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
session1_24jun_shift1 20 session1_24jun_shift2 25 session1_25jun_shift1 14 session1_25jun_shift2 17 session1_26jun_shift1 26 session1_26jun_shift2 23 session1_27jun_shift1 4 session1_27jun_shift2 29 session1_28jun_shift1 13 session1_29jun_shift1 20 session1_29jun_shift2 5 session2_25jul_shift1 29 session2_25jul_shift2 22 session2_26jul_shift1 29 session2_26jul_shift2 24 session2_27jul_shift1 26 session2_27jul_shift2 29 session2_28jul_shift1 12 session2_28jul_shift2 29 session2_29jul_shift1 18 session2_29jul_shift2 17
2021
session1_24feb_shift1 10 session1_24feb_shift2 7 session1_25feb_shift1 29 session1_25feb_shift2 29 session1_26feb_shift2 17 session2_16mar_shift1 29 session2_16mar_shift2 15 session2_17mar_shift1 20 session2_17mar_shift2 24 session2_18mar_shift1 12 session2_18mar_shift2 11 session3_20jul_shift1 30 session3_20jul_shift2 29 session3_22jul_shift1 7 session3_25jul_shift1 2 session3_25jul_shift2 15 session3_27jul_shift1 3 session3_27jul_shift2 4 session4_01sep_shift2 11 session4_26aug_shift1 5 session4_26aug_shift2 2 session4_27aug_shift1 3 session4_27aug_shift2 28 session4_31aug_shift1 28 session4_31aug_shift2 4
2020
session1_07jan_shift1 26 session1_07jan_shift2 17 session1_08jan_shift1 5 session1_08jan_shift2 12 session1_09jan_shift1 22 session1_09jan_shift2 18 session2_02sep_shift1 19 session2_02sep_shift2 17 session2_03sep_shift1 21 session2_03sep_shift2 9 session2_04sep_shift1 10 session2_04sep_shift2 24 session2_05sep_shift1 23 session2_05sep_shift2 27 session2_06sep_shift1 13 session2_06sep_shift2 10
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
2023 session2_08apr_shift1

29 maths questions

Q61 Roots of polynomials Vieta's formulas: compute symmetric functions of roots View
Let $\alpha , \beta , \gamma$ be the three roots of the equation $x ^ { 3 } + b x + c = 0$ if $\beta \gamma = 1 = - \alpha$ then $b ^ { 3 } + 2 c ^ { 3 } - 3 \alpha ^ { 3 } - 6 \beta ^ { 3 } - 8 \gamma ^ { 3 }$ is equal to
(1) $\frac { 155 } { 8 }$
(2) 21
(3) $\frac { 169 } { 8 }$
(4) 19
Q62 Complex Numbers Arithmetic Systems of Equations via Real and Imaginary Part Matching View
If for $z = \alpha + i \beta , | z + 2 | = z + 4 ( 1 + i )$, then $\alpha + \beta$ and $\alpha \beta$ are the roots of the equation
(1) $x ^ { 2 } + 3 x - 4 = 0$
(2) $x ^ { 2 } + 7 x + 12 = 0$
(3) $x ^ { 2 } + x - 12 = 0$
(4) $x ^ { 2 } + 2 x - 3 = 0$
Q63 Permutations & Arrangements Word Permutations with Repeated Letters View
The number of arrangements of the letters of the word "INDEPENDENCE" in which all the vowels always occur together is
(1) 16800
(2) 33600
(3) 18000
(4) 14800
Q64 Permutations & Arrangements Circular Arrangement View
The number of ways, in which 5 girls and 7 boys can be seated at a round table so that no two girls sit together is
(1) 720
(2) $126 ( 5 ! ) ^ { 2 }$
(3) $7 ( 360 ) ^ { 2 }$
(4) $7 ( 720 ) ^ { 2 }$
Q65 Number Theory Divisibility and Divisor Analysis View
The largest natural number $n$ such that $3 n$ divides 66! is $\_\_\_\_$
Q66 Arithmetic Sequences and Series Summation of Derived Sequence from AP View
Let $S _ { K } = \frac { 1 + 2 + \ldots + K } { K }$ and $\sum _ { j = 1 } ^ { n } S ^ { 2 } { } _ { j } = \frac { n } { A } \left( B n ^ { 2 } + C n + D \right)$ where $A , B , C , D \in N$ and $A$ has least value then
(1) $A + C + D$ is not divisible by $D$
(2) $A + B = 5 ( D - C )$
(3) $A + B + C + D$ is divisible by 5
(4) $A + B$ is divisible by $D$
Q67 Binomial Theorem (positive integer n) Determine Parameters from Conditions on Coefficients or Terms View
If the coefficients of three consecutive terms in the expansion of $( 1 + x ) ^ { n }$ are in the ratio $1 : 5 : 20$ then the coefficient of the fourth term is
(1) 2436
(2) 5481
(3) 1827
(4) 3654
Q68 Binomial Theorem (positive integer n) Find a Specific Coefficient in a Single Binomial Expansion View
Let $[ t ]$ denote the greatest integer $\leq t$. If the constant term in the expansion of $\left( 3 x ^ { 2 } - \frac { 1 } { 2 x ^ { 5 } } \right) ^ { 7 }$ is $\alpha$ then $[ \alpha ]$ is equal to $\_\_\_\_$
Q69 Straight Lines & Coordinate Geometry Triangle Properties and Special Points View
Let $C ( \alpha , \beta )$ be the circumcentre of the triangle formed by the lines $4 x + 3 y = 69$, $4 y - 3 x = 17$, and $x + 7 y = 61$. Then $( \alpha - \beta ) ^ { 2 } + \alpha + \beta$ is equal to
(1) 18
(2) 17
(3) 15
(4) 16
Q70 Circles Circle Equation Derivation View
Consider a circle $C _ { 1 } : x ^ { 2 } + y ^ { 2 } - 4 x - 2 y = \alpha - 5$. Let its mirror image in the line $y = 2 x + 1$ be another circle $C _ { 2 } : 5 x ^ { 2 } + 5 y ^ { 2 } - 10 f x - 10 g y + 36 = 0$. Let $r$ be the radius of $C _ { 2 }$. Then $\alpha + r$ is equal to $\_\_\_\_$
Q71 Conic sections Focal Chord and Parabola Segment Relations View
Let $R$ be the focus of the parabola $y ^ { 2 } = 20 x$ and the line $y = m x + c$ intersect the parabola at two points $P$ and $Q$. Let the point $G ( 10 , 10 )$ be the centroid of the triangle $P Q R$. If $c - m = 6$, then $P Q ^ { 2 }$ is
(1) 296
(2) 325
(3) 317
(4) 346
Q72 Chain Rule Limit Evaluation Involving Composition or Substitution View
$\lim _ { x \rightarrow 0 } \left( \left( \frac { 1 - \cos ^ { 2 } ( 3 x ) } { \cos ^ { 3 } ( 4 x ) } \right) \left( \frac { \sin ^ { 3 } ( 4 x ) } { \left( \log _ { e } ( 2 x + 1 ) \right) ^ { 5 } } \right) \right)$ is equal to
(1) 15
(2) 9
(3) 18
(4) 24
Q74 Measures of Location and Spread View
Let the mean and variance of 8 numbers $x , y , 10 , 12 , 6 , 12 , 4 , 8$ be 9 and 9.25 respectively. If $x > y$, then $3 x - 2 y$ is equal to $\_\_\_\_$
Q75 Combinations & Selection Subset Counting with Set-Theoretic Conditions View
Let the number of elements in sets $A$ and $B$ be five and two respectively. Then the number of subsets of $A \times B$ each having at least 3 and at most 6 elements is
(1) 752
(2) 782
(3) 792
(4) 772
Q76 Independent Events View
Let $A = \{ 0 , 3 , 4 , 6 , 7 , 8 , 9 , 10 \}$ and $R$ be the relation defined on $A$ such that $R = \{ ( x , y ) \in A \times A : x - y$ is odd positive integer or $x - y = 2 \}$. The minimum number of elements that must be added to the relation $R$, so that it is a symmetric relation, is equal to $\_\_\_\_$
Q77 Matrices Determinant and Rank Computation View
Let $A = \left[ \begin{array} { c c c } 2 & 1 & 0 \\ 1 & 2 & - 1 \\ 0 & - 1 & 2 \end{array} \right]$. If $| \mathrm{adj} ( \mathrm{adj} ( \mathrm{adj}\, 2 A ) ) | = ( 16 ) ^ { n }$, then $n$ is equal to
(1) 8
(2) 10
(3) 9
(4) 12
Q78 Matrices Matrix Power Computation and Application View
Let $P = \left[ \begin{array} { c c } \frac { \sqrt { 3 } } { 2 } & \frac { 1 } { 2 } \\ - \frac { 1 } { 2 } & \frac { \sqrt { 3 } } { 2 } \end{array} \right] , A = \left[ \begin{array} { l l } 1 & 1 \\ 0 & 1 \end{array} \right]$ and $Q = P A P ^ { T }$. If $P ^ { T } Q ^ { 2007 } P = \left[ \begin{array} { l l } a & b \\ c & d \end{array} \right]$ then $2 a + b - 3 c - 4 d$ is equal to
(1) 2004
(2) 2005
(3) 2007
(4) 2006
Q79 Differentiating Transcendental Functions Higher-order or nth derivative computation View
Let $f ( x ) = \frac { \sin x + \cos x - \sqrt { 2 } } { \sin x - \cos x } , x \in [ 0 , \pi ] - \left\{ \frac { \pi } { 4 } \right\}$, then $f \left( \frac { 7 \pi } { 12 } \right) f ^ { \prime \prime } \left( \frac { 7 \pi } { 12 } \right)$ is equal to
(1) $\frac { 2 } { 9 }$
(2) $\frac { - 2 } { 3 }$
(3) $\frac { - 1 } { 3 \sqrt { 3 } }$
(4) $\frac { 2 } { 3 \sqrt { 3 } }$
Q80 Sequences and series, recurrence and convergence Monotonicity and boundedness analysis View
If $a _ { \alpha }$ is the greatest term in the sequence $a _ { n } = \frac { n ^ { 3 } } { n ^ { 4 } + 147 } , n = 1 , 2 , 3 \ldots$, then $\alpha$ is equal to $\_\_\_\_$
Q81 Integration by Substitution Substitution to Compute an Indefinite Integral with Initial Condition View
Let $I ( x ) = \int \frac { x + 1 } { x \left( 1 + x e ^ { x } \right) ^ { 2 } } d x , x > 0$. If $\lim _ { x \rightarrow \infty } I ( x ) = 0$ then $I ( 1 )$ is equal to
(1) $\frac { e + 2 } { e + 1 } - \log _ { e } ( e + 1 )$
(2) $\frac { e + 1 } { e + 2 } + \log _ { e } ( e + 1 )$
(3) $\frac { e + 1 } { e + 2 } - \log _ { e } ( e + 1 )$
(4) $\frac { e + 2 } { e + 1 } + \log _ { e } ( e + 1 )$
Q82 Indefinite & Definite Integrals Piecewise/Periodic Function Integration View
Let $[ t ]$ denote the greatest integer $\leq t$. Then $\frac { 2 } { \pi } \int _ { \frac { \pi } { 6 } } ^ { \frac { 5 \pi } { 6 } } ( 8 [ \operatorname { cosec } x ] - 5 [ \cot x ] ) d x$ is equal to $\_\_\_\_$
Q83 Areas by integration View
The area of the region $\left\{ ( x , y ) : x ^ { 2 } \leq y \leq 8 - x ^ { 2 } , y \leq 7 \right\}$ is
(1) 27
(2) 18
(3) 20
(4) 21
Q84 Differential equations Solving Separable DEs with Initial Conditions View
If the solution curve of the differential equation $\left( y - 2 \log _ { e } x \right) d x + \left( x \log _ { e } x ^ { 2 } \right) d y = 0 , x > 1$ passes through the points $\left( e , \frac { 4 } { 3 } \right)$ and $\left( e ^ { 4 } , \alpha \right)$, then $\alpha$ is equal to $\_\_\_\_$
Q85 Vectors 3D & Lines Section Division and Coordinate Computation View
If the points with position vectors $\alpha \hat { \mathrm { i } } + 10 \hat { \mathrm { j } } + 13 \hat { \mathrm { k } } , 6 \hat { \mathrm { i } } + 11 \hat { \mathrm { j } } + 11 \hat { \mathrm { k } } , \frac { 9 } { 2 } \hat { \mathrm { i } } + \beta \hat { \mathrm { j } } - 8 \hat { \mathrm { k } }$ are collinear, then $( 19 \alpha - 6 \beta ) ^ { 2 }$ is equal to
(1) 36
(2) 25
(3) 49
(4) 16
Q86 Vectors 3D & Lines Vector Algebra and Triple Product Computation View
Let $\vec { a } = 6 \hat { i } + 9 \hat { j } + 12 \hat { k } , \vec { b } = \alpha \hat { i } + 11 \hat { j } - 2 \hat { k }$ and $\vec { c }$ be vectors such that $\vec { a } \times \vec { c } = \vec { a } \times \vec { b }$. If $\vec { a } \cdot \vec { c } = - 12$, and $\vec { c } \cdot ( \hat { i } - 2 \hat { j } + \hat { k } ) = 5$ then $\vec { c } \cdot ( \hat { i } + \hat { j } + \hat { k } )$ is equal to $\_\_\_\_$
Q87 Vectors 3D & Lines Shortest Distance Between Two Lines View
The shortest distance between the lines $\frac { x - 4 } { 4 } = \frac { y + 2 } { 5 } = \frac { z + 3 } { 3 }$ and $\frac { x - 1 } { 3 } = \frac { y - 3 } { 4 } = \frac { z - 4 } { 2 }$ is
(1) $6 \sqrt { 3 }$
(2) $2 \sqrt { 6 }$
(3) $6 \sqrt { 2 }$
(4) $3 \sqrt { 6 }$
Q88 Vectors 3D & Lines Normal Vector and Plane Equation View
If the equation of the plane containing the line $x + 2 y + 3 z - 4 = 0 = 2 x + y - z + 5$ and perpendicular to the plane $\vec { r } = ( \hat { i } - \hat { j } ) + \lambda ( \hat { i } + \hat { j } + \hat { k } ) + \mu ( \hat { i } - 2 \hat { j } + 3 \hat { k } )$ is $a x + b y + c z = 4$ then $( a - b + c )$ is equal to
(1) 18
(2) 22
(3) 20
(4) 24
Q89 Vectors 3D & Lines Distance from a Point to a Line (Show/Compute) View
Let $\lambda _ { 1 } , \lambda _ { 2 }$ be the values of $\lambda$ for which the points $\left( \frac { 5 } { 2 } , 1 , \lambda \right)$ and $( - 2 , 0 , 1 )$ are at equal distance from the plane $2 x + 3 y - 6 z + 7 = 0$. If $\lambda _ { 1 } > \lambda _ { 2 }$, then the distance of the point $\left( \lambda _ { 1 } - \lambda _ { 2 } , \lambda _ { 2 } , \lambda _ { 1 } \right)$ from the line $\frac { x - 5 } { 1 } = \frac { y - 1 } { 2 } = \frac { z + 7 } { 2 }$ is $\_\_\_\_$
Q90 Conditional Probability Bayes' Theorem with Production/Source Identification View
In a bolt factory, machines $A , B$ and $C$ manufacture respectively $20 \% , 30 \%$ and $50 \%$ of the total bolts. Of their output 3, 4 and 2 percent are respectively defective bolts. A bolt is drawn at random from the product. If the bolt drawn is found to be defective then the probability that it is manufactured by the machine $C$ is
(1) $\frac { 5 } { 14 }$
(2) $\frac { 9 } { 28 }$
(3) $\frac { 3 } { 7 }$
(4) $\frac { 2 } { 7 }$