jee-main

Papers (169)
2025
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2024
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2023
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2022
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2021
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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
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
2014 19apr

29 maths questions

Q61 Solving quadratics and applications Solving an equation via substitution to reduce to quadratic form View
The equation $\sqrt { 3 x ^ { 2 } + x + 5 } = x - 3$, where $x$ is real, has
(1) no solution
(2) exactly four solutions
(3) exactly one solution
(4) exactly two solutions
Q62 Complex Numbers Arithmetic Systems of Equations via Real and Imaginary Part Matching View
For all complex numbers $z$ of the form $1 + i \alpha , \alpha \in R$, if $z ^ { 2 } = x + i y$, then
(1) $y ^ { 2 } - 4 x + 4 = 0$
(2) $y ^ { 2 } + 4 x - 4 = 0$
(3) $y ^ { 2 } - 4 x + 2 = 0$
(4) $y ^ { 2 } + 4 x + 2 = 0$
Q63 Combinations & Selection Selection with Group/Category Constraints View
Two women and some men participated in a chess tournament in which every participant played two games with each of the other participants. If the number of games that the men played between them-selves exceeds the number of games that the men played with the women by 66 , then the number of men who participated in the tournament lies in the interval
(1) $( 11,13 ]$
(2) $( 14,17 )$
(3) $[ 10,12 )$
(4) $[ 8,9 ]$
Q64 Arithmetic Sequences and Series Compute Partial Sum of an Arithmetic Sequence View
Let $f ( n ) = \left[ \frac { 1 } { 3 } + \frac { 3 n } { 100 } \right] n$, where $[ n ]$ denotes the greatest integer less than or equal to $n$. Then $\sum _ { n = 1 } ^ { 56 } f ( n )$ is equal to
(1) 56
(2) 1287
(3) 1399
(4) 689
Q65 Arithmetic Sequences and Series Counting or Combinatorial Problems on APs View
The number of terms in an $A.P$. is even, the sum of the odd terms in it is 24 and that the even terms is 30 . If the last term exceeds the first term by $10 \frac { 1 } { 2 }$, then the number of terms in the $A.P$. is
(1) 4
(2) 8
(3) 16
(4) 12
Q66 Binomial Theorem (positive integer n) Find a Specific Coefficient in a Single Binomial Expansion View
The coefficient of $x ^ { 1012 }$ in the expansion of $\left( 1 + x ^ { n } + x ^ { 253 } \right) ^ { 10 }$, (where $n \leq 22$ is any positive integer), is
(1) ${ } ^ { 253 } C _ { 4 }$
(2) ${ } ^ { 10 } C _ { 4 }$
(3) $4 n$
(4) 1
Q67 Straight Lines & Coordinate Geometry Point-to-Line Distance Computation View
If a line $L$ is perpendicular to the line $5 x - y = 1$, and the area of the triangle formed by the line $L$ and the coordinate axes is 5 sq units, then the distance of the line $L$ from the line $x + 5 y = 0$ is
(1) $\frac { 7 } { \sqrt { 13 } }$ units
(2) $\frac { 7 } { \sqrt { 5 } }$ units
(3) $\frac { 5 } { \sqrt { 13 } }$ units
(4) $\frac { 5 } { \sqrt { 7 } }$ units
Q68 Straight Lines & Coordinate Geometry Triangle Properties and Special Points View
The circumcentre of a triangle lies at the origin and its centroid is the midpoint of the line segment joining the points $\left( a ^ { 2 } + 1 , a ^ { 2 } + 1 \right)$ and $( 2 a , - 2 a ) , a \neq 0$. Then for any $a$, the orthocentre of this triangle lies on the line
(1) $y - \left( a ^ { 2 } + 1 \right) x = 0$
(2) $y - 2 a x = 0$
(3) $y + x = 0$
(4) $( a - 1 ) ^ { 2 } x - ( a + 1 ) ^ { 2 } y = 0$
Q69 Circles Circle Equation Derivation View
The equation of the circle described on the chord $3 x + y + 5 = 0$ of the circle $x ^ { 2 } + y ^ { 2 } = 16$ as the diameter is
(1) $x ^ { 2 } + y ^ { 2 } + 3 x + y + 1 = 0$
(2) $x ^ { 2 } + y ^ { 2 } + 3 x + y - 22 = 0$
(3) $x ^ { 2 } + y ^ { 2 } + 3 x + y - 11 = 0$
(4) $x ^ { 2 } + y ^ { 2 } + 3 x + y - 2 = 0$
Q70 Conic sections Focal Chord and Parabola Segment Relations View
A chord is drawn through the focus of the parabola $y ^ { 2 } = 6 x$ such that its distance from the vertex of this parabola is $\frac { \sqrt { 5 } } { 2 }$, then its slope can be
(1) $\frac { \sqrt { 5 } } { 2 }$
(2) $\frac { 2 } { \sqrt { 3 } }$
(3) $\frac { \sqrt { 3 } } { 2 }$
(4) $\frac { 2 } { \sqrt { 5 } }$
Q71 Conic sections Tangent and Normal Line Problems View
The tangent at an extremity (in the first quadrant) of the latus rectum of the hyperbola $\frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 5 } = 1$, meets the $x$-axis and $y$-axis at $A$ and $B$, respectively. Then $OA ^ { 2 } - OB ^ { 2 }$, where $O$ is the origin, equals
(1) $- \frac { 20 } { 9 }$
(2) $\frac { 16 } { 9 }$
(3) 4
(4) $- \frac { 4 } { 3 }$
Q73 Measures of Location and Spread View
Let $\bar { x } , M$ and $\sigma ^ { 2 }$ be respectively the mean, mode and variance of $n$ observations $x _ { 1 } , x _ { 2 } , \ldots , x _ { n }$ and $d _ { i } = - x _ { i } - a , i = 1,2 , \ldots , n$, where $a$ is any number. Statement I: Variance of $d _ { 1 } , d _ { 2 } , \ldots , d _ { n }$ is $\sigma ^ { 2 }$. Statement II: Mean and mode of $d _ { 1 } , d _ { 2 } , \ldots , d _ { n }$ are $- \bar { x } - a$ and $- M - a$, respectively.
(1) Statement I and Statement II are both true
(2) Statement I and Statement II are both false
(3) Statement I is true and Statement II is false
(4) Statement I is false and Statement II is true
Q74 Matrices Matrix Algebra and Product Properties View
Let $A$ and $B$ be any two $3 \times 3$ matrices. If $A$ is symmetric and $B$ is skew symmetric, then the matrix $AB - BA$ is
(1) skew symmetric
(2) $I$ or $- I$, where $I$ is an identity matrix
(3) symmetric
(4) neither symmetric nor skew symmetric
Q75 Polynomial Division & Manipulation View
If $\Delta _ { r } = \left| \begin{array} { c c c } r & 2 r - 1 & 3 r - 2 \\ \frac { n } { 2 } & n - 1 & a \\ \frac { 1 } { 2 } n ( n - 1 ) & ( n - 1 ) ^ { 2 } & \frac { 1 } { 2 } ( n - 1 ) ( 3 n + 4 ) \end{array} \right|$, then the value of $\sum _ { r = 1 } ^ { n - 1 } \Delta _ { r }$
(1) Is independent of both $a$ and $n$
(2) Depends only on $a$
(3) Depends only on $n$
(4) Depends both on $a$ and $n$
Q76 Standard trigonometric equations Inverse trigonometric equation View
The principal value of $\tan ^ { - 1 } \left( \cot \frac { 43 \pi } { 4 } \right)$ is
(1) $\frac { \pi } { 4 }$
(2) $- \frac { \pi } { 4 }$
(3) $\frac { 3 \pi } { 4 }$
(4) $- \frac { 3 \pi } { 4 }$
Q77 Curve Sketching Function Properties from Symmetry or Parity View
The function $f ( x ) = | \sin 4 x | + | \cos 2 x |$, is a periodic function with a fundamental period
(1) $\pi$
(2) $2 \pi$
(3) $\frac { \pi } { 4 }$
(4) $\frac { \pi } { 2 }$
Q78 Composite & Inverse Functions Injectivity, Surjectivity, or Bijectivity Classification View
Let $f : R \rightarrow R$ be defined by $f ( x ) = \frac { | x | - 1 } { | x | + 1 }$, then $f$ is
(1) one-one but not onto
(2) neither one-one nor onto
(3) both one-one and onto
(4) onto but not one-one
Q79 Composite & Inverse Functions Find or Apply an Inverse Function Formula View
If the function $f ( x ) = \left\{ \begin{array} { c l } \frac { \sqrt { 2 + \cos x } - 1 } { ( \pi - x ) ^ { 2 } } , & x \neq \pi \\ k , & x = \pi \end{array} \right.$ is continuous at $x = \pi$, then $k$ equals
(1) $\frac { 1 } { 4 }$
(2) 0
(3) 2
(4) $\frac { 1 } { 2 }$
Q80 Differentiation from First Principles View
Let $f : R \rightarrow R$ be a function such that $| f ( x ) | \leq x ^ { 2 }$, for all $x \in R$. Then, at $x = 0 , f$ is
(1) differentiable but not continuous
(2) neither continuous nor differentiable
(3) continuous as well as differentiable
(4) continuous but not differentiable
Q81 Applied differentiation Applied modeling with differentiation View
If the volume of a spherical ball is increasing at the rate of $4 \pi \mathrm { cc } / \mathrm { sec }$ then the rate of increase of its radius (in $\mathrm { cm } / \mathrm { sec }$), when the volume is $288 \pi \mathrm { cc }$ is
(1) $\frac { 1 } { 9 }$
(2) $\frac { 1 } { 6 }$
(3) $\frac { 1 } { 24 }$
(4) $\frac { 1 } { 36 }$
Q82 Completing the square and sketching Two quadratic functions: intersection, tangency, or equality conditions View
If non-zero real numbers $b$ and $c$ are such that $\min f ( x ) > \max g ( x )$, where $f ( x ) = x ^ { 2 } + 2 b x + 2 c ^ { 2 }$ and $g ( x ) = - x ^ { 2 } - 2 c x + b ^ { 2 } , ( x \in R )$; then $\left| \frac { c } { b } \right|$ lies in the interval
(1) $( \sqrt { 2 } , \infty )$
(2) $\left[ \frac { 1 } { 2 } , \frac { 1 } { \sqrt { 2 } } \right)$
(3) $\left( 0 , \frac { 1 } { 2 } \right)$
(4) $\left[ \frac { 1 } { \sqrt { 2 } } , \sqrt { 2 } \right]$
Q83 Integration by Substitution Substitution to Transform Integral Form (Show Transformed Expression) View
If $m$ is a non-zero number and $\int \frac { x ^ { 5 m - 1 } + 2 x ^ { 4 m - 1 } } { \left( x ^ { 2 m } + x ^ { m } + 1 \right) ^ { 3 } } d x = f ( x ) + c$, then $f ( x )$ is equal to
(1) $\frac { \left( x ^ { 5 m } - x ^ { 4 m } \right) } { 2 m \left( x ^ { 2 m } + x ^ { m } + 1 \right) ^ { 2 } }$
(2) $\frac { 1 } { 2 m } \frac { x ^ { 4 m } } { \left( x ^ { 2 m } + x ^ { m } + 1 \right) ^ { 2 } }$
(3) $\frac { x ^ { 5 m } } { 2 m \left( x ^ { 2 m } + x ^ { m } + 1 \right) ^ { 2 } }$
(4) $\frac { 2 m \left( x ^ { 5 m } + x ^ { 4 m } \right) } { \left( x ^ { 2 m } + x ^ { m } + 1 \right) ^ { 2 } }$
Q84 Integration by Substitution Substitution to Compute an Indefinite Integral with Initial Condition View
Let, the function $F$ be defined as $F ( x ) = \int _ { 1 } ^ { x } \frac { e ^ { t } } { t } d t , x > 0$, then the value of the integral $\int _ { 1 } ^ { x } \frac { e ^ { t } } { t + a } d t$, where $a > 0$, is
(1) $e ^ { a } [ F ( x ) - F ( 1 + a ) ]$
(2) $e ^ { - a } [ F ( x + a ) - F ( a ) ]$
(3) $e ^ { a } [ F ( x + a ) - F ( 1 + a ) ]$
(4) $e ^ { - a } [ F ( x + a ) - F ( 1 + a ) ]$
Q85 Areas by integration View
The area of the region (in square units) above the $x$-axis bounded by the curve $y = \tan x , 0 \leq x \leq \frac { \pi } { 2 }$ and the tangent to the curve at $x = \frac { \pi } { 4 }$ is
(1) $\frac { 1 } { 2 } \left( \log 2 - \frac { 1 } { 2 } \right)$
(2) $\frac { 1 } { 2 } ( 1 + \log 2 )$
(3) $\frac { 1 } { 2 } ( 1 - \log 2 )$
(4) $\frac { 1 } { 2 } \left( \log 2 + \frac { 1 } { 2 } \right)$
Q86 First order differential equations (integrating factor) View
If $\frac { d y } { d x } + y \tan x = \sin 2 x$ and $y ( 0 ) = 1$, then $y ( \pi )$ is equal to
(1) - 1
(2) 5
(3) 1
(4) - 5
Q87 Vectors: Cross Product & Distances View
If $\vec { x } = 3 \hat { i } - 6 \hat { j } - \widehat { k } , \vec { y } = \hat { i } + 4 \hat { j } - 3 \widehat { k }$ and $\vec { z } = 3 \hat { i } - 4 \hat { j } - 12 \widehat { k }$, then the magnitude of the projection of $\vec { x } \times \vec { y }$ on $\vec { z }$ is
(1) 14
(2) 12
(3) 15
(4) 10
Q88 Vectors: Lines & Planes Dihedral Angle or Angle Between Planes/Lines View
If the angle between the line $2 ( x + 1 ) = y = z + 4$ and the plane $2 x - y + \sqrt { \lambda } z + 4 = 0$ is $\frac { \pi } { 6 }$, then the value of $\lambda$ is
(1) $\frac { 45 } { 7 }$
(2) $\frac { 135 } { 11 }$
(3) $\frac { 135 } { 7 }$
(4) $\frac { 45 } { 11 }$
Q89 Vectors: Lines & Planes Find Parametric Representation of a Line View
Equation of the line of the shortest distance between the lines $\frac { x } { 1 } = \frac { y } { - 1 } = \frac { z } { 1 }$ and $\frac { x - 1 } { 0 } = \frac { y + 1 } { - 2 } = \frac { z } { 1 }$ is
(1) $\frac { x } { - 2 } = \frac { y } { 1 } = \frac { z } { 2 }$
(2) $\frac { x } { 1 } = \frac { y } { - 1 } = \frac { z } { - 2 }$
(3) $\frac { x - 1 } { 1 } = \frac { y + 1 } { - 1 } = \frac { z } { - 2 }$
(4) $\frac { x - 1 } { 1 } = \frac { y + 1 } { - 1 } = \frac { z } { 1 }$
Q90 Conditional Probability Direct Conditional Probability Computation from Definitions View
Let $A$ and $E$ be any two events with positive probabilities Statement I: $P ( E / A ) \geq P ( A / E ) P ( E )$. Statement II: $P ( A / E ) \geq P ( A \cap E )$.
(1) Both the statements are false
(2) Both the statements are true
(3) Statement-I is false, Statement-II is true
(4) Statement - I is true, Statement - II is false