jee-main

Papers (169)
2025
session1_22jan_shift1 25 session1_22jan_shift2 25 session1_23jan_shift1 25 session1_23jan_shift2 25 session1_24jan_shift1 25 session1_24jan_shift2 25 session1_28jan_shift1 25 session1_28jan_shift2 25 session1_29jan_shift1 29 session1_29jan_shift2 25
2024
session1_01feb_shift1 4 session1_01feb_shift2 22 session1_27jan_shift1 28 session1_27jan_shift2 30 session1_29jan_shift1 30 session1_29jan_shift2 23 session1_30jan_shift1 17 session1_30jan_shift2 30 session1_31jan_shift1 16 session1_31jan_shift2 15 session2_04apr_shift1 4 session2_04apr_shift2 30 session2_05apr_shift1 4 session2_05apr_shift2 30 session2_06apr_shift1 22 session2_06apr_shift2 30 session2_08apr_shift1 30 session2_08apr_shift2 30 session2_09apr_shift1 5 session2_09apr_shift2 30
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
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
2021 session1_25feb_shift1

29 maths questions

Q61 Discriminant and conditions for roots Parameter range for no real roots (positive definite) View
The integer $k$, for which the inequality $x ^ { 2 } - 2 ( 3 k - 1 ) x + 8 k ^ { 2 } - 7 > 0$ is valid for every $x$ in $R$ is:
(1) 4
(2) 2
(3) 3
(4) 0
Q62 Circles Distance from Center to Line View
Let the lines $( 2 - i ) z = ( 2 + i ) \bar { z }$ and $( 2 + i ) z + ( i - 2 ) \bar { z } - 4 i = 0$, (here $i ^ { 2 } = - 1$ ) be normal to a circle $C$. If the line $i z + \bar { z } + 1 + i = 0$ is tangent to this circle $C$, then its radius is :
(1) $\frac { 3 } { \sqrt { 2 } }$
(2) $3 \sqrt { 2 }$
(3) $\frac { 3 } { 2 \sqrt { 2 } }$
(4) $\frac { 1 } { 2 \sqrt { 2 } }$
Q63 Permutations & Arrangements Distribution of Objects into Bins/Groups View
The total number of positive integral solutions $( x , y , z )$ such that $x y z = 24$ is :
(1) 45
(2) 30
(3) 36
(4) 24
Q64 Geometric Sequences and Series Geometric Series with Trigonometric or Functional Terms View
If $0 < \theta , \phi < \frac { \pi } { 2 } , x = \sum _ { n = 0 } ^ { \infty } \cos ^ { 2 n } \theta , y = \sum _ { n = 0 } ^ { \infty } \sin ^ { 2 n } \phi$ and $z = \sum _ { n = 0 } ^ { \infty } \cos ^ { 2 n } \theta \cdot \sin ^ { 2 n } \phi$ then :
(1) $x y - z = ( x + y ) z$
(2) $x y + y z + z x = z$
(3) $x y + z = ( x + y ) z$
(4) $x y z = 4$
Q65 Standard trigonometric equations Solve trigonometric inequality View
All possible values of $\theta \in [ 0,2 \pi ]$ for which $\sin 2 \theta + \tan 2 \theta > 0$ lie in :
(1) $\left( 0 , \frac { \pi } { 2 } \right) \cup \left( \pi , \frac { 3 \pi } { 2 } \right)$
(2) $\left( 0 , \frac { \pi } { 2 } \right) \cup \left( \frac { \pi } { 2 } , \frac { 3 \pi } { 4 } \right) \cup \left( \pi , \frac { 7 \pi } { 6 } \right)$
(3) $\left( 0 , \frac { \pi } { 4 } \right) \cup \left( \frac { \pi } { 2 } , \frac { 3 \pi } { 4 } \right) \cup \left( \pi , \frac { 5 \pi } { 4 } \right) \cup \left( \frac { 3 \pi } { 2 } , \frac { 7 \pi } { 4 } \right)$
(4) $\left( 0 , \frac { \pi } { 4 } \right) \cup \left( \frac { \pi } { 2 } , \frac { 3 \pi } { 4 } \right) \cup \left( \frac { 3 \pi } { 2 } , \frac { 11 \pi } { 6 } \right)$
Q66 Circles Circle Equation Derivation View
The image of the point $( 3,5 )$ in the line $x - y + 1 = 0$, lies on :
(1) $( x - 2 ) ^ { 2 } + ( y - 4 ) ^ { 2 } = 4$
(2) $( x - 4 ) ^ { 2 } + ( y - 4 ) ^ { 2 } = 8$
(3) $( x - 4 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 16$
(4) $( x - 2 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 12$
Q67 Circles Tangent Lines and Tangent Lengths View
A tangent is drawn to the parabola $y ^ { 2 } = 6 x$ which is perpendicular to the line $2 x + y = 1$. Which of the following points does NOT lie on it?
(1) $( 0,3 )$
(2) $( 4,5 )$
(3) $( 5,4 )$
(4) $( - 6,0 )$
Q68 Circles Intersection of Circles or Circle with Conic View
If the curves, $\frac { x ^ { 2 } } { a } + \frac { y ^ { 2 } } { b } = 1$ and $\frac { x ^ { 2 } } { c } + \frac { y ^ { 2 } } { d } = 1$ intersect each other at an angle of $90 ^ { \circ }$, then which of the following relations is TRUE?
(1) $a - c = b + d$
(2) $a - b = c - d$
(3) $a + b = c + d$
(4) $a b = \frac { c + d } { a + b }$
Q69 Curve Sketching Limit Computation from Algebraic Expressions View
$\lim _ { n \rightarrow \infty } \left( 1 + \frac { 1 + \frac { 1 } { 2 } + \ldots\ldots + \frac { 1 } { n } } { n ^ { 2 } } \right) ^ { n }$ is equal to
(1) $\frac { 1 } { e }$
(2) 0
(3) $\frac { 1 } { 2 }$
(4) 1
Q71 Sine and Cosine Rules Heights and distances / angle of elevation problem View
A man is observing, from the top of a tower, a boat speeding towards the tower from a certain point $A$, with uniform speed. At that point, angle of depression of the boat with the man's eye is $30 ^ { \circ }$ (Ignore man's height). After sailing for 20 seconds, towards the base of the tower (which is at the level of water), the boat has reached a point $B$, where the angle of depression is $45 ^ { \circ }$. Then the time taken (in seconds) by the boat from $B$ to reach the base of the tower is :
(1) 10
(2) $10 ( \sqrt { 3 } - 1 )$
(3) $10 \sqrt { 3 }$
(4) $10 ( \sqrt { 3 } + 1 )$
Q72 Composite & Inverse Functions Injectivity, Surjectivity, or Bijectivity Classification View
Let $f , g : N \rightarrow N$ such that $f ( n + 1 ) = f ( n ) + f ( 1 ) \forall n \in N$ and $g$ be any arbitrary function. Which of the following statements is NOT true?
(1) If $f$ is onto, then $f ( n ) = n \forall n \in N$
(2) If $g$ is onto, then $f o g$ is one-one
(3) $f$ is one-one
(4) If $f \circ g$ is one-one, then $g$ is one-one
Q73 Stationary points and optimisation Determine parameters from given extremum conditions View
If Rolle's theorem holds for the function $f ( x ) = x ^ { 3 } - a x ^ { 2 } + b x - 4 , x \in [ 1,2 ]$ with $f ^ { \prime } \left( \frac { 4 } { 3 } \right) = 0$, then ordered pair $( a , b )$ is equal to :
(1) $( - 5 , - 8 )$
(2) $( - 5,8 )$
(3) $( 5,8 )$
(4) $( 5 , - 8 )$
Q74 Integration by Substitution Substitution to Transform Integral Form (Show Transformed Expression) View
The value of the integral $\int \frac { \sin \theta \cdot \sin 2 \theta \left( \sin ^ { 6 } \theta + \sin ^ { 4 } \theta + \sin ^ { 2 } \theta \right) \sqrt { 2 \sin ^ { 4 } \theta + 3 \sin ^ { 2 } \theta + 6 } } { 1 - \cos 2 \theta } d \theta$ is (where $c$ is a constant of integration)
(1) $\frac { 1 } { 18 } \left[ 11 - 18 \sin ^ { 2 } \theta + 9 \sin ^ { 4 } \theta - 2 \sin ^ { 6 } \theta \right] ^ { \frac { 3 } { 2 } } + c$
(2) $\frac { 1 } { 18 } \left[ 9 - 2 \sin ^ { 6 } \theta - 3 \sin ^ { 4 } \theta - 6 \sin ^ { 2 } \theta \right] ^ { \frac { 3 } { 2 } } + c$
(3) $\frac { 1 } { 18 } \left[ 11 - 18 \cos ^ { 2 } \theta + 9 \cos ^ { 4 } \theta - 2 \cos ^ { 6 } \theta \right] ^ { \frac { 3 } { 2 } } + c$
(4) $\frac { 1 } { 18 } \left[ 9 - 2 \cos ^ { 6 } \theta - 3 \cos ^ { 4 } \theta - 6 \cos ^ { 2 } \theta \right] ^ { - \frac { 3 } { 2 } } + c$
Q75 Indefinite & Definite Integrals Piecewise/Periodic Function Integration View
The value of $\int _ { - 1 } ^ { 1 } x ^ { 2 } e ^ { \left[ x ^ { 3 } \right] } d x$, where $[ t ]$ denotes the greatest integer $\leq t$, is :
(1) $\frac { e + 1 } { 3 }$
(2) $\frac { e - 1 } { 3 e }$
(3) $\frac { 1 } { 3 e }$
(4) $\frac { e + 1 } { 3 e }$
Q76 Differential equations Solving Separable DEs with Initial Conditions View
If a curve passes through the origin and the slope of the tangent to it at any point $( x , y )$ is $\frac { x ^ { 2 } - 4 x + y + 8 } { x - 2 }$, then this curve also passes through the point:
(1) $( 5,4 )$
(2) $( 4,4 )$
(3) $( 4,5 )$
(4) $( 5,5 )$
Q77 Vectors: Lines & Planes Dihedral Angle or Angle Between Planes/Lines View
Let $\alpha$ be the angle between the lines whose direction cosines satisfy the equations $l + m - n = 0$ and $l ^ { 2 } + m ^ { 2 } - n ^ { 2 } = 0$. Then the value of $\sin ^ { 4 } \alpha + \cos ^ { 4 } \alpha$ is :
(1) $\frac { 5 } { 8 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 3 } { 8 }$
(4) $\frac { 3 } { 4 }$
Q78 Vectors 3D & Lines Parametric Representation of a Line View
The equation of the line through the point $( 0,1,2 )$ and perpendicular to the line $\frac { x - 1 } { 2 } = \frac { y + 1 } { 3 } = \frac { z - 1 } { - 2 }$ is :
(1) $\frac { x } { 3 } = \frac { y - 1 } { - 4 } = \frac { z - 2 } { 3 }$
(2) $\frac { x } { 3 } = \frac { y - 1 } { 4 } = \frac { z - 2 } { 3 }$
(3) $\frac { x } { - 3 } = \frac { y - 1 } { 4 } = \frac { z - 2 } { 3 }$
(4) $\frac { x } { 3 } = \frac { y - 1 } { 4 } = \frac { z - 2 } { - 3 }$
Q79 Discriminant and conditions for roots Probability involving discriminant conditions View
The coefficients $a , b$ and $c$ of the quadratic equation, $a x ^ { 2 } + b x + c = 0$ are obtained by throwing a dice three times. The probability that this equation has equal roots is:
(1) $\frac { 1 } { 72 }$
(2) $\frac { 1 } { 36 }$
(3) $\frac { 1 } { 54 }$
(4) $\frac { 5 } { 216 }$
Q80 Conditional Probability Sequential/Multi-Stage Conditional Probability View
When a missile is fired from a ship, the probability that it is intercepted is $\frac { 1 } { 3 }$ and the probability that the missile hits the target, given that it is not intercepted, is $\frac { 3 } { 4 }$. If three missiles are fired independently from the ship, then the probability that all three hit the target, is:
(1) $\frac { 3 } { 8 }$
(2) $\frac { 1 } { 27 }$
(3) $\frac { 1 } { 8 }$
(4) $\frac { 3 } { 4 }$
Q81 Permutations & Arrangements Forming Numbers with Digit Constraints View
The total number of numbers, lying between 100 and 1000 that can be formed with the digits $1,2,3,4,5$, if the repetition of digits is not allowed and numbers are divisible by either 3 or 5, is
Q82 Geometric Sequences and Series Find a Threshold Index (Algorithm or Calculation) View
Let $A _ { 1 } , A _ { 2 } , A _ { 3 } , \ldots$ be squares such that for each $n \geqslant 1$, the length of the side of $A _ { n }$ equals the length of diagonal of $A _ { n + 1 }$. If the length of $A _ { 1 }$ is 12 cm, then the smallest value of $n$ for which area of $A _ { n }$ is less than one, is
Q83 Conic sections Locus and Trajectory Derivation View
The locus of the point of intersection of the lines $( \sqrt { 3 } ) k x + k y - 4 \sqrt { 3 } = 0$ and $\sqrt { 3 } x - y - 4 ( \sqrt { 3 } ) k = 0$ is a conic, whose eccentricity is
Q84 Matrices Matrix Algebra and Product Properties View
If $A = \left[ \begin{array} { c c } 0 & - \tan \left( \frac { \theta } { 2 } \right) \\ \tan \left( \frac { \theta } { 2 } \right) & 0 \end{array} \right]$ and $\left( I _ { 2 } + A \right) \left( I _ { 2 } - A \right) ^ { - 1 } = \left[ \begin{array} { c c } a & - b \\ b & a \end{array} \right]$, then $13 \left( a ^ { 2 } + b ^ { 2 } \right)$ is equal to
Q85 3x3 Matrices Matrix Algebraic Properties and Abstract Reasoning View
Let $A = \left[ \begin{array} { l l l } x & y & z \\ y & z & x \\ z & x & y \end{array} \right]$, where $x , y$ and $z$ are real numbers such that $x + y + z > 0$ and $x y z = 2$. If $A ^ { 2 } = I _ { 3 }$, then the value of $x ^ { 3 } + y ^ { 3 } + z ^ { 3 }$ is
Q86 Simultaneous equations View
If the system of equations $$\begin{aligned} & k x + y + 2 z = 1 \\ & 3 x - y - 2 z = 2 \\ & - 2 x - 2 y - 4 z = 3 \end{aligned}$$ has infinitely many solutions, then $k$ is equal to
Q87 Modulus function Differentiability of functions involving modulus View
The number of points, at which the function $f ( x ) = | 2 x + 1 | - 3 | x + 2 | + \left| x ^ { 2 } + x - 2 \right| , x \in R$ is not differentiable, is
Q88 Stationary points and optimisation Determine parameters from given extremum conditions View
Let $f ( x )$ be a polynomial of degree 6 in $x$, in which the coefficient of $x ^ { 6 }$ is unity and it has extrema at $x = - 1$ and $x = 1$. If $\lim _ { x \rightarrow 0 } \frac { f ( x ) } { x ^ { 3 } } = 1$, then $5 \cdot f ( 2 )$ is equal to
Q89 Areas by integration View
The graphs of sine and cosine functions, intersect each other at a number of points and between two consecutive points of intersection, the two graphs enclose the same area $A$. Then $A ^ { 4 }$ is equal to
Q90 Vectors 3D & Lines Vector Algebra and Triple Product Computation View
Let $\vec { a } = \hat { i } + 2 \hat { j } - \widehat { k } , \vec { b } = \hat { i } - \hat { j }$ and $\vec { c } = \hat { i } - \hat { j } - \hat { k }$ be three given vectors. If $\vec { r }$ is a vector such that $\vec { r } \times \vec { a } = \vec { c } \times \vec { a }$ and $\vec { r } \cdot \vec { b } = 0$, then $\vec { r } \cdot \vec { a }$ is equal to