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
session1_24jun_shift1 20 session1_24jun_shift2 25 session1_25jun_shift1 14 session1_25jun_shift2 17 session1_26jun_shift1 26 session1_26jun_shift2 23 session1_27jun_shift1 4 session1_27jun_shift2 29 session1_28jun_shift1 13 session1_29jun_shift1 20 session1_29jun_shift2 5 session2_25jul_shift1 29 session2_25jul_shift2 22 session2_26jul_shift1 29 session2_26jul_shift2 24 session2_27jul_shift1 26 session2_27jul_shift2 29 session2_28jul_shift1 12 session2_28jul_shift2 29 session2_29jul_shift1 18 session2_29jul_shift2 17
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
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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
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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
2023 session1_25jan_shift2

27 maths questions

Q61 Roots of polynomials Determine coefficients or parameters from root conditions View
Let $a \in R$ and let $\alpha , \beta$ be the roots of the equation $x ^ { 2 } + 60 ^ { \frac { 1 } { 4 } } x + a = 0$. If $\alpha ^ { 4 } + \beta ^ { 4 } = - 30$, then the product of all possible values of $a$ is $\_\_\_\_$.
Q62 Complex Numbers Argand & Loci Circle Equation and Properties via Complex Number Manipulation View
Let z be a complex number such that $\left| \frac { z - 2 i } { z + i } \right| = 2 , z \neq - i$. Then $z$ lies on the circle of radius 2 and centre
(1) $( 2,0 )$
(2) $( 0,2 )$
(3) $( 0,0 )$
(4) $( 0 , - 2 )$
Q63 Permutations & Arrangements Forming Numbers with Digit Constraints View
The number of numbers, strictly between 5000 and 10000 can be formed using the digits $1,3,5,7,9$ without repetition, is
(1) 6
(2) 12
(3) 120
(4) 72
Q64 Combinations & Selection Selection with Group/Category Constraints View
Suppose Anil's mother wants to give 5 whole fruits to Anil from a basket of 7 red apples, 5 white apples and 8 oranges. If in the selected 5 fruits, at least 2 orange, at least one red apple and at least one white apple must be given, then the number of ways, Anil's mother can offer 5 fruits to Anil is $\_\_\_\_$.
Q65 Arithmetic Sequences and Series Find Specific Term from Given Conditions View
Let $f ( x ) = 2 x ^ { n } + \lambda , \lambda \in \mathbb { R } , \mathrm { n } \in \mathbb { N }$, and $f ( 4 ) = 133 , f ( 5 ) = 255$. Then the sum of all the positive integer divisors of $( f ( 3 ) - f ( 2 ) )$ is
(1) 61
(2) 60
(3) 58
(4) 59
Q66 Geometric Sequences and Series Arithmetic-Geometric Sequence Interplay View
For the two positive numbers $a , b$, if $a , b$ and $\frac { 1 } { 18 }$ are in a geometric progression, while $\frac { 1 } { a } , 10$ and $\frac { 1 } { b }$ are in an arithmetic progression, then $16 a + 12 b$ is equal to $\_\_\_\_$.
Q67 Combinations & Selection Combinatorial Identity or Bijection Proof View
$\sum _ { k = 0 } ^ { 6 } { } ^ { 51 - k } C _ { 3 }$ is equal to
(1) ${ } ^ { 51 } C _ { 4 } - { } ^ { 45 } C _ { 4 }$
(2) ${ } ^ { 51 } C _ { 3 } - { } ^ { 45 } C _ { 3 }$
(3) ${ } ^ { 52 } C _ { 4 } - { } ^ { 45 } C _ { 4 }$
(4) ${ } ^ { 52 } C _ { 3 } - { } ^ { 45 } C _ { 3 }$
Q69 Standard trigonometric equations Solve trigonometric equation for solutions in an interval View
If $m$ and $n$ respectively are the numbers of positive and negative value of $\theta$ in the interval $[ - \pi , \pi ]$ that satisfy the equation $\cos 2 \theta \cos \frac { \theta } { 2 } = \cos 3 \theta \cos \frac { 9 \theta } { 2 }$, then $m n$ is equal to $\_\_\_\_$.
Q71 Circles Tangent Lines and Tangent Lengths View
Points $P ( - 3,2 ) , Q ( 9,10 )$ and $R ( \alpha , 4 )$ lie on a circle $C$ with $P R$ as its diameter. The tangents to $C$ at the points $Q$ and $R$ intersect at the point $S$. If $S$ lies on the line $2 x - k y = 1$, then $k$ is equal to $\_\_\_\_$.
Q72 Conic sections Locus and Trajectory Derivation View
The equations of two sides of a variable triangle are $x = 0$ and $y = 3$, and its third side is a tangent to the parabola $y ^ { 2 } = 6 x$. The locus of its circumcentre is:
(1) $4 y ^ { 2 } - 18 y - 3 x - 18 = 0$
(2) $4 y ^ { 2 } + 18 y + 3 x + 18 = 0$
(3) $4 y ^ { 2 } - 18 y + 3 x + 18 = 0$
(4) $4 y ^ { 2 } - 18 y - 3 x + 18 = 0$
Q74 Matrices Matrix Algebra and Product Properties View
Let $A , B , C$ be $3 \times 3$ matrices such that $A$ is symmetric and $B$ and $C$ are skew-symmetric. Consider the statements $( S 1 ) A ^ { 13 } B ^ { 26 } - B ^ { 26 } A ^ { 13 }$ is symmetric (S2) $A ^ { 26 } C ^ { 13 } - C ^ { 13 } A ^ { 26 }$ is symmetric Then,
(1) Only $S 2$ is true
(2) Only S1 is true
(3) Both $S 1$ and $S 2$ are false
(4) Both $S 1$ and $S 2$ are true
Q75 Matrices Matrix Power Computation and Application View
Let $\mathrm { A } = \left[ \begin{array} { c c } \frac { 1 } { \sqrt { 10 } } & \frac { 3 } { \sqrt { 10 } } \\ \frac { - 3 } { \sqrt { 10 } } & \frac { 1 } { \sqrt { 10 } } \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { c c } 1 & - \mathrm { i } \\ 0 & 1 \end{array} \right]$, where $\mathrm { i } = \sqrt { - 1 }$. If $\mathrm { M } = \mathrm { A } ^ { \mathrm { T } } \mathrm { BA }$, then the inverse of the matrix $\mathrm { AM } ^ { 2023 } \mathrm {~A} ^ { \mathrm { T } }$ is
(1) $\left[ \begin{array} { c c } 1 & - 2023 i \\ 0 & 1 \end{array} \right]$
(2) $\left[ \begin{array} { l l } 1 & 0 \\ - 2023 i & 1 \end{array} \right]$
(3) $\left[ \begin{array} { l l } 1 & 0 \\ 2023 i & 1 \end{array} \right]$
(4) $\left[ \begin{array} { c c } 1 & 2023 i \\ 0 & 1 \end{array} \right]$
Q76 Exponential Functions Parameter Determination from Conditions View
Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a function defined by $f ( x ) = \log _ { \sqrt { m } } \{ \sqrt { 2 } ( \sin x - \cos x ) + m - 2 \}$, for some $m$, such that the range of $f$ is $[ 0,2 ]$. Then the value of $m$ is $\_\_\_\_$.
(1) 5
(2) 3
(3) 2
(4) 4
Q77 Composite & Inverse Functions Counting Functions with Composition or Mapping Constraints View
The number of functions $f : \{ 1,2,3,4 \} \rightarrow \{ \mathrm { a } \in \mathbb { Z } : | \mathrm { a } | \leq 8 \}$ satisfying $f ( \mathrm { n } ) + \frac { 1 } { \mathrm { n } } f ( \mathrm { n } + 1 ) = 1 , \forall \mathrm { n } \in \{ 1,2,3 \}$ is
(1) 3
(2) 4
(3) 1
(4) 2
Q78 Composite & Inverse Functions Find or Apply an Inverse Function Formula View
If the function $f ( x ) = \left\{ \begin{array} { c l } ( 1 + | \cos x | ) \frac { \lambda } { | \cos x | } , & 0 < x < \frac { \pi } { 2 } \\ \mu , & x = \frac { \pi } { 2 } \\ e ^ { \frac { \cot 6 x } { \cot 4 x } } , & \frac { \pi } { 2 } < x < \pi \end{array} \right.$ is continuous at $x = \frac { \pi } { 2 }$, then $9 \lambda + 6 \log _ { e } \mu + \mu ^ { 6 } - e ^ { 6 \lambda }$ is equal to
(1) 11
(2) 8
(3) $2 e ^ { 4 } + 8$
(4) 10
Q79 Stationary points and optimisation Determine parameters from given extremum conditions View
Let the function $f ( x ) = 2 x ^ { 3 } + ( 2 p - 7 ) x ^ { 2 } + 3 ( 2 p - 9 ) x - 6$ have a maxima for some value of $x < 0$ and a minima for some value of $x > 0$. Then, the set of all values of $p$ is
(1) $\left( \frac { 9 } { 2 } , \infty \right)$
(2) $\left( 0 , \frac { 9 } { 2 } \right)$
(3) $\left( - \infty , \frac { 9 } { 2 } \right)$
(4) $\left( - \frac { 9 } { 2 } , \frac { 9 } { 2 } \right)$
Q80 Integration by Substitution Substitution to Evaluate a Definite Integral (Numerical Answer) View
The integral $16 \int _ { 1 } ^ { 2 } \frac { d x } { x ^ { 3 } \left( x ^ { 2 } + 2 \right) ^ { 2 } }$ is equal to
(1) $\frac { 11 } { 6 } + \log _ { e } 4$
(2) $\frac { 11 } { 12 } + \log _ { e } 4$
(3) $\frac { 11 } { 12 } - \log _ { e } 4$
(4) $\frac { 11 } { 6 } - \log _ { e } 4$
Q81 Integration by Substitution Substitution to Evaluate a Definite Integral (Numerical Answer) View
If $\int _ { \frac { 1 } { 3 } } ^ { 3 } \left| \log _ { e } x \right| dx = \frac { m } { n } \log _ { e } \left( \frac { n ^ { 2 } } { e } \right)$, where m and n are coprime natural numbers, then $m ^ { 2 } + n ^ { 2 } - 5$ is equal to $\_\_\_\_$.
Q82 Areas by integration View
Let T and C respectively, be the transverse and conjugate axes of the hyperbola $16 x ^ { 2 } - y ^ { 2 } + 64 x + 4 y + 44 = 0$. Then the area of the region above the parabola $x ^ { 2 } = y + 4$, below the transverse axis T and on the right of the conjugate axis C is:
(1) $4 \sqrt { 6 } + \frac { 44 } { 3 }$
(2) $4 \sqrt { 6 } + \frac { 28 } { 3 }$
(3) $4 \sqrt { 6 } - \frac { 44 } { 3 }$
(4) $4 \sqrt { 6 } - \frac { 28 } { 3 }$
Q83 First order differential equations (integrating factor) View
Let $y = y ( t )$ be a solution of the differential equation $\frac { d y } { d t } + \alpha y = \gamma e ^ { - \beta t }$ where $\alpha > 0 , \beta > 0$ and $\gamma > 0$. Then $\lim _ { t \rightarrow \infty } y ( t )$
(1) is 0
(2) does not exist
(3) is 1
(4) is $-1$
Q84 Vectors 3D & Lines Vector Algebra and Triple Product Computation View
If the four points, whose position vectors are $3 \hat { i } - 4 \hat { j } + 2 \widehat { k } , \hat { i } + 2 \hat { j } - \widehat { k } , - 2 \hat { i } - \hat { j } + 3 \widehat { k }$ and $5 \hat { i } - 2 \alpha \hat { j } + 4 \widehat { k }$ are coplanar, then $\alpha$ is equal to
(1) $\frac { 73 } { 17 }$
(2) $- \frac { 107 } { 17 }$
(3) $- \frac { 73 } { 17 }$
(4) $\frac { 107 } { 17 }$
Q85 Vectors 3D & Lines Vector Algebra and Triple Product Computation View
Let $\vec { a } = - \hat { i } - \hat { j } + \hat { k } , \vec { a } \cdot \vec { b } = 1$ and $\vec { a } \times \vec { b } = \hat { i } - \hat { j }$. Then $\vec { a } - 6 \vec { b }$ is equal to
(1) $3 ( \hat { i } - \hat { j } - \widehat { k } )$
(2) $3 ( \hat { i } + \hat { j } + \hat { k } )$
(3) $3 ( \hat { i } - \hat { j } + \widehat { k } )$
(4) $3 ( \hat { i } + \hat { j } - \widehat { k } )$
Q86 Vectors: Lines & Planes Distance Computation (Point-to-Plane or Line-to-Line) View
The shortest distance between the lines $x + 1 = 2 y = - 12 z$ and $x = y + 2 = 6 z - 6$ is
(1) 2
(2) 3
(3) $\frac { 5 } { 2 }$
(4) $\frac { 3 } { 2 }$
Q87 Vectors: Lines & Planes Perpendicular/Orthogonal Projection onto a Plane View
The foot of perpendicular of the point $( 2,0,5 )$ on the line $\frac { x + 1 } { 2 } = \frac { y - 1 } { 5 } = \frac { z + 1 } { - 1 }$ is $( \alpha , \beta , \gamma )$. Then, which of the following is NOT correct?
(1) $\frac { \alpha \beta } { \gamma } = \frac { 4 } { 15 }$
(2) $\frac { \alpha } { \beta } = - 8$
(3) $\frac { \beta } { \gamma } = - 5$
(4) $\frac { \gamma } { \alpha } = \frac { 5 } { 8 }$
Q88 Vectors: Lines & Planes Distance Computation (Point-to-Plane or Line-to-Line) View
If the shortest distance between the line joining the points $( 1,2,3 )$ and $( 2,3,4 )$, and the line $\frac { x - 1 } { 2 } = \frac { y + 1 } { - 1 } = \frac { z - 2 } { 0 }$ is $\alpha$, then $28 \alpha ^ { 2 }$ is equal to $\_\_\_\_$.
Q89 Probability Definitions Finite Equally-Likely Probability Computation View
Let $N$ be the sum of the numbers appeared when two fair dice are rolled and let the probability that $N - 2 , \sqrt { 3 N } , N + 2$ are in geometric progression be $\frac { k } { 48 }$. Then the value of $k$ is
(1) 2
(2) 4
(3) 16
(4) 8
Q90 Conditional Probability Bayes' Theorem with Diagnostic/Screening Test View
$25\%$ of the population are smokers. A smoker has 27 times more chances to develop lung cancer than a non-smoker. A person is diagnosed with lung cancer and the probability that this person is a smoker is $\frac { k } { 10 }$. Then the value of $k$ is $\_\_\_\_$.