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
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2017
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2016
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2015
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2014
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2013
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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
2024 session1_27jan_shift1

28 maths questions

Q61 Complex Numbers Arithmetic Modulus Computation View
If $\mathrm { S } = \mathrm { z } \in \mathrm { C } : | \mathrm { z } - \mathrm { i } | = | \mathrm { z } + \mathrm { i } | = | \mathrm { z } - 1 |$, then, $\mathrm { n } ( \mathrm { S } )$ is:
(1) 1
(2) 0
(3) 3
(4) 2
Q62 Arithmetic Sequences and Series Counting or Combinatorial Problems on APs View
The number of common terms in the progressions $4,9,14,19 , \ldots$. up to $25 ^ { \text {th} }$ term and $3,6,9,12 , \ldots$. up to $37 ^ { \text {th} }$ term is:
(1) 9
(2) 5
(3) 7
(4) 8
Q63 Binomial Theorem (positive integer n) Evaluate a Summation Involving Binomial Coefficients View
If $A$ denotes the sum of all the coefficients in the expansion of $\left( 1 - 3 x + 10 x ^ { 2 } \right) ^ { n }$ and $B$ denotes the sum of all the coefficients in the expansion of $\left( 1 + x ^ { 2 } \right) ^ { n }$, then:
(1) $\mathrm { A } = \mathrm { B } ^ { 3 }$
(2) $3 \mathrm {~A} = \mathrm { B }$
(3) $\mathrm { B } = \mathrm { A } ^ { 3 }$
(4) $\mathrm { A } = 3 \mathrm {~B}$
Q64 Binomial Theorem (positive integer n) Determine Parameters from Conditions on Coefficients or Terms View
${ } ^ { n - 1 } C _ { r } = \left( k ^ { 2 } - 8 \right) ^ { n } C _ { r + 1 }$ if and only if:
(1) $2 \sqrt { 2 } < k \leq 3$
(2) $2 \sqrt { 3 } < \mathrm { k } \leq 3 \sqrt { 2 }$
(3) $2 \sqrt { 3 } < \mathrm { k } < 3 \sqrt { 3 }$
(4) $2 \sqrt { 2 } < \mathrm { k } < 2 \sqrt { 3 }$
Q65 Straight Lines & Coordinate Geometry Slope and Angle Between Lines View
The portion of the line $4 x + 5 y = 20$ in the first quadrant is trisected by the lines $\mathrm { L } _ { 1 }$ and $\mathrm { L } _ { 2 }$ passing through the origin. The tangent of an angle between the lines $L _ { 1 }$ and $L _ { 2 }$ is:
(1) $\frac { 8 } { 5 }$
(2) $\frac { 25 } { 41 }$
(3) $\frac { 2 } { 5 }$
(4) $\frac { 30 } { 41 }$
Q66 Circles Circle Equation Derivation View
Four distinct points $( 2 \mathrm { k } , 3 \mathrm { k } ) , ( 1,0 ) , ( 0,1 )$ and $( 0,0 )$ lie on a circle for $k$ equal to:
(1) $\frac { 2 } { 13 }$
(2) $\frac { 3 } { 13 }$
(3) $\frac { 5 } { 13 }$
(4) $\frac { 1 } { 13 }$
Q67 Circles Optimization on a Circle View
If the shortest distance of the parabola $y ^ { 2 } = 4 x$ from the centre of the circle $x ^ { 2 } + y ^ { 2 } - 4 x - 16 y + 64 = 0$ is $d$ , then $\mathrm { d } ^ { 2 }$ is equal to:
(1) 16
(2) 24
(3) 20
(4) 36
Q68 Conic sections Chord Properties and Midpoint Problems View
The length of the chord of the ellipse $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 16 } = 1$, whose mid point is $\left( 1 , \frac { 2 } { 5 } \right)$, is equal to:
(1) $\frac { \sqrt { 1691 } } { 5 }$
(2) $\frac { \sqrt { 2009 } } { 5 }$
(3) $\frac { \sqrt { 1741 } } { 5 }$
(4) $\frac { \sqrt { 1541 } } { 5 }$
Q69 Chain Rule Limit Evaluation Involving Composition or Substitution View
If $a = \lim _ { x \rightarrow 0 } \frac { \sqrt { 1 + \sqrt { 1 + x ^ { 4 } } } - \sqrt { 2 } } { x ^ { 4 } }$ and $b = \lim _ { x \rightarrow 0 } \frac { \sin ^ { 2 } x } { \sqrt { 2 } - \sqrt { 1 + \cos x } }$, then the value of $a b ^ { 3 }$ is:
(1) 36
(2) 32
(3) 25
(4) 30
Q70 Measures of Location and Spread View
Let $\mathrm { a } _ { 1 } , \mathrm { a } _ { 2 } , \ldots , \mathrm { a } _ { 10 }$ be 10 observations such that $\sum _ { \mathrm { k } = 1 } ^ { 10 } \mathrm { a } _ { \mathrm { k } } = 50$ and $\sum _ { \forall \mathrm { k } < \mathrm { j } } \mathrm { a } _ { \mathrm { k } } \cdot \mathrm { a } _ { \mathrm { j } } = 1100$. Then the standard deviation of $a _ { 1 } , a _ { 2 } , \ldots , a _ { 10 }$ is equal to:
(1) 5
(2) $\sqrt { 5 }$
(3) 10
(4) $\sqrt { 115 }$
Q72 Matrices Matrix Algebra and Product Properties View
Consider the matrix $f ( x ) = \left[ \begin{array} { c c c } \cos x & - \sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{array} \right]$. Given below are two statements: Statement I: $f ( - x )$ is the inverse of the matrix $f ( x )$. Statement II: $f ( x ) f ( y ) = f ( x + y )$. In the light of the above statements, choose the correct answer from the options given below
(1) Statement I is false but Statement II is true
(2) Both Statement I and Statement II are false
(3) Statement I is true but Statement II is false
(4) Both Statement I and Statement II are true
Q74 Curve Sketching Finding Parameters for Continuity View
Consider the function $\mathrm { f } ( \mathrm { x } ) = \left\{ \begin{array} { c l } \frac { \mathrm { a } \left( 7 \mathrm { x } - 12 - \mathrm { x } ^ { 2 } \right) } { \mathrm { b } \left| \mathrm { x } ^ { 2 } - 7 \mathrm { x } + 12 \right| } & , \mathrm { x } < 3 \\ 2 ^ { \frac { \sin ( \mathrm { x } - 3 ) } { \mathrm { x } - [ \mathrm { x } ] } } & , \mathrm { x } > 3 \\ \mathrm {~b} & , \mathrm { x } = 3 \end{array} \right.$, where $[ \mathrm { x } ]$ denotes the greatest integer less than or equal to x. If S denotes the set of all ordered pairs $( \mathrm { a } , \mathrm { b } )$ such that $\mathrm { f } ( \mathrm { x } )$ is continuous at $x = 3$, then the number of elements in S is:
(1) 2
(2) Infinitely many
(3) 4
(4) 1
Q75 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
If $\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 3 + x } + \sqrt { 1 + x } } d x = a + b \sqrt { 2 } + c \sqrt { 3 }$, where $a , b , c$ are rational numbers, then $2 a + 3 b - 4 c$ is equal to:
(1) 4
(2) 10
(3) 7
(4) 8
Q76 Integration by Substitution Substitution Combined with Symmetry or Companion Integral View
If $( a , b )$ be the orthocentre of the triangle whose vertices are $( 1,2 ) , ( 2,3 )$ and $( 3,1 )$, and $I _ { 1 } = \int _ { \mathrm { a } } ^ { \mathrm { b } } \mathrm { x } \sin \left( 4 \mathrm { x } - \mathrm { x } ^ { 2 } \right) \mathrm { dx } , \mathrm { I } _ { 2 } = \int _ { \mathrm { a } } ^ { \mathrm { b } } \sin \left( 4 \mathrm { x } - \mathrm { x } ^ { 2 } \right) \mathrm { dx }$, then $36 \frac { I _ { 1 } } { I _ { 2 } }$ is equal to:
(1) 72
(2) 88
(3) 80
(4) 66
Q77 First order differential equations (integrating factor) View
Let $x = x ( t )$ and $y = y ( t )$ be solutions of the differential equations $\frac { \mathrm { dx } } { \mathrm { dt } } + \mathrm { ax } = 0$ and $\frac { \mathrm { dy } } { \mathrm { dt } } + $ by $= 0$ respectively, $\mathrm { a } , \mathrm { b } \in \mathrm { R }$. Given that $x ( 0 ) = 2 ; y ( 0 ) = 1$ and $3 y ( 1 ) = 2 x ( 1 )$, the value of $t$, for which $x ( t ) = y ( t )$, is:
(1) $\log _ { \frac { 2 } { 3 } } 2$
(2) $\log _ { 4 } 3$
(3) $\log _ { 3 } 4$
(4) $\log _ { \frac { 4 } { 3 } } 2$
Q78 Vectors: Cross Product & Distances View
If $\vec { a } = \hat { i } + 2 \hat { j } + \hat { k } , \vec { b } = 3 ( \hat { i } - \hat { j } + \hat { k } )$ and $\overrightarrow { \mathrm { c } }$ be the vector such that $\vec { a } \times \vec { c } = \vec { b }$ and $\vec { a } \cdot \vec { c } = 3$, then $\vec { a } \cdot ( ( \vec { c } \times \vec { b } ) - \vec { b } - \vec { c } )$ is equal to
(1) 32
(2) 24
(3) 20
(4) 36
Q79 Vectors: Lines & Planes Distance Computation (Point-to-Plane or Line-to-Line) View
The distance, of the point $( 7 , - 2,11 )$ from the line $\frac { x - 6 } { 1 } = \frac { y - 4 } { 0 } = \frac { z - 8 } { 3 }$ along the line $\frac { x - 5 } { 2 } = \frac { y - 1 } { - 3 } = \frac { z - 5 } { 6 }$, is:
(1) 12
(2) 14
(3) 18
(4) 21
Q80 Vectors: Lines & Planes Distance Computation (Point-to-Plane or Line-to-Line) View
If the shortest distance between the lines $\frac { x - 4 } { 1 } = \frac { y + 1 } { 2 } = \frac { z } { - 3 }$ and $\frac { x - \lambda } { 2 } = \frac { y + 1 } { 4 } = \frac { z - 2 } { - 5 }$ is $\frac { 6 } { \sqrt { 5 } }$, then the sum of all possible values of $\lambda$ is:
(1) 5
(2) 8
(3) 7
(4) 10
Q81 Generalised Binomial Theorem View
If $\alpha$ satisfies the equation $x ^ { 2 } + x + 1 = 0$ and $( 1 + \alpha ) ^ { 7 } = \mathrm { A } + \mathrm { B } \alpha + \mathrm { C } \alpha ^ { 2 } , \mathrm {~A} , \mathrm {~B} , \mathrm { C } \geq 0$, then $5 ( 3 \mathrm {~A} - 2 \mathrm {~B} - \mathrm { C } )$ is equal to
Q82 Arithmetic Sequences and Series Arithmetic-Geometric Hybrid Problem View
If $8 = 3 + \frac { 1 } { 4 } ( 3 + p ) + \frac { 1 } { 4 ^ { 2 } } ( 3 + 2 p ) + \frac { 1 } { 4 ^ { 3 } } ( 3 + 3 p ) + \ldots \infty$, then the value of $p$ is
Q83 Standard trigonometric equations Solve trigonometric equation for solutions in an interval View
Let the set of all $a \in R$ such that the equation $\cos 2 x + a \sin x = 2 a - 7$ has a solution be $[ p , q ]$ and $r = \tan 9 ^ { \circ } - \tan 27 ^ { \circ } - \frac { 1 } { \cot 63 ^ { \circ } } + \tan 81 ^ { \circ }$, then $p q r$ is equal to $\_\_\_\_$.
Q84 Matrices Linear System and Inverse Existence View
Let $\mathrm { A } = \left[ \begin{array} { l l l } 2 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{array} \right] , \mathrm { B } = \left[ \begin{array} { l l l } \mathrm { B } _ { 1 } & \mathrm {~B} _ { 2 } & \mathrm {~B} _ { 3 } \end{array} \right]$, where $\mathrm { B } _ { 1 } , \mathrm {~B} _ { 2 } , \mathrm {~B} _ { 3 }$ are column matrices, and $\mathrm { AB } _ { 1 } = \left[ \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right]$, $\mathrm { AB } _ { 2 } = \left[ \begin{array} { l } 2 \\ 3 \\ 0 \end{array} \right] , \mathrm { AB } _ { 3 } = \left[ \begin{array} { l } 3 \\ 2 \\ 1 \end{array} \right]$ If $\alpha = | \mathrm { B } |$ and $\beta$ is the sum of all the diagonal elements of B, then $\alpha ^ { 3 } + \beta ^ { 3 }$ is equal to
Q85 Chain Rule Higher-Order Derivatives of Products/Compositions View
Let $f ( x ) = x ^ { 3 } + x ^ { 2 } f ^ { \prime } ( 1 ) + x f ^ { \prime \prime } ( 2 ) + f ^ { \prime \prime \prime } ( 3 ) , x \in R$. Then $f ^ { \prime } ( 10 )$ is equal to
Q86 Chain Rule Straightforward Polynomial or Basic Differentiation View
Let for a differentiable function $f : ( 0 , \infty ) \rightarrow R , f ( x ) - f ( y ) \geq \log _ { e } \left( \frac { x } { y } \right) + x - y , \forall x , y \in ( 0 , \infty )$. Then $\sum _ { n = 1 } ^ { 20 } f ^ { \prime } \left( \frac { 1 } { n ^ { 2 } } \right)$ is equal to
Q87 Areas by integration View
Let the area of the region $\left\{ ( x , y ) : x - 2 y + 4 \geq 0 , x + 2 y ^ { 2 } \geq 0 , x + 4 y ^ { 2 } \leq 8 , y \geq 0 \right\}$ be $\frac { m } { n }$, where $m$ and $n$ are coprime numbers. Then $\mathrm { m } + \mathrm { n }$ is equal to $\_\_\_\_$.
Q88 Differential equations Solving Separable DEs with Initial Conditions View
If the solution of the differential equation $( 2 x + 3 y - 2 ) d x + ( 4 x + 6 y - 7 ) d y = 0 , y ( 0 ) = 3$, is $\alpha x + \beta y + 3 \log _ { e } | 2 x + 3 y - \gamma | = 6$, then $\alpha + 2 \beta + 3 \gamma$ is equal to $\_\_\_\_$.
Q89 Vectors Introduction & 2D Angle or Cosine Between Vectors View
The least positive integral value of $\alpha$, for which the angle between the vectors $\alpha \hat { \mathrm { i } } - 2 \hat { \mathrm { j } } + 2 \widehat { \mathrm { k } }$ and $\alpha \hat { \mathrm { i } } + 2 \alpha \hat { \mathrm { j } } - 2 \widehat { \mathrm { k } }$ is acute, is $\_\_\_\_$.
Q90 Discrete Probability Distributions Properties of Named Discrete Distributions (Non-Binomial) View
A fair die is tossed repeatedly until a six is obtained. Let $X$ denote the number of tosses required and let $\mathrm { a } = \mathrm { P } ( \mathrm { X } = 3 ) , \mathrm { b } = \mathrm { P } ( \mathrm { X } \geq 3 )$ and $\mathrm { c } = \mathrm { P } ( \mathrm { X } \geq 6 \mid \mathrm { X } > 3 )$. Then $\frac { \mathrm { b } + \mathrm { c } } { \mathrm { a } }$ is equal to