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
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
2017 09apr

30 maths questions

Q61 Complex numbers 2 Roots of Unity and Cyclotomic Properties View
Let $\omega$ be a complex number such that $2 \omega + 1 = z$ where $z = \sqrt { - 3 }$. If $$\left| \begin{array} { c c c } { 1 } & { 1 } & { 1 } \\ { 1 } & { - \omega ^ { 2 } - 1 } & { \omega ^ { 2 } } \\ { 1 } & { \omega ^ { 2 } } & { \omega ^ { 7 } } \end{array} \right| = 3 k$$ then $k$ is equal to:
(1) $z$
(2) $- z$
(3) $- 1$
(4) 1
Q62 Matrices Linear System and Inverse Existence View
The set of all values of $\lambda$ for which the system of linear equations $$x - 2 y - 2 z = \lambda x$$ $$x + 2 y + z = \lambda y$$ $$- x - y = \lambda z$$ has a non-trivial solution:
(1) is an empty set
(2) is a singleton
(3) contains two elements
(4) contains more than two elements
Q63 Circles Circles Tangent to Each Other or to Axes View
The radius of a circle, having minimum area, which touches the curve $y = 4 - x ^ { 2 }$ and the lines $y = | x |$ is:
(1) $2 ( \sqrt { 2 } + 1 )$
(2) $2 ( \sqrt { 2 } - 1 )$
(3) $4 ( \sqrt { 2 } - 1 )$
(4) $4 ( \sqrt { 2 } + 1 )$
Q64 Stationary points and optimisation Geometric or applied optimisation problem View
Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, is:
(1) 12.5
(2) 10
(3) 25
(4) 30
Q65 Standard Integrals and Reverse Chain Rule Reverse Chain Rule Antiderivative (MCQ) View
The integral $\int \frac { 2 x ^ { 3 } - 1 } { x ^ { 4 } + x } d x$ is equal to (here $C$ is a constant of integration)
(1) $\frac { 1 } { 2 } \ln \frac { | x ^ { 3 } + 1 | } { x ^ { 2 } } + C$
(2) $\frac { 1 } { 2 } \ln \frac { ( x ^ { 3 } + 1 ) ^ { 2 } } { | x ^ { 3 } | } + C$
(3) $\ln \frac { | x ^ { 3 } + 1 | } { x ^ { 2 } } + C$
(4) $\ln \frac { | x ^ { 3 } + 1 | } { x ^ { 3 } } + C$
Q66 Areas Between Curves Area Between Curves with Parametric or Implicit Region Definition View
The area (in sq. units) of the region $\{ ( x , y ) : x \geq 0 , x + y \leq 3 , x ^ { 2 } \leq 4 y$ and $y \leq 1 + \sqrt { x } \}$ is:
(1) $\frac { 59 } { 12 }$
(2) $\frac { 3 } { 2 }$
(3) $\frac { 7 } { 3 }$
(4) $\frac { 5 } { 2 }$
Q67 Differentiating Transcendental Functions Evaluate derivative at a point or find tangent slope View
If $y = \left( \frac { x } { x + 1 } \right) ^ { x } + x ^ { \left( \frac { x } { x + 1 } \right) }$, find $\frac { d y } { d x }$ at $x = 1$.
(1) $\frac { 1 } { 2 } + \ln 2$
(2) $1 + \frac { 1 } { 2 } \ln 2$
(3) $1 - \frac { 1 } { 2 } \ln 2$
(4) $\frac { 1 } { 2 } - \ln 2$
Q68 Indefinite & Definite Integrals Accumulation Function Analysis View
If $f : \mathbb { R } \to \mathbb { R }$ is a differentiable function and $f ( 2 ) = 6$, then $\lim _ { x \to 2 } \int _ { 6 } ^ { f ( x ) } \frac { 2 t \, d t } { ( x - 2 ) }$ is:
(1) $2 f ^ { \prime } ( 2 )$
(2) $12 f ^ { \prime } ( 2 )$
(3) $0$
(4) $24 f ^ { \prime } ( 2 )$
Q69 Composite & Inverse Functions Evaluate Composition from Algebraic Definitions View
For $x \in \mathbb { R }$, $f ( x ) = | \log 2 - \sin x |$ and $g ( x ) = f ( f ( x ) )$, then:
(1) $g$ is not differentiable at $x = 0$
(2) $g ^ { \prime } ( 0 ) = \cos ( \log 2 )$
(3) $g ^ { \prime } ( 0 ) = - \cos ( \log 2 )$
(4) $g$ is differentiable at $x = 0$ and $g ^ { \prime } ( 0 ) = - \sin ( \log 2 )$
Q70 Composite & Inverse Functions Injectivity, Surjectivity, or Bijectivity Classification View
The function $f : \mathbb { R } \to \left[ - \frac { 1 } { 2 } , \frac { 1 } { 2 } \right]$ defined as $f ( x ) = \frac { x } { 1 + x ^ { 2 } }$, is:
(1) Surjective but not injective
(2) Neither injective nor surjective
(3) Invertible
(4) Injective but not surjective
Q71 Tangents, normals and gradients Normal or perpendicular line problems View
The normal to the curve $y ( x - 2 ) ( x - 3 ) = x + 6$ at the point where the curve intersects the $y$-axis passes through the point:
(1) $\left( \frac { 1 } { 2 } , - \frac { 1 } { 3 } \right)$
(2) $\left( \frac { 1 } { 2 } , \frac { 1 } { 3 } \right)$
(3) $\left( - \frac { 1 } { 2 } , - \frac { 1 } { 2 } \right)$
(4) $\left( \frac { 1 } { 2 } , \frac { 1 } { 2 } \right)$
Q72 Conic sections Tangent and Normal Line Problems View
The eccentricity of an ellipse whose centre is at the origin is $\frac { 1 } { 2 }$. If one of its directrices is $x = - 4$, then the equation of the normal to it at $\left( 1 , \frac { 3 } { 2 } \right)$ is:
(1) $4 x - 2 y = 1$
(2) $4 x + 2 y = 7$
(3) $x + 2 y = 4$
(4) $2 y - x = 2$
Q73 Conic sections Tangent and Normal Line Problems View
A hyperbola passes through the point $P ( \sqrt { 2 } , \sqrt { 3 } )$ and has foci at $( \pm 2 , 0 )$. Then the tangent to this hyperbola at $P$ also passes through the point:
(1) $( 3 \sqrt { 2 } , 2 \sqrt { 3 } )$
(2) $( 2 \sqrt { 2 } , 3 \sqrt { 3 } )$
(3) $( \sqrt { 3 } , \sqrt { 2 } )$
(4) $( - \sqrt { 2 } , - \sqrt { 3 } )$
Q74 Vectors: Lines & Planes Distance Computation (Point-to-Plane or Line-to-Line) View
The distance of the point $(1, 3, -7)$ from the plane passing through the point $(1, -1, -1)$, having normal perpendicular to both the lines $$\frac { x - 1 } { 1 } = \frac { y + 2 } { - 2 } = \frac { z - 4 } { 3 } \quad \text{and} \quad \frac { x - 2 } { 2 } = \frac { y + 1 } { - 1 } = \frac { z + 7 } { - 1 }$$ is:
(1) $\frac { 10 } { \sqrt { 74 } }$
(2) $\frac { 20 } { \sqrt { 74 } }$
(3) $\frac { 5 } { \sqrt { 83 } }$
(4) $\frac { 10 } { \sqrt { 83 } }$
Q75 Straight Lines & Coordinate Geometry Locus Determination View
If a variable line drawn through the intersection of the lines $\frac { x } { 3 } + \frac { y } { 4 } = 1$ and $\frac { x } { 4 } + \frac { y } { 3 } = 1$, meets the coordinate axes at $A$ and $B$, $( A \neq B )$, then the locus of the midpoint of $AB$ is:
(1) $7 x y = 6 ( x + y )$
(2) $4 ( x + y ) ^ { 2 } - 28 ( x + y ) + 49 = 0$
(3) $6 x y = 7 ( x + y )$
(4) $14 ( x + y ) ^ { 2 } - 97 ( x + y ) + 168 = 0$
Q76 Probability Definitions Probability Involving Algebraic or Number-Theoretic Conditions View
If two different numbers are taken from the set $\{ 0 , 1 , 2 , 3 , \ldots , 10 \}$; then the probability that their sum as well as absolute difference are both multiple of 4, is:
(1) $\frac { 6 } { 55 }$
(2) $\frac { 12 } { 55 }$
(3) $\frac { 14 } { 45 }$
(4) $\frac { 7 } { 55 }$
Q77 Binomial Distribution Compute Expectation, Variance, or Standard Deviation View
A box contains 15 green and 10 yellow balls. If 10 balls are randomly drawn, one-by-one, with replacement, then the variance of the number of green balls drawn is:
(1) $\frac { 6 } { 25 }$
(2) 6
(3) 4
(4) $\frac { 12 } { 5 }$
Q78 Measures of Location and Spread View
The mean age of 25 teachers in a school is 40 years. A teacher retires at the age of 60 years and a new teacher is appointed in his place. If the mean age of the teachers in this school now is 39 years, then the age (in years) of the newly appointed teacher is:
(1) 25
(2) 30
(3) 35
(4) 40
Q79 Proof True/False Justification View
The following statement $(p \to q) \to [ ( \sim p \to q ) \to q ]$ is:
(1) a fallacy
(2) a tautology
(3) equivalent to $\sim p \to q$
(4) equivalent to $p \to \sim q$
Q80 First order differential equations (integrating factor) View
If $( 2 + \sin x ) \frac { d y } { d x } + ( y + 1 ) \cos x = 0$ and $y ( 0 ) = 1$, then $y \left( \frac { \pi } { 2 } \right)$ is equal to:
(1) $\frac { 1 } { 3 }$
(2) $- \frac { 2 } { 3 }$
(3) $- \frac { 1 } { 3 }$
(4) $\frac { 4 } { 3 }$
Q81 Reduction Formulae Derive a Reduction/Recurrence Formula via Integration by Parts View
Let $I _ { n } = \int _ { 0 } ^ { 1 } ( 1 - x ^ { 3 } ) ^ { n } d x$, where $n \in \mathbb { N }$. Then $\frac { 3 n + 1 } { 3 n } \cdot \frac { I _ { n + 1 } } { I _ { n } }$ is equal to:
(1) 1
(2) $\frac { n } { n + 1 }$
(3) $\frac { n + 1 } { n }$
(4) $\frac { 3 n + 1 } { 3 n - 2 }$
Q82 Solving quadratics and applications Evaluating an algebraic expression given a constraint View
Let $a , b , c \in \mathbb { R }$. If $f ( x ) = a x ^ { 2 } + b x + c$ is such that $a + b + c = 3$ and $f ( x + y ) = f ( x ) + f ( y ) + x y$, $\forall x , y \in \mathbb { R }$, then $\sum _ { n = 1 } ^ { 10 } f ( n )$ is equal to:
(1) 330
(2) 165
(3) 190
(4) 255
Q83 Arithmetic Sequences and Series Properties of AP Terms under Transformation View
For any three positive real numbers $a$, $b$ and $c$, $9 ( 25 a ^ { 2 } + b ^ { 2 } ) + 25 ( c ^ { 2 } - 3 a c ) = 15 b ( 3 a + c )$. Then:
(1) $b$, $c$ and $a$ are in G.P.
(2) $b$, $c$ and $a$ are in A.P.
(3) $a$, $b$ and $c$ are in A.P.
(4) $a$, $b$ and $c$ are in G.P.
Q84 Measures of Location and Spread View
If $\sum _ { i = 1 } ^ { 9 } ( x _ { i } - 5 ) = 9$ and $\sum _ { i = 1 } ^ { 9 } ( x _ { i } - 5 ) ^ { 2 } = 45$, then the standard deviation of the 9 items $x _ { 1 } , x _ { 2 } , \ldots , x _ { 9 }$ is:
(1) 9
(2) 4
(3) 2
(4) 3
Q85 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
The integral $\int _ { \frac { \pi } { 4 } } ^ { \frac { 3 \pi } { 4 } } \frac { d x } { 1 + \cos x }$ is equal to:
(1) $- 1$
(2) $- 2$
(3) 2
(4) 4
Q86 Vectors: Lines & Planes Perpendicular/Orthogonal Projection onto a Plane View
If the image of the point $P ( 1 , - 2 , 3 )$ in the plane $2 x + 3 y - 4 z + 22 = 0$ measured parallel to the line $\frac { x } { 1 } = \frac { y } { 4 } = \frac { z } { 5 }$ is $Q$, then $P Q$ is equal to:
(1) $3 \sqrt { 5 }$
(2) $2 \sqrt { 42 }$
(3) $\sqrt { 42 }$
(4) $6 \sqrt { 5 }$
Q87 Vectors 3D & Lines Vector Algebra and Triple Product Computation View
Let $\vec { u } = \hat { i } + \hat { j }$, $\vec { v } = \hat { i } - \hat { j }$ and $\vec { w } = \hat { i } + 2 \hat { j } + 3 \hat { k }$. If $\hat { n }$ is a unit vector such that $\vec { u } \cdot \hat { n } = 0$ and $\vec { v } \cdot \hat { n } = 0$, then $| \vec { w } \cdot \hat { n } |$ is equal to:
(1) 0
(2) 1
(3) 2
(4) 3
Q88 3x3 Matrices Linear System with Parameter — Infinite Solutions View
The number of real values of $\lambda$ for which the system of linear equations $$2 x + 4 y - \lambda z = 0$$ $$4 x + \lambda y + 2 z = 0$$ $$\lambda x + 2 y + 2 z = 0$$ has infinitely many solutions, is:
(1) 0
(2) 1
(3) 2
(4) 3
Q89 Complex numbers 2 Roots of Unity and Cyclotomic Properties View
The value of $\sum _ { k = 1 } ^ { 10 } \left( \sin \frac { 2 k \pi } { 11 } + i \cos \frac { 2 k \pi } { 11 } \right)$ is:
(1) 1
(2) $- 1$
(3) $- i$
(4) $i$
Q90 Quadratic trigonometric equations View
If $5 ( \tan ^ { 2 } x - \cos ^ { 2 } x ) = 2 \cos 2 x + 9$, then the value of $\cos 4 x$ is:
(1) $- \frac { 7 } { 9 }$
(2) $- \frac { 3 } { 5 }$
(3) $\frac { 1 } { 3 }$
(4) $\frac { 2 } { 9 }$