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
2016 10apr

28 maths questions

Q61 Indices and Surds Solving Equations Involving Surds View
If $x$ is a solution of the equation $\sqrt { 2 x + 1 } - \sqrt { 2 x - 1 } = 1 , \left( x \geq \frac { 1 } { 2 } \right)$, then $\sqrt { 4 x ^ { 2 } - 1 }$ is equal to :
(1) $\frac { 3 } { 4 }$
(2) $\frac { 1 } { 2 }$
(3) $2 \sqrt { 2 }$
(4) 2
Q62 Complex Numbers Arithmetic Powers of i or Complex Number Integer Powers View
Let $z = 1 + a i$, be a complex number, $a > 0$, such that $z ^ { 3 }$ is a real number. Then, the sum $1 + z + z ^ { 2 } + \ldots + z ^ { 11 }$ is equal to :
(1) $1365 \sqrt { 3 } i$
(2) $- 1365 \sqrt { 3 } i$
(3) $- 1250 \sqrt { 3 } i$
(4) $1250 \sqrt { 3 } i$
Q63 Combinations & Selection Basic Combination Computation View
If $\frac { { } ^ { n + 2 } C _ { 6 } } { { } ^ { n - 2 } P _ { 2 } } = 11$, then $n$ satisfies the equation:
(1) $n ^ { 2 } + n - 110 = 0$
(2) $n ^ { 2 } + 2 n - 80 = 0$
(3) $n ^ { 2 } + 3 n - 108 = 0$
(4) $n ^ { 2 } + 5 n - 84 = 0$
Q64 Arithmetic Sequences and Series Compute Partial Sum of an Arithmetic Sequence View
Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots a _ { n } , \ldots$, be in A.P. If $a _ { 3 } + a _ { 7 } + a _ { 11 } + a _ { 15 } = 72$, then the sum of its first 17 terms is equal to :
(1) 306
(2) 204
(3) 153
(4) 612
Q65 Sequences and series, recurrence and convergence Summation of sequence terms View
The sum $\sum _ { r = 1 } ^ { 10 } \left( r ^ { 2 } + 1 \right) \times ( r ! )$, is equal to:
(1) $11 \times ( 11 ! )$
(2) $10 \times ( 11 ! )$
(3) $(11)!$
(4) $101 \times ( 10 ! )$
Q66 Binomial Theorem (positive integer n) Find a Specific Coefficient in a Single Binomial Expansion View
If the coefficients of $x ^ { - 2 }$ and $x ^ { - 4 }$, in the expansion of $\left( x ^ { \frac { 1 } { 3 } } + \frac { 1 } { 2 x ^ { \frac { 1 } { 3 } } } \right) ^ { 18 } , ( x > 0 )$, are $m$ and $n$ respectively, then $\frac { m } { n }$ is equal to
(1) 27
(2) 182
(3) $\frac { 5 } { 4 }$
(4) $\frac { 4 } { 5 }$
Q67 Addition & Double Angle Formulae Function Analysis via Identity Transformation View
If $A > 0 , B > 0$ and $A + B = \frac { \pi } { 6 }$, then the minimum positive value of $( \tan A + \tan B )$ is :
(1) $\sqrt { 3 } - \sqrt { 2 }$
(2) $4 - 2 \sqrt { 3 }$
(3) $\frac { 2 } { \sqrt { 3 } }$
(4) $2 - \sqrt { 3 }$
Q68 Standard trigonometric equations Locus or solution set characterization of a trigonometric relation View
Let $P = \{ \theta : \sin \theta - \cos \theta = \sqrt { 2 } \cos \theta \}$ and $Q = \{ \theta : \sin \theta + \cos \theta = \sqrt { 2 } \sin \theta \}$, be two sets. Then
(1) $P \subset Q$ and $Q - P \neq \phi$
(2) $Q \not \subset P$
(3) $P = Q$
(4) $P \not \subset Q$
Q69 Straight Lines & Coordinate Geometry Section Ratio and Division of Segments View
A straight line through origin $O$ meets the lines $3 y = 10 - 4 x$ and $8 x + 6 y + 5 = 0$ at points $A$ and $B$ respectively. Then, $O$ divides the segment $A B$ in the ratio
(1) $2 : 3$
(2) $1 : 2$
(3) $4 : 1$
(4) $3 : 4$
Q70 Straight Lines & Coordinate Geometry Reflection and Image in a Line View
A ray of light is incident along a line which meets another line $7 x - y + 1 = 0$ at the point $( 0,1 )$. The ray is then reflected from this point along the line $y + 2 x = 1$. Then the equation of the line of incidence of the ray of light is :
(1) $41 x - 25 y + 25 = 0$
(2) $41 x + 25 y - 25 = 0$
(3) $41 x - 38 y + 38 = 0$
(4) $41 x + 38 y - 38 = 0$
Q71 Circles Tangent Lines and Tangent Lengths View
Equation of the tangent to the circle, at the point $( 1 , - 1 )$, whose center, is the point of intersection of the straight lines $x - y = 1$ and $2 x + y = 3$ is:
(1) $x + 4 y + 3 = 0$
(2) $3 x - y - 4 = 0$
(3) $x - 3 y - 4 = 0$
(4) $4 x + y - 3 = 0$
Q72 Circles Optimization on a Circle View
$P$ and $Q$ are two distinct points on the parabola, $y ^ { 2 } = 4 x$, with parameters $t$ and $t _ { 1 }$, respectively. If the normal at $P$ passes through $Q$, then the minimum value of $t _ { 1 } ^ { 2 }$, is
(1) 8
(2) 4
(3) 6
(4) 2
Q73 Conic sections Equation Determination from Geometric Conditions View
A hyperbola whose transverse axis is along the major axis of the conic $\frac { x ^ { 2 } } { 3 } + \frac { y ^ { 2 } } { 4 } = 4$ and has vertices at the foci of the conic. If the eccentricity of the hyperbola is $\frac { 3 } { 2 }$, then which of the following points does not lie on the hyperbola?
(1) $( \sqrt { 5 } , 2 \sqrt { 2 } )$
(2) $( 0,2 )$
(3) $( 5,2 \sqrt { 3 } )$
(4) $( \sqrt { 10 } , 2 \sqrt { 3 } )$
Q74 Chain Rule Limit Evaluation Involving Composition or Substitution View
$\lim _ { x \rightarrow 0 } \frac { ( 1 - \cos 2 x ) ^ { 2 } } { 2 x \tan x - x \tan 2 x }$ is
(1) 2
(2) $- \frac { 1 } { 2 }$
(3) $- 2$
(4) $\frac { 1 } { 2 }$
Q76 Measures of Location and Spread View
The mean of 5 observations is 5 and their variance is 12.4 . If three of the observations are $1,2 \& 6$; then the value of the remaining two is :
(1) 1,11
(2) 5,5
(3) 5,11
(4) None of these
Q78 Matrices Matrix Algebra and Product Properties View
Let $A$, be a $3 \times 3$ matrix, such that $A ^ { 2 } - 5 A + 7 I = O$. Statement - I : $A ^ { - 1 } = \frac { 1 } { 7 } ( 5 I - A )$. Statement - II : The polynomial $A ^ { 3 } - 2 A ^ { 2 } - 3 A + I$, can be reduced to $5 ( A - 4 I )$. Then :
(1) Both the statements are true
(2) Both the statements are false
(3) Statement - I is true, but Statement - II is false
(4) Statement - I is false, but Statement - II is true
Q79 Matrices Matrix Power Computation and Application View
If $A = \left[ \begin{array} { c c } - 4 & - 1 \\ 3 & 1 \end{array} \right]$, then the determinant of the matrix $\left( A ^ { 2016 } - 2 A ^ { 2015 } - A ^ { 2014 } \right)$ is :
(1) $- 175$
(2) 2014
(3) 2016
(4) $- 25$
Q80 Curve Sketching Finding Parameters for Continuity View
Let $a , b \in R , ( a \neq 0 )$. If the function $f$, defined as $$f ( x ) = \left\{ \begin{array} { c } \frac { 2 x ^ { 2 } } { a } , 0 \leq x < 1 \\ a , 1 \leq x < \sqrt { 2 } \\ \frac { 2 b ^ { 2 } - 4 b } { x ^ { 3 } } , \sqrt { 2 } \leq x < 8 \end{array} \right.$$ is continuous in the interval $[ 0 , \infty )$, then an ordered pair $( a , b )$ can be
(1) $( - \sqrt { 2 } , 1 - \sqrt { 3 } )$
(2) $( \sqrt { 2 } , - 1 + \sqrt { 3 } )$
(3) $( \sqrt { 2 } , 1 - \sqrt { 3 } )$
(4) $( - \sqrt { 2 } , 1 + \sqrt { 3 } )$
Q81 Tangents, normals and gradients Normal or perpendicular line problems View
Let C be a curve given by $y ( x ) = 1 + \sqrt { 4 x - 3 } , x > \frac { 3 } { 4 }$. If $P$ is a point on C , such that the tangent at $P$ has slope $\frac { 2 } { 3 }$, then a point through which the normal at $P$ passes, is :
(1) $( 1,7 )$
(2) $( 3 , - 4 )$
(3) $( 4 , - 3 )$
(4) $( 2,3 )$
Q82 Stationary points and optimisation Determine intervals of increase/decrease or monotonicity conditions View
Let $f ( x ) = \sin ^ { 4 } x + \cos ^ { 4 } x$. Then, $f$ is an increasing function in the interval:
(1) $]\frac { 5 \pi } { 8 } , \frac { 3 \pi } { 4 } [$
(2) $]\frac { \pi } { 2 } , \frac { 5 \pi } { 8 } [$
(3) $]\frac { \pi } { 4 } , \frac { \pi } { 2 } [$
(4) $]0 , \frac { \pi } { 4 } [$
Q83 Integration by Substitution Substitution to Transform Integral Form (Show Transformed Expression) View
The integral $\int \frac { d x } { ( 1 + \sqrt { x } ) \sqrt { x - x ^ { 2 } } }$ is equal to
(1) $- 2 \sqrt { \frac { 1 + \sqrt { x } } { 1 - \sqrt { x } } } + c$
(2) $- \sqrt { \frac { 1 - \sqrt { x } } { 1 + \sqrt { x } } } + c$
(3) $- 2 \sqrt { \frac { 1 - \sqrt { x } } { 1 + \sqrt { x } } } + c$
(4) $\sqrt { \frac { 1 + \sqrt { x } } { 1 - \sqrt { x } } } + c$
Q84 Differential equations Integral Equations Reducible to DEs View
For $x \in R , x \neq 0$, if $y ( x )$ is a differentiable function such that $x \int _ { 1 } ^ { x } y ( t ) d t = ( x + 1 ) \int _ { 1 } ^ { x } t y ( t ) d t$, then $y ( x )$ equals (where $C$ is a constant)
(1) $C x ^ { 3 } e ^ { \frac { 1 } { x } }$
(2) $\frac { C } { x ^ { 2 } } e ^ { - \frac { 1 } { x } }$
(3) $\frac { C } { x } e ^ { - \frac { 1 } { x } }$
(4) $\frac { C } { x ^ { 3 } } e ^ { - \frac { 1 } { x } }$
Q85 Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution View
The value of the integral $\int _ { 4 } ^ { 10 } \frac { \left[ x ^ { 2 } \right] } { \left[ x ^ { 2 } - 28 x + 196 \right] + \left[ x ^ { 2 } \right] } d x$, where $[ x ]$ denotes the greatest integer less than or equal to $x$, is
(1) $\frac { 1 } { 3 }$
(2) 6
(3) 7
(4) 3
Q86 Second order differential equations Second-order ODE with initial or boundary value conditions View
The solution of the differential equation $\frac { d y } { d x } + \frac { y } { 2 } \sec x = \frac { \tan x } { 2 y }$, where $0 \leq x < \frac { \pi } { 2 }$ and $y ( 0 ) = 1$, is given by
(1) $y ^ { 2 } = 1 + \frac { x } { \sec x + \tan x }$
(2) $y = 1 + \frac { x } { \sec x + \tan x }$
(3) $y = 1 - \frac { x } { \sec x + \tan x }$
(4) $y ^ { 2 } = 1 - \frac { x } { \sec x + \tan x }$
Q87 Vectors: Lines & Planes Coplanarity and Relative Position of Planes View
The number of distinct real values of $\lambda$, for which the lines $\frac { x - 1 } { 1 } = \frac { y - 2 } { 2 } = \frac { z + 3 } { \lambda ^ { 2 } }$ and $\frac { x - 3 } { 1 } = \frac { y - 2 } { \lambda ^ { 2 } } = \frac { z - 1 } { 2 }$, are coplanar is
(1) 2
(2) 4
(3) 3
(4) 1
Q88 Vectors 3D & Lines Section Division and Coordinate Computation View
$ABC$ is a triangle in a plane with vertices $A ( 2,3,5 ) , B ( - 1,3,2 )$ and $C ( \lambda , 5 , \mu )$. If the median through $A$ is equally inclined to the coordinate axes, then the value of $\left( \lambda ^ { 3 } + \mu ^ { 3 } + 5 \right)$ is
(1) 1130
(2) 1348
(3) 1077
(4) 676
Q89 Vectors 3D & Lines Section Division and Coordinate Computation View
Let $ABC$ be a triangle whose circumcentre is at $P$. If the position vectors $A , B , C$ and $P$ are $\vec{a}, \vec{b}, \vec{c}$ and $\frac { \vec { a } + \vec { b } + \vec { c } } { 4 }$ respectively, then the position vector of the orthocentre of this triangle, is :
(1) $- \left( \frac { \vec { a } + \vec { b } + \vec { c } } { 2 } \right)$
(2) $\vec { a } + \vec { b } + \vec { c }$
(3) $\frac { ( \vec { a } + \vec { b } + \vec { c } ) } { 2 }$
(4) $\overrightarrow { 0 }$
Q90 Binomial Distribution Compute Cumulative or Complement Binomial Probability View
An experiment succeeds twice as often as it fails. The probability of at least 5 successes in the six trials of this experiment is
(1) $\frac { 496 } { 729 }$
(2) $\frac { 192 } { 729 }$
(3) $\frac { 240 } { 729 }$
(4) $\frac { 256 } { 729 }$