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
02apr 28 08apr 29 09apr 30
2016
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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
2019 session1_10jan_shift1

30 maths questions

Q61 Discriminant and conditions for roots Parameter range for specific root conditions (location/count) View
Consider the quadratic equation $( c - 5 ) x ^ { 2 } - 2 c x + ( c - 4 ) = 0 , c \neq 5$. Let $S$ be the set of all integral values of $c$ for which one root of the equation lies in the interval $( 0,2 )$ and its other root lies in the interval $( 2,3 )$. Then the number of elements in $S$ is
(1) 11
(2) 12
(3) 18
(4) 10
Q62 Complex Numbers Arithmetic Modulus Computation View
Let $z _ { 1 }$ and $z _ { 2 }$ be any two non-zero complex numbers such that $3 \left| z _ { 1 } \right| = 4 \left| z _ { 2 } \right|$. If $z = \frac { 3 z _ { 1 } } { 2 z _ { 2 } } + \frac { 2 z _ { 2 } } { 3 z _ { 1 } }$ then maximum value of $| z |$ is
(1) $\frac { 7 } { 2 }$
(2) $\frac { 9 } { 2 }$
(3) $\frac { 5 } { 2 }$
(4) $\frac { 1 } { 2 } \sqrt { \frac { 17 } { 2 } }$
Q63 Inequalities Solve Polynomial/Rational Inequality for Solution Set View
If $5,5 r , 5 r ^ { 2 }$ are the lengths of the sides of a triangle, then $r$ can not be equal to:
(1) $\frac { 3 } { 4 }$
(2) $\frac { 3 } { 2 }$
(3) $\frac { 5 } { 4 }$
(4) $\frac { 7 } { 4 }$
Q64 Arithmetic Sequences and Series Compute Partial Sum of an Arithmetic Sequence View
The sum of all two digit positive numbers which when divided by 7 yield 2 or 5 as remainder is
(1) 1356
(2) 1365
(3) 1256
(4) 1465
Q65 Binomial Theorem (positive integer n) Evaluate a Summation Involving Binomial Coefficients View
If $\sum _ { i = 1 } ^ { 20 } \left( \frac { { } ^ { 20 } C _ { i - 1 } } { { } ^ { 20 } C _ { i } + { } ^ { 20 } C _ { i - 1 } } \right) ^ { 3 } = \frac { k } { 21 }$, then $k$ equals
(1) 200
(2) 100
(3) 50
(4) 400
Q66 Generalised Binomial Theorem View
If the third term in the binomial expansion of $\left( 1 + x ^ { \log _ { 2 } x } \right) ^ { 5 }$ equals 2560, then a possible value of $x$ is
(1) $4 \sqrt { 2 }$
(2) $\frac { 1 } { 8 }$
(3) $2 \sqrt { 2 }$
(4) $\frac { 1 } { 4 }$
Q67 Quadratic trigonometric equations View
The sum of all values of $\theta \in \left( 0 , \frac { \pi } { 2 } \right)$ satisfying $\sin ^ { 2 } 2 \theta + \cos ^ { 4 } 2 \theta = \frac { 3 } { 4 }$ is
(1) $\frac { \pi } { 2 }$
(2) $\frac { 3 \pi } { 8 }$
(3) $\frac { 5 \pi } { 4 }$
(4) $\pi$
Q68 Straight Lines & Coordinate Geometry Triangle Properties and Special Points View
If the line $3 x + 4 y - 24 = 0$ intersects the $x$-axis is at the point $A$ and the $y$-axis at the point $B$, then the incentre of the triangle $O A B$, where $O$ is the origin, is:
(1) $( 4,4 )$
(2) $( 3,4 )$
(3) $( 4,3 )$
(4) $( 2,2 )$
Q69 Straight Lines & Coordinate Geometry Locus Determination View
A point $P$ moves on the line $2 x - 3 y + 4 = 0$. If $Q ( 1,4 )$ and $R ( 3 , - 2 )$ are fixed points, then the locus of the centroid of $\triangle P Q R$ is a line:
(1) with slope $\frac { 2 } { 3 }$
(2) with slope $\frac { 3 } { 2 }$
(3) parallel to $y$-axis
(4) parallel to $x$-axis
Q70 Circles Circles Tangent to Each Other or to Axes View
If a circle $C$ passing through the point $( 4,0 )$ touches the circle $x ^ { 2 } + y ^ { 2 } + 4 x - 6 y = 12$ externally at the point $( 1 , - 1 )$, then the radius of $C$ is:
(1) 4 units
(2) 5 units
(3) $2 \sqrt { 5 }$ units
(4) $\sqrt { 57 }$ units
Q71 Circles Intersection of Circles or Circle with Conic View
If the parabolas $y ^ { 2 } = 4 b ( x - c )$ and $y ^ { 2 } = 8 a x$ have a common normal, then which one of the following is a valid choice for the ordered triad $( a , b , c )$
(1) $( 1,1,3 )$
(2) $\left( \frac { 1 } { 2 } , 2,0 \right)$
(3) $\left( \frac { 1 } { 2 } , 2,3 \right)$
(4) All of above
Q72 Conic sections Tangent and Normal Line Problems View
The equation of a tangent to the hyperbola, $4 x ^ { 2 } - 5 y ^ { 2 } = 20$, parallel to the line $x - y = 2$, is
(1) $x - y + 7 = 0$
(2) $x - y - 3 = 0$
(3) $x - y + 1 = 0$
(4) $x - y + 9 = 0$
Q73 Sign Change & Interval Methods View
For each $t \in R$, let $[ t ]$ be the greatest integer less than or equal to $t$. Then, $\lim _ { x \rightarrow 1 ^ { + } } \frac { ( 1 - | x | + \sin | 1 - x | ) \sin \left( [ 1 - x ] \frac { \pi } { 2 } \right) } { | 1 - x | [ 1 - x ] }$
(1) equals 0
(2) equals - 1
(3) does not exist
(4) equal 1
Q74 Proof True/False Justification View
Consider the statement: ``$P ( n ) : n ^ { 2 } - n + 41$ is prime''. Then which one of the following is true?
(1) $P ( 3 )$ is false but $P ( 5 )$ is true
(2) Both $P ( 3 )$ and $P ( 5 )$ are false
(3) Both $P ( 3 )$ and $P ( 5 )$ are true
(4) $P ( 5 )$ is false but $P ( 3 )$ is true
Q75 Measures of Location and Spread View
The mean of five observations is 5 and their variance is 9.20. If three of the given five observations are 1, 3 and 8, then a ratio of other two observations is
(1) $10 : 3$
(2) $4 : 9$
(3) $6 : 7$
(4) $5 : 8$
Q76 Sine and Cosine Rules Heights and distances / angle of elevation problem View
Consider a triangular plot $A B C$ with sides $A B = 7 m , B C = 5 m$ and $C A = 6 m$. A vertical lamp-post at the mid-point $D$ of $A C$ subtends an angle $30 ^ { \circ }$ at $B$. The height (in $m$) of the lamp-post is:
(1) $2 \sqrt { 21 }$
(2) $\frac { 2 } { 3 } \sqrt { 21 }$
(3) $\frac { 3 } { 2 } \sqrt { 21 }$
(4) $7 \sqrt { 3 }$
Q77 Principle of Inclusion/Exclusion View
In a class of 140 students numbered 1 to 140, all even numbered students opted Mathematics course, those whose number is divisible by 3 opted Physics course and those whose number is divisible by 5 opted Chemistry course. Then the number of students who did not opt for any of the three courses is:
(1) 42
(2) 1
(3) 38
(4) 102
Q78 Simultaneous equations View
If the system of equations $x + y + z = 5 , x + 2 y + 3 z = 9 , x + 3 y + \alpha z = \beta$ has infinitely many solutions, then $\beta - \alpha$ equals
(1) 8
(2) 21
(3) 5
(4) 18
Q79 3x3 Matrices Determinant of Parametric or Structured Matrix View
Let $d \in R$, and $A = \left[ \begin{array} { c c c } - 2 & 4 + d & ( \sin \theta ) - 2 \\ 1 & ( \sin \theta ) + 2 & d \\ 5 & ( 2 \sin \theta ) - d & ( - \sin \theta ) + 2 + 2 d \end{array} \right] , \theta \in [ 0,2 \pi ]$. If the minimum value of $\operatorname { det } ( A )$ is 8, then a value of $d$ is:
(1) $2 ( \sqrt { 2 } + 2 )$
(2) $2 ( \sqrt { 2 } + 1 )$
(3) $- 5$
(4) $- 7$
Q80 Curve Sketching Continuity and Discontinuity Analysis of Piecewise Functions View
Let $f ( x ) = \left\{ \begin{array} { c c } \max \left( | x | , x ^ { 2 } \right) , & | x | \leq 2 \\ 8 - 2 | x | , & 2 < | x | \leq 4 \end{array} \right.$. Let $S$ be the set of points in the interval $( - 4,4 )$ at which $f$ is not differentiable. Then $S$
(1) equals $\{ - 2 , - 1,0,1,2 \}$
(2) equals $\{ - 2,2 \}$
(3) is an empty set
(4) equal $\{ - 2 , - 1,1,2 \}$
Q81 Chain Rule Higher-Order Derivatives of Products/Compositions View
Let, $f : R \rightarrow R$ be a function such that $f ( x ) = x ^ { 3 } + x ^ { 2 } f \prime ( 1 ) + x f \prime \prime ( 2 ) + f \prime \prime \prime ( 3 ) , \forall x \in R$. Then $f ( 2 )$ equals
(1) 30
(2) 8
(3) $- 4$
(4) $- 2$
Q82 Stationary points and optimisation Geometric or applied optimisation problem View
The shortest distance between the point $\left( \frac { 3 } { 2 } , 0 \right)$ and the curve $y = \sqrt { x } , ( x > 0 )$, is
(1) $\frac { \sqrt { 3 } } { 2 }$
(2) $\frac { 5 } { 4 }$
(3) $\frac { 3 } { 2 }$
(4) $\frac { \sqrt { 5 } } { 2 }$
Q83 Integration by Substitution Substitution to Transform Integral Form (Show Transformed Expression) View
Let, $n \geq 2$ be a natural number and $0 < \theta < \frac { \pi } { 2 }$. Then $\int \frac { \left( \sin ^ { n } \theta - \sin \theta \right) ^ { \frac { 1 } { n } } \cos \theta } { \sin ^ { n + 1 } \theta } d \theta$, is equal to
(1) $\frac { n } { n ^ { 2 } - 1 } \left( 1 - \frac { 1 } { \sin ^ { n + 1 } \theta } \right) ^ { \frac { n + 1 } { n } } + c$
(2) $\frac { n } { n ^ { 2 } + 1 } \left( 1 - \frac { 1 } { \sin ^ { n - 1 } \theta } \right) ^ { \frac { n + 1 } { n } } + c$
(3) $\frac { n } { n ^ { 2 } - 1 } \left( 1 - \frac { 1 } { \sin ^ { n - 1 } \theta } \right) ^ { \frac { n + 1 } { n } } + c$
(4) $\frac { n } { n ^ { 2 } - 1 } \left( 1 + \frac { 1 } { \sin ^ { n - 1 } \theta } \right) ^ { \frac { n + 1 } { n } } + c$
Q84 Stationary points and optimisation Find absolute extrema on a closed interval or domain View
Let $I = \int _ { a } ^ { b } \left( x ^ { 4 } - 2 x ^ { 2 } \right) d x$. If $I$ is minimum then the ordered pair $( a , b )$ is
(1) $( 0 , \sqrt { 2 } )$
(2) $( \sqrt { 2 } , - \sqrt { 2 } )$
(3) $( - \sqrt { 2 } , 0 )$
(4) $( - \sqrt { 2 } , \sqrt { 2 } )$
Q85 Areas by integration View
If the area enclosed between the curves $y = k x ^ { 2 }$ and $x = k y ^ { 2 } , ( k > 0 )$, is 1 sq. unit. Then $k$ is
(1) $\sqrt { 3 }$
(2) $\frac { 1 } { \sqrt { 3 } }$
(3) $\frac { \sqrt { 3 } } { 2 }$
(4) $\frac { 2 } { \sqrt { 3 } }$
Q86 First order differential equations (integrating factor) View
If $\frac { d y } { d x } + \frac { 3 } { \cos ^ { 2 } x } y = \frac { 1 } { \cos ^ { 2 } x } , x \in \left( - \frac { \pi } { 3 } , \frac { \pi } { 3 } \right)$, and $y \left( \frac { \pi } { 4 } \right) = \frac { 4 } { 3 }$, then $y \left( - \frac { \pi } { 4 } \right)$ equals
(1) $\frac { 1 } { 3 }$
(2) $\frac { 1 } { 3 } + e ^ { 3 }$
(3) $\frac { 1 } { 3 } + e ^ { 6 }$
(4) $- \frac { 4 } { 3 }$
Q87 Vectors Introduction & 2D Perpendicularity or Parallel Condition View
Let $\vec { a } = 2 \hat { i } + \lambda _ { 1 } \hat { j } + 3 \hat { k } , \vec { b } = 4 \hat { i } + \left( 3 - \lambda _ { 2 } \right) \hat { j } + 6 \hat { k }$ and $\vec { c } = 3 \hat { i } + 6 \hat { j } + \left( \lambda _ { 3 } - 1 \right) \hat { k }$ be three vectors such that $\vec { b } = 2 \vec { a }$ and $\vec { a }$ is perpendicular to $\vec { c }$. Then a possible value of $\left( \lambda _ { 1 } , \lambda _ { 2 } , \lambda _ { 3 } \right)$ is
(1) $\left( - \frac { 1 } { 2 } , 4,0 \right)$
(2) $( 1,5,1 )$
(3) $\left( \frac { 1 } { 2 } , 4 , - 2 \right)$
(4) $( 1,3,1 )$
Q88 Vectors: Lines & Planes Parallelism Between Line and Plane or Constraint on Parameters View
Let $A$ be a point on the line $\vec { r } = ( 1 - 3 \mu ) \hat { i } + ( \mu - 1 ) \hat { j } + ( 2 + 5 \mu ) \hat { k }$ and $B ( 3,2,6 )$ be a point in the space. Then the value of $\mu$ for which the vector $\overrightarrow { A B }$ is parallel to the plane $x - 4 y + 3 z = 1$ is
(1) $\frac { 1 } { 2 }$
(2) $\frac { 1 } { 4 }$
(3) $- \frac { 1 } { 4 }$
(4) $\frac { 1 } { 8 }$
Q89 Vectors: Lines & Planes Find Cartesian Equation of a Plane View
The plane passing through the point $( 4 , - 1,2 )$ and parallel to the lines $\frac { x + 2 } { 3 } = \frac { y - 2 } { - 1 } = \frac { z + 1 } { 2 }$ and $\frac { x - 2 } { 1 } = \frac { y - 3 } { 2 } = \frac { z - 4 } { 3 }$ also passes through the point
(1) $( 1,1 , - 1 )$
(2) $( - 1 , - 1 , - 1 )$
(3) $( - 1 , - 1,1 )$
(4) $( 1,1,1 )$
Q90 Probability Definitions Conditional Probability and Bayes' Theorem View
An unbiased coin is tossed. If the outcome is a head then a pair of unbiased dice is rolled and the sum of the numbers obtained on them is noted. If the toss of the coin results in tail then a card from a well-shuffled pack of nine cards numbered $1,2,3 , \ldots , 9$ is randomly picked and the number on the card is noted. The probability that the noted number is either 7 or 8 is
(1) $\frac { 13 } { 36 }$
(2) $\frac { 19 } { 72 }$
(3) $\frac { 15 } { 72 }$
(4) $\frac { 19 } { 36 }$