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Papers (191)
2026
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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
07apr 29 09apr 12 22apr 5 23apr 14 25apr 13
2012
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2011
jee-main_2011.pdf 18
2010
jee-main_2010.pdf 6
2009
jee-main_2009.pdf 2
2008
jee-main_2008.pdf 4
2007
jee-main_2007.pdf 38
2006
jee-main_2006.pdf 15
2005
jee-main_2005.pdf 25
2004
jee-main_2004.pdf 22
2003
jee-main_2003.pdf 8
2002
jee-main_2002.pdf 12
2025 session2_08apr_shift1

36 maths questions

Q2 Travel graphs View
Q2. A particle moving in a straight line covers half the distance with speed $6 \mathrm {~m} / \mathrm { s }$. The other half is covered in two equal time intervals with speeds $9 \mathrm {~m} / \mathrm { s }$ and $15 \mathrm {~m} / \mathrm { s }$ respectively. The average speed of the particle during the motion is :
(1) $10 \mathrm {~m} / \mathrm { s }$
(2) $8 \mathrm {~m} / \mathrm { s }$
(3) $9.2 \mathrm {~m} / \mathrm { s }$
(4) $8.8 \mathrm {~m} / \mathrm { s }$
Q3 Pulley systems View
Q3. A light unstretchable string passing over a smooth light pulley connects two blocks of masses $m _ { 1 }$ and $m _ { 2 }$. If the acceleration of the system is $\frac { g } { 8 }$, then the ratio of the masses $\frac { m _ { 2 } } { m _ { 1 } }$ is :
(1) $8 : 1$
(2) $5 : 3$
(3) $4 : 3$
(4) $9 : 7$
Q4 Work done and energy Work-energy theorem: finding speed or kinetic energy from net work View
Q4. A particle of mass $m$ moves on a straight line with its velocity increasing with distance according to the equation $v = \alpha \sqrt { x }$, where $\alpha$ is a constant. The total work done by all the forces applied on the particle during its displacement from $x = 0$ to $x = \mathrm { d }$, will be :
(1) $\frac { m } { 2 \alpha ^ { 2 } d }$
(2) $\frac { \mathrm { md } } { 2 \alpha ^ { 2 } }$
(3) $2 m \alpha ^ { 2 } d$
(4) $\frac { m \alpha ^ { 2 } d } { 2 }$
Q5 Moments View
Q5. A heavy iron bar, of weight $W$ is having its one end on the ground and the other on the shoulder of a person. The bar makes an angle $\theta$ with the horizontal. The weight experienced by the person is :
(1) $\mathrm { W } \cos \theta$
(2) $\frac { W } { 2 }$
(3) W
(4) $W \sin \theta$
Q6 Advanced work-energy problems View
Q6. An astronaut takes a ball of mass $m$ from earth to space. He throws the ball into a circular orbit about earth at an altitude of 318.5 km . From earth's surface to the orbit, the change in total mechanical energy of the ball is $x \frac { \mathrm { GM } _ { \mathrm { e } } \mathrm { m } } { 21 \mathrm { R } _ { \mathrm { e } } }$. The value of $x$ is (take $\mathrm { R } _ { \mathrm { e } } = 6370 \mathrm {~km}$ ) :
(1) 10
(2) 12
(3) 9
(4) 11
Q21 Vectors Introduction & 2D Perpendicularity or Parallel Condition View
Q21. If $\vec { a }$ and $\vec { b }$ makes an angle $\cos ^ { - 1 } \left( \frac { 5 } { 9 } \right)$ with each other, then $| \vec { a } + \vec { b } | = \sqrt { 2 } | \vec { a } - \vec { b } |$ for $| \vec { a } | = n | \vec { b } |$ The integer value of n is $\_\_\_\_$
Q22 Power and driving force View
Q22. A string is wrapped around the rim of a wheel of moment of inertia $0.40 \mathrm { kgm } ^ { 2 }$ and radius 10 cm . The wheel is free to rotate about its axis. Initially the wheel is at rest. The string is now pulled by a force of 40 N . The angular velocity of the wheel after 10 s is $x \mathrm { rad } / \mathrm { s }$, where $x$ is $\_\_\_\_$
Q61 Roots of polynomials Vieta's formulas: compute symmetric functions of roots View
Q61. Let $\alpha , \beta$ be the roots of the equation $x ^ { 2 } + 2 \sqrt { 2 } x - 1 = 0$. The quadratic equation, whose roots are $\alpha ^ { 4 } + \beta ^ { 4 }$ and $\frac { 1 } { 10 } \left( \alpha ^ { 6 } + \beta ^ { 6 } \right)$, is :
(1) $x ^ { 2 } - 190 x + 9466 = 0$
(2) $x ^ { 2 } - 180 x + 9506 = 0$
(3) $x ^ { 2 } - 195 x + 9506 = 0$
(4) $x ^ { 2 } - 195 x + 9466 = 0$
Q62 Arithmetic Sequences and Series Telescoping or Non-Standard Summation Involving an AP View
Q62. If the sum of the series $\frac { 1 } { 1 \cdot ( 1 + \mathrm { d } ) } + \frac { 1 } { ( 1 + \mathrm { d } ) ( 1 + 2 \mathrm {~d} ) } + \ldots + \frac { 1 } { ( 1 + 9 \mathrm {~d} ) ( 1 + 10 \mathrm {~d} ) }$ is equal to 5 , then 50 d is equal to :
(1) 10
(2) 5
(3) 15
(4) 20
Q63 Binomial Theorem (positive integer n) Find a Specific Coefficient in a Product of Binomial/Polynomial Expressions View
Q63. The coefficient of $x ^ { 70 }$ in $x ^ { 2 } ( 1 + x ) ^ { 98 } + x ^ { 3 } ( 1 + x ) ^ { 97 } + x ^ { 4 } ( 1 + x ) ^ { 96 } + \ldots + x ^ { 54 } ( 1 + x ) ^ { 46 }$ is ${ } ^ { 99 } \mathrm { C } _ { \mathrm { p } } - { } ^ { 46 } \mathrm { C } _ { \mathrm { q } }$. Then a possible value of $p + q$ is :
(1) 55
(2) 83
(3) 61
(4) 68
Q64 Standard trigonometric equations Solve trigonometric inequality View
Q64. Let $| \cos \theta \cos ( 60 - \theta ) \cos ( 60 + \theta ) | \leq \frac { 1 } { 8 } , \theta \epsilon [ 0,2 \pi ]$. Then, the sum of all $\theta \epsilon [ 0,2 \pi ]$, where $\cos 3 \theta$ attains its maximum value, is :
(1) $15 \pi$
(2) $18 \pi$
(3) $6 \pi$
(4) $9 \pi$
Q65 Straight Lines & Coordinate Geometry Reflection and Image in a Line View
Q65. A ray of light coming from the point $P ( 1,2 )$ gets reflected from the point $Q$ on the $x$-axis and then passes through the point $R ( 4,3 )$. If the point $S ( h , k )$ is such that PQRS is a parallelogram, then $h k ^ { 2 }$ is equal to :
(1) 70
(2) 80
(3) 60
(4) 90
Q66 Circles Circle Equation Derivation View
Q66. Let a circle passing through $( 2,0 )$ have its centre at the point $( h , k )$. Let $\left( x _ { c } , y _ { c } \right)$ be the point of intersection of the lines $3 x + 5 y = 1$ and $( 2 + c ) x + 5 c ^ { 2 } y = 1$. If $\mathrm { h } = \lim _ { \mathrm { c } \rightarrow 1 } x _ { \mathrm { c } }$ and $\mathrm { k } = \lim _ { \mathrm { c } \rightarrow 1 } y _ { \mathrm { c } }$, then the equation of the circle is :
(1) $25 x ^ { 2 } + 25 y ^ { 2 } - 2 x + 2 y - 60 = 0$
(2) $5 x ^ { 2 } + 5 y ^ { 2 } - 4 x + 2 y - 12 = 0$
(3) $5 x ^ { 2 } + 5 y ^ { 2 } - 4 x - 2 y - 12 = 0$
(4) $25 x ^ { 2 } + 25 y ^ { 2 } - 20 x + 2 y - 60 = 0$
Q67 Composite & Inverse Functions Evaluate Composition from Algebraic Definitions View
Q67. Let $f ( x ) = x ^ { 2 } + 9 , g ( x ) = \frac { x } { x - 9 }$ and $\mathrm { a } = f \circ g ( 10 ) , \mathrm { b } = g \circ f ( 3 )$. If e and $l$ denote the eccentricity and the length of the latus rectum of the ellipse $\frac { x ^ { 2 } } { a } + \frac { y ^ { 2 } } { b } = 1$, then $8 \mathrm { e } ^ { 2 } + l ^ { 2 }$ is equal to.
(1) 8
(2) 16
(3) 6
(4) 12
Q68 Measures of Location and Spread View
Q68. The frequency distribution of the age of students in a class of 40 students is given below.
Age151617181920
No of Students58512$x$$y$

If the mean deviation about the median is 1.25 , then $4 x + 5 y$ is equal to :
(1) 46
(2) 43
(3) 44
(4) 47
Q69 Simultaneous equations View
Q69. $3 x + 5 y + \lambda z = 3$ Let $\lambda , \mu \in \mathbf { R }$. If the system of equations $7 x + 11 y - 9 z = 2$ has infinitely many solutions, then $\mu + 2 \lambda$ is $97 x + 155 y - 189 z = \mu$ equal to :
(1) 24
(2) 25
(3) 22
(4) 27
Q70 Composite & Inverse Functions Determine Domain or Range of a Composite Function View
Q70. If the domain of the function $f ( x ) = \sin ^ { - 1 } \left( \frac { x - 1 } { 2 x + 3 } \right)$ is $\mathbf { R } - ( \alpha , \beta )$, then $12 \alpha \beta$ is equal to :
(1) 32
(2) 40
(3) 24
(4) 36
Q71 Chain Rule Straightforward Polynomial or Basic Differentiation View
Q71. Let $f ( x ) = a x ^ { 3 } + b x ^ { 2 } + c x + 41$ be such that $f ( 1 ) = 40 , f ^ { \prime } ( 1 ) = 2$ and $f ^ { \prime } ( 1 ) = 4$. Then $\mathrm { a } ^ { 2 } + \mathrm { b } ^ { 2 } + \mathrm { c } ^ { 2 }$ is equal to:
(1) 73
(2) 62
(3) 51
(4) 54
Q72 Stationary points and optimisation Geometric or applied optimisation problem View
Q72. A variable line $L$ passes through the point $( 3,5 )$ and intersects the positive coordinate axes at the points A and B . The minimum area of the triangle OAB , where O is the origin, is :
(1) 30
(2) 25
(3) 40
(4) 35
Q73 Integration by Substitution Substitution to Evaluate a Definite Integral (Numerical Answer) View
Q73. Let $\int \frac { 2 - \tan x } { 3 + \tan x } \mathrm {~d} x = \frac { 1 } { 2 } \left( \alpha x + \log _ { \mathrm { e } } | \beta \sin x + \gamma \cos x | \right) + C$, where $C$ is the constant of integration. Then $\alpha + \frac { \gamma } { \beta }$ is equal to :
(1) 7
(2) 4
(3) 1
(4) 3
Q74 Areas by integration View
Q74. The parabola $y ^ { 2 } = 4 x$ divides the area of the circle $x ^ { 2 } + y ^ { 2 } = 5$ in two parts. The area of the smaller part is equal to:
(1) $\frac { 1 } { 3 } + 5 \sin ^ { - 1 } \left( \frac { 2 } { \sqrt { 5 } } \right)$
(2) $\frac { 1 } { 3 } + \sqrt { 5 } \sin ^ { - 1 } \left( \frac { 2 } { \sqrt { 5 } } \right)$
(3) $\frac { 2 } { 3 } + 5 \sin ^ { - 1 } \left( \frac { 2 } { \sqrt { 5 } } \right)$
(4) $\frac { 2 } { 3 } + \sqrt { 5 } \sin ^ { - 1 } \left( \frac { 2 } { \sqrt { 5 } } \right)$
Q75 Differential equations Solving Separable DEs with Initial Conditions View
Q75. The solution curve, of the differential equation $2 y \frac { \mathrm {~d} y } { \mathrm {~d} x } + 3 = 5 \frac { \mathrm {~d} y } { \mathrm {~d} x }$, passing through the point $( 0,1 )$ is a conic, whose vertex lies on the line:
(1) $2 x + 3 y = 9$
(2) $2 x + 3 y = - 9$
(3) $2 x + 3 y = - 6$
(4) $2 x + 3 y = 6$
Q76 Differential equations Solving Separable DEs with Initial Conditions View
Q76. The solution of the differential equation $\left( x ^ { 2 } + y ^ { 2 } \right) \mathrm { d } x - 5 x y \mathrm {~d} y = 0 , y ( 1 ) = 0$, is :
(1) $\left| x ^ { 2 } - 2 y ^ { 2 } \right| ^ { 6 } = x$
(2) $\left| x ^ { 2 } - 4 y ^ { 2 } \right| ^ { 6 } = x$
(3) $\left| x ^ { 2 } - 4 y ^ { 2 } \right| ^ { 5 } = x ^ { 2 }$
(4) $\left| x ^ { 2 } - 2 y ^ { 2 } \right| ^ { 5 } = x ^ { 2 }$
Q77 Vectors 3D & Lines Vector Algebra and Triple Product Computation View
Q77. Let three vectors $\overrightarrow { \mathrm { a } } = \alpha \hat { i } + 4 \hat { j } + 2 \hat { k } , \overrightarrow { \mathrm {~b} } = 5 \hat { i } + 3 \hat { j } + 4 \hat { k } , \overrightarrow { \mathrm { c } } = x \hat { i } + y \hat { j } + z \hat { k }$ form a triangle such that $\vec { c } = \vec { a } - \vec { b }$ and the area of the triangle is $5 \sqrt { 6 }$. If $\alpha$ is a positive real number, then $| \vec { c } | ^ { 2 }$ is equal to:
(1) 16
(2) 14
(3) 12
(4) 10
Q78 Vectors 3D & Lines Vector Algebra and Triple Product Computation View
Q78. Let $\overrightarrow { O A } = 2 \vec { a } , \overrightarrow { O B } = 6 \vec { a } + 5 \vec { b }$ and $\overrightarrow { O C } = 3 \vec { b }$, where $O$ is the origin. If the area of the parallelogram with adjacent sides $\overrightarrow { \mathrm { OA } }$ and $\overrightarrow { \mathrm { OC } }$ is 15 sq. units, then the area (in sq. units) of the quadrilateral OABC is equal to :
(1) 32
(2) 40
(3) 38
(4) 35
Q79 Vectors: Lines & Planes Find Parametric Representation of a Line View
Q79. Let the line L intersect the lines $x - 2 = - y = z - 1,2 ( x + 1 ) = 2 ( y - 1 ) = z + 1$ and be parallel to the line $\frac { x - 2 } { 3 } = \frac { y - 1 } { 1 } = \frac { z - 2 } { 2 }$. Then which of the following points lies on L ?
(1) $\left( - \frac { 1 } { 3 } , 1 , - 1 \right)$
(2) $\left( - \frac { 1 } { 3 } , - 1,1 \right)$
(3) $\left( - \frac { 1 } { 3 } , 1,1 \right)$
(4) $\left( - \frac { 1 } { 3 } , - 1 , - 1 \right)$
Q80 Vectors: Lines & Planes Distance Computation (Point-to-Plane or Line-to-Line) View
Q80. The shortest distance between the lines $\frac { x - 3 } { 4 } = \frac { y + 7 } { - 11 } = \frac { z - 1 } { 5 }$ and $\frac { x - 5 } { 3 } = \frac { y - 9 } { - 6 } = \frac { z + 2 } { 1 }$ is:
(1) $\frac { 178 } { \sqrt { 563 } }$
(2) $\frac { 187 } { \sqrt { 563 } }$
(3) $\frac { 185 } { \sqrt { 563 } }$
(4) $\frac { 179 } { \sqrt { 563 } }$
Q81 Complex Numbers Arithmetic Modulus Computation View
Q81. The sum of the square of the modulus of the elements in the set $\{ z = \mathrm { a } + \mathrm { ib } : \mathrm { a } , \mathrm { b } \in \mathbf { Z } , z \in \mathbf { C } , | z - 1 | \leq 1 , | z - 5 | \leq | z - 5 \mathrm { i } | \}$ is
Q82 Number Theory Modular Arithmetic Computation View
Q82. The remainder when $428 ^ { 2024 }$ is divided by 21 is
Q83 Circles Circle Equation Derivation View
Q83. Let the centre of a circle, passing through the points $( 0,0 ) , ( 1,0 )$ and touching the circle $x ^ { 2 } + y ^ { 2 } = 9$, be $( h , k )$ - Then for all possible values of the coordinates of the centre $( h , k ) , 4 \left( h ^ { 2 } + k ^ { 2 } \right)$ is equal to
Q84 Sequences and series, recurrence and convergence Summation of sequence terms View
Q84. Let $\lim _ { n \rightarrow \infty } \left( \frac { n } { \sqrt { n ^ { 4 } + 1 } } - \frac { 2 n } { \left( n ^ { 2 } + 1 \right) \sqrt { n ^ { 4 } + 1 } } + \frac { n } { \sqrt { n ^ { 4 } + 16 } } - \frac { 8 n } { \left( n ^ { 2 } + 4 \right) \sqrt { n ^ { 4 } + 16 } } + \ldots + \frac { n } { \sqrt { n ^ { 4 } + n ^ { 4 } } } - \frac { 2 n \cdot n ^ { 2 } } { \left( n ^ { 2 } + n ^ { 2 } \right) \sqrt { n ^ { 4 } + n ^ { 4 } } } \right)$ be $\frac { \pi } { k }$, using only the principal values of the inverse trigonometric functions. Then $\mathrm { k } ^ { 2 }$ is equal to $\_\_\_\_$
Q85 Probability Definitions Combinatorial Counting (Non-Probability) View
Q85. Let $A = \{ 2,3,6,7 \}$ and $B = \{ 4,5,6,8 \}$. Let $R$ be a relation defined on $A \times B$ by ( $\left. a _ { 1 } , b _ { 1 } \right) R \left( a _ { 2 } , b _ { 2 } \right)$ if and only if $a _ { 1 } + a _ { 2 } = b _ { 1 } + b _ { 2 }$. Then the number of elements in $R$ is $\_\_\_\_$
Q86 3x3 Matrices Determinant of Parametric or Structured Matrix View
Q86. Let $A$ be a non-singular matrix of order 3 . If $\operatorname { det } ( 3 \operatorname { adj } ( 2 \operatorname { adj } ( ( \operatorname { det } A ) A ) ) ) = 3 ^ { - 13 } \cdot 2 ^ { - 10 }$ and $\operatorname { det } ( 3 \operatorname { adj } ( 2 \mathrm {~A} ) ) = 2 ^ { \mathrm { m } } \cdot 3 ^ { \mathrm { n } }$, then $| 3 \mathrm {~m} + 2 \mathrm { n } |$ is equal to
Q87 Arithmetic Sequences and Series Compute Partial Sum of an Arithmetic Sequence View
Q87. If a function $f$ satisfies $f ( \mathrm {~m} + \mathrm { n } ) = f ( \mathrm {~m} ) + f ( \mathrm { n } )$ for all $\mathrm { m } , \mathrm { n } \in \mathbf { N }$ and $f ( 1 ) = 1$, then the largest natural number $\lambda$ such that $\sum _ { k = 1 } ^ { 2022 } f ( \lambda + k ) \leq ( 2022 ) ^ { 2 }$ is equal to $\_\_\_\_$
Q88. Let $f : ( 0 , \pi ) \rightarrow \mathbf { R }$ be a function given by $f ( x ) = \left\{ \begin{array} { c c } \left( \frac { 8 } { 7 } \right) ^ { \frac { \tan 8 x } { \tan 7 x } } , & 0 < x < \frac { \pi } { 2 } \\ \mathrm { a } - 8 , & x = \frac { \pi } { 2 } \\ ( 1 + | \cot x | ) ^ { \mathrm { b } } | \tan x | , & \frac { \pi } { 2 } < x < \pi \end{array} \right.$ where $\mathrm { a } , \mathrm { b } \in \mathbf { Z }$. If $f$ is continuous at $x = \frac { \pi } { 2 }$, then $\mathrm { a } ^ { 2 } + \mathrm { b } ^ { 2 }$ is equal to
Q89 Stationary points and optimisation Determine parameters from given extremum conditions View
Q89. Let the set of all positive values of $\lambda$, for which the point of local minimum of the function $\left( 1 + x \left( \lambda ^ { 2 } - x ^ { 2 } \right) \right)$ satisfies $\frac { x ^ { 2 } + x + 2 } { x ^ { 2 } + 5 x + 6 } < 0$, be $( \alpha , \beta )$. Then $\alpha ^ { 2 } + \beta ^ { 2 }$ is equal to $\_\_\_\_$
Q90 Discriminant and conditions for roots Probability involving discriminant conditions View
Q90. Let $\mathrm { a } , \mathrm { b }$ and c denote the outcome of three independent rolls of a fair tetrahedral die, whose four faces are marked $1,2,3,4$. If the probability that $a x ^ { 2 } + b x + c = 0$ has all real roots is $\frac { m } { n } , \operatorname { gcd } ( \mathrm {~m} , \mathrm { n } ) = 1$, then $\mathrm { m } + \mathrm { n }$ is equal to $\_\_\_\_$
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$57 . ( 8 )$58. (2)
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81. (9)82. (1)
89. (39)90. (19)

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  62. (1)
  63. (1)
  64. (2)
  65. (1010)
  66. (81)