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
session1_07jan_shift1 28 session1_07jan_shift2 20 session1_08jan_shift1 5 session1_08jan_shift2 11 session1_09jan_shift1 26 session1_09jan_shift2 16 session2_02sep_shift1 18 session2_02sep_shift2 16 session2_03sep_shift1 23 session2_03sep_shift2 8 session2_04sep_shift1 14 session2_04sep_shift2 27 session2_05sep_shift1 22 session2_05sep_shift2 29 session2_06sep_shift1 11 session2_06sep_shift2 10
2019
session1_09jan_shift1 6 session1_09jan_shift2 29 session1_10jan_shift1 29 session1_10jan_shift2 14 session1_11jan_shift1 6 session1_11jan_shift2 5 session1_12jan_shift1 10 session1_12jan_shift2 29 session2_08apr_shift1 29 session2_08apr_shift2 29 session2_09apr_shift1 29 session2_09apr_shift2 29 session2_10apr_shift1 2 session2_10apr_shift2 5 session2_12apr_shift1 3 session2_12apr_shift2 9
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
04apr 29 10apr 29 11apr 8
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
2019 session1_10jan_shift1

29 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 Binomial Theorem (positive integer n) Determine Parameters from Conditions on Coefficients or Terms 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 Trig Proofs Trigonometric Equation Constraint Deduction 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 Conic sections 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) $x - y + 9 = 0$
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 Applied differentiation 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 3D & Lines 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 3D & Lines 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 3D & Lines 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) $\frac { 19 } { 36 }$