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_28jan_shift1

25 maths questions

Q1 Complex Numbers Argand & Loci Geometric Properties of Triangles/Polygons from Affixes View
Let $O$ be the origin, the point $A$ be $z _ { 1 } = \sqrt { 3 } + 2 \sqrt { 2 } i$, the point $B \left( z _ { 2 } \right)$ be such that $\sqrt { 3 } \left| z _ { 2 } \right| = \left| z _ { 1 } \right|$ and $\arg \left( z _ { 2 } \right) = \arg \left( z _ { 1 } \right) + \frac { \pi } { 6 }$. Then
(1) area of triangle ABO is $\frac { 11 } { \sqrt { 3 } }$
(2) ABO is an obtuse angled isosceles triangle
(3) area of triangle ABO is $\frac { 11 } { 4 }$
(4) ABO is a scalene triangle
Q2 Solving quadratics and applications Determining quadratic function from given conditions View
Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a function defined by $f ( x ) = ( 2 + 3 a ) x ^ { 2 } + \left( \frac { a + 2 } { a - 1 } \right) x + b , a \neq 1$. If $f ( x + \mathrm { y } ) = f ( x ) + f ( \mathrm { y } ) + 1 - \frac { 2 } { 7 } x \mathrm { y }$, then the value of $28 \sum _ { i = 1 } ^ { 5 } | f ( i ) |$ is
(1) 545
(2) 715
(3) 735
(4) 675
Q3 Conic sections Triangle or Quadrilateral Area and Perimeter with Foci View
Let $ABCD$ be a trapezium whose vertices lie on the parabola $y ^ { 2 } = 4 x$. Let the sides $AD$ and $BC$ of the trapezium be parallel to y-axis. If the diagonal AC is of length $\frac { 25 } { 4 }$ and it passes through the point $( 1,0 )$, then the area of $ABCD$ is
(1) $\frac { 75 } { 4 }$
(2) $\frac { 25 } { 2 }$
(3) $\frac { 125 } { 8 }$
(4) $\frac { 75 } { 8 }$
Q4 Stationary points and optimisation Composite or piecewise function extremum analysis View
The sum of all local minimum values of the function
$$f ( x ) = \left\{ \begin{array} { l r } 1 - 2 x , & x < - 1 \\ \frac { 1 } { 3 } ( 7 + 2 | x | ) , & - 1 \leq x \leq 2 \\ \frac { 11 } { 18 } ( x - 4 ) ( x - 5 ) , & x > 2 \end{array} \right.$$
is
(1) $\frac { 157 } { 72 }$
(2) $\frac { 131 } { 72 }$
(3) $\frac { 171 } { 72 }$
(4) $\frac { 167 } { 72 }$
Q5 Combinations & Selection Basic Combination Computation View
Let ${ } ^ { \mathrm { n } } \mathrm { C } _ { \mathrm { r } - 1 } = 28 , { } ^ { \mathrm { n } } \mathrm { C } _ { \mathrm { r } } = 56$ and ${ } ^ { \mathrm { n } } \mathrm { C } _ { \mathrm { r } + 1 } = 70$. Let $\mathrm { A } ( 4 \cos t , 4 \sin t ) , \mathrm { B } ( 2 \sin t , - 2 \cos \mathrm { t } )$ and $C \left( 3 r - n , r ^ { 2 } - n - 1 \right)$ be the vertices of a triangle $ABC$, where $t$ is a parameter. If $( 3 x - 1 ) ^ { 2 } + ( 3 y ) ^ { 2 } = \alpha$, is the locus of the centroid of triangle ABC, then $\alpha$ equals
(1) 6
(2) 18
(3) 8
(4) 20
Q6 Circles Circle Equation Derivation View
Let the equation of the circle, which touches $x$-axis at the point $( a , 0 ) , a > 0$ and cuts off an intercept of length $b$ on $y$-axis be $x ^ { 2 } + y ^ { 2 } - \alpha x + \beta y + \gamma = 0$. If the circle lies below $x$-axis, then the ordered pair $( 2 a , b ^ { 2 } )$ is equal to
(1) $\left( \gamma , \beta ^ { 2 } - 4 \alpha \right)$
(2) $\left( \alpha , \beta ^ { 2 } + 4 \gamma \right)$
(3) $\left( \gamma , \beta ^ { 2 } + 4 \alpha \right)$
(4) $\left( \alpha , \beta ^ { 2 } - 4 \gamma \right)$
Q7 Exponential Functions Algebraic Simplification and Expression Manipulation View
If $f ( x ) = \frac { 2 ^ { x } } { 2 ^ { x } + \sqrt { 2 } } , \mathrm { x } \in \mathbb { R }$, then $\sum _ { \mathrm { k } = 1 } ^ { 81 } f \left( \frac { \mathrm { k } } { 82 } \right)$ is equal to
(1) $1.81 \sqrt { 2 }$
(2) 41
(3) 82
(4) $\frac { 81 } { 2 }$
Q8 Probability Definitions Probability Involving Algebraic or Number-Theoretic Conditions View
Two numbers $\mathrm { k } _ { 1 }$ and $\mathrm { k } _ { 2 }$ are randomly chosen from the set of natural numbers. Then, the probability that the value of $\mathrm { i } ^ { \mathrm { k } _ { 1 } } + \mathrm { i } ^ { \mathrm { k } _ { 2 } } , ( \mathrm { i } = \sqrt { - 1 } )$ is non-zero, equals
(1) $\frac { 1 } { 2 }$
(2) $\frac { 3 } { 4 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 2 } { 3 }$
Q9 Vectors 3D & Lines Distance from a Point to a Line (Show/Compute) View
If the image of the point $( 4,4,3 )$ in the line $\frac { x - 1 } { 2 } = \frac { y - 2 } { 1 } = \frac { z - 1 } { 3 }$ is $( \alpha , \beta , \gamma )$, then $\alpha + \beta + \gamma$ is equal to
(1) 9
(2) 12
(3) 7
(4) 8
Q10 Addition & Double Angle Formulae Addition/Subtraction Formula Evaluation View
$\cos \left( \sin ^ { - 1 } \frac { 3 } { 5 } + \sin ^ { - 1 } \frac { 5 } { 13 } + \sin ^ { - 1 } \frac { 33 } { 65 } \right)$ is equal to:
(1) 1
(2) 0
(3) $\frac { 32 } { 65 }$
(4) $\frac { 33 } { 65 }$
Q11 Vectors 3D & Lines Section Division and Coordinate Computation View
Let $\mathrm { A } ( x , y , z )$ be a point in $xy$-plane, which is equidistant from three points $( 0,3,2 ) , ( 2,0,3 )$ and $( 0,0,1 )$. Let $\mathrm { B } = ( 1,4 , - 1 )$ and $\mathrm { C } = ( 2,0 , - 2 )$. Then among the statements (S1) : $\triangle \mathrm { ABC }$ is an isosceles right angled triangle, and (S2) : the area of $\triangle \mathrm { ABC }$ is $\frac { 9 \sqrt { 2 } } { 2 }$,
(1) both are true
(2) only (S2) is true
(3) only (S1) is true
(4) both are false
Q12 Areas Between Curves Area Involving Piecewise or Composite Functions View
The area (in sq. units) of the region $\left\{ ( x , y ) : 0 \leq y \leq 2 | x | + 1,0 \leq y \leq x ^ { 2 } + 1 , | x | \leq 3 \right\}$ is
(1) $\frac { 80 } { 3 }$
(2) $\frac { 64 } { 3 }$
(3) $\frac { 32 } { 3 }$
(4) $\frac { 17 } { 3 }$
Q13 Modulus function Solving equations involving modulus View
The sum of the squares of all the roots of the equation $x ^ { 2 } + | 2 x - 3 | - 4 = 0$, is
(1) $3 ( 3 - \sqrt { 2 } )$
(2) $6 ( 3 - \sqrt { 2 } )$
(3) $6 ( 2 - \sqrt { 2 } )$
(4) $3 ( 2 - \sqrt { 2 } )$
Q14 Arithmetic Sequences and Series Compute Partial Sum of an Arithmetic Sequence View
Let $\mathrm { T } _ { \mathrm { r } }$ be the $\mathrm { r } ^ { \text {th} }$ term of an A.P. If for some $\mathrm { m } , T _ { m } = \frac { 1 } { 25 } , T _ { 25 } = \frac { 1 } { 20 }$, and $20 \sum _ { \mathrm { r } = 1 } ^ { 25 } T _ { \mathrm { r } } = 13$, then $5 \mathrm { m } \sum _ { \mathrm { r } = \mathrm { m } } ^ { 2 \mathrm { m } } T _ { \mathrm { r } }$ is equal to
(1) 98
(2) 126
(3) 142
(4) 112
Q15 Discrete Probability Distributions Probability Distribution Table Completion and Expectation Calculation View
Three defective oranges are accidently mixed with seven good ones and on looking at them, it is not possible to differentiate between them. Two oranges are drawn at random from the lot. If $x$ denote the number of defective oranges, then the variance of $x$ is
(1) $28/75$
(2) $18/25$
(3) $26/75$
(4) $14/25$
Q16 Indefinite & Definite Integrals Finding a Function from an Integral Equation View
Let for some function $\mathrm { y } = f ( x ) , \int _ { 0 } ^ { x } t f ( t ) d t = x ^ { 2 } f ( x ) , x > 0$ and $f ( 2 ) = 3$. Then $f ( 6 )$ is equal to
(1) 1
(2) 3
(3) 6
(4) 2
Q17 Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution View
If $\int _ { - \frac { \pi } { 2 } } ^ { \frac { \pi } { 2 } } \frac { 96 x ^ { 2 } \cos ^ { 2 } x } { \left( 1 + e ^ { x } \right) } \mathrm { d } x = \pi \left( \alpha \pi ^ { 2 } + \beta \right) , \alpha , \beta \in \mathbb { Z }$, then $( \alpha + \beta ) ^ { 2 }$ equals
(1) 64
(2) 196
(3) 144
(4) 100
Q18 Sequences and series, recurrence and convergence Summation of sequence terms View
Let $\left\langle a _ { \mathrm { n } } \right\rangle$ be a sequence such that $a _ { 0 } = 0 , a _ { 1 } = \frac { 1 } { 2 }$ and $2 a _ { \mathrm { n } + 2 } = 5 a _ { \mathrm { n } + 1 } - 3 a _ { \mathrm { n } } , \mathrm { n } = 0,1,2,3 , \ldots$ Then $\sum _ { \mathrm { k } = 1 } ^ { 100 } a _ { k }$ is equal to
(1) $3 a _ { 99 } - 100$
(2) $3 \mathrm { a } _ { 100 } - 100$
(3) $3 a _ { 99 } + 100$
(4) $3 \mathrm { a } _ { 100 } + 100$
Q19 Permutations & Arrangements Forming Numbers with Digit Constraints View
The number of different 5 digit numbers greater than 50000 that can be formed using the digits $0, 1, 2, 3, 4, 5, 6, 7$, such that the sum of their first and last digits should not be more than 8, is
(1) 4608
(2) 5720
(3) 5719
(4) 4607
Q20 Proof True/False Justification View
The relation $R = \{ ( x , y ) : x , y \in \mathbb { Z }$ and $x + y$ is even $\}$ is:
(1) reflexive and symmetric but not transitive
(2) an equivalence relation
(3) symmetric and transitive but not reflexive
(4) reflexive and transitive but not symmetric
Q21 Curve Sketching Continuity and Discontinuity Analysis of Piecewise Functions View
Let $\mathrm { f } ( x ) = \left\{ \begin{array} { l l } 3 x , & x < 0 \\ \min \{ 1 + x + [ x ] , x + 2 [ x ] \} , & 0 \leq x \leq 2 \\ 5 , & x > 2 , \end{array} \right.$ where [.] denotes greatest integer function. If $\alpha$ and $\beta$ are the number of points, where f is not continuous and is not differentiable, respectively, then $\alpha + \beta$ equals
Q22 Matrices Matrix Entry and Coefficient Identities View
Let M denote the set of all real matrices of order $3 \times 3$ and let $\mathrm { S } = \{ - 3 , - 2 , - 1,1,2 \}$. Let
$$\begin{aligned} & \mathrm { S } _ { 1 } = \left\{ \mathrm { A } = \left[ a _ { \mathrm { ij } } \right] \in \mathrm { M } : \mathrm { A } = \mathrm { A } ^ { \mathrm { T } } \text { and } a _ { \mathrm { ij } } \in \mathrm { S } , \forall \mathrm { i } , \mathrm { j } \right\} , \\ & \mathrm { S } _ { 2 } = \left\{ \mathrm { A } = \left[ a _ { \mathrm { ij } } \right] \in \mathrm { M } : \mathrm { A } = - \mathrm { A } ^ { \mathrm { T } } \text { and } a _ { \mathrm { ij } } \in \mathrm { S } , \forall \mathrm { i } , \mathrm { j } \right\} , \\ & \mathrm { S } _ { 3 } = \left\{ \mathrm { A } = \left[ a _ { \mathrm { ij } } \right] \in \mathrm { M } : a _ { 11 } + a _ { 22 } + a _ { 33 } = 0 \text { and } a _ { \mathrm { ij } } \in \mathrm { S } , \forall \mathrm { i } , \mathrm { j } \right\} . \end{aligned}$$
If $n \left( \mathrm { S } _ { 1 } \cup \mathrm { S } _ { 2 } \cup \mathrm { S } _ { 3 } \right) = 125 \alpha$, then $\alpha$ equals
Q23 Binomial Theorem (positive integer n) Evaluate a Summation Involving Binomial Coefficients View
If $\alpha = 1 + \sum _ { r = 1 } ^ { 6 } ( - 3 ) ^ { r - 1 } \quad { } ^ { 12 } \mathrm { C } _ { 2 r - 1 }$, then the distance of the point $( 12 , \sqrt { 3 } )$ from the line $\alpha x - \sqrt { 3 } y + 1 = 0$ is $\_\_\_\_$.
Q24 Geometric Sequences and Series Sum of an Infinite Geometric Series (Direct Computation) View
Let $\mathrm { E } _ { 1 } : \frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$ be an ellipse. Ellipses $\mathrm { E } _ { i }$ are constructed such that their centres and eccentricities are same as that of $E _ { 1 }$, and the length of minor axis of $E _ { i }$ is the length of major axis of $E _ { i + 1 } ( i \geq 1 )$. If $A _ { i }$ is the area of the ellipse $E _ { i }$, then $\frac { 5 } { \pi } \left( \sum _ { i = 1 } ^ { \infty } A _ { i } \right)$, is equal to
Q25 Vectors Introduction & 2D Magnitude of Vector Expression View
Let $\vec { a } = \hat { \mathrm { i } } + \hat { \mathrm { j } } + \hat { \mathrm { k } } , \overrightarrow { \mathrm { b } } = 2 \hat { \mathrm { i } } + 2 \hat { \mathrm { j } } + \hat { \mathrm { k } }$ and $\overrightarrow { \mathrm { d } } = \vec { a } \times \overrightarrow { \mathrm { b } }$. If $\overrightarrow { \mathrm { c } }$ is a vector such that $\vec { a } \cdot \overrightarrow { \mathrm { c } } = | \overrightarrow { \mathrm { c } } | , | \overrightarrow { \mathrm { c } } - 2 \vec { a } | ^ { 2 } = 8$ and the angle between $\overrightarrow { \mathrm { d } }$ and $\overrightarrow { \mathrm { c } }$ is $\frac { \pi } { 4 }$, then $| 10 - 3 \overrightarrow { \mathrm { b } } \cdot \overrightarrow { \mathrm { c } } | + | \overrightarrow { \mathrm { d } } \times \overrightarrow { \mathrm { c } } | ^ { 2 }$ is equal to