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
07apr 29 09apr 14 22apr 5 23apr 14 25apr 13
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_29jan_shift2

25 maths questions

Q1 Indefinite & Definite Integrals Accumulation Function Analysis View
Let $f ( x ) = \int _ { 0 } ^ { x } t \left( t ^ { 2 } - 9 t + 20 \right) d t , 1 \leq x \leq 5$. If the range of $f$ is $[ \alpha , \beta ]$, then $4 ( \alpha + \beta )$ equals:
(1) 253
(2) 154
(3) 125
(4) 157
Q2 Vectors 3D & Lines Vector Algebra and Triple Product Computation View
Let $\hat { a }$ be a unit vector perpendicular to the vector $\overrightarrow { \mathrm { b } } = \hat { i } - 2 \hat { j } + 3 \hat { k }$ and $\overrightarrow { \mathrm { c } } = 2 \hat { i } + 3 \hat { j } - \hat { k }$, and makes an angle of $\cos ^ { - 1 } \left( - \frac { 1 } { 3 } \right)$ with the vector $\hat { i } + \hat { j } + \hat { k }$. If $\hat { a }$ makes an angle of $\frac { \pi } { 3 }$ with the vector $\hat { i } + \alpha \hat { j } + \hat { k }$, then the value of $\alpha$ is :
(1) $\sqrt { 6 }$
(2) $- \sqrt { 6 }$
(3) $- \sqrt { 3 }$
(4) $\sqrt { 3 }$
Q3 First order differential equations (integrating factor) View
If for the solution curve $y = f ( x )$ of the differential equation $\frac { \mathrm { d } y } { \mathrm {~d} x } + ( \tan x ) y = \frac { 2 + \sec x } { ( 1 + 2 \sec x ) ^ { 2 } }$, $x \in \left( \frac { - \pi } { 2 } , \frac { \pi } { 2 } \right) , f \left( \frac { \pi } { 3 } \right) = \frac { \sqrt { 3 } } { 10 }$, then $f \left( \frac { \pi } { 4 } \right)$ is equal to :
(1) $\frac { \sqrt { 3 } + 1 } { 10 ( 4 + \sqrt { 3 } ) }$
(2) $\frac { 5 - \sqrt { 3 } } { 2 \sqrt { 2 } }$
(3) $\frac { 9 \sqrt { 3 } + 3 } { 10 ( 4 + \sqrt { 3 } ) }$
(4) $\frac { 4 - \sqrt { 2 } } { 14 }$
Q4 Vectors 3D & Lines Distance from a Point to a Line (Show/Compute) View
Let P be the foot of the perpendicular from the point $( 1,2,2 )$ on the line $\mathrm { L } : \frac { x - 1 } { 1 } = \frac { y + 1 } { - 1 } = \frac { z - 2 } { 2 }$. Let the line $\vec { r } = ( - \hat { i } + \hat { j } - 2 \hat { k } ) + \lambda ( \hat { i } - \hat { j } + \hat { k } ) , \lambda \in \mathbf { R }$, intersect the line L at Q . Then $2 ( \mathrm { PQ } ) ^ { 2 }$ is equal to :
(1) 25
(2) 19
(3) 29
(4) 27
Q5 Matrices Matrix Power Computation and Application View
Let $A = \left[ a _ { i j } \right]$ be a matrix of order $3 \times 3$, with $a _ { i j } = ( \sqrt { 2 } ) ^ { i + j }$. If the sum of all the elements in the third row of $A ^ { 2 }$ is $\alpha + \beta \sqrt { 2 } , \alpha , \beta \in \mathbf { Z }$, then $\alpha + \beta$ is equal to :
(1) 280
(2) 224
(3) 210
(4) 168
Q6 Straight Lines & Coordinate Geometry Area Computation in Coordinate Geometry View
Let the line $x + y = 1$ meet the axes of $x$ and $y$ at A and B , respectively. A right angled triangle AMN is inscribed in the triangle OAB , where O is the origin and the points M and N lie on the lines $O B$ and $A B$, respectively. If the area of the triangle $A M N$ is $\frac { 4 } { 9 }$ of the area of the triangle $O A B$ and $\mathrm { AN } : \mathrm { NB } = \lambda : 1$, then the sum of all possible value(s) of $\lambda$ is:
(1) 2
(2) $\frac { 5 } { 2 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 13 } { 6 }$
Q7 Permutations & Arrangements Dictionary Order / Rank of a Permutation View
If all the words with or without meaning made using all the letters of the word ``KANPUR'' are arranged as in a dictionary, then the word at $440 ^ { \text {th} }$ position in this arrangement, is :
(1) PRNAUK
(2) PRKANU
(3) PRKAUN
(4) PRNAKU
Q8 Discriminant and conditions for roots Parameter range for no real roots (positive definite) View
If the set of all $\mathrm { a } \in \mathbf { R }$, for which the equation $2 x ^ { 2 } + ( a - 5 ) x + 15 = 3 \mathrm { a }$ has no real root, is the interval $( \alpha , \beta )$, and $X = \{ x \in Z : \alpha < x < \beta \}$, then $\sum _ { x \in X } x ^ { 2 }$ is equal to :
(1) 2109
(2) 2129
(3) 2119
(4) 2139
Q9 Discrete Random Variables Probability Distribution Construction and Parameter Determination View
Let $\mathrm { A } = \left[ \mathrm { a } _ { i j } \right]$ be a $2 \times 2$ matrix such that $\mathrm { a } _ { i j } \in \{ 0,1 \}$ for all $i$ and $j$. Let the random variable X denote the possible values of the determinant of the matrix $A$. Then, the variance of $X$ is :
(1) $\frac { 3 } { 4 }$
(2) $\frac { 5 } { 8 }$
(3) $\frac { 3 } { 8 }$
(4) $\frac { 1 } { 4 }$
Q10 Implicit equations and differentiation Piecewise differentiability and continuity conditions View
Let the function $f ( x ) = \left( x ^ { 2 } + 1 \right) \left| x ^ { 2 } - a x + 2 \right| + \cos | x |$ be not differentiable at the two points $x = \alpha = 2$ and $x = \beta$. Then the distance of the point $( \alpha , \beta )$ from the line $12 x + 5 y + 10 = 0$ is equal to :
(1) 5
(2) 4
(3) 3
(4) 2
Q11 Areas Between Curves Area Involving Conic Sections or Circles View
Let the area enclosed between the curves $| y | = 1 - x ^ { 2 }$ and $x ^ { 2 } + y ^ { 2 } = 1$ be $\alpha$. If $9 \alpha = \beta \pi + \gamma ; \beta , \gamma$ are integers, then the value of $| \beta - \gamma |$ equals.
(1) 27
(2) 33
(3) 15
(4) 18
Q12 Number Theory Modular Arithmetic Computation View
The remainder, when $7 ^ { 103 }$ is divided by 23 , is equal to :
(1) 6
(2) 17
(3) 9
(4) 14
Q13 Conic sections Chord Properties and Midpoint Problems View
If $\alpha x + \beta y = 109$ is the equation of the chord of the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$, whose mid point is $\left( \frac { 5 } { 2 } , \frac { 1 } { 2 } \right)$, then $\alpha + \beta$ is equal to :
(1) 58
(2) 46
(3) 37
(4) 72
Q14 Curve Sketching Range and Image Set Determination View
If the domain of the function $\log _ { 5 } \left( 18 x - x ^ { 2 } - 77 \right)$ is $( \alpha , \beta )$ and the domain of the function $\log _ { ( x - 1 ) } \left( \frac { 2 x ^ { 2 } + 3 x - 2 } { x ^ { 2 } - 3 x - 4 } \right)$ is $( \gamma , \delta )$, then $\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }$ is equal to :
(1) 195
(2) 179
(3) 186
(4) 174
Q15 Circles Chord Length and Chord Properties View
Let a circle $C$ pass through the points $( 4,2 )$ and $( 0,2 )$, and its centre lie on $3 x + 2 y + 2 = 0$. Then the length of the chord, of the circle $C$, whose mid-point is $( 1,2 )$, is :
(1) $\sqrt { 3 }$
(2) $2 \sqrt { 2 }$
(3) $2 \sqrt { 3 }$
(4) $4 \sqrt { 2 }$
Q16 Vectors 3D & Lines Line-Plane Intersection View
Let a straight line $L$ pass through the point $P ( 2 , - 1,3 )$ and be perpendicular to the lines $\frac { x - 1 } { 2 } = \frac { y + 1 } { 1 } = \frac { z - 3 } { - 2 }$ and $\frac { x - 3 } { 1 } = \frac { y - 2 } { 3 } = \frac { z + 2 } { 4 }$. If the line $L$ intersects the $y z$-plane at the point $Q$, then the distance between the points $P$ and $Q$ is :
(1) $\sqrt { 10 }$
(2) $2 \sqrt { 3 }$
(3) 2
(4) 3
Q17 Probability Definitions Conditional Probability and Bayes' Theorem View
Bag 1 contains 4 white balls and 5 black balls, and Bag 2 contains $n$ white balls and 3 black balls. One ball is drawn randomly from Bag 1 and transferred to Bag 2. A ball is then drawn randomly from Bag 2. If the probability, that the ball drawn is white, is $29 / 45$, then $n$ is equal to :
(1) 6
(2) 3
(3) 5
(4) 4
Q18 Simultaneous equations View
Let $\alpha , \beta ( \alpha \neq \beta )$ be the values of m , for which the equations $x + y + z = 1 ; x + 2 y + 4 z = \mathrm { m }$ and $x + 4 y + 10 z = m ^ { 2 }$ have infinitely many solutions. Then the value of $\sum _ { n = 1 } ^ { 10 } \left( n ^ { \alpha } + n ^ { \beta } \right)$ is equal to :
(1) 3080
(2) 560
(3) 3410
(4) 440
Q19 Geometric Sequences and Series Sum of an Infinite Geometric Series (Direct Computation) View
Let $S = \mathbf { N } \cup \{ 0 \}$. Define a relation $R$ from $S$ to $\mathbf { R }$ by : $\mathbf { R } = \left\{ ( x , y ) : \log _ { \mathrm { e } } y = x \log _ { \mathrm { e } } \left( \frac { 2 } { 5 } \right) , x \in \mathrm {~S} , y \in \mathbf { R } \right\}$ Then, the sum of all the elements in the range of $R$ is equal to :
(1) $\frac { 10 } { 9 }$
(2) $\frac { 3 } { 2 }$
(3) $\frac { 5 } { 2 }$
(4) $\frac { 5 } { 3 }$
Q20 Trig Proofs Power-Sum Evaluation via Trigonometric Constraint View
If $\sin x + \sin ^ { 2 } x = 1 , x \in \left( 0 , \frac { \pi } { 2 } \right)$, then $\left( \cos ^ { 12 } x + \tan ^ { 12 } x \right) + 3 \left( \cos ^ { 10 } x + \tan ^ { 10 } x + \cos ^ { 8 } x + \tan ^ { 8 } x \right) + \left( \cos ^ { 6 } x + \tan ^ { 6 } x \right)$ is equal to :
(1) 4
(2) 1
(3) 3
(4) 2
Q21 Indefinite & Definite Integrals Piecewise/Periodic Function Integration View
If $24 \int _ { 0 } ^ { \frac { \pi } { 4 } } \left( \sin \left| 4 x - \frac { \pi } { 12 } \right| + [ 2 \sin x ] \right) \mathrm { d } x = 2 \pi + \alpha$, where $[ \cdot ]$ denotes the greatest integer function, then $\alpha$ is equal to $\_\_\_\_$ .
Q22 Arithmetic Sequences and Series Compute Partial Sum of an Arithmetic Sequence View
Let $a _ { 1 } , a _ { 2 } , \ldots , a _ { 2024 }$ be an Arithmetic Progression such that $a _ { 1 } + \left( a _ { 5 } + a _ { 10 } + a _ { 15 } + \ldots + a _ { 2020 } \right) + a _ { 2024 } = 2233$. Then $a _ { 1 } + a _ { 2 } + a _ { 3 } + \ldots + a _ { 2024 }$ is equal to $\_\_\_\_$
Q23 Laws of Logarithms Simplify or Evaluate a Logarithmic Expression View
If $\lim _ { \mathrm { t } \rightarrow 0 } \left( \int _ { 0 } ^ { 1 } ( 3 x + 5 ) ^ { \mathrm { t } } \mathrm { d } x \right) ^ { \frac { 1 } { t } } = \frac { \alpha } { 5 \mathrm { e } } \left( \frac { 8 } { 5 } \right) ^ { \frac { 2 } { 3 } }$, then $\alpha$ is equal to $\_\_\_\_$
Q24 Circles Circle Equation Derivation View
Let $y ^ { 2 } = 12 x$ be the parabola and $S$ be its focus. Let PQ be a focal chord of the parabola such that $( \mathrm { SP } ) ( \mathrm { SQ } ) = \frac { 147 } { 4 }$. Let C be the circle described taking PQ as a diameter. If the equation of a circle $C$ is $64 x ^ { 2 } + 64 y ^ { 2 } - \alpha x - 64 \sqrt { 3 } y = \beta$, then $\beta - \alpha$ is equal to $\_\_\_\_$ .
Q25 Complex Numbers Arithmetic Roots of Unity and Cyclotomic Expressions View
Let integers $\mathrm { a } , \mathrm { b } \in [ - 3,3 ]$ be such that $\mathrm { a } + \mathrm { b } \neq 0$. Then the number of all possible ordered pairs $( \mathrm { a } , \mathrm { b } )$, for which $\left| \frac { z - \mathrm { a } } { z + \mathrm { b } } \right| = 1$ and $\left| \begin{array} { c c c } z + 1 & \omega & \omega ^ { 2 } \\ \omega & z + \omega ^ { 2 } & 1 \\ \omega ^ { 2 } & 1 & z + \omega \end{array} \right| = 1 , z \in \mathrm { C }$, where $\omega$ and $\omega ^ { 2 }$ are the roots of $x ^ { 2 } + x + 1 = 0$, is equal to $\_\_\_\_$ .