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
session1_24feb_shift1 9 session1_24feb_shift2 4 session1_25feb_shift1 29 session1_25feb_shift2 29 session1_26feb_shift2 15 session2_16mar_shift1 29 session2_16mar_shift2 18 session2_17mar_shift1 21 session2_17mar_shift2 27 session2_18mar_shift1 18 session2_18mar_shift2 9 session3_20jul_shift1 29 session3_20jul_shift2 29 session3_22jul_shift1 9 session3_25jul_shift1 8 session3_25jul_shift2 14 session3_27jul_shift1 4 session3_27jul_shift2 7 session4_01sep_shift2 14 session4_26aug_shift1 5 session4_26aug_shift2 2 session4_27aug_shift1 3 session4_27aug_shift2 29 session4_31aug_shift1 28 session4_31aug_shift2 4
2020
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2019
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2018
08apr 30 15apr 28 15apr_shift1 28 15apr_shift2 6 16apr 19
2017
02apr 30 08apr 30 09apr 34
2016
03apr 28 09apr 29 10apr 30
2015
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2014
06apr 28 09apr 28 11apr 4 12apr 5 19apr 29
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
2023 session1_29jan_shift1

29 maths questions

Q61 Roots of polynomials Determine coefficients or parameters from root conditions View
Let $\lambda \neq 0$ be a real number. Let $\alpha , \beta$ be the roots of the equation $14 x ^ { 2 } - 31 x + 3 \lambda = 0$ and $\alpha , \gamma$ be the roots of the equation $35 x ^ { 2 } - 53 x + 4 \lambda = 0$. Then $\frac { 3 \alpha } { \beta }$ and $\frac { 4 \alpha } { \gamma }$ are the roots of the equation :
(1) $7 x ^ { 2 } + 245 x - 250 = 0$
(2) $7 x ^ { 2 } - 245 x + 250 = 0$
(3) $49 x ^ { 2 } - 245 x + 250 = 0$
(4) $49 x ^ { 2 } + 245 x + 250 = 0$
Q62 Complex Numbers Arithmetic Systems of Equations via Real and Imaginary Part Matching View
For two non-zero complex number $z _ { 1 }$ and $z _ { 2 }$, if $\operatorname { Re } \left( z _ { 1 } z _ { 2 } \right) = 0$ and $\operatorname { Re } \left( z _ { 1 } + z _ { 2 } \right) = 0$, then which of the following are possible?
(A) $\operatorname { Im } \left( \mathrm { z } _ { 1 } \right) > 0$ and $\operatorname { Im } \left( \mathrm { z } _ { 2 } \right) > 0$
(B) $\operatorname { Im } \left( \mathrm { z } _ { 1 } \right) < 0$ and $\operatorname { Im } \left( \mathrm { z } _ { 2 } \right) > 0$
(C) $\operatorname { Im } \left( \mathrm { z } _ { 1 } \right) > 0$ and $\operatorname { Im } \left( \mathrm { z } _ { 2 } \right) < 0$
(D) $\operatorname { Im } \left( \mathrm { z } _ { 1 } \right) < 0$ and $\operatorname { Im } \left( \mathrm { z } _ { 2 } \right) < 0$ Choose the correct answer from the options given below:
(1) B and D
(2) B and C
(3) A and B
(4) A and C
Q63 Combinations & Selection Basic Combination Computation View
If all the six digit numbers $\mathrm { x } _ { 1 } \mathrm { x } _ { 2 } \mathrm { x } _ { 3 } \mathrm { x } _ { 4 } \mathrm { x } _ { 5 } \mathrm { x } _ { 6 }$ with $0 < \mathrm { x } _ { 1 } < \mathrm { x } _ { 2 } < \mathrm { x } _ { 3 } < \mathrm { x } _ { 4 } < \mathrm { x } _ { 5 } < \mathrm { x } _ { 6 }$ are arranged in the increasing order, then the sum of the digits in the $72 ^ { \text {th} }$ number is $\_\_\_\_$ .
Q64 Permutations & Arrangements Basic Combination Computation View
Five digit numbers are formed using the digits $1,2,3,5,7$ with repetitions and are written in descending order with serial numbers. For example, the number 77777 has serial number 1. Then the serial number of 35337 is
Q65 Geometric Sequences and Series Finite Geometric Sum and Term Relationships View
Let $\mathrm { a } _ { 1 } , \mathrm { a } _ { 2 } , \mathrm { a } _ { 3 } , \ldots$ be a GP of increasing positive numbers. If the product of fourth and sixth terms is 9 and the sum of fifth and seventh terms is 24 , then $\mathbf { a } _ { 1 } \mathbf { a } _ { 9 } + \mathbf { a } _ { 2 } \mathbf { a } _ { 4 } \mathbf { a } _ { 9 } + \mathbf { a } _ { 5 } + \mathbf { a } _ { 7 }$ is equal to
Q66 Binomial Theorem (positive integer n) Determine Parameters from Conditions on Coefficients or Terms View
Let the coefficients of three consecutive terms in the binomial expansion of $( 1 + 2 x ) ^ { \mathrm { n } }$ be in the ratio $2 : 5 : 8$. Then the coefficient of the term, which is in the middle of these three terms, is
Q67 Binomial Theorem (positive integer n) Determine Parameters from Conditions on Coefficients or Terms View
If the co-efficient of $x ^ { 9 }$ in $\left( \alpha x ^ { 3 } + \frac { 1 } { \beta x } \right) ^ { 11 }$ and the co-efficient of $x ^ { - 9 }$ in $\left( \alpha x - \frac { 1 } { \beta x ^ { 3 } } \right) ^ { 11 }$ are equal, then $( \alpha \beta ) ^ { 2 }$ is equal to
Q68 Trig Proofs Trigonometric Identity Simplification View
Let $f ( \theta ) = 3 \left( \sin ^ { 4 } \left( \frac { 3 \pi } { 2 } - \theta \right) + \sin ^ { 4 } ( 3 \pi + \theta ) \right) - 2 \left( 1 - \sin ^ { 2 } 2 \theta \right)$ and $S = \left\{ \theta \in [ 0 , \pi ] : f ^ { \prime } ( \theta ) = - \frac { \sqrt { 3 } } { 2 } \right\}$. If $4 \beta = \sum _ { \theta \in S } \theta$ then $f ( \beta )$ is equal to
(1) $\frac { 11 } { 8 }$
(2) $\frac { 5 } { 4 }$
(3) $\frac { 9 } { 8 }$
(4) $\frac { 3 } { 2 }$
Q69 Straight Lines & Coordinate Geometry Reflection and Image in a Line View
A light ray emits from the origin making an angle $30 ^ { \circ }$ with the positive $x$-axis. After getting reflected by the line $\mathrm { x } + \mathrm { y } = 1$, if this ray intersects x-axis at Q , then the abscissa of Q is
(1) $\frac { 2 } { ( \sqrt { 3 } - 1 ) }$
(2) $\frac { 2 } { 3 + \sqrt { 3 } }$
(3) $\frac { 2 } { 3 - \sqrt { 3 } }$
(4) $\frac { \sqrt { 3 } } { 2 ( \sqrt { 3 } + 1 ) }$
Q70 Straight Lines & Coordinate Geometry Area Computation in Coordinate Geometry View
Let $B$ and $C$ be the two points on the line $y + x = 0$ such that $B$ and $C$ are symmetric with respect to the origin. Suppose $A$ is a point on $\mathrm { y } - 2 \mathrm { x } = 2$ such that $\triangle A B C$ is an equilateral triangle. Then, the area of the $\triangle A B C$ is
(1) $3 \sqrt { 3 }$
(2) $2 \sqrt { 3 }$
(3) $\frac { 8 } { \sqrt { 3 } }$
(4) $\frac { 10 } { \sqrt { 3 } }$
Q71 Circles Tangent Lines and Tangent Lengths View
Let the tangents at the points $A ( 4 , - 11 )$ and $B ( 8 , - 5 )$ on the circle $x ^ { 2 } + y ^ { 2 } - 3 x + 10 y - 15 = 0$, intersect at the point $C$. Then the radius of the circle, whose centre is $C$ and the line joining $A$ and $B$ is its tangent, is equal to
(1) $\frac { 3 \sqrt { 3 } } { 4 }$
(2) $2 \sqrt { 13 }$
(3) $\sqrt { 13 }$
(4) $\frac { 2 \sqrt { 13 } } { 3 }$
Q72 Chain Rule Limit Evaluation Involving Composition or Substitution View
Let $x = 2$ be a root of the equation $x ^ { 2 } + p x + q = 0$ and $f ( x ) = \left\{ \begin{array} { c l } \frac { 1 - \cos \left( x ^ { 2 } - 4 p x + q ^ { 2 } + 8 q + 16 \right) } { ( x - 2 p ) ^ { 4 } } , & x \neq 2 p \\ 0 , & x = 2 p \end{array} \right.$. Then $\lim _ { x \rightarrow 2 p ^ { + } } [ f ( x ) ]$ where $[ \cdot ]$ denotes greatest integer function, is
(1) 2
(2) 1
(3) 0
(4) $- 1$
Q74 Matrices Matrix Algebraic Properties and Abstract Reasoning View
Let $\alpha$ and $\beta$ be real numbers. Consider a $3 \times 3$ matrix $A$ such that $A ^ { 2 } = 3 A + \alpha I$. If $A ^ { 4 } = 21 A + \beta I$, then
(1) $\alpha = 1$
(2) $\alpha = 4$
(3) $\beta = 8$
(4) $\beta = - 8$
Q75 Simultaneous equations View
Consider the following system of questions
$$\begin{aligned} & \alpha x + 2 y + z = 1 \\ & 2 \alpha x + 3 y + z = 1 \\ & 3 x + \alpha y + 2 z = \beta \end{aligned}$$
For some $\alpha , \beta \in \mathbb { R }$. Then which of the following is NOT correct.
(1) It has no solution if $\alpha = - 1$ and $\beta \neq 2$
(2) It has no solution for $\alpha = - 1$ and for all $\beta \in \mathbb { R }$
(3) It has no solution for $\alpha = 3$ and for all $\beta \neq 2$
(4) It has a solution for all $\alpha \neq - 1$ and $\beta = 2$
Q76 Laws of Logarithms Determine Domain or Range of a Composite Function View
The domain of $f ( x ) = \frac { \log _ { ( x + 1 ) } ( x - 2 ) } { e ^ { 2 \log _ { e } x } - ( 2 x + 3 ) } , x \in R$ is
(1) $\mathbb { R } - \{ - 1,3 \}$
(2) $( 2 , \infty ) - \{ 3 \}$
(3) $( - 1 , \infty ) - \{ 3 \}$
(4) $\mathbb { R } - \{ 3 \}$
Q77 Composite & Inverse Functions Injectivity, Surjectivity, or Bijectivity Classification View
Let $f : R \rightarrow R$ be a function such that $f ( x ) = \frac { x ^ { 2 } + 2 x + 1 } { x ^ { 2 } + 1 }$. Then
(1) $f ( x )$ is many-one in $( - \infty , - 1 )$
(2) $f ( x )$ is many-one in $( 1 , \infty )$
(3) $f ( x )$ is one-one in $[ 1 , \infty )$ but not in $( - \infty , \infty )$
(4) $f ( x )$ is one-one in $( - \infty , \infty )$
Q78 Differential equations Recover a Function from a Composition or Functional Equation View
Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a differentiable function that satisfies the relation $f ( x + y ) = f ( x ) + f ( y ) - 1 , \forall x , y \in \mathbb { R }$. If $f ^ { \prime } ( 0 ) = 2$, then $| f ( - 2 ) |$ is equal to
Q79 Arithmetic Sequences and Series Telescoping or Non-Standard Summation Involving an AP View
Suppose f is a function satisfying $\mathrm { f } ( \mathrm { x } + \mathrm { y } ) = \mathrm { f } ( \mathrm { x } ) + \mathrm { f } ( \mathrm { y } )$ for all $\mathrm { x } , \mathrm { y } \in \mathbb { N }$ and $\mathrm { f } ( 1 ) = \frac { 1 } { 5 }$. If $\sum _ { n = 1 } ^ { m } \frac { f ( n ) } { n ( n + 1 ) ( n + 2 ) } = \frac { 1 } { 12 }$ then m is equal to $\_\_\_\_$ .
Q80 Integration by Substitution Finding a Function from an Integral Equation View
Let $f ( x ) = x + \frac { a } { \pi ^ { 2 } - 4 } \sin x + \frac { b } { \pi ^ { 2 } - 4 } \cos x , x \in \mathbb { R }$ be a function which satisfies $f ( x ) = x + \int _ { 0 } ^ { \pi / 2 } \sin ( x + y ) f ( y ) d y$. Then $( a + b )$ is equal to
(1) $- \pi ( \pi + 2 )$
(2) $- 2 \pi ( \pi + 2 )$
(3) $- 2 \pi ( \pi - 2 )$
(4) $- \pi ( \pi - 2 )$
Q81 Indefinite & Definite Integrals Piecewise/Periodic Function Integration View
Let $[ \mathrm { x } ]$ denote the greatest integer $\leq \mathrm { x }$. Consider the function $\mathrm { f } ( \mathrm { x } ) = \max \left\{ \mathrm { x } ^ { 2 } , 1 + [ x ] \right\}$. Then the value of the integral $\int _ { 0 } ^ { 2 } f ( x ) d x$ is :
(1) $\frac { 5 + 4 \sqrt { 2 } } { 3 }$
(2) $\frac { 8 + 4 \sqrt { 2 } } { 3 }$
(3) $\frac { 1 + 5 \sqrt { 2 } } { 3 }$
(4) $\frac { 4 + 5 \sqrt { 2 } } { 3 }$
Q82 Areas by integration View
Let $A = \left\{ ( x , y ) \in \mathbb { R } ^ { 2 } : y \geq 0,2 x \leq y \leq \sqrt { 4 - ( x - 1 ) ^ { 2 } } \right\}$ and $B = \left\{ ( x , y ) \in \mathbb { R } \times \mathbb { R } : 0 \leq y \leq \min \left\{ 2 x , \sqrt { 4 - ( x - 1 ) ^ { 2 } } \right\} \right\}$. Then the ratio of the area of $A$ to the area of $B$ is
(1) $\frac { \pi - 1 } { \pi + 1 }$
(2) $\frac { \pi } { \pi - 1 }$
(3) $\frac { \pi } { \pi + 1 }$
(4) $\frac { \pi + 1 } { \pi - 1 }$
Q83 Areas by integration View
Let $\Delta$ be the area of the region $\left\{ ( x , y ) \in \mathbb { R } ^ { 2 } : x ^ { 2 } + y ^ { 2 } \leq 21 , y ^ { 2 } \leq 4 x , x \geq 1 \right\}$. Then $\frac { 1 } { 2 } \left( \Delta - 21 \sin ^ { - 1 } \frac { 2 } { \sqrt { 7 } } \right)$ is equal to
(1) $2 \sqrt { 3 } - \frac { 1 } { 3 }$
(2) $\sqrt { 3 } - \frac { 2 } { 3 }$
(3) $2 \sqrt { 3 } - \frac { 2 } { 3 }$
(4) $\sqrt { 3 } - \frac { 4 } { 3 }$
Q84 First order differential equations (integrating factor) Solving Separable DEs with Initial Conditions View
Let $\mathrm { y } = \mathrm { f } ( \mathrm { x } )$ be the solution of the differential equation $y ( x + 1 ) d x - x ^ { 2 } d y = 0 , y ( 1 ) = e$. Then $\lim _ { x \rightarrow 0 ^ { + } } f ( x )$ is equal to
(1) 0
(2) $\frac { 1 } { e }$
(3) $e ^ { 2 }$
(4) $\frac { 1 } { e ^ { 2 } }$
Q85 Vectors 3D & Lines Vector Algebra and Triple Product Computation View
If the vectors $\vec { a } = \lambda \hat { i } + \mu \hat { j } + 4 \widehat { k } , \vec { b } = - 2 \hat { i } + 4 \hat { j } - 2 \widehat { k }$ and $\vec { c } = 2 \hat { i } + 3 \hat { j } + \widehat { k }$ are coplanar and the projection of $\vec { a }$ on the vector $\vec { b }$ is $\sqrt { 54 }$ units, then the sum of all possible values of $\lambda + \mu$ is equal to
(1) 0
(2) 6
(3) 24
(4) 18
Q86 Vectors 3D & Lines Vector Algebra and Triple Product Computation View
Let $\vec { a } , \vec { b }$ and $\vec { c }$ be three non-zero non-coplanar vectors. Let the position vectors of four points $A , B , C$ and $D$ be $\overrightarrow { \mathrm { a } } - \overrightarrow { \mathrm { b } } + \overrightarrow { \mathrm { c } } , \lambda \overrightarrow { \mathrm { a } } - 3 \overrightarrow { \mathrm {~b} } + 4 \overrightarrow { \mathrm { c } } , - \vec { a } + 2 \vec { b } - 3 \vec { c }$ and $2 \vec { a } - 4 \vec { b } + 6 \vec { c }$ respectively. If $\overrightarrow { A B } , \overrightarrow { A C }$ and $\overrightarrow { A D }$ are coplanar, then $\lambda$ is :
Q87 Vectors 3D & Lines Find Cartesian Equation of a Plane View
Let the equation of the plane P containing the line $x + 10 = \frac { 8 - y } { 2 } = z$ be $a x + b y + 3 z = 2 ( a + b )$ and the distance of the plane P from the point $( 1,27,7 )$ be $c$. Then $\mathrm { a } ^ { 2 } + \mathrm { b } ^ { 2 } + \mathrm { c } ^ { 2 }$ is equal to
Q88 Vectors 3D & Lines Distance from a Point to a Line (Show/Compute) View
Let the co-ordinates of one vertex of $\triangle A B C$ be $A ( 0,2 , \alpha )$ and the other two vertices lie on the line $\frac { \mathrm { x } + \alpha } { 5 } = \frac { \mathrm { y } - 1 } { 2 } = \frac { \mathrm { z } + 4 } { 3 }$. For $\alpha \in \mathbb { Z }$, if the area of $\triangle A B C$ is 21 sq. units and the line segment $B C$ has length $2 \sqrt { 21 }$ units, then $\alpha ^ { 2 }$ is equal to $\_\_\_\_$ .
Q89 Probability Definitions Finite Equally-Likely Probability Computation View
Fifteen football players of a club-team are given 15 T-shirts with their names written on the backside. If the players pick up the T-shirts randomly, then the probability that at least 3 players pick the correct T-shirt is
(1) $\frac { 5 } { 24 }$
(2) $\frac { 2 } { 15 }$
(3) $\frac { 1 } { 6 }$
(4) $\frac { 5 } { 36 }$
Q90 Hypergeometric Distribution View
There rotten apples are mixed accidently with seven good apples and four apples are drawn one by one without replacement. Let the random variable X denote the number of rotten apples. If $\mu$ and $\sigma ^ { 2 }$ represent mean and variance of X , respectively, then $10 \left( \mu ^ { 2 } + \sigma ^ { 2 } \right)$ is equal to
(1) 20
(2) 250
(3) 25
(4) 30