jee-main

Papers (169)
2025
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2024
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2023
session1_01feb_shift1 24 session1_01feb_shift2 3 session1_24jan_shift1 13 session1_24jan_shift2 12 session1_25jan_shift1 28 session1_25jan_shift2 27 session1_29jan_shift1 29 session1_29jan_shift2 28 session1_30jan_shift1 2 session1_30jan_shift2 29 session1_31jan_shift1 28 session1_31jan_shift2 17 session2_06apr_shift1 5 session2_06apr_shift2 17 session2_08apr_shift1 29 session2_08apr_shift2 14 session2_10apr_shift1 29 session2_10apr_shift2 15 session2_11apr_shift1 5 session2_11apr_shift2 4 session2_12apr_shift1 26 session2_13apr_shift1 25 session2_13apr_shift2 20 session2_15apr_shift1 20
2022
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2021
session1_24feb_shift1 10 session1_24feb_shift2 7 session1_25feb_shift1 29 session1_25feb_shift2 29 session1_26feb_shift2 17 session2_16mar_shift1 29 session2_16mar_shift2 15 session2_17mar_shift1 20 session2_17mar_shift2 24 session2_18mar_shift1 12 session2_18mar_shift2 11 session3_20jul_shift1 30 session3_20jul_shift2 29 session3_22jul_shift1 7 session3_25jul_shift1 2 session3_25jul_shift2 15 session3_27jul_shift1 3 session3_27jul_shift2 4 session4_01sep_shift2 11 session4_26aug_shift1 5 session4_26aug_shift2 2 session4_27aug_shift1 3 session4_27aug_shift2 28 session4_31aug_shift1 28 session4_31aug_shift2 4
2020
session1_07jan_shift1 26 session1_07jan_shift2 17 session1_08jan_shift1 5 session1_08jan_shift2 12 session1_09jan_shift1 22 session1_09jan_shift2 18 session2_02sep_shift1 19 session2_02sep_shift2 17 session2_03sep_shift1 21 session2_03sep_shift2 9 session2_04sep_shift1 10 session2_04sep_shift2 24 session2_05sep_shift1 23 session2_05sep_shift2 27 session2_06sep_shift1 13 session2_06sep_shift2 10
2019
session1_09jan_shift1 6 session1_09jan_shift2 29 session1_10jan_shift1 30 session1_10jan_shift2 12 session1_11jan_shift1 6 session1_11jan_shift2 5 session1_12jan_shift1 10 session1_12jan_shift2 20 session2_08apr_shift1 29 session2_08apr_shift2 29 session2_09apr_shift1 29 session2_09apr_shift2 29 session2_10apr_shift1 2 session2_10apr_shift2 3 session2_12apr_shift1 3 session2_12apr_shift2 9
2018
08apr 29 15apr 28 15apr_shift1 28 15apr_shift2 2 16apr 15
2017
02apr 28 08apr 29 09apr 30
2016
03apr 30 09apr 30 10apr 28
2015
04apr 29 10apr 30
2014
06apr 28 09apr 28 11apr 4 12apr 5 19apr 29
2013
07apr 29 09apr 14 22apr 5 23apr 14 25apr 13
2012
07may 18 12may 22 19may 13 26may 17 offline 30
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
2023 session1_29jan_shift2

28 maths questions

Q61 Roots of polynomials Vieta's formulas: compute symmetric functions of roots View
Let $\alpha _ { 1 } , \alpha _ { 2 } , \ldots , \alpha _ { 7 }$ be the roots of the equation $x ^ { 7 } + 3 x ^ { 5 } - 13 x ^ { 3 } - 15 x = 0$ and $\left| \alpha _ { 1 } \right| \geq \left| \alpha _ { 2 } \right| \geq \ldots \geq \left| \alpha _ { 7 } \right|$.
Then, $\alpha _ { 1 } \alpha _ { 2 } - \alpha _ { 3 } \alpha _ { 4 } + \alpha _ { 5 } \alpha _ { 6 }$ is equal to $\_\_\_\_$
Q62 Complex Numbers Arithmetic Systems of Equations via Real and Imaginary Part Matching View
Let $\alpha = 8 - 14 \mathrm { i } , \mathrm { A } = \left\{ \mathrm { z } \in \mathbb { C } : \frac { \alpha \mathrm { z } - \bar { \alpha } \overline { \mathrm { z } } } { \mathrm { z } ^ { 2 } - ( \overline { \mathrm { z } } ) ^ { 2 } - 112 \mathrm { i } } = 1 \right\}$ and $B = \{ z \in \mathbb { C } : | z + 3 i | = 4 \}$
Then, $\sum _ { z \in A \cap B } ( \operatorname { Re } z - \operatorname { Im } z )$ is equal to $\_\_\_\_$
Q63 Permutations & Arrangements Dictionary Order / Rank of a Permutation View
The letters of the word OUGHT are written in all possible ways and these words are arranged as in a dictionary, in a series. Then the serial number of the word TOUGH is : (1) 89 (2) 84 (3) 86 (4) 79
Q64 Permutations & Arrangements Forming Numbers with Digit Constraints View
The total number of 4-digit numbers whose greatest common divisor with 54 is 2 , is $\_\_\_\_$
Q65 Sequences and series, recurrence and convergence Summation of sequence terms View
Let $a _ { 1 } = b _ { 1 } = 1$ and $a _ { n } = a _ { n - 1 } + ( n - 1 ) , b _ { n } = b _ { n - 1 } + \mathrm { a } _ { n - 1 } , \forall n \geq 2$. If $\mathrm { S } = \sum _ { \mathrm { n } = 1 } ^ { 10 } \left( \frac { b _ { n } } { 2 ^ { n } } \right)$ and $\mathrm { T } = \sum _ { n = 1 } ^ { 8 } \frac { \mathrm { n } } { 2 ^ { n - 1 } }$ then $2 ^ { 7 } ( 2 S - T )$ is equal to $\_\_\_\_$
Q66 Geometric Sequences and Series Sum of an Infinite Geometric Series (Direct Computation) View
Let $\left\{ a _ { k } \right\}$ and $\left\{ b _ { k } \right\} , k \in \mathbb { N }$, be two G.P.s with common ratio $r _ { 1 }$ and $r _ { 2 }$ respectively such that $\mathrm { a } _ { 1 } = \mathrm { b } _ { 1 } = 4$ and $\mathrm { r } _ { 1 } < \mathrm { r } _ { 2 }$. Let $\mathrm { c } _ { \mathrm { k } } = \mathrm { a } _ { \mathrm { k } } + \mathrm { b } _ { \mathrm { k } } , \mathrm { k } \in \mathbb { N }$. If $\mathrm { c } _ { 2 } = 5$ and $\mathrm { c } _ { 3 } = \frac { 13 } { 4 }$ then $\sum _ { \mathrm { k } = 1 } ^ { \infty } \mathrm { c } _ { \mathrm { k } } - \left( 12 \mathrm { a } _ { 6 } + 8 \mathrm {~b} _ { 4 } \right)$ is equal to $\_\_\_\_$
Q67 Permutations & Arrangements Forming Numbers with Digit Constraints View
The number of 3 digit numbers, that are divisible by either 3 or 4 but not divisible by 48 , is (1) 472 (2) 432 (3) 507 (4) 400
Q68 Binomial Theorem (positive integer n) Evaluate a Summation Involving Binomial Coefficients View
Let $K$ be the sum of the coefficients of the odd powers of $x$ in the expansion of $( 1 + x ) ^ { 99 }$. Let a be the middle term in the expansion of $\left( 2 + \frac { 1 } { \sqrt { 2 } } \right) ^ { 200 }$. If $\frac { { } ^ { 200 } C _ { 99 } K } { a } = \frac { 2 ^ { l } m } { n }$, where $m$ and $n$ are odd numbers, then the ordered pair $( l , \mathrm { n } )$ is equal to: (1) $( 50,51 )$ (2) $( 51,99 )$ (3) $( 50,101 )$ (4) $( 51,101 )$
Q69 Trig Graphs & Exact Values View
The set of all values of $\lambda$ for which the equation $\cos ^ { 2 } 2 x - 2 \sin ^ { 4 } x - 2 \cos ^ { 2 } x = \lambda$ has a solution is: (1) $[ - 2 , - 1 ]$ (2) $\left[ - 2 , - \frac { 3 } { 2 } \right]$ (3) $\left[ - 1 , - \frac { 1 } { 2 } \right]$ (4) $\left[ - \frac { 3 } { 2 } , - 1 \right]$
Q70 Circles Tangent Lines and Tangent Lengths View
A circle with centre $( 2,3 )$ and radius 4 intersects the line $x + y = 3$ at the points $P$ and $Q$. If the tangents at $P$ and $Q$ intersect at the point $S ( \alpha , \beta )$, then $4 \alpha - 7 \beta$ is equal to $\_\_\_\_$
Q71 Circles Inscribed/Circumscribed Circle Computations View
A triangle is formed by the tangents at the point $( 2,2 )$ on the curves $y ^ { 2 } = 2 x$ and $x ^ { 2 } + y ^ { 2 } = 4 x$, and the line $\mathrm { x } + \mathrm { y } + 2 = 0$. If $r$ is the radius of its circumcircle, then $r ^ { 2 }$ is equal to $\_\_\_\_$
Q72 Trig Graphs & Exact Values View
If the tangent at a point P on the parabola $\mathrm { y } ^ { 2 } = 3 \mathrm { x }$ is parallel to the line $x + 2 y = 1$ and the tangents at the points $Q$ and $R$ on the ellipse $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 1 } = 1$ are perpendicular to the line $x - y = 2$, then the area of the triangle $P Q R$ is: (1) $\frac { 9 } { \sqrt { 5 } }$ (2) $5 \sqrt { 3 }$ (3) $\frac { 3 } { 2 } \sqrt { 5 }$ (4) $3 \sqrt { 5 }$
Q74 Measures of Location and Spread View
Let $X = \{ 11,12,13 , \ldots , 40,41 \}$ and $Y = \{ 61,62,63 , \ldots , 90,91 \}$ be the two sets of observations. If $\overline { \mathrm { x } }$ and $\overline { \mathrm { y } }$ are their respective means and $\sigma ^ { 2 }$ is the variance of all the observations in $\mathrm { X } \cup \mathrm { Y }$, then $\left| \overline { \mathrm { x } } + \overline { \mathrm { y } } - \sigma ^ { 2 } \right|$ is equal to $\_\_\_\_$
Q76 Matrices Determinant and Rank Computation View
The set of all values of $t \in \mathbb { R }$, for which the matrix $\left[ \begin{array} { c c c } \mathrm { e } ^ { t } & \mathrm { e } ^ { - t } ( \sin t - 2 \cos t ) & \mathrm { e } ^ { - t } ( - 2 \sin t - \cos t ) \\ \mathrm { e } ^ { t } & \mathrm { e } ^ { - t } ( 2 \sin t + \cos t ) & \mathrm { e } ^ { - t } ( \sin t - 2 \cos t ) \\ \mathrm { e } ^ { t } & \mathrm { e } ^ { - t } \cos t & \mathrm { e } ^ { - t } \sin t \end{array} \right]$ is invertible, is (1) $\left\{ ( 2 k + 1 ) \frac { \pi } { 2 } , k \in \mathbb { Z } \right\}$ (2) $\left\{ k \pi + \frac { \pi } { 4 } , k \in \mathbb { Z } \right\}$ (3) $\{ k \pi , k \in \mathbb { Z } \}$ (4) $\mathbb { R }$
Q77 Matrices Linear System and Inverse Existence View
Let A be a symmetric matrix such that $| A | = 2$ and $\left[ \begin{array} { l l } 2 & 1 \\ 3 & \frac { 3 } { 2 } \end{array} \right] A = \left[ \begin{array} { l l } 1 & 2 \\ \alpha & \beta \end{array} \right]$. If the sum of the diagonal elements of A is $s$, then $\frac { \beta s } { \alpha ^ { 2 } }$ is equal to $\_\_\_\_$.
Q78 Differential equations Finding a DE from a Limit or Implicit Condition View
Consider a function $\mathrm { f } : \mathbb { N } \rightarrow \mathbb { R }$, satisfying $f ( 1 ) + 2 f ( 2 ) + 3 f ( 3 ) + \ldots + x f ( x ) = x ( x + 1 ) f ( x ) ; x \geq 2$ with $f ( 1 ) = 1$. Then $\frac { 1 } { f ( 2022 ) } + \frac { 1 } { f ( 2028 ) }$ is equal to (1) 8200 (2) 8000 (3) 8400 (4) 8100
Q79 Differentiating Transcendental Functions Determine parameters from function or curve conditions View
Let $f$ and $g$ be twice differentiable functions on $R$ such that $f ^ { \prime \prime } ( x ) = g ^ { \prime \prime } ( x ) + 6 x$ $f ^ { \prime } ( 1 ) = 4 g ^ { \prime } ( 1 ) - 3 = 9$ $f ( 2 ) = 3 g ( 2 ) = 12$ Then which of the following is NOT true ? (1) $g ( - 2 ) - f ( - 2 ) = 20$ (2) If $- 1 < x < 2$, then $| f ( x ) - g ( x ) | < 8$ (3) $\left| f ^ { \prime } ( x ) - g ^ { \prime } ( x ) \right| < 6 \Rightarrow - 1 < x < 1$ (4) There exists $x _ { 0 } \in \left( 1 , \frac { 3 } { 2 } \right)$ such that $f \left( x _ { 0 } \right) = g \left( x _ { 0 } \right)$
Q80 Tangents, normals and gradients Normal or perpendicular line problems View
If the equation of the normal to the curve $y = \frac { x - a } { ( x + b ) ( x - 2 ) }$ at the point $( 1 , - 3 )$ is $x - 4 y = 13$ then the value of $a + b$ is equal to $\_\_\_\_$
Q81 Integration using inverse trig and hyperbolic functions View
The value of the integral $\int _ { 1 } ^ { 2 } \left( \frac { t ^ { 4 } + 1 } { t ^ { 6 } + 1 } \right) d t$ is : (1) $\tan ^ { - 1 } \frac { 1 } { 2 } + \frac { 1 } { 3 } \tan ^ { - 1 } 8 - \frac { \pi } { 3 }$ (2) $\tan ^ { - 1 } 2 - \frac { 1 } { 3 } \tan ^ { - 1 } 8 + \frac { \pi } { 3 }$ (3) $\tan ^ { - 1 } 2 + \frac { 1 } { 3 } \tan ^ { - 1 } 8 - \frac { \pi } { 3 }$ (4) $\tan ^ { - 1 } \frac { 1 } { 2 } - \frac { 1 } { 3 } \tan ^ { - 1 } 8 + \frac { \pi } { 3 }$
Q82 Integration using inverse trig and hyperbolic functions View
The value of the integral $\int _ { 1 / 2 } ^ { 2 } \frac { \tan ^ { - 1 } x } { x } d x$ is equal to (1) $\pi \log _ { e } 2$ (2) $\frac { 1 } { 2 } \log _ { e } 2$ (3) $\frac { \pi } { 4 } \log _ { \mathrm { e } } 2$ (4) $\frac { \pi } { 2 } \log _ { \mathrm { e } } 2$
Q83 Areas by integration View
The area of the region $A = \left\{ ( x , y ) : | \cos x - \sin x | \leq y \leq \sin x , 0 \leq x \leq \frac { \pi } { 2 } \right\}$ is: (1) $1 - \frac { 3 } { \sqrt { 2 } } + \frac { 4 } { \sqrt { 5 } }$ (2) $\sqrt { 5 } + 2 \sqrt { 2 } - 4.5$ (3) $\frac { 3 } { \sqrt { 5 } } - \frac { 3 } { \sqrt { 2 } } + 1$ (4) $\sqrt { 5 } - 2 \sqrt { 2 } + 1$
Q84 First order differential equations (integrating factor) View
Let $y = y ( x )$ be the solution of the differential equation $x \log _ { e } x \frac { d y } { d x } + y = x ^ { 2 } \log _ { e } x , ( x > 1 )$. If $y ( 2 ) = 2$, then $y ( e )$ is equal to (1) $\frac { 4 + e ^ { 2 } } { 4 }$ (2) $\frac { 1 + e ^ { 2 } } { 4 }$ (3) $\frac { 2 + e ^ { 2 } } { 2 }$ (4) $\frac { 1 + e ^ { 2 } } { 2 }$
Q85 Vectors Introduction & 2D Dot Product Computation View
If $\vec { a } = \hat { i } + 2 \widehat { k } , \vec { b } = \hat { i } + \hat { j } + \widehat { k } , \vec { c } = 7 \hat { i } - 3 \hat { j } + 4 \widehat { k } , \vec { r } \times \vec { b } + \vec { b } \times \vec { c } = \overrightarrow { 0 }$ and $\vec { r } \cdot \vec { a } = 0$ then $\vec { r } \cdot \vec { c }$ is equal to: (1) 34 (2) 12 (3) 36 (4) 30
Q86 Vectors Introduction & 2D Dot Product Computation View
Let $\vec { a } = 4 \hat { i } + 3 \hat { j }$ and $\vec { b } = 3 \hat { i } - 4 \hat { j } + 5 \hat { k }$ and $\overrightarrow { \mathrm { c } }$ is a vector such that $\vec { c } \cdot ( \vec { a } \times \vec { b } ) + 25 = 0 , \vec { c } \cdot ( \hat { i } + \hat { j } + \hat { k } ) = 4$ and projection of $\vec { c }$ on $\overrightarrow { \mathrm { a } }$ is 1 , then the projection of $\vec { c }$ on $\vec { b }$ equals: (1) $\frac { 5 } { \sqrt { 2 } }$ (2) $\frac { 1 } { 5 }$ (3) $\frac { 1 } { \sqrt { 2 } }$ (4) $\frac { 3 } { \sqrt { 2 } }$
Q87 Vectors: Lines & Planes Distance Computation (Point-to-Plane or Line-to-Line) View
Shortest distance between the lines $\frac { x - 1 } { 2 } = \frac { y + 8 } { - 7 } = \frac { z - 4 } { 5 }$ and $\frac { x - 1 } { 2 } = \frac { y - 2 } { 1 } = \frac { z - 6 } { - 3 }$ is (1) $2 \sqrt { 3 }$ (2) $4 \sqrt { 3 }$ (3) $3 \sqrt { 3 }$ (4) $5 \sqrt { 3 }$
Q88 Vectors: Lines & Planes Find Intersection of a Line and a Plane View
The plane $2 x - y + z = 4$ intersects the line segment joining the points $\mathrm { A } ( \mathrm { a } , - 2,4 )$ and $\mathrm { B } ( 2 , \mathrm {~b} , - 3 )$ at the point C in the ratio 2 : 1 and the distance of the point C from the origin is $\sqrt { 5 }$. If $a b < 0$ and P is the point $( \mathrm { a } - \mathrm { b } , \mathrm { b } , 2 \mathrm {~b} - \mathrm { a } )$ then $\mathrm { CP } ^ { 2 }$ is equal to: (1) $\frac { 17 } { 3 }$ (2) $\frac { 16 } { 3 }$ (3) $\frac { 73 } { 3 }$ (4) $\frac { 97 } { 3 }$
Q89 Vectors: Lines & Planes Find Intersection of a Line and a Plane View
If the lines $\frac { x - 1 } { 1 } = \frac { y - 2 } { 2 } = \frac { z + 3 } { 1 }$ and $\frac { x - a } { 2 } = \frac { y + 2 } { 3 } = \frac { z - 3 } { 1 }$ intersects at the point $P$, then the distance of the point $P$ from the plane $z = a$ is : (1) 16 (2) 28 (3) 10 (4) 22
Q90 Probability Definitions Probability Distribution and Sampling View
Let $S = \left\{ w _ { 1 } , w _ { 2 } , \ldots \right\}$ be the sample space associated to a random experiment. Let $P \left( w _ { n } \right) = \frac { P \left( w _ { n - 1 } \right) } { 2 } , n \geq 2$ . Let $A = \{ 2 k + 3 l ; k , l \in \mathbb { N } \}$ and $B = \left\{ w _ { n } ; n \in A \right\}$. Then $P ( B )$ is equal to (1) $\frac { 3 } { 32 }$ (2) $\frac { 3 } { 64 }$ (3) $\frac { 1 } { 16 }$ (4) $\frac { 1 } { 32 }$