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 09apr

28 maths questions

Q61 Solving quadratics and applications Finding roots or coefficients of a quadratic using Vieta's relations View
If $\frac { 1 } { \sqrt { \alpha } } , \frac { 1 } { \sqrt { \beta } }$ are the roots of the equation $a x ^ { 2 } + b x + 1 = 0 , ( a \neq 0 , a , b \in R )$, then the equation $x \left( x + b ^ { 3 } \right) + \left( a ^ { 3 } - 3 a b x \right) = 0$ has roots:
(1) $\sqrt { \alpha \beta }$ and $\alpha \beta$
(2) $\alpha ^ { - \frac { 3 } { 2 } }$ and $\beta ^ { - \frac { 3 } { 2 } }$
(3) $\alpha \beta ^ { \frac { 1 } { 2 } }$ and $\alpha ^ { \frac { 1 } { 2 } } \beta$
(4) $\alpha ^ { \frac { 3 } { 2 } }$ and $\beta ^ { \frac { 3 } { 2 } }$
Q62 Solving quadratics and applications Finding roots or coefficients of a quadratic using Vieta's relations View
If equations $a x ^ { 2 } + b x + c = 0 , ( a , b , c \in R , a \neq 0 )$ and $2 x ^ { 2 } + 3 x + 4 = 0$ have a common root, then $a : b : c$ equals :
(1) $2 : 3 : 4$
(2) $4 : 3 : 2$
(3) $1 : 2 : 3$
(4) $3 : 2 : 1$
Q63 Complex Numbers Arithmetic Solving Equations for Unknown Complex Numbers View
Let $w ( \operatorname { Im } w \neq 0 )$ be a complex number. Then, the set of all complex numbers $z$ satisfying the equation $w - \bar { w } z = k ( 1 - z )$, for some real number $k$, is
(1) $\{ z : z \neq 1 \}$
(2) $\{ z : | z | = 1 , z \neq 1 \}$
(3) $\{ z : z = \bar { z } \}$
(4) $\{ z : | z | = 1 \}$
Q64 Permutations & Arrangements Forming Numbers with Digit Constraints View
The sum of the digits in the unit's place of all the 4-digit numbers formed by using the numbers $3,4,5$ and $6$, without repetition is:
(1) 18
(2) 36
(3) 108
(4) 432
Q65 Arithmetic Sequences and Series Find Specific Term from Given Conditions View
Given an A.P. whose terms are all positive integers. The sum of its first nine terms is greater than 200 and less than 220. If the second term in it is 12, then its $4^{\text{th}}$ term is:
(1) 8
(2) 24
(3) 20
(4) 16
Q66 Arithmetic Sequences and Series Telescoping or Non-Standard Summation Involving an AP View
If the sum $\frac { 3 } { 1 ^ { 2 } } + \frac { 5 } { 1 ^ { 2 } + 2 ^ { 2 } } + \frac { 7 } { 1 ^ { 2 } + 2 ^ { 2 } + 3 ^ { 2 } } + \ldots + $ up to 20 terms is equal to $\frac { k } { 21 }$, then $k$ is equal to
(1) 240
(2) 120
(3) 60
(4) 180
Q67 Binomial Theorem (positive integer n) Find a Specific Coefficient in a Single Binomial Expansion View
The number of terms in the expansion of $( 1 + x ) ^ { 101 } \left( 1 - x + x ^ { 2 } \right) ^ { 100 }$ in powers of $x$ is
(1) 301
(2) 302
(3) 101
(4) 202
Q68 Reciprocal Trig & Identities View
If $\operatorname { cosec } \theta = \frac { \mathrm { p } + \mathrm { q } } { \mathrm { p } - \mathrm { q } } ( \mathrm { p } \neq \mathrm { q } , \mathrm { p } \neq 0 )$, then $\left| \cot \left( \frac { \pi } { 4 } + \frac { \theta } { 2 } \right) \right|$ is equals to:
(1) $p q$
(2) $\sqrt { \frac { p } { q } }$
(3) $\sqrt { \frac { q } { p } }$
(4) $\sqrt { \mathrm { pq } }$
Q69 Standard trigonometric equations Solve trigonometric equation for solutions in an interval View
The number of values of $\alpha$ in $[ 0,2 \pi ]$ for which $2 \sin ^ { 3 } \alpha - 7 \sin ^ { 2 } \alpha + 7 \sin \alpha = 2$, is:
(1) 3
(2) 1
(3) 6
(4) 4
Q70 Straight Lines & Coordinate Geometry Triangle Properties and Special Points View
Given three points $P , Q , R$ with $P ( 5,3 )$ and $R$ lies on the $x$-axis. If the equation of $RQ$ is $x - 2 y = 2$ and $PQ$ is parallel to the $x$-axis, then the centroid of $\triangle PQR$ lies on the line
(1) $x - 2 y + 1 = 0$
(2) $2 x + y - 9 = 0$
(3) $2 x - 5 y = 0$
(4) $5 x - 2 y = 0$
Q71 Circles Circle-Related Locus Problems View
Let $a$ and $b$ be any two numbers satisfying $\frac { 1 } { a ^ { 2 } } + \frac { 1 } { b ^ { 2 } } = \frac { 1 } { 4 }$. Then, the foot of perpendicular from the origin on the variable line $\frac { x } { a } + \frac { y } { b } = 1$ lies on:
(1) A circle of radius $= 2$
(2) A hyperbola with each semi-axis $= \sqrt { 2 }$.
(3) A hyperbola with each semi-axis $= 2$
(4) A circle of radius $= \sqrt { 2 }$
Q72 Circles Circle-Line Intersection and Point Conditions View
If the point $( 1,4 )$ lies inside the circle $x ^ { 2 } + y ^ { 2 } - 6 x + 10 y + p = 0$ and the circle does not touch or intersect the coordinate axes, then the set of all possible values of $p$ is the interval
(1) $( 25,39 )$
(2) $( 25,29 )$
(3) $( 0,25 )$
(4) $( 9,25 )$
Q73 Conic sections Eccentricity or Asymptote Computation View
If $OB$ is the semi-minor axis of an ellipse, $F _ { 1 }$ and $F _ { 2 }$ are its focii and the angle between $F _ { 1 } B$ and $F _ { 2 } B$ is a right angle, then the square of the eccentricity of the ellipse is
(1) $\frac { 1 } { 4 }$
(2) $\frac { 1 } { \sqrt { 2 } }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 1 } { 2 \sqrt { 2 } }$
Q74 Composite & Inverse Functions Evaluate Composition from Algebraic Definitions View
If $f ( x )$ is continuous and $f \left( \frac { 9 } { 2 } \right) = \frac { 2 } { 9 }$, then $\lim _ { x \rightarrow 0 } f \left( \frac { 1 - \cos 3 x } { x ^ { 2 } } \right)$ equals to
(1) $\frac { 8 } { 9 }$
(2) 0
(3) $\frac { 2 } { 9 }$
(4) $\frac { 9 } { 2 }$
Q76 Measures of Location and Spread View
In a set of $2n$ distinct observations, each of the observation below the median of all the observations is increased by 5 and each of the remaining observations is decreased by 3. Then, the mean of the new set of observations:
(1) Increases by 2.
(2) Increase by 1.
(3) Decreases by 2.
(4) Decreases by 1.
Q78 3x3 Matrices Matrix Algebraic Properties and Abstract Reasoning View
If $B$ is a $3 \times 3$ matrix such that $B ^ { 2 } = 0$, then det. $\left[ ( I + B ) ^ { 50 } - 50 B \right]$ is equal to:
(1) 1
(2) 2
(3) 3
(4) 50
Q79 Simultaneous equations View
If $a , b , c$ are non-zero real numbers and if the system of equations $$( a - 1 ) x = y + z$$ $$( b - 1 ) y = x + z$$ $$( c - 1 ) z = x + y$$ has a non-trivial solution, then $ab + bc + ca$ equals:
(1) $-1$
(2) $a + b + c$
(3) $abc$
(4) 1
Q80 Chain Rule Higher-Order Derivatives of Products/Compositions View
If $y = e ^ { n x }$, then $\frac { d ^ { 2 } y } { d x ^ { 2 } } \cdot \frac { d ^ { 2 } x } { d y ^ { 2 } }$ is equal to:
(1) $n e ^ { - n x }$
(2) $- n e ^ { - n x }$
(3) $n e ^ { n x }$
(4) 1
Q81 Exponential Functions Exponential Equation Solving View
If $f ( x ) = \left( \frac { 3 } { 5 } \right) ^ { x } + \left( \frac { 4 } { 5 } \right) ^ { x } - 1 , x \in R$, then the equation $f ( x ) = 0$ has:
(1) No solution
(2) More than two solutions
(3) One solution
(4) Two solutions
Q82 Stationary points and optimisation Determine parameters from given extremum conditions View
If the Rolle's theorem holds for the function $f ( x ) = 2 x ^ { 3 } + a x ^ { 2 } + b x$ in the interval $[ - 1,1 ]$ for the point $c = \frac { 1 } { 2 }$, then the value of $2 a + b$ is:
(1) $-1$
(2) 2
(3) 1
(4) $-2$
Q83 Standard Integrals and Reverse Chain Rule Definite Integral Evaluation via Substitution or Standard Forms View
$\int \frac { \sin ^ { 8 } x - \cos ^ { 8 } x } { \left( 1 - 2 \sin ^ { 2 } x \cos ^ { 2 } x \right) } d x$ is equal to
(1) $- \frac { 1 } { 2 } \sin 2 x + c$
(2) $- \sin ^ { 2 } x + c$
(3) $- \frac { 1 } { 2 } \sin x + c$
(4) $\frac { 1 } { 2 } \sin 2 x + c$
Q84 Integration by Substitution Substitution to Evaluate a Definite Integral (Numerical Answer) View
The integral $\int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { \ln ( 1 + 2 x ) } { 1 + 4 x ^ { 2 } } d x$ equals
(1) $\frac { \pi } { 4 } \ln 2$
(2) $\frac { \pi } { 16 } \ln 2$
(3) $\frac { \pi } { 8 } \ln 2$
(4) $\frac { \pi } { 32 } \ln 2$
Q85 Areas by integration View
Let $A = \left\{ ( x , y ) : y ^ { 2 } \leq 4 x , y - 2 x \geq - 4 \right\}$. The area of the region $A$ in square units is
(1) 10
(2) 8
(3) 9
(4) 11
Q86 Differential equations Higher-Order and Special DEs (Proof/Theory) View
If the differential equation representing the family of all circles touching $x$-axis at the origin is $\left( x ^ { 2 } - y ^ { 2 } \right) \frac { d y } { d x } = g ( x ) y$, then $g ( x )$ equals
(1) $\frac { 1 } { 2 } x ^ { 2 }$
(2) $2 x$
(3) $\frac { 1 } { 2 } x$
(4) $2 x ^ { 2 }$
Q87 Vectors Introduction & 2D Magnitude of Vector Expression View
If $| \vec { a } | = 2 , | \vec { b } | = 3$ and $| \overrightarrow { 2 a } - \vec { b } | = 5$, then $| \overrightarrow { 2 a } + \vec { b } |$ equals:
(1) 5
(2) 7
(3) 17
(4) 1
Q88 Vectors: Lines & Planes Find Cartesian Equation of a Plane View
Equation of the plane which passes through the point of intersection of lines $\frac { x - 1 } { 3 } = \frac { y - 2 } { 1 } = \frac { z - 3 } { 2 }$ and $\frac { x - 3 } { 1 } = \frac { y - 1 } { 2 } = \frac { z - 2 } { 3 }$ and has the largest distance from the origin is:
(1) $4 x + 3 y + 5 z = 50$
(2) $3 x + 4 y + 5 z = 49$
(3) $5 x + 4 y + 3 z = 57$
(4) $7 x + 2 y + 4 z = 54$
Q89 Vectors: Lines & Planes Dihedral Angle or Angle Between Planes/Lines View
A line in the 3-dimensional space makes an angle $\theta \left( 0 < \theta \leq \frac { \pi } { 2 } \right)$ with both the $X$ and $Y$-axes. Then, the set of all values of $\theta$ is in the interval:
(1) $\left( \frac { \pi } { 3 } , \frac { \pi } { 2 } \right]$
(2) $\left( 0 , \frac { \pi } { 4 } \right]$
(3) $\left[ \frac { \pi } { 4 } , \frac { \pi } { 2 } \right]$
(4) $\left[ \frac { \pi } { 6 } , \frac { \pi } { 3 } \right]$
Q90 Probability Definitions Probability Using Set/Event Algebra View
If $A$ and $B$ are two events such that $P ( A \cup B ) = P ( A \cap B )$, then the incorrect statement amongst the following statements is:
(1) $P ( A ) + P ( B ) = 1$
(2) $P \left( A \cap B ^ { \prime } \right) = 0$
(3) $A \& B$ are equally likely
(4) $P \left( A ^ { \prime } \cap B \right) = 0$