jee-main

Papers (169)
2025
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2024
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2023
session1_01feb_shift1 24 session1_01feb_shift2 3 session1_24jan_shift1 13 session1_24jan_shift2 12 session1_25jan_shift1 28 session1_25jan_shift2 27 session1_29jan_shift1 29 session1_29jan_shift2 28 session1_30jan_shift1 2 session1_30jan_shift2 29 session1_31jan_shift1 28 session1_31jan_shift2 17 session2_06apr_shift1 5 session2_06apr_shift2 17 session2_08apr_shift1 29 session2_08apr_shift2 14 session2_10apr_shift1 29 session2_10apr_shift2 15 session2_11apr_shift1 5 session2_11apr_shift2 4 session2_12apr_shift1 26 session2_13apr_shift1 25 session2_13apr_shift2 20 session2_15apr_shift1 20
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
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2016
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2015
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2014
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2013
07apr 29 09apr 14 22apr 5 23apr 14 25apr 13
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
2021 session4_31aug_shift1

28 maths questions

Q61 Arithmetic Sequences and Series Telescoping or Non-Standard Summation Involving an AP View
The sum of 10 terms of the series $\frac { 3 } { 1 ^ { 2 } \times 2 ^ { 2 } } + \frac { 5 } { 2 ^ { 2 } \times 3 ^ { 2 } } + \frac { 7 } { 3 ^ { 2 } \times 4 ^ { 2 } } + \ldots$ is :
(1) $\frac { 143 } { 144 }$
(2) $\frac { 99 } { 100 }$
(3) 1
(4) $\frac { 120 } { 121 }$
Q62 Geometric Sequences and Series Arithmetic-Geometric Sequence Interplay View
Three numbers are in an increasing geometric progression with common ratio $r$. If the middle number is doubled, then the new numbers are in an arithmetic progression with common difference $d$. If the fourth term of GP is $3 r ^ { 2 }$, then $r ^ { 2 } - d$ is equal to :
(1) $7 - \sqrt { 3 }$
(2) $7 + 3 \sqrt { 3 }$
(3) $7 - 7 \sqrt { 3 }$
(4) $7 + \sqrt { 3 }$
Q63 Solving quadratics and applications Finding roots or coefficients of a quadratic using Vieta's relations View
$\operatorname { cosec } 18 ^ { \circ }$ is a root of the equation:
(1) $x ^ { 2 } - 2 x - 4 = 0$
(2) $4 x ^ { 2 } + 2 x - 1 = 0$
(3) $x ^ { 2 } + 2 x - 4 = 0$
(4) $x ^ { 2 } - 2 x + 4 = 0$
Q64 Straight Lines & Coordinate Geometry Point-to-Line Distance Computation View
If $p$ and $q$ are the lengths of the perpendiculars from the origin on the lines, $x \operatorname { cosec } \alpha - y \sec \alpha = k \cot 2 \alpha$ and $x \sin \alpha + y \cos \alpha = k \sin 2 \alpha$ respectively, then $k ^ { 2 }$ is equal to $:$
(1) $2 p ^ { 2 } + q ^ { 2 }$
(2) $p ^ { 2 } + 2 q ^ { 2 }$
(3) $4 q ^ { 2 } + p ^ { 2 }$
(4) $4 p ^ { 2 } + q ^ { 2 }$
Q65 Circles Circle Equation Derivation View
The length of the latus rectum of a parabola, whose vertex and focus are on the positive $x$-axis at a distance $R$ and $S ( > \mathrm { R } )$ respectively from the origin, is :
(1) $2 ( S - R )$
(2) $2 ( S + R )$
(3) $4 ( S - R )$
(4) $4 ( S + R )$
Q66 Circles Tangent Lines and Tangent Lengths View
The line $12 x \cos \theta + 5 y \sin \theta = 60$ is tangent to which of the following curves ?
(1) $x ^ { 2 } + y ^ { 2 } = 30$
(2) $144 x ^ { 2 } + 25 y ^ { 2 } = 3600$
(3) $x ^ { 2 } + y ^ { 2 } = 169$
(4) $25 x ^ { 2 } + 12 y ^ { 2 } = 3600$
Q67 Chain Rule Limit Evaluation Involving Composition or Substitution View
$\lim _ { x \rightarrow 0 } \frac { \sin ^ { 2 } \left( \pi \cos ^ { 4 } x \right) } { x ^ { 4 } }$ is equal to :
(1) $2 \pi ^ { 2 }$
(2) $\pi ^ { 2 }$
(3) $4 \pi ^ { 2 }$
(4) $4 \pi$
Q69 Sine and Cosine Rules Heights and distances / angle of elevation problem View
A vertical pole fixed to the horizontal ground is divided in the ratio 3 : 7 by a mark on it with lower part shorter than the upper part. If the two parts subtend equal angles at a point on the ground 18 m away from the base of the pole, then the height of the pole (in meters) is :
(1) $8 \sqrt { 10 }$
(2) $6 \sqrt { 10 }$
(3) $12 \sqrt { 10 }$
(4) $12 \sqrt { 15 }$
Q71 3x3 Matrices Determinant of Parametric or Structured Matrix View
If $\mathrm { a } _ { \mathrm { r } } = \cos \frac { 2 \mathrm { r } \pi } { 9 } + i \sin \frac { 2 \mathrm { r } \pi } { 9 } , \mathrm { r } = 1,2,3 , \ldots , i = \sqrt { - 1 }$, then the determinant $\left| \begin{array} { l l l } a _ { 1 } & a _ { 2 } & a _ { 3 } \\ a _ { 4 } & a _ { 5 } & a _ { 6 } \\ a _ { 7 } & a _ { 8 } & a _ { 9 } \end{array} \right|$ is equal to $:$
(1) $\mathrm { a } _ { 9 }$
(2) $a _ { 1 } a _ { 9 } - a _ { 3 } a _ { 7 }$
(3) $a _ { 5 }$
(4) $a _ { 2 } a _ { 6 } - a _ { 4 } a _ { 8 }$
Q72 Simultaneous equations View
If the following system of linear equations $2 x + y + z = 5$ $x - y + z = 3$ $x + y + a z = b$ has no solution, then :
(1) $a = - \frac { 1 } { 3 } , b \neq \frac { 7 } { 3 }$
(2) $a \neq \frac { 1 } { 3 } , b = \frac { 7 } { 3 }$
(3) $a \neq - \frac { 1 } { 3 } , b = \frac { 7 } { 3 }$
(4) $a = \frac { 1 } { 3 } , b \neq \frac { 7 } { 3 }$
Q73 Chain Rule Piecewise Function Differentiability Analysis View
The function $f ( x ) = \left| x ^ { 2 } - 2 x - 3 \right| \cdot \mathrm { e } ^ { 9 x ^ { 2 } - 12 x + 4 }$ is not differentiable at exactly :
(1) Four points
(2) Two points
(3) three points
(4) one point
Q74 Chain Rule Continuity Conditions via Composition View
If the function $f ( x ) = \begin{cases} \frac { 1 } { x } \log _ { \mathrm { e } } \left( \frac { 1 + \frac { x } { a } } { 1 - \frac { x } { b } } \right) & , x < 0 \\ k & , x = 0 \\ \frac { \cos ^ { 2 } x - \sin ^ { 2 } x - 1 } { \sqrt { x ^ { 2 } + 1 } - 1 } & , x > 0 \end{cases}$ is continuous at $x = 0$, then $\frac { 1 } { a } + \frac { 1 } { b } + \frac { 4 } { k }$ is equal to:
(1) 4
(2) 5
(3) $- 4$
(4) $- 5$
Q75 Exponential Equations & Modelling Solve Exponential Equation for Unknown Variable View
The number of real roots of the equation $e ^ { 4 x } + 2 e ^ { 3 x } - e ^ { x } - 6 = 0$ is :
(1) 0
(2) 1
(3) 4
(4) 2
Q76 Integration by Substitution Substitution to Transform Integral Form (Show Transformed Expression) View
The integral $\int \frac { 1 } { \sqrt [ 4 ] { ( x - 1 ) ^ { 3 } ( x + 2 ) ^ { 5 } } } \mathrm {~d} x$ is equal to : (where $C$ is a constant of integration)
(1) $\frac { 3 } { 4 } \left( \frac { x + 2 } { x - 1 } \right) ^ { \frac { 5 } { 4 } } + C$
(2) $\frac { 4 } { 3 } \left( \frac { x - 1 } { x + 2 } \right) ^ { \frac { 1 } { 4 } } + C$
(3) $\frac { 4 } { 3 } \left( \frac { x - 1 } { x + 2 } \right) ^ { \frac { 5 } { 4 } } + \mathrm { C }$
(4) $\frac { 3 } { 4 } \left( \frac { x + 2 } { x - 1 } \right) ^ { \frac { 1 } { 4 } } + \mathrm { C }$
Q77 Chain Rule Chain Rule Combined with Fundamental Theorem of Calculus View
Let $f$ be a non-negative function in $[ 0,1 ]$ and twice differentiable in $( 0,1 )$. If $\int _ { 0 } ^ { x } \sqrt { 1 - \left( f ^ { \prime } ( t ) \right) ^ { 2 } } \mathrm { dt } = \int _ { 0 } ^ { x } f ( \mathrm { t } ) \mathrm { dt } , 0 \leq x \leq 1$ and $f ( 0 ) = 0$, then $\lim _ { x \rightarrow 0 } \frac { 1 } { x ^ { 2 } } \int _ { 0 } ^ { x } f ( \mathrm { t } ) \mathrm { dt } :$
(1) does not exist
(2) equals 0
(3) equals 1
(4) equals $\frac { 1 } { 2 }$
Q78 Differential equations Solving Separable DEs with Initial Conditions View
If $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 ^ { x + y } - 2 ^ { x } } { 2 ^ { y } } , y ( 0 ) = 1$, then $y ( 1 )$ is equal to :
(1) $\log _ { 2 } \left( 1 + \mathrm { e } ^ { 2 } \right)$
(2) $\log _ { 2 } ( 2 \mathrm { e } )$
(3) $\log _ { 2 } ( 2 + e )$
(4) $\log _ { 2 } ( 1 + e )$
Q79 Vectors Introduction & 2D Magnitude of Vector Expression View
Let $\vec { a }$ and $\vec { b }$ be two vectors such that $| 2 \vec { a } + 3 \vec { b } | = | 3 \vec { a } + \vec { b } |$ and the angle between $\vec { a }$ and $\vec { b }$ is $60 ^ { \circ }$. If $\frac { 1 } { 8 } \vec { a }$ is a unit vector, then $| \vec { b } |$ is equal to :
(1) 8
(2) 4
(3) 6
(4) 5
Q80 Vectors: Lines & Planes Find Cartesian Equation of a Plane View
Let the equation of the plane, that passes through the point $( 1,4 , - 3 )$ and contains the line of intersection of the planes $3 x - 2 y + 4 z - 7 = 0$ and $x + 5 y - 2 z + 9 = 0$, be $\alpha x + \beta y + \gamma z + 3 = 0$, then $\alpha + \beta + \gamma$ is equal to :
(1) $- 15$
(2) 15
(3) $- 23$
(4) 23
Q81 Complex Numbers Argand & Loci Distance and Region Optimization on Loci View
A point $z$ moves in the complex plane such that $\arg \left( \frac { z - 2 } { z + 2 } \right) = \frac { \pi } { 4 }$, then the minimum value of $| z - 9 \sqrt { 2 } - 2 i | ^ { 2 }$ is equal to
Q82 Combinations & Selection Counting Arrangements with Run or Pattern Constraints View
The number of six letter words (with or without meaning), formed using all the letters of the word 'VOWELS', so that all the consonants never come together, is
Q83 Generalised Binomial Theorem View
If $\left( \frac { 3 ^ { 6 } } { 4 ^ { 4 } } \right) k$ is the term, independent of $x$, in the binomial expansion of $\left( \frac { x } { 4 } - \frac { 12 } { x ^ { 2 } } \right) ^ { 12 }$, then $k$ is equal to
Q84 Circles Circle-Line Intersection and Point Conditions View
If the variable line $3 x + 4 y = \alpha$ lies between the two circles $( x - 1 ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 1$ and $( x - 9 ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 4$, without intercepting a chord on either circle, then the sum of all the integral values of $\alpha$ is
Q85 Measures of Location and Spread View
The mean of 10 numbers $7 \times 8, 10 \times 10, 13 \times 12, 16 \times 14 , \ldots$ is
Q86 Stationary points and optimisation Determine intervals of increase/decrease or monotonicity conditions View
If $R$ is the least value of $a$ such that the function $f ( x ) = x ^ { 2 } + \mathrm { a } x + 1$ is increasing on $[ 1,2 ]$ and $S$ is the greatest value of $a$ such that the function $f ( x ) = x ^ { 2 } + a x + 1$ is decreasing on $[ 1,2 ]$, then the value of $| R - S |$ is
Q87 Indefinite & Definite Integrals Piecewise/Periodic Function Integration View
Let $[ t ]$ denote the greatest integer $\leq \mathrm { t }$. Then the value of $8 \cdot \int _ { - \frac { 1 } { 2 } } ^ { 1 } ( [ 2 x ] + | x | ) \mathrm { d } x$ is
Q88 Indefinite & Definite Integrals Finding a Function from an Integral Equation View
If $x \phi ( x ) = \int _ { 5 } ^ { x } \left( 3 t ^ { 2 } - 2 \phi ^ { \prime } ( t ) \right) d t , x > - 2 , \phi ( 0 ) = 4$, then $\phi ( 2 )$ is
Q89 Vectors 3D & Lines Line-Plane Intersection View
The square of the distance of the point of intersection of the line $\frac { x - 1 } { 2 } = \frac { y - 2 } { 3 } = \frac { z + 1 } { 6 }$ and the plane $2 x - y + z = 6$ from the point $( - 1 , - 1,2 )$ is
Q90 Probability Definitions Conditional Probability and Bayes' Theorem View
An electric instrument consists of two units. Each unit must function independently for the instrument to operate. The probability that the first unit functions is 0.9 and that of the second unit is 0.8 . The instrument is switched on and it fails to operate. If the probability that only the first unit failed and second unit is functioning is $p$, then $98p$ is equal to