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
2024 session2_06apr_shift2

30 maths questions

Q61 Complex Numbers Argand & Loci Circle Equation and Properties via Complex Number Manipulation View
If $z _ { 1 } , z _ { 2 }$ are two distinct complex number such that $\left| \frac { z _ { 1 } - 2 z _ { 2 } } { \frac { 1 } { 2 } - z _ { 1 } \bar { z } _ { 2 } } \right| = 2$, then
(1) $z _ { 1 }$ lies on a circle of radius $\frac { 1 } { 2 }$ and $z _ { 2 }$ lies on a circle of radius 1.
(2) both $z _ { 1 }$ and $z _ { 2 }$ lie on the same circle.
(3) either $z _ { 1 }$ lies on a circle of radius $\frac { 1 } { 2 }$ or $z _ { 2 }$ lies on a circle of radius 1.
(4) either $z _ { 1 }$ lies on a circle of radius 1 or $z _ { 2 }$ lies on a circle of radius $\frac { 1 } { 2 }$.
Q62 Binomial Theorem (positive integer n) Determine Parameters from Conditions on Coefficients or Terms View
Let $0 \leq \mathrm { r } \leq \mathrm { n }$. If ${ } ^ { \mathrm { n } + 1 } \mathrm { C } _ { \mathrm { r } + 1 } : { } ^ { n } \mathrm { C } _ { \mathrm { r } } : { } ^ { \mathrm { n } - 1 } \mathrm { C } _ { \mathrm { r } - 1 } = 55 : 35 : 21$, then $2 \mathrm { n } + 5 \mathrm { r }$ is equal to:
(1) 50
(2) 62
(3) 55
(4) 60
Q63 Permutations & Arrangements Dictionary Order / Rank of a Permutation View
If all the words with or without meaning made using all the letters of the word "NAGPUR" are arranged as in a dictionary, then the word at $315 ^ { \text {th} }$ position in this arrangement is:
(1) NRAGUP
(2) NRAPUG
(3) NRAPGU
(4) NRAGPU
Q64 Geometric Sequences and Series Fractal/Iterative Geometric Construction (Area, Length, or Perimeter Series) View
Let $ABC$ be an equilateral triangle. A new triangle is formed by joining the middle points of all sides of the triangle $ABC$ and the same process is repeated infinitely many times. If P is the sum of perimeters and $Q$ is the sum of areas of all the triangles formed in this process, then:
(1) $\mathrm { P } ^ { 2 } = 6 \sqrt { 3 } \mathrm { Q }$
(2) $\mathrm { P } ^ { 2 } = 36 \sqrt { 3 } \mathrm { Q }$
(3) $P = 36 \sqrt { 3 } Q ^ { 2 }$
(4) $\mathrm { P } ^ { 2 } = 72 \sqrt { 3 } \mathrm { Q }$
Q65 Arithmetic Sequences and Series Compute Partial Sum of an Arithmetic Sequence View
A software company sets up $m$ number of computer systems to finish an assignment in 17 days. If 4 computer systems crashed on the start of the second day, 4 more computer systems crashed on the start of the third day and so on, then it took 8 more days to finish the assignment. The value of $m$ is equal to:
(1) 150
(2) 180
(3) 160
(4) 125
Q66 Circles Circle Equation Derivation View
If $\mathrm { P } ( 6,1 )$ be the orthocentre of the triangle whose vertices are $\mathrm { A } ( 5 , - 2 ) , \mathrm { B } ( 8,3 )$ and $\mathrm { C } ( \mathrm { h } , \mathrm { k } )$, then the point $C$ lies on the circle:
(1) $x ^ { 2 } + y ^ { 2 } - 61 = 0$
(2) $x ^ { 2 } + y ^ { 2 } - 52 = 0$
(3) $x ^ { 2 } + y ^ { 2 } - 65 = 0$
(4) $x ^ { 2 } + y ^ { 2 } - 74 = 0$
Q67 Circles Circle-Related Locus Problems View
If the locus of the point, whose distances from the point $( 2,1 )$ and $( 1,3 )$ are in the ratio $5 : 4$, is $a x ^ { 2 } + b y ^ { 2 } + c x y + d x + e y + 170 = 0$, then the value of $a ^ { 2 } + 2 b + 3 c + 4 d + e$ is equal to:
(1) 37
(2) 437
(3) $- 27$
(4) 5
Q68 Sequences and series, recurrence and convergence Convergence proof and limit determination View
$\lim _ { n \rightarrow \infty } \frac { \left( 1 ^ { 2 } - 1 \right) ( n - 1 ) + \left( 2 ^ { 2 } - 2 \right) ( n - 2 ) + \cdots + \left( ( n - 1 ) ^ { 2 } - ( n - 1 ) \right) \cdot 1 } { \left( 1 ^ { 3 } + 2 ^ { 3 } + \cdots \cdots + n ^ { 3 } \right) - \left( 1 ^ { 2 } + 2 ^ { 2 } + \cdots \cdots + n ^ { 2 } \right) }$ is equal to:
(1) $\frac { 2 } { 3 }$
(2) $\frac { 1 } { 3 }$
(3) $\frac { 3 } { 4 }$
(4) $\frac { 1 } { 2 }$
Q69 Probability Definitions Combinatorial Counting (Non-Probability) View
Let $A = \{ 1,2,3,4,5 \}$. Let R be a relation on A defined by $x \mathrm { R } y$ if and only if $4 x \leq 5 \mathrm { y }$. Let m be the number of elements in R and n be the minimum number of elements from $\mathrm { A } \times \mathrm { A }$ that are required to be added to R to make it a symmetric relation. Then $\mathrm { m } + \mathrm { n }$ is equal to:
(1) 25
(2) 24
(3) 26
(4) 23
Q70 Matrices Determinant and Rank Computation View
If $A$ is a square matrix of order 3 such that $\operatorname { det } ( A ) = 3$ and $\operatorname { det } \left( \operatorname { adj } \left( - 4 \operatorname { adj } \left( - 3 \operatorname { adj } \left( 3 \operatorname { adj } \left( ( 2 \mathrm {~A} ) ^ { - 1 } \right) \right) \right) \right) \right) = 2 ^ { \mathrm { m } } 3 ^ { \mathrm { n } }$, then $\mathrm { m } + 2 \mathrm { n }$ is equal to:
(1) 2
(2) 3
(3) 6
(4) 4
Q71 Function Transformations View
Let $f ( x ) = \frac { 1 } { 7 - \sin 5 x }$ be a function defined on $\mathbf { R }$. Then the range of the function $f ( x )$ is equal to:
(1) $\left[ \frac { 1 } { 7 } , \frac { 1 } { 6 } \right]$
(2) $\left[ \frac { 1 } { 8 } , \frac { 1 } { 5 } \right]$
(3) $\left[ \frac { 1 } { 7 } , \frac { 1 } { 5 } \right]$
(4) $\left[ \frac { 1 } { 8 } , \frac { 1 } { 6 } \right]$
Q72 Chain Rule Chain Rule with Composition of Explicit Functions View
Suppose for a differentiable function $h , h ( 0 ) = 0 , h ( 1 ) = 1$ and $h ^ { \prime } ( 0 ) = h ^ { \prime } ( 1 ) = 2$. If $\mathrm { g } ( x ) = h \left( \mathrm { e } ^ { x } \right) \mathrm { e } ^ { h ( x ) }$, then $g ^ { \prime } ( 0 )$ is equal to:
(1) 5
(2) 4
(3) 8
(4) 3
Q73 Stationary points and optimisation Find absolute extrema on a closed interval or domain View
If the function $f ( x ) = \left( \frac { 1 } { x } \right) ^ { 2 x } ; x > 0$ attains the maximum value at $x = \frac { 1 } { \mathrm { e } }$ then:
(1) $\mathrm { e } ^ { \pi } < \pi ^ { \mathrm { e } }$
(2) $\mathrm { e } ^ { \pi } > \pi ^ { \mathrm { e } }$
(3) $( 2 e ) ^ { \pi } > \pi ^ { ( 2 e ) }$
(4) $\mathrm { e } ^ { 2 \pi } < ( 2 \pi ) ^ { \mathrm { e } }$
Q74 Standard Integrals and Reverse Chain Rule Integral Equation to Determine a Function Value View
If $\int \frac { 1 } { \mathrm { a } ^ { 2 } \sin ^ { 2 } x + \mathrm { b } ^ { 2 } \cos ^ { 2 } x } \mathrm {~d} x = \frac { 1 } { 12 } \tan ^ { - 1 } ( 3 \tan x ) +$ constant, then the maximum value of $\mathrm { a } \sin x + \mathrm { b } \cos x$, is:
(1) $\sqrt { 40 }$
(2) $\sqrt { 41 }$
(3) $\sqrt { 39 }$
(4) $\sqrt { 42 }$
Q75 Areas by integration View
If the area of the region $\left\{ ( x , y ) : \frac { \mathrm { a } } { x ^ { 2 } } \leq y \leq \frac { 1 } { x } , 1 \leq x \leq 2,0 < \mathrm { a } < 1 \right\}$ is $\left( \log _ { \mathrm { e } } 2 \right) - \frac { 1 } { 7 }$ then the value of $7 \mathrm { a } - 3$ is equal to:
(1) 0
(2) 2
(3) $- 1$
(4) 1
Q76 Differential equations Solving Separable DEs with Initial Conditions View
Suppose the solution of the differential equation $\frac { d y } { d x } = \frac { ( 2 + \alpha ) x - \beta y + 2 } { \beta x - 2 \alpha y - ( \beta \gamma - 4 \alpha ) }$ represents a circle passing through origin. Then the radius of this circle is:
(1) 2
(2) $\sqrt { 17 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { \sqrt { 17 } } { 2 }$
Q77 Vectors Introduction & 2D Dot Product Computation View
Let $\overrightarrow { \mathrm { a } } = 2 \hat { i } + \hat { j } - \hat { k } , \overrightarrow { \mathrm {~b} } = ( ( \overrightarrow { \mathrm { a } } \times ( \hat { i } + \hat { j } ) ) \times \hat { i } ) \times \hat { i }$. Then the square of the projection of $\overrightarrow { \mathrm { a } }$ on $\overrightarrow { \mathrm { b } }$ is:
(1) $\frac { 1 } { 3 }$
(2) $\frac { 2 } { 3 }$
(3) 2
(4) $\frac { 1 } { 5 }$
Q78 Vectors Introduction & 2D Dot Product Computation View
Let $\overrightarrow { \mathrm { a } } = 6 \hat { i } + \hat { j } - \hat { k }$ and $\overrightarrow { \mathrm { b } } = \hat { i } + \hat { j }$. If $\overrightarrow { \mathrm { c } }$ is a vector such that $| \overrightarrow { \mathrm { c } } | \geq 6 , \overrightarrow { \mathrm { a } } \cdot \overrightarrow { \mathrm { c } } = 6 | \overrightarrow { \mathrm { c } } | , | \overrightarrow { \mathrm { c } } - \overrightarrow { \mathrm { a } } | = 2 \sqrt { 2 }$ and the angle between $\vec { a } \times \vec { b }$ and $\vec { c }$ is $60 ^ { \circ }$, then $| ( \vec { a } \times \vec { b } ) \times \vec { c } |$ is equal to:
(1) $\frac { 9 } { 2 } ( 6 - \sqrt { 6 } )$
(2) $\frac { 3 } { 2 } \sqrt { 6 }$
(3) $\frac { 9 } { 2 } ( 6 + \sqrt { 6 } )$
(4) $\frac { 3 } { 2 } \sqrt { 3 }$
Q79 Vectors 3D & Lines Distance from a Point to a Line (Show/Compute) View
Let $\mathrm { P } ( \alpha , \beta , \gamma )$ be the image of the point $\mathrm { Q } ( 3 , - 3,1 )$ in the line $\frac { x - 0 } { 1 } = \frac { y - 3 } { 1 } = \frac { z - 1 } { - 1 }$ and R be the point $( 2,5 , - 1 )$. If the area of the triangle $PQR$ is $\lambda$ and $\lambda ^ { 2 } = 14 K$, then $K$ is equal to:
(1) 36
(2) 81
(3) 72
(4) 18
Q80 Probability Definitions Finite Equally-Likely Probability Computation View
If three letters can be posted to any one of the 5 different addresses, then the probability that the three letters are posted to exactly two addresses is:
(1) $\frac { 18 } { 25 }$
(2) $\frac { 12 } { 25 }$
(3) $\frac { 6 } { 25 }$
(4) $\frac { 4 } { 25 }$
Q81 Sequences and series, recurrence and convergence Direct term computation from recurrence View
Let $\alpha , \beta$ be roots of $x ^ { 2 } + \sqrt { 2 } x - 8 = 0$. If $\mathrm { U } _ { \mathrm { n } } = \alpha ^ { \mathrm { n } } + \beta ^ { n }$, then $\frac { \mathrm { U } _ { 10 } + \sqrt { 2 } \mathrm { U } _ { 9 } } { 2 \mathrm { U } _ { 8 } }$ is equal to $\_\_\_\_$
Q82 Binomial Theorem (positive integer n) Evaluate a Summation Involving Binomial Coefficients View
If $\mathrm { S } ( x ) = ( 1 + x ) + 2 ( 1 + x ) ^ { 2 } + 3 ( 1 + x ) ^ { 3 } + \cdots + 60 ( 1 + x ) ^ { 60 } , x \neq 0$, and $( 60 ) ^ { 2 } \mathrm {~S} ( 60 ) = \mathrm { a } ( \mathrm { b } ) ^ { \mathrm { b } } + \mathrm { b }$, where $a , b \in N$, then $( a + b )$ equal to $\_\_\_\_$
Q83 Conic sections Focal Distance and Point-on-Conic Metric Computation View
The length of the latus rectum and directrices of a hyperbola with eccentricity e are 9 and $x = \pm \frac { 4 } { \sqrt { 13 } }$, respectively. Let the line $y - \sqrt { 3 } x + \sqrt { 3 } = 0$ touch this hyperbola at $( x _ { 0 } , y _ { 0 } )$. If m is the product of the focal distances of the point $\left( x _ { 0 } , y _ { 0 } \right)$, then $4 \mathrm { e } ^ { 2 } + \mathrm { m }$ is equal to $\_\_\_\_$
Q84 Sine and Cosine Rules Find an angle using the cosine rule View
In a triangle $\mathrm { ABC } , \mathrm { BC } = 7 , \mathrm { AC } = 8 , \mathrm { AB } = \alpha \in \mathrm { N }$ and $\cos \mathrm { A } = \frac { 2 } { 3 }$. If $49 \cos ( 3 \mathrm { C } ) + 42 = \frac { \mathrm { m } } { \mathrm { n } }$, where $\operatorname { gcd } ( \mathrm { m } , \mathrm { n } ) = 1$, then $\mathrm { m } + \mathrm { n }$ is equal to $\_\_\_\_$
Q85 Simultaneous equations View
If the system of equations $$2 x + 7 y + \lambda z = 3$$ $$3 x + 2 y + 5 z = 4$$ $$x + \mu y + 32 z = - 1$$ has infinitely many solutions, then $( \lambda - \mu )$ is equal to $\_\_\_\_$
Q86 Curve Sketching Continuity and Discontinuity Analysis of Piecewise Functions View
Let $[ \mathrm { t } ]$ denote the greatest integer less than or equal to t. Let $f : [ 0 , \infty ) \rightarrow \mathbf { R }$ be a function defined by $f ( x ) = \left[ \frac { x } { 2 } + 3 \right] - [ \sqrt { x } ]$. Let $S$ be the set of all points in the interval $[ 0,8 ]$ at which $f$ is not continuous. Then $\sum _ { \mathrm { a } \in \mathrm { S } } \mathrm { a }$ is equal to $\_\_\_\_$
Q87 Indefinite & Definite Integrals Piecewise/Periodic Function Integration View
Let $[ t ]$ denote the largest integer less than or equal to $t$. If $$\int _ { 0 } ^ { 3 } \left( \left[ x ^ { 2 } \right] + \left[ \frac { x ^ { 2 } } { 2 } \right] \right) \mathrm { d } x = \mathrm { a } + \mathrm { b } \sqrt { 2 } - \sqrt { 3 } - \sqrt { 5 } + \mathrm { c } \sqrt { 6 } - \sqrt { 7 }$$ where $\mathrm { a } , \mathrm { b } , \mathrm { c } \in \mathbf { Z }$, then $\mathrm { a } + \mathrm { b } + \mathrm { c }$ is equal to $\_\_\_\_$
Q88 First order differential equations (integrating factor) View
If the solution $y ( x )$ of the given differential equation $\left( \mathrm { e } ^ { y } + 1 \right) \cos x \mathrm {~d} x + \mathrm { e } ^ { y } \sin x \mathrm {~d} y = 0$ passes through the point $\left( \frac { \pi } { 2 } , 0 \right)$, then the value of $\mathrm { e } ^ { y \left( \frac { \pi } { 6 } \right) }$ is equal to $\_\_\_\_$
Q89 Vectors 3D & Lines Shortest Distance Between Two Lines View
If the shortest distance between the lines $\frac { x - \lambda } { 3 } = \frac { y - 2 } { - 1 } = \frac { z - 1 } { 1 }$ and $\frac { x + 2 } { - 3 } = \frac { y + 5 } { 2 } = \frac { z - 4 } { 4 }$ is $\frac { 44 } { \sqrt { 30 } }$, then the largest possible value of $| \lambda |$ is equal to $\_\_\_\_$
Q90 Hypergeometric Distribution View
From a lot of 12 items containing 3 defectives, a sample of 5 items is drawn at random. Let the random variable X denote the number of defective items in the sample. Let items in the sample be drawn one by one without replacement. If variance of $X$ is $\frac { m } { n }$, where $\operatorname { gcd } ( m , n ) = 1$, then $n - m$ is equal to $\_\_\_\_$