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
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2017
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2016
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2015
04apr 29 10apr 30
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
2022 session2_27jul_shift2

29 maths questions

Q61 Solving quadratics and applications Finding roots or coefficients of a quadratic using Vieta's relations View
If $\alpha , \beta$ are the roots of the equation $x ^ { 2 } - \left( 5 + 3 ^ { \sqrt { \log _ { 3 } 5 } } - 5 ^ { \sqrt { \log _ { 5 } 3 } } \right) x + 3 \left( 3 ^ { \left( \log _ { 3 } 5 \right) ^ { \frac { 1 } { 3 } } } - 5 ^ { \left( \log _ { 5 } 3 \right) ^ { \frac { 2 } { 3 } } } - 1 \right) = 0$ then the equation, whose roots are $\alpha + \frac { 1 } { \beta }$ and $\beta + \frac { 1 } { \alpha }$,
(1) $3 x ^ { 2 } - 20 x - 12 = 0$
(2) $3 x ^ { 2 } - 10 x - 4 = 0$
(3) $3 x ^ { 2 } - 10 x + 2 = 0$
(4) $3 x ^ { 2 } - 20 x + 16 = 0$
Q62 Complex Numbers Arithmetic Systems of Equations via Real and Imaginary Part Matching View
Let $S$ be the set of all $( \alpha , \beta ) , \pi < \alpha , \beta < 2 \pi$, for which the complex number $\frac { 1 - i \sin \alpha } { 1 + 2 i \sin \alpha }$ is purely imaginary and $\frac { 1 + i \cos \beta } { 1 - 2 i \cos \beta }$ is purely real. Let $Z _ { \alpha \beta } = \sin 2 \alpha + i \cos 2 \beta , ( \alpha , \beta ) \in S$. Then $\sum _ { ( \alpha , \beta ) \in S } \left( i Z _ { \alpha \beta } + \frac { 1 } { i \bar { Z } _ { \alpha \beta } } \right)$ is equal to
(1) 3
(2) $3 i$
(3) 1
(4) $2 - i$
Q63 Arithmetic Sequences and Series Compute Partial Sum of an Arithmetic Sequence View
Let the sum of an infinite G.P., whose first term is $a$ and the common ratio is $r$, be 5 . Let the sum of its first five terms be $\frac { 98 } { 25 }$. Then the sum of the first 21 terms of an AP, whose first term is $10 a r , n ^ { \text {th } }$ term is $a _ { n }$ and the common difference is $10 a r ^ { 2 }$, is equal to
(1) $21 a _ { 11 }$
(2) $22 a _ { 11 }$
(3) $15 a _ { 16 }$
(4) $14 a _ { 16 }$
Q64 Standard trigonometric equations Solve trigonometric equation for solutions in an interval View
Let $S = \left\{ \theta \in \left( 0 , \frac { \pi } { 2 } \right) : \sum _ { m = 1 } ^ { 9 } \sec \left( \theta + ( m - 1 ) \frac { \pi } { 6 } \right) \sec \left( \theta + \frac { m \pi } { 6 } \right) = - \frac { 8 } { \sqrt { 3 } } \right\}$. Then
(1) $\mathrm { S } = \left\{ \frac { \pi } { 12 } \right\}$
(2) $S = \left\{ \frac { 2 \pi } { 3 } \right\}$
(3) $\sum _ { \theta \in S } \theta = \frac { \pi } { 2 }$
(4) $\sum _ { \theta \in S } \theta = \frac { 3 \pi } { 4 }$
Q65 Straight Lines & Coordinate Geometry Triangle Properties and Special Points View
The equations of the sides $A B , B C$ and $C A$ of a triangle $A B C$ are $2 x + y = 0 , x + p y = 39$ and $x - y = 3$ respectively and $P ( 2,3 )$ is its circumcentre. Then which of the following is NOT true
(1) $( A C ) ^ { 2 } = 9 p$
(2) $( A C ) ^ { 2 } + p ^ { 2 } = 136$
(3) $32 <$ area $( \triangle A B C ) < 36$
(4) $34 <$ area $( \triangle A B C ) < 38$
Q66 Circles Tangent Lines and Tangent Lengths View
A circle $C _ { 1 }$ passes through the origin $O$ and has diameter 4 on the positive $x$-axis. The line $y = 2 x$ gives a chord $O A$ of a circle $C _ { 1 }$. Let $C _ { 2 }$ be the circle with $O A$ as a diameter. If the tangent to $C _ { 2 }$ at the point $A$ meets the $x$-axis at $P$ and $y$-axis at $Q$, then $Q A : A P$ is equal to
(1) $1 : 4$
(2) $1 : 5$
(3) $2 : 5$
(4) $1 : 3$
Q67 Circles Circle Equation Derivation View
If the length of the latus rectum of a parabola, whose focus is $( a , a )$ and the tangent at its vertex is $x + y = a$, is 16 , then $| a |$ is equal to
(1) $2 \sqrt { 2 }$
(2) $2 \sqrt { 3 }$
(3) $4 \sqrt { 2 }$
(4) 4
Q69 SUVAT in 2D & Gravity View
The angle of elevation of the top $P$ of a vertical tower $P Q$ of height 10 from a point $A$ on the horizontal ground is $45 ^ { \circ }$. Let $R$ be a point on $A Q$ and from a point $B$, vertically above $R$, the angle of elevation of $P$ is $60 ^ { \circ }$. If $\angle B A Q = 30 ^ { \circ } , A B = d$ and the area of the trapezium $P Q R B$ is $\alpha$, then the ordered pair ( $d , \alpha$ ) is
(1) $( 10 ( \sqrt { 3 } - 1 ) , 25 )$
(2) $\left( 10 ( \sqrt { 3 } - 1 ) , \frac { 25 } { 2 } \right)$
(3) $( 10 ( \sqrt { 3 } + 1 ) , 25 )$
(4) $\left( 10 ( \sqrt { 3 } + 1 ) , \frac { 25 } { 2 } \right)$
Q70 Matrices Determinant and Rank Computation View
Let $A = \left( \begin{array} { c c } 4 & - 2 \\ \alpha & \beta \end{array} \right)$. If $A ^ { 2 } + \gamma A + 18 I = O$, then $\operatorname { det } ( A )$ is equal to
(1) - 18
(2) 18
(3) - 50
(4) 50
Q71 Composite & Inverse Functions Determine Domain or Range of a Composite Function View
The domain of the function $f ( x ) = \sin ^ { - 1 } \left[ 2 x ^ { 2 } - 3 \right] + \log _ { 2 } \left( \log _ { \frac { 1 } { 2 } } \left( x ^ { 2 } - 5 x + 5 \right) \right)$, where $[ t ]$ is the greatest integer function, is
(1) $\left( - \sqrt { \frac { 5 } { 2 } } , \frac { 5 - \sqrt { 5 } } { 2 } \right)$
(2) $\left( \frac { 5 - \sqrt { 5 } } { 2 } , \frac { 5 + \sqrt { 5 } } { 2 } \right)$
(3) $\left( 1 , \frac { 5 - \sqrt { 5 } } { 2 } \right)$
(4) $\left[ 1 , \frac { 5 + \sqrt { } 5 } { 2 } \right)$
Q72 Composite & Inverse Functions Find or Apply an Inverse Function Formula View
If for $p \neq q \neq 0$, then function $f ( x ) = \frac { \sqrt [ 7 ] { p ( 729 + x ) } - 3 } { \sqrt [ 3 ] { 729 + q x } - 9 }$ is continuous at $x = 0$, then
(1) $7 p q f ( 0 ) - 1 = 0$
(2) $63 q f ( 0 ) - p ^ { 2 } = 0$
(3) $21 q f ( 0 ) - p ^ { 2 } = 0$
(4) $7 p q f ( 0 ) - 9 = 0$
Q73 Indefinite & Definite Integrals Piecewise/Periodic Function Integration View
Let $f ( x ) = 2 + | x | - | x - 1 | + | x + 1 | , x \in R$. Consider $( S 1 ) : f ^ { \prime } \left( - \frac { 3 } { 2 } \right) + f ^ { \prime } \left( - \frac { 1 } { 2 } \right) + f ^ { \prime } \left( \frac { 1 } { 2 } \right) + f ^ { \prime } \left( \frac { 3 } { 2 } \right) = 2$ $( S 2 ) : \int _ { - 2 } ^ { 2 } f ( x ) d x = 12$ Then,
(1) both ( $S 1$ ) and ( $S 2$ ) are correct
(2) both $( S 1 )$ and $( S 2 )$ are wrong
(3) only ( $S 1$ ) is correct
(4) only ( $S 2$ ) is correct
Q74 Indefinite & Definite Integrals Piecewise/Periodic Function Integration View
$\int _ { 0 } ^ { 2 } \left( \left| 2 x ^ { 2 } - 3 x \right| + \left[ x - \frac { 1 } { 2 } \right] \right) d x$, where $[ t ]$ is the greatest integer function, is equal to
(1) $\frac { 7 } { 6 }$
(2) $\frac { 19 } { 12 }$
(3) $\frac { 31 } { 12 }$
(4) $\frac { 3 } { 2 }$
Q75 Areas by integration View
The area of the region enclosed by $y \leq 4 x ^ { 2 } , x ^ { 2 } \leq 9 y$ and $y \leq 4$, is equal to
(1) $\frac { 40 } { 3 }$
(2) $\frac { 56 } { 3 }$
(3) $\frac { 112 } { 3 }$
(4) $\frac { 80 } { 3 }$
Q76 Tangents, normals and gradients Normal or perpendicular line problems View
Consider a curve $y = y ( x )$ in the first quadrant as shown in the figure. Let the area $A _ { 1 }$ is twice the area $A _ { 2 }$. Then the normal to the curve perpendicular to the line $2 x - 12 y = 15$ does NOT pass through the point
(1) $( 6,21 )$
(2) $( 8,9 )$
(3) $( 10 , - 4 )$
(4) $( 12 , - 15 )$
Q77 Vectors 3D & Lines Distance from a Point to a Line (Show/Compute) View
If the length of the perpendicular drawn from the point $P ( a , 4,2 ) , a > 0$ on the line $\frac { x + 1 } { 2 } = \frac { y - 3 } { 3 } = \frac { z - 1 } { - 1 }$ is $2 \sqrt { 6 }$ units and $Q \left( \alpha _ { 1 } , \alpha _ { 2 } , \alpha _ { 3 } \right)$ is the image of the point $P$ in this line, then $a + \sum _ { i = 1 } ^ { 3 } \alpha _ { i }$ is equal to
(1) 7
(2) 8
(3) 12
(4) 14
Q78 Vectors: Lines & Planes Dihedral Angle or Angle Between Planes/Lines View
If the line of intersection of the planes $a x + b y = 3$ and $a x + b y + c z = 0 , a > 0$ makes an angle $30 ^ { \circ }$ with the plane $y - z + 2 = 0$, then the direction cosines of the line are
(1) $\frac { 1 } { \sqrt { 2 } } , \frac { 1 } { \sqrt { 2 } } , 0$
(2) $\frac { 1 } { \sqrt { 2 } } , \frac { - 1 } { \sqrt { 2 } } , 0$
(3) $\frac { 1 } { \sqrt { 5 } } , - \frac { 2 } { \sqrt { 5 } } , 0$
(4) $\frac { 1 } { 2 } , - \frac { \sqrt { 3 } } { 2 } , 0$
Q79 Binomial Distribution Find Parameters from Moment Conditions View
Let $X$ have a binomial distribution $B ( n , p )$ such that the sum and the product of the mean and variance of $X$ are 24 and 128 respectively. If $P ( X > n - 3 ) = \frac { k } { 2 ^ { n } }$, then $k$ is equal to
(1) 528
(2) 529
(3) 629
(4) 630
Q80 Discrete Probability Distributions Binomial Distribution Identification and Application View
A six faced die is biased such that $3 \times P ($ a prime number $) = 6 \times P ($ a composite number $) = 2 \times P ( 1 )$. Let $X$ be a random variable that counts the number of times one gets a perfect square on some throws of this die. If the die is thrown twice, then the mean of $X$ is
(1) $\frac { 3 } { 11 }$
(2) $\frac { 5 } { 11 }$
(3) $\frac { 7 } { 11 }$
(4) $\frac { 8 } { 11 }$
Q81 Arithmetic Sequences and Series Telescoping or Non-Standard Summation Involving an AP View
$\frac { 2 ^ { 3 } - 1 ^ { 3 } } { 1 \times 7 } + \frac { 4 ^ { 3 } - 3 ^ { 3 } + 2 ^ { 3 } - 1 ^ { 3 } } { 2 \times 11 } + \frac { 6 ^ { 3 } - 5 ^ { 3 } + 4 ^ { 3 } - 3 ^ { 3 } + 2 ^ { 3 } - 1 ^ { 3 } } { 3 \times 15 } + \ldots \ldots + \frac { 30 ^ { 3 } - 29 ^ { 3 } + 28 ^ { 3 } - 27 ^ { 3 } + \ldots + 2 ^ { 3 } - 1 ^ { 3 } } { 15 \times 63 }$ is equal to $\_\_\_\_$ .
Q82 Binomial Theorem (positive integer n) Find the Largest Term or Coefficient in a Binomial Expansion View
Let for the $9 ^ { \text {th } }$ term in the binomial expansion of $( 3 + 6 x ) ^ { n }$, in the increasing powers of $6 x$, to be the greatest for $x = \frac { 3 } { 2 }$, the least value of $n$ is $n _ { 0 }$. If $k$ is the ratio of the coefficient of $x ^ { 6 }$ to the coefficient of $x ^ { 3 }$, then $k + n _ { 0 }$ is equal to $\_\_\_\_$ .
Q83 Conic sections Tangent and Normal Line Problems View
A common tangent $T$ to the curves $C _ { 1 } : \frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 9 } = 1$ and $C _ { 2 } : \frac { x ^ { 2 } } { 42 } - \frac { y ^ { 2 } } { 143 } = 1$ does not pass through the fourth quadrant. If $T$ touches $C _ { 1 }$ at $\left( x _ { 1 } , y _ { 1 } \right)$ and $C _ { 2 }$ at $\left( x _ { 2 } , y _ { 2 } \right)$, then $\left| 2 x _ { 1 } + x _ { 2 } \right|$ is equal to $\_\_\_\_$ .
Q84 Matrices Determinant and Rank Computation View
Consider a matrix $\mathrm { A } = \left[ \begin{array} { c c c } \alpha & \beta & \gamma \\ \alpha ^ { 2 } & \beta ^ { 2 } & \gamma ^ { 2 } \\ \beta + \gamma & \gamma + \alpha & \alpha + \beta \end{array} \right]$, where $\alpha , \beta , \gamma$ are three distinct natural numbers. If $\frac { \operatorname { det } ( \operatorname { adj } ( \operatorname { adj } ( \operatorname { adj } ( \operatorname { adj } A ) ) ) } { ( \alpha - \beta ) ^ { 16 } ( \beta - \gamma ) ^ { 16 } ( \gamma - \alpha ) ^ { 16 } } = 2 ^ { 32 } \times 3 ^ { 16 }$, then the number of such 3-tuples $( \alpha , \beta , \gamma )$ is $\_\_\_\_$ .
Q85 Composite & Inverse Functions Counting Functions with Composition or Mapping Constraints View
The number of functions $f$, from the set $A = \left\{ x \in N : x ^ { 2 } - 10 x + 9 \leq 0 \right\}$ to the set $B = \left\{ n ^ { 2 } : n \in N \right\}$ such that $f ( x ) \leq ( x - 3 ) ^ { 2 } + 1$, for every $x \in A$, is $\_\_\_\_$ .
Q86 Implicit equations and differentiation Second derivative via implicit differentiation View
For the curve $C : \left( x ^ { 2 } + y ^ { 2 } - 3 \right) + \left( x ^ { 2 } - y ^ { 2 } - 1 \right) ^ { 5 } = 0$, the value of $3 y ^ { \prime } - y ^ { 3 } y ^ { \prime \prime }$, at the point $( \alpha , \alpha ) , \alpha > 0$, on $C$, is equal to $\_\_\_\_$ .
Q87 Applied differentiation Applied modeling with differentiation View
A water tank has the shape of a right circular cone with axis vertical and vertex downwards. Its semivertical angle is $\tan ^ { - 1 } \frac { 3 } { 4 }$. Water is poured in it at a constant rate of 6 cubic meter per hour. The rate (in square meter per hour), at which the wet curved surface area of the tank is increasing, when the depth of water in the tank is 4 meters, is $\_\_\_\_$ .
Q88 Indefinite & Definite Integrals Piecewise/Periodic Function Integration View
Let $f ( x ) = \min \{ [ x - 1 ] , [ x - 2 ] , \ldots , [ x - 10 ] \}$ where $[ t ]$ denotes the greatest integer $\leq t$. Then $\int _ { 0 } ^ { 10 } f ( x ) d x + \int _ { 0 } ^ { 10 } ( f ( x ) ) ^ { 2 } d x + \int _ { 0 } ^ { 10 } | f ( x ) | d x$ is equal to $\_\_\_\_$ .
Q89 Differential equations Integral Equations Reducible to DEs View
Let $f$ be a differentiable function satisfying $f ( x ) = \frac { 2 } { \sqrt { 3 } } \int _ { 0 } ^ { \sqrt { 3 } } f \left( \frac { \lambda ^ { 2 } x } { 3 } \right) d \lambda , x > 0$ and $f ( 1 ) = \sqrt { 3 }$. If $y = f ( x )$ passes through the point $( \alpha , 6 )$, then $\alpha$ is equal to $\_\_\_\_$ .
Q90 Vectors Introduction & 2D Dot Product Computation View
Let $\vec { a } , \vec { b } , \vec { c }$ be three non-coplanar vectors such that $\vec { a } \times \vec { b } = \overrightarrow { 4 c } , \vec { b } \times \vec { c } = 9 \vec { a }$ and $\vec { c } \times \vec { a } = \alpha \vec { b } , \alpha > 0$. If $| \vec { a } | + | \vec { b } | + | \vec { c } | = 36$, then $\alpha$ is equal to $\_\_\_\_$ .