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_22jan_shift1

25 maths questions

Q1 Geometric Sequences and Series Finite Geometric Sum and Term Relationships View
Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ be a G.P. of increasing positive terms. If $a _ { 1 } a _ { 5 } = 28$ and $a _ { 2 } + a _ { 4 } = 29$, then $a _ { 6 }$ is equal to:
(1) 628
(2) 812
(3) 526
(4) 784
Q2 First order differential equations (integrating factor) View
Let $x = x ( y )$ be the solution of the differential equation $y ^ { 2 } \mathrm {~d} x + \left( x - \frac { 1 } { y } \right) \mathrm { d } y = 0$. If $x ( 1 ) = 1$, then $x \left( \frac { 1 } { 2 } \right)$ is :
(1) $\frac { 1 } { 2 } + \mathrm { e }$
(2) $3 + e$
(3) $3 - e$
(4) $\frac { 3 } { 2 } + e$
Q3 Conditional Probability Sequential/Multi-Stage Conditional Probability View
Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black balls. If the probability that the first selected ball is black, given that the second selected ball is also black, is $\frac { m } { n }$, where $\operatorname { gcd } ( m , n ) = 1$, then $m + n$ is equal to:
(1) 4
(2) 14
(3) 13
(4) 11
Q4 Exponential Equations & Modelling Solve Exponential Equation for Unknown Variable View
The product of all solutions of the equation $\mathrm { e } ^ { 5 \left( \log _ { \mathrm { e } } x \right) ^ { 2 } + 3 } = x ^ { 8 } , x > 0$, is:
(1) $e ^ { 8 / 5 }$
(2) $e ^ { 6 / 5 }$
(3) $e ^ { 2 }$
(4) e
Q5 Straight Lines & Coordinate Geometry Reflection and Image in a Line View
Let the triangle PQR be the image of the triangle with vertices $( 1,3 ) , ( 3,1 )$ and $( 2,4 )$ in the line $x + 2 y = 2$. If the centroid of $\triangle \mathrm { PQR }$ is the point $( \alpha , \beta )$, then $15 ( \alpha - \beta )$ is equal to:
(1) 19
(2) 24
(3) 21
(4) 22
Q6 Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution View
Let for $f ( x ) = 7 \tan ^ { 8 } x + 7 \tan ^ { 6 } x - 3 \tan ^ { 4 } x - 3 \tan ^ { 2 } x , \quad \mathrm { I } _ { 1 } = \int _ { 0 } ^ { \pi / 4 } f ( x ) \mathrm { d } x$ and $\mathrm { I } _ { 2 } = \int _ { 0 } ^ { \pi / 4 } x f ( x ) \mathrm { d } x$. Then $7 \mathrm { I } _ { 1 } + 12 \mathrm { I } _ { 2 }$ is equal to:
(1) 2
(2) 1
(3) $2 \pi$
(4) $\pi$
Q7 Circles Inscribed/Circumscribed Circle Computations View
Let the parabola $y = x ^ { 2 } + \mathrm { p } x - 3$, meet the coordinate axes at the points $\mathrm { P } , \mathrm { Q }$ and R. If the circle C with centre at $( - 1 , - 1 )$ passes through the points $P , Q$ and $R$, then the area of $\triangle P Q R$ is:
(1) 7
(2) 4
(3) 6
(4) 5
Q8 Vectors: Cross Product & Distances View
Let $\mathrm { L } _ { 1 } : \frac { x - 1 } { 2 } = \frac { y - 2 } { 3 } = \frac { z - 3 } { 4 }$ and $\mathrm { L } _ { 2 } : \frac { x - 2 } { 3 } = \frac { y - 4 } { 4 } = \frac { z - 5 } { 5 }$ be two lines. Then which of the following points lies on the line of the shortest distance between $L _ { 1 }$ and $L _ { 2 }$?
(1) $\left( \frac { 14 } { 3 } , - 3 , \frac { 22 } { 3 } \right)$
(2) $\left( - \frac { 5 } { 3 } , - 7,1 \right)$
(3) $\left( 2,3 , \frac { 1 } { 3 } \right)$
(4) $\left( \frac { 8 } { 3 } , - 1 , \frac { 1 } { 3 } \right)$
Q9 Differential equations Finding a DE from a Limit or Implicit Condition View
Let $f ( x )$ be a real differentiable function such that $f ( 0 ) = 1$ and $f ( x + y ) = f ( x ) f ^ { \prime } ( y ) + f ^ { \prime } ( x ) f ( y )$ for all $x , y \in \mathbf { R }$. Then $\sum _ { \mathrm { n } = 1 } ^ { 100 } \log _ { \mathrm { e } } f ( \mathrm { n } )$ is equal to:
(1) 2525
(2) 5220
(3) 2384
(4) 2406
Q10 Combinations & Selection Selection with Adjacency or Spacing Constraints View
From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is 'M', is:
(1) 5148
(2) 6084
(3) 4356
(4) 14950
Q11 Reciprocal Trig & Identities View
Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum values of $16 \left( \left( \sec ^ { - 1 } x \right) ^ { 2 } + \left( \operatorname { cosec } ^ { - 1 } x \right) ^ { 2 } \right)$ is:
(1) $24 \pi ^ { 2 }$
(2) $22 \pi ^ { 2 }$
(3) $31 \pi ^ { 2 }$
(4) $18 \pi ^ { 2 }$
Q12 Areas by integration View
Let $f : \mathbf { R } \rightarrow \mathbf { R }$ be a twice differentiable function such that $f ( x + y ) = f ( x ) f ( y )$ for all $x , y \in \mathbf { R }$. If $f ^ { \prime } ( 0 ) = 4 \mathrm { a }$ and $f$ satisfies $f ^ { \prime \prime } ( x ) - 3 \mathrm { a } f ^ { \prime } ( x ) - f ( x ) = 0 , \mathrm { a } > 0$, then the area of the region $\mathrm { R } = \{ ( x , y ) \mid 0 \leq y \leq f ( \mathrm { a } x ) , 0 \leq x \leq 2 \}$ is:
(1) $e ^ { 2 } - 1$
(2) $\mathrm { e } ^ { 2 } + 1$
(3) $e ^ { 4 } + 1$
(4) $e ^ { 4 } - 1$
Q13 Areas Between Curves Area Involving Conic Sections or Circles View
The area of the region, inside the circle $( x - 2 \sqrt { 3 } ) ^ { 2 } + y ^ { 2 } = 12$ and outside the parabola $y ^ { 2 } = 2 \sqrt { 3 } x$ is:
(1) $3 \pi + 8$
(2) $6 \pi - 16$
(3) $3 \pi - 8$
(4) $6 \pi - 8$
Q14 Conic sections Eccentricity or Asymptote Computation View
Let the foci of a hyperbola be $( 1,14 )$ and $( 1 , - 12 )$. If it passes through the point $( 1,6 )$, then the length of its latus-rectum is:
(1) $\frac { 24 } { 5 }$
(2) $\frac { 25 } { 6 }$
(3) $\frac { 144 } { 5 }$
(4) $\frac { 288 } { 5 }$
Q15 Sequences and Series Evaluation of a Finite or Infinite Sum View
If $\sum _ { r = 1 } ^ { n } T _ { r } = \frac { ( 2 n - 1 ) ( 2 n + 1 ) ( 2 n + 3 ) ( 2 n + 5 ) } { 64 }$, then $\lim _ { n \rightarrow \infty } \sum _ { r = 1 } ^ { n } \left( \frac { 1 } { T _ { r } } \right)$ is equal to:
(1) 0
(2) $\frac { 2 } { 3 }$
(3) 1
(4) $\frac { 1 } { 3 }$
Q16 Discrete Probability Distributions Expectation and Variance from Context-Based Random Variables View
A coin is tossed three times. Let $X$ denote the number of times a tail follows a head. If $\mu$ and $\sigma ^ { 2 }$ denote the mean and variance of $X$, then the value of $64 \left( \mu + \sigma ^ { 2 } \right)$ is:
(1) 51
(2) 64
(3) 32
(4) 48
Q17 Groups Group Order and Structure Theorems View
The number of non-empty equivalence relations on the set $\{ 1,2,3 \}$ is:
(1) 6
(2) 5
(3) 7
(4) 4
Q18 Circles Circles Tangent to Each Other or to Axes View
A circle $C$ of radius 2 lies in the second quadrant and touches both the coordinate axes. Let $r$ be the radius of a circle that has centre at the point $( 2,5 )$ and intersects the circle $C$ at exactly two points. If the set of all possible values of r is the interval $( \alpha , \beta )$, then $3 \beta - 2 \alpha$ is equal to:
(1) 10
(2) 15
(3) 12
(4) 14
Q19 Number Theory GCD, LCM, and Coprimality View
Let $A = \{ 1,2,3 , \ldots , 10 \}$ and $B = \left\{ \frac { m } { n } : m , n \in A , m < n \right.$ and $\left. \operatorname { gcd } ( m , n ) = 1 \right\}$. Then $n ( B )$ is equal to:
(1) 36
(2) 31
(3) 37
(4) 29
Q20 Complex Numbers Argand & Loci Powers and Roots of Unity with Geometric Consequences View
Let $z _ { 1 } , z _ { 2 }$ and $z _ { 3 }$ be three complex numbers on the circle $| z | = 1$ with $\arg \left( z _ { 1 } \right) = \frac { - \pi } { 4 } , \arg \left( z _ { 2 } \right) = 0$ and $\arg \left( z _ { 3 } \right) = \frac { \pi } { 4 }$. If $\left| z _ { 1 } \bar { z } _ { 2 } + z _ { 2 } \bar { z } _ { 3 } + z _ { 3 } \bar { z } _ { 1 } \right| ^ { 2 } = \alpha + \beta \sqrt { 2 } , \alpha , \beta \in \mathbf { Z }$, then the value of $\alpha ^ { 2 } + \beta ^ { 2 }$ is:
(1) 24
(2) 29
(3) 41
(4) 31
Q21 Matrices Determinant and Rank Computation View
Let $A$ be a square matrix of order 3 such that $\operatorname { det } ( A ) = - 2$ and $\operatorname { det } ( 3 \operatorname { adj } ( - 6 \operatorname { adj } ( 3 A ) ) ) = 2 ^ { \mathrm { m } + \mathrm { n } } \cdot 3 ^ { \mathrm { mn } } , \mathrm { m } > \mathrm { n }$. Then $4 \mathrm {~m} + 2 \mathrm { n }$ is equal to $\_\_\_\_$
Q22 Binomial Theorem (positive integer n) Evaluate a Summation Involving Binomial Coefficients View
If $\sum _ { r = 0 } ^ { 5 } \frac { { } ^ { 11 } C _ { 2r } } { 2 r + 2 } = \frac { \mathrm { m } } { \mathrm { n } } , \operatorname { gcd } ( \mathrm { m } , \mathrm { n } ) = 1$, then $\mathrm { m } - \mathrm { n }$ is equal to $\_\_\_\_$
Q23 Vectors Introduction & 2D Area Computation Using Vectors View
Let $\vec { c }$ be the projection vector of $\vec { b } = \lambda \hat { i } + 4 \hat { k } , \lambda > 0$, on the vector $\vec { a } = \hat { i } + 2 \hat { j } + 2 \hat { k }$. If $| \vec { a } + \vec { c } | = 7$, then the area of the parallelogram formed by the vectors $\vec { b }$ and $\vec { c }$ is $\_\_\_\_$
Q24 Areas by integration View
Let the function, $f ( x ) = \left\{ \begin{array} { l l } - 3 a x ^ { 2 } - 2 , & x < 1 \\ a ^ { 2 } + b x , & x \geqslant 1 \end{array} \right.$ be differentiable for all $x \in \mathbf { R }$, where $\mathbf { a } > 1 , \mathbf { b } \in \mathbf { R }$. If the area of the region enclosed by $y = f ( x )$ and the line $y = - 20$ is $\alpha + \beta \sqrt { 3 } , \alpha , \beta \in Z$, then the value of $\alpha + \beta$ is $\_\_\_\_$
Q25 Vectors 3D & Lines Line-Plane Intersection View
Let $\mathrm { L } _ { 1 } : \frac { x - 1 } { 3 } = \frac { y - 1 } { - 1 } = \frac { z + 1 } { 0 }$ and $\mathrm { L } _ { 2 } : \frac { x - 2 } { 2 } = \frac { y } { 0 } = \frac { z + 4 } { \alpha } , \alpha \in \mathbf { R }$, be two lines, which intersect at the point $B$. If $P$ is the foot of perpendicular from the point $A ( 1,1 , - 1 )$ on $L _ { 2 }$, then the value of $26 \alpha ( \mathrm {~PB} ) ^ { 2 }$ is $\_\_\_\_$