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
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2018
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2017
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2016
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2015
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2014
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2013
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2012
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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
2007 jee-main_2007.pdf

38 maths questions

Q83 Discriminant and conditions for roots Parameter range for specific root conditions (location/count) View
If the difference between the roots of the equation $x ^ { 2 } + a x + 1 = 0$ is less than $\sqrt { 5 }$, then the set of possible values of $a$ is
(1) $( - 3,3 )$
(2) $( - 3 , \infty )$
(3) $( 3 , \infty )$
(4) $( - \infty , - 3 )$
Q84 Complex Numbers Argand & Loci Distance and Region Optimization on Loci View
If $| z + 4 | \leq 3$, then the maximum value of $| z + 1 |$ is
(1) 4
(2) 10
(3) 6
(4) 0
Q85 Combinations & Selection Partitioning into Teams or Groups View
The set $S = \{ 1,2,3 , \ldots , 12 )$ is to be partitioned into three sets $A , B , C$ of equal size. Thus, $A \cup B \cup C = S , A \cap B = B \cap C = A \cap C = \phi$. The number of ways to partition $S$ is
(1) $\frac { 12 ! } { 3 ! ( 4 ! ) ^ { 3 } }$
(2) $\frac { 12 ! } { 3 ! ( 3 ! ) ^ { 4 } }$
(3) $\frac { 12 ! } { ( 4 ! ) ^ { 3 } }$
(4) $\frac { 12 ! } { ( 3 ! ) ^ { 4 } }$
Q86 Geometric Sequences and Series Finite Geometric Sum and Term Relationships View
In a geometric progression consisting of positive terms, each term equals the sum of the next two terms. Then the common ratio of this progression equals
(1) $\frac { 1 } { 2 } ( 1 - \sqrt { 5 } )$
(2) $\frac { 1 } { 2 } \sqrt { 5 }$
(3) $\sqrt { 5 }$
(4) $\frac { 1 } { 2 } ( \sqrt { 5 } - 1 )$
Q87 Stationary points and optimisation Find absolute extrema on a closed interval or domain View
If $p$ and $q$ are positive real numbers such that $p ^ { 2 } + q ^ { 2 } = 1$, then the maximum value of ( $p + q$ ) is
(1) 2
(2) $1 / 2$
(3) $\frac { 1 } { \sqrt { 2 } }$
(4) $\sqrt { 2 }$
Q88 Exponential Functions Algebraic Simplification and Expression Manipulation View
The sum of the series $\frac { 1 } { 2 ! } - \frac { 1 } { 3 ! } + \frac { 1 } { 4 ! } - \ldots$ upto infinity is
(1) $e ^ { - 2 }$
(2) $e ^ { - 1 }$
(3) $e ^ { - 1 / 2 }$
(4) $e ^ { 1 / 2 }$
Q89 Binomial Theorem (positive integer n) Determine Parameters from Conditions on Coefficients or Terms View
In the binomial expansion of $( a - b ) ^ { n } , n \geq 5$, the sum of $5 ^ { \text {th } }$ and $6 ^ { \text {th } }$ terms is zero, then $\frac { a } { b }$ equals
(1) $\frac { 5 } { n - 4 }$
(2) $\frac { 6 } { n - 5 }$
(3) $\frac { n - 5 } { 6 }$
(4) $\frac { n - 4 } { 5 }$
Q90 Binomial Theorem (positive integer n) Evaluate a Summation Involving Binomial Coefficients View
The sum of the series ${ } ^ { 20 } \mathrm { C } _ { 0 } - { } ^ { 20 } \mathrm { C } _ { 1 } + { } ^ { 20 } \mathrm { C } _ { 2 } - { } ^ { 20 } \mathrm { C } _ { 3 } + \ldots - \ldots + { } ^ { 20 } \mathrm { C } _ { 10 }$ is
(1) $- { } ^ { 20 } \mathrm { C } _ { 10 }$
(2) $\frac { 1 } { 2 } { } ^ { 20 } \mathrm { C } _ { 10 }$
(3) 0
(4) ${ } ^ { 20 } \mathrm { C } _ { 10 }$
Q91 Straight Lines & Coordinate Geometry Triangle Properties and Special Points View
Let $A ( h , k ) , B ( 1,1 )$ and $C ( 2,1 )$ be the vertices of a right angled triangle with $A C$ as its hypotenuse. If the area of the triangle is 1 , then the set of values which ' k ' can take is given by
(1) $\{ 1,3 \}$
(2) $\{ 0,2 \}$
(3) $\{ - 1,3 \}$
(4) $\{ - 3 , - 2 \}$
Q92 Straight Lines & Coordinate Geometry Slope and Angle Between Lines View
Let $P = ( - 1,0 ) , Q = ( 0,0 )$ and $R = ( 3,3 \sqrt { 3 } )$ be three points. The equation of the bisector of the angle PQR
(1) $\sqrt { 3 } x + y = 0$
(2) $x + \frac { \sqrt { 3 } } { 2 } y = 0$
(3) $\frac { \sqrt { 3 } } { 2 } x + y = 0$
(4) $x + \sqrt { 3 } y = 0$
Q93 Straight Lines & Coordinate Geometry Slope and Angle Between Lines View
If one of the lines of $m y ^ { 2 } + \left( 1 - m ^ { 2 } \right) x y - m x ^ { 2 } = 0$ is a bisector of the angle between the lines $x y = 0$, then $m$ is
(1) $- 1 / 2$
(2) - 2
(3) 1
(4) 2
Q94 Circles Circles Tangent to Each Other or to Axes View
Consider a family of circles which are passing through the point $( - 1,1 )$ and are tangent to $x -$ axis. If $( h , k )$ are the co-ordinates of the centre of the circles, then the set of values of $k$ is given by the interval
(1) $0 < \mathrm { k } < 1 / 2$
(2) $k \geq 1 / 2$
(3) $- 1 / 2 \leq k \leq 1 / 2$
(4) $k \leq 1 / 2$
Q95 Circles Tangent Lines and Tangent Lengths View
The equation of a tangent to the parabola $y ^ { 2 } = 8 x$ is $y = x + 2$. The point on this line from which the other tangent to the parabola is perpendicular to the given tangent is
(1) $( - 1,1 )$
(2) $( 0,2 )$
(3) $( 2,4 )$
(4) $( - 2,0 )$
Q96 Conic sections Conic Identification and Conceptual Properties View
For the hyperbola $\frac { x ^ { 2 } } { \cos ^ { 2 } \alpha } - \frac { y ^ { 2 } } { \sin ^ { 2 } \alpha } = 1$, which of the following remains constant when $\alpha$ varies?
(1) eccentricity
(2) directrix
(3) abscissae of vertices
(4) abscissae of foci
Q97 Composite & Inverse Functions Find or Apply an Inverse Function Formula View
The function $f : R \sim \{ 0 \} \rightarrow R$ given by $f ( x ) = \frac { 1 } { x } - \frac { 2 } { e ^ { 2 x } - 1 }$ can be made continuous at $x = 0$ by defining $f ( 0 )$ as
(1) 2
(2) - 1
(3) 0
(4) 1
Q98 Measures of Location and Spread View
The average marks of boys in a class is 52 and that of girls is 42 . The average marks of boys and girls combined is 50 . The percentage of boys in the class is
(1) 40
(2) 20
(3) 80
(4) 60
Q99 Radians, Arc Length and Sector Area View
A tower stands at the centre of a circular park. $A$ and $B$ are two points on the boundary of the park such that $A B ( = a )$ subtends an angle of $60 ^ { \circ }$ at the foot of the tower, and the angle of elevation of the top of the tower from $A$ or $B$ is $30 ^ { \circ }$. The height of the tower is
(1) $\frac { 2 a } { \sqrt { 3 } }$
(2) $2 a \sqrt { 3 }$
(3) $\frac { a } { \sqrt { 3 } }$
(4) $a \sqrt { 3 }$
Q100 3x3 Matrices Determinant of Parametric or Structured Matrix View
Let $A = \left[ \begin{array} { c c c } 5 & 5 \alpha & \alpha \\ 0 & \alpha & 5 \alpha \\ 0 & 0 & 5 \end{array} \right]$. If $\left| A ^ { 2 } \right| = 25$, then $| \alpha |$ equals
(1) $5 ^ { 2 }$
(2) 1
(3) $1 / 5$
(4) 5
Q101 3x3 Matrices Determinant of Parametric or Structured Matrix View
If $D = \left| \begin{array} { c c c } 1 & 1 & 1 \\ 1 & 1 + x & 1 \\ 1 & 1 & 1 + y \end{array} \right|$ for $x \neq 0 , y \neq 0$ then $D$ is
(1) divisible by neither $x$ nor $y$
(2) divisible by both $x$ and $y$
(3) divisible by $x$ but not $y$
(4) divisible by $y$ but not $x$
Q102 Standard trigonometric equations Inverse trigonometric equation View
If $\sin ^ { - 1 } \left( \frac { x } { 5 } \right) + \operatorname { cosec } ^ { - 1 } \left( \frac { 5 } { 4 } \right) = \frac { \pi } { 2 }$ then a value of $x$ is
(1) 1
(2) 3
(3) 4
(4) 5
Q103 Composite & Inverse Functions Determine Domain or Range of a Composite Function View
The largest interval lying in $\left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$ for which the function $\left[ f ( x ) = 4 ^ { - x ^ { 2 } } + \cos ^ { - 1 } \left( \frac { x } { 2 } - 1 \right) + \log ( \cos x ) \right]$ is defined, is
(1) $[ 0 , \pi ]$
(2) $\left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$
(3) $\left[ - \frac { \pi } { 4 } , \frac { \pi } { 2 } \right)$
(4) $\left[ 0 , \frac { \pi } { 2 } \right)$
Q104 Curve Sketching Continuity and Differentiability of Special Functions View
Let $f : R \rightarrow R$ be a function defined by $f ( x ) = \operatorname { Min } \{ x + 1 , | x | + 1 \}$. Then which of the following is true?
(1) $f ( x ) \geq 1$ for all $x \in R$
(2) $f ( x )$ is not differentiable at $x = 1$
(3) $f ( x )$ is differentiable everywhere
(4) $f ( x )$ is not differentiable at $x = 0$
Q105 Differential equations Finding a DE from a Limit or Implicit Condition View
The normal to a curve at $P ( x , y )$ meets the $x$-axis at $G$. If the distance of $G$ from the origin is twice the abscissa of $P$, then the curve is a
(1) ellipse
(2) parabola
(3) circle
(4) pair of straight lines
Q106 Applied differentiation Properties of differentiable functions (abstract/theoretical) View
A value of $C$ for which the conclusion of Mean Value Theorem holds for the function $f ( x ) = \log _ { \mathrm { e } } x$ on the interval $[ 1,3 ]$ is
(1) $2 \log _ { 3 } e$
(2) $\frac { 1 } { 2 } \log _ { e } 3$
(3) $\log _ { 3 } e$
(4) $\log _ { e } 3$
Q107 Stationary points and optimisation Determine intervals of increase/decrease or monotonicity conditions View
The function $f ( x ) = \tan ^ { - 1 } ( \sin x + \cos x )$ is an increasing function in
(1) $\left( \frac { \pi } { 4 } , \frac { \pi } { 2 } \right)$
(2) $\left( - \frac { \pi } { 2 } , \frac { \pi } { 4 } \right)$
(3) $\left( 0 , \frac { \pi } { 2 } \right)$
(4) $\left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$
Q108 Integration by Substitution Substitution to Transform Integral Form (Show Transformed Expression) View
$\int \frac { d x } { \cos x + \sqrt { 3 } \sin x }$ equals
(1) $\frac { 1 } { 2 } \log \tan \left( \frac { x } { 2 } + \frac { \pi } { 12 } \right) + c$
(2) $\frac { 1 } { 2 } \log \tan \left( \frac { x } { 2 } - \frac { \pi } { 12 } \right) + c$
(3) $\log \tan \left( \frac { x } { 2 } + \frac { \pi } { 12 } \right) + c$
(4) $\log \tan \left( \frac { x } { 2 } - \frac { \pi } { 12 } \right) + c$
Q109 Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution View
Let $F ( x ) = f ( x ) + f \left( \frac { 1 } { x } \right)$, where $f ( x ) = \int _ { 1 } ^ { x } \frac { \log t } { 1 + t } d t$. Then $F ( e )$ equals
(1) $\frac { 1 } { 2 }$
(2) 0
(3) 1
(4) 2
Q110 Integration using inverse trig and hyperbolic functions View
The solution for $x$ of the equation $\int _ { \sqrt { 2 } } ^ { x } \frac { d t } { t \sqrt { t ^ { 2 } - 1 } } = \frac { \pi } { 2 }$ is
(1) 2
(2) $\pi$
(3) $\frac { \sqrt { 3 } } { 2 }$
(4) None of these
Q111 Areas by integration View
The area enclosed between the curves $y ^ { 2 } = x$ and $y = | x |$ is
(1) $2 / 3$
(2) 1
(3) $1 / 6$
(4) $1 / 3$
Q112 Differential equations Higher-Order and Special DEs (Proof/Theory) View
The differential equation of all circles passing through the origin and having their centres on the $x$-axis is
(1) $x ^ { 2 } = y ^ { 2 } + x y \frac { d y } { d x }$
(2) $x ^ { 2 } = y ^ { 2 } + 3 x y \frac { d y } { d x }$
(3) $y ^ { 2 } = x ^ { 2 } + 2 x y \frac { d y } { d x }$
(4) $y ^ { 2 } = x ^ { 2 } - 2 x y \frac { d y } { d x }$
Q113 Vectors Introduction & 2D Magnitude of Vector Expression View
The resultant of two forces P N and 3 N is a force of 7 N . If the direction of 3 N force were reversed, the resultant would be $\sqrt { 19 } \mathrm {~N}$. The value of P is
(1) 5 N
(2) 6 N
(3) 3 N
(4) 4 N
Q114 Vectors Introduction & 2D Angle or Cosine Between Vectors View
If $\hat { u }$ and $\hat { v }$ are unit vectors and $\theta$ is the acute angle between them, then $2 \hat { u } \times 3 \hat { v }$ is a unit vector for
(1) exactly two values of $\theta$
(2) more than two values of $\theta$
(3) no value of $\theta$
(4) exactly one value of $\theta$
Q115 Vectors 3D & Lines Vector Algebra and Triple Product Computation View
Let $\overline { \mathrm { a } } = \hat { \mathrm { i } } + \hat { \mathrm { j } } + \hat { \mathrm { k } } , \overline { \mathrm { b } } = \hat { \mathrm { i } } - \hat { \mathrm { j } } + 2 \hat { \mathrm { k } }$ and $\overline { \mathrm { c } } = \mathrm { x } \hat { \mathrm { i } } + ( \mathrm { x } - 2 ) \hat { \mathrm { j } } - \hat { \mathrm { k } }$. If the vector $\overline { \mathrm { c } }$ lies in the plane of $\bar { a }$ and $\bar { b }$, then $x$ equals
(1) 0
(2) 1
(3) - 4
(4) - 2
Q116 Vectors 3D & Lines MCQ: Relationship Between Two Lines View
Let $L$ be the line of intersection of the planes $2 x + 3 y + z = 1$ and $x + 3 y + 2 z = 2$. If $L$ makes an angles $\alpha$ with the positive $x$-axis, then $\cos \alpha$ equals
(1) $\frac { 1 } { \sqrt { 3 } }$
(2) $\frac { 1 } { 2 }$
(3) 1
(4) $\frac { 1 } { \sqrt { 2 } }$
Q117 Vectors 3D & Lines MCQ: Relationship Between Two Lines View
If a line makes an angle of $\frac { \pi } { 4 }$ with the positive directions of each of $x$-axis and $y$-axis, then the angle that the line makes with the positive direction of the $z$-axis is
(1) $\frac { \pi } { 6 }$
(2) $\frac { \pi } { 3 }$
(3) $\frac { \pi } { 4 }$
(4) $\frac { \pi } { 2 }$
Q118 Circles Sphere and 3D Circle Problems View
If ( $2,3,5$ ) is one end of a diameter of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 6 x - 12 y - 2 z + 20 = 0$, then the coordinates of the other end of the diameter are
(1) $( 4,9 , - 3 )$
(2) $( 4 , - 3,3 )$
(3) $( 4,3,5 )$
(4) $( 4,3 , - 3 )$
Q119 Binomial Distribution Compute Exact Binomial Probability View
A pair of fair dice is thrown independently three times. The probability of getting a score of exactly 9 twice is
(1) $1 / 729$
(2) $8 / 9$
(3) $8 / 729$
(4) $8 / 243$
Q120 Independent Events View
Two aeroplanes I and II bomb a target in succession. The probabilities of I and II scoring a hit correctly are 0.3 and 0.2 , respectively. The second plane will bomb only if the first misses the target. The probability that the target is hit by the second plane is
(1) 0.06
(2) 0.14
(3) 0.2
(4) None of these