { OA } } | ^ { 2 } = 10$, then the value of r is :
(1) $\sqrt { 7 }$
(2) 14
(3) 3
(4) 7
Q9 Conic sections Chord Properties and Midpoint Problems View
The length of the chord of the ellipse $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 2 } = 1$, whose mid-point is $\left( 1 , \frac { 1 } { 2 } \right)$, is :
(1) $\frac { 5 } { 3 } \sqrt { 15 }$
(2) $\frac { 1 } { 3 } \sqrt { 15 }$
(3) $\frac { 2 } { 3 } \sqrt { 15 }$
(4) $\sqrt { 15 }$
Q10 Matrices Linear System and Inverse Existence View
The system of equations $$x + y + z = 6$$ $$x + 2 y + 5 z = 9$$ $$x + 5 y + \lambda z = \mu$$ has no solution if
(1) $\lambda = 15 , \mu \neq 17$
(2) $\lambda \neq 17 , \mu \neq 18$
(3) $\lambda = 17 , \mu \neq 18$
(4) $\lambda = 17 , \mu = 18$
Q11 Addition & Double Angle Formulae Function Analysis via Identity Transformation View
Let the range of the function $f ( x ) = 6 + 16 \cos x \cdot \cos \left( \frac { \pi } { 3 } - x \right) \cdot \cos \left( \frac { \pi } { 3 } + x \right) \cdot \sin 3 x \cdot \cos 6 x , x \in \mathbf { R }$ be $[ \alpha , \beta ]$. Then the distance of the point $( \alpha , \beta )$ from the line $3 x + 4 y + 12 = 0$ is :
(1) 11
(2) 8
(3) 10
(4) 9
Q12 Differential equations Solving Separable DEs with Initial Conditions View
Let $x = x ( y )$ be the solution of the differential equation $y = \left( x - y \frac { \mathrm {~d} x } { \mathrm {~d} y } \right) \sin \left( \frac { x } { y } \right) , y > 0$ and $x ( 1 ) = \frac { \pi } { 2 }$. Then $\cos ( x ( 2 ) )$ is equal to :
(1) $1 - 2 \left( \log _ { e } 2 \right) ^ { 2 }$
(2) $1 - 2 \left( \log _ { \mathrm { e } } 2 \right)$
(3) $2 \left( \log _ { e } 2 \right) - 1$
(4) $2 \left( \log _ { e } 2 \right) ^ { 2 } - 1$
Q13 Connected Rates of Change Reverse-Engineering a Geometric Quantity from Given Rates View
A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm, the ice-cream melts at the rate of $81 \mathrm {~cm} ^ { 3 } / \mathrm { min }$ and the thickness of the ice-cream layer decreases at the rate of $\frac { 1 } { 4 \pi } \mathrm {~cm} / \mathrm { min }$. The surface area (in $\mathrm { cm } ^ { 2 }$) of the chocolate ball (without the ice-cream layer) is :
(1) $196 \pi$
(2) $256 \pi$
(3) $225 \pi$
(4) $128 \pi$
Q14 Complex Numbers Arithmetic Solving Equations for Unknown Complex Numbers View
The number of complex numbers $z$, satisfying $| z | = 1$ and $\left| \frac { z } { \bar { z } } + \frac { \bar { z } } { z } \right| = 1$, is :
(1) 4
(2) 8
(3) 10
(4) 6
Q15 Matrices Linear System and Inverse Existence View
Let $A = \left[ a _ { i j } \right]$ be $3 \times 3$ matrix such that $A \left[ \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right] = \left[ \begin{array} { l } 0 \\ 0 \\ 1 \end{array} \right] , A \left[ \begin{array} { l } 4 \\ 1 \\ 3 \end{array} \right] = \left[ \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right]$ and $A \left[ \begin{array} { l } 2 \\ 1 \\ 2 \end{array} \right] = \left[ \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right]$, then $a _ { 23 }$ equals :
(1) $- 1$
(2) 2
(3) 1
(4) 0
Q16 Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution View
If $\mathrm { I } = \int _ { 0 } ^ { \frac { \pi } { 2 } } \frac { \sin ^ { \frac { 3 } { 2 } } x } { \sin ^ { \frac { 3 } { 2 } } x + \cos ^ { \frac { 3 } { 2 } } x } \mathrm {~d} x$, then $\int _ { 0 } ^ { 2\mathrm{I} } \frac { x \sin x \cos x } { \sin ^ { 4 } x + \cos ^ { 4 } x } \mathrm {~d} x$ equals :
(1) $\frac { \pi ^ { 2 } } { 12 }$
(2) $\frac { \pi ^ { 2 } } { 4 }$
(3) $\frac { \pi ^ { 2 } } { 16 }$
(4) $\frac { \pi ^ { 2 } } { 8 }$
Q17 Probability Definitions Finite Equally-Likely Probability Computation View
A board has 16 squares as shown in the figure (a $4 \times 4$ grid of squares). Out of these 16 squares, two squares are chosen at random. The probability that they have no side in common is :
(1) $7/10$
(2) $4/5$
(3) $23/30$
(4) $3/5$
Q18 Circles Circle Equation Derivation View
Let the shortest distance from $( \mathrm { a } , 0 )$, $\mathrm { a } > 0$, to the parabola $y ^ { 2 } = 4 x$ be 4. Then the equation of the circle passing through the point $( a , 0 )$ and the focus of the parabola, and having its centre on the axis of the parabola is :
(1) $x ^ { 2 } + y ^ { 2 } - 10 x + 9 = 0$
(2) $x ^ { 2 } + y ^ { 2 } - 6 x + 5 = 0$
(3) $x ^ { 2 } + y ^ { 2 } - 4 x + 3 = 0$
(4) $x ^ { 2 } + y ^ { 2 } - 8 x + 7 = 0$
Q19 Binomial Theorem (positive integer n) Determine Parameters from Conditions on Coefficients or Terms View
If in the expansion of $( 1 + x ) ^ { \mathrm { p } } ( 1 - x ) ^ { \mathrm { q } }$, the coefficients of $x$ and $x ^ { 2 }$ are 1 and $-2$, respectively, then $\mathrm { p } ^ { 2 } + \mathrm { q } ^ { 2 }$ is equal to :
(1) 18
(2) 13
(3) 8
(4) 20
Q20 Areas Between Curves Find Parameter Given Area Condition View
If the area of the region $\left\{ ( x , y ) : - 1 \leq x \leq 1 , 0 \leq y \leq a + \mathrm { e } ^ { | x | } - \mathrm { e } ^ { - x } , \mathrm { a } > 0 \right\}$ is $\frac { \mathrm { e } ^ { 2 } + 8 \mathrm { e } + 1 } { \mathrm { e } }$, then the value of $a$ is :
(1) 8
(2) 7
(3) 5
(4) 6
Q21 Measures of Location and Spread View
The variance of the numbers $8, 21, 34, 47, \ldots, 320$ is
Q22 Arithmetic Sequences and Series Find Specific Term from Given Conditions View
The roots of the quadratic equation $3 x ^ { 2 } - \mathrm { p } x + \mathrm { q } = 0$ are $10 ^ { \text {th} }$ and $11 ^ { \text {th} }$ terms of an arithmetic progression with common difference $\frac { 3 } { 2 }$. If the sum of the first 11 terms of this arithmetic progression is 88, then $q - 2 p$ is equal to
Q23 Permutations & Arrangements Linear Arrangement with Constraints View
The number of ways, 5 boys and 4 girls can sit in a row so that either all the boys sit together or no two boys sit together, is
Q24 Circles Circle Equation Derivation View
The focus of the parabola $y ^ { 2 } = 4 x + 16$ is the centre of the circle $C$ of radius 5. If the values of $\lambda$, for which $C$ passes through the point of intersection of the lines $3 x - y = 0$ and $x + \lambda y = 4$, are $\lambda _ { 1 }$ and $\lambda _ { 2 }$, $\lambda _ { 1 } < \lambda _ { 2 }$, then $12 \lambda _ { 1 } + 29 \lambda _ { 2 }$ is equal to
Q25 Complex Numbers Arithmetic Powers of i or Complex Number Integer Powers View
Let $\alpha , \beta$ be the roots of the equation $x ^ { 2 } - a x - b = 0$ with $\operatorname { Im } ( \alpha ) < \operatorname { Im } ( \beta )$. Let $P _ { n } = \alpha ^ { n } - \beta ^ { n }$. If $\mathrm { P } _ { 3 } = - 5 \sqrt { 7 } i , \mathrm { P } _ { 4 } = - 3 \sqrt { 7 } i , \mathrm { P } _ { 5 } = 11 \sqrt { 7 } i$ and $\mathrm { P } _ { 6 } = 45 \sqrt { 7 } i$, then $\left| \alpha ^ { 4 } + \beta ^ { 4 } \right|$ is equal to