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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
03apr 28 09apr 29 10apr 30
2015
04apr 29 10apr 29 11apr 8
2014
06apr 28 09apr 28 11apr 4 12apr 5 19apr 29
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_03apr_shift1

35 maths questions

Q1 Vectors Introduction & 2D Angle or Cosine Between Vectors View
Q1. The angle between vector $\vec { Q }$ and the resultant of $( 2 \vec { Q } + 2 \vec { P } )$ and $( 2 \vec { Q } - 2 \vec { P } )$ is :
(1) $\tan ^ { - 1 } \frac { ( 2 \vec { Q } - 2 \vec { P } ) } { 2 \vec { Q } + 2 \vec { P } }$
(2) $0 ^ { \circ }$
(3) $\tan ^ { - 1 } ( \mathrm { P } / \mathrm { Q } )$
(4) $\tan ^ { - 1 } ( 2 Q / P )$
Q4 Forces, equilibrium and resultants View
Q4. A wooden block of mass 5 kg rests on a soft horizontal floor. When an iron cylinder of mass 25 kg is placed on the top of the block, the floor yields and the block and the cylinder together go down with an acceleration of $0.1 \mathrm {~ms} ^ { - 2 }$. The action force of the system on the floor is equal to:
(1) 196 N
(2) 291 N
(3) 294 N
(4) 297 N
Q5 Work done and energy Work done by gravity in specific scenarios View
Q5. A body of mass 50 kg is lifted to a height of 20 m from the ground in the two different ways as shown in the figures. The ratio of work done against the gravity in both the respective cases, will be :
[Figure]
(1) $1 : 2$
(3) $2 : 1$
[Figure]
Case $- 2 \rightarrow$ Along the ramp
(2) $\sqrt { 3 } : 2$
(4) $1 : 1$
Q21 Constant acceleration (SUVAT) Distance in successive equal time intervals View
Q21. A body moves on a frictionless plane starting from rest. If $S _ { n }$ is distance moved between $t = n - 1$ and $\mathrm { t } = \mathrm { n }$ and $\mathrm { S } _ { \mathrm { n } - 1 }$ is distance moved between $\mathrm { t } = \mathrm { n } - 2$ and $\mathrm { t } = \mathrm { n } - 1$, then the ratio $\frac { \mathrm { S } _ { \mathrm { n } - 1 } } { \mathrm {~S} _ { \mathrm { n } } }$ is $\left( 1 - \frac { 2 } { x } \right)$ for $\mathrm { n } = 10$. The value of $x$ is $\_\_\_\_$ .
Q22 Newton's laws and connected particles Tension in strings connecting blocks in linear arrangement View
Q22. Three blocks $M _ { 1 } , M _ { 2 } , M _ { 3 }$ having masses $4 \mathrm {~kg} , 6 \mathrm {~kg}$ and 10 kg respectively are hanging from a smooth pully using rope 1,2 and 3 as shown in figure. The tension in the rope $1 , T _ { 1 }$ when they are moving upward with [Figure] acceleration of $2 \mathrm {~ms} ^ { - 2 }$ is $\_\_\_\_$ $\mathrm { N } \left( \right.$ if $\left. \mathrm { g } = 10 \mathrm {~m} / \mathrm { s } ^ { 2 } \right)$.
Q61 Complex Numbers Arithmetic True/False or Property Verification Statements View
Q61. Consider the following two statements : Statement I : For any two non-zero complex numbers $z _ { 1 } , z _ { 2 }$, $\left( \left| z _ { 1 } \right| + \left| z _ { 2 } \right| \right) \left| \frac { z _ { 1 } } { \left| z _ { 1 } \right| } + \frac { z _ { 2 } } { \left| z _ { 2 } \right| } \right| \leq 2 \left( \left| z _ { 1 } \right| + \left| z _ { 2 } \right| \right)$, and Statement II : If $x , y , z$ are three distinct complex numbers and $\mathrm { a } , \mathrm { b } , \mathrm { c }$ are three positive real numbers such that $\frac { \mathrm { a } } { | y - z | } = \frac { \mathrm { b } } { | z - x | } = \frac { \mathrm { c } } { | x - y | }$, then $\frac { \mathrm { a } ^ { 2 } } { y - z } + \frac { \mathrm { b } ^ { 2 } } { z - x } + \frac { \mathrm { c } ^ { 2 } } { x - y } = 1$. Between the above two statements,
(1) Statement I is correct but Statement II is
(2) both Statement I and Statement II are correct. incorrect.
(3) both Statement I and Statement II are incorrect.
(4) Statement I is incorrect but Statement II is correct.
Q62 Arithmetic Sequences and Series Telescoping or Non-Standard Summation Involving an AP View
Q62. If $\frac { 1 } { \sqrt { 1 } + \sqrt { 2 } } + \frac { 1 } { \sqrt { 2 } + \sqrt { 3 } } + \ldots + \frac { 1 } { \sqrt { 99 } + \sqrt { 100 } } = m$ and $\frac { 1 } { 1 \cdot 2 } + \frac { 1 } { 2 \cdot 3 } + \ldots + \frac { 1 } { 99 \cdot 100 } = n$, then the point $( \mathrm { m } , \mathrm { n } )$ lies on the line
(1) $11 ( x - 1 ) - 100 ( y - 2 ) = 0$
(2) $11 x - 100 y = 0$
(3) $11 ( x - 2 ) - 100 ( y - 1 ) = 0$
(4) $11 ( x - 1 ) - 100 y = 0$
Q63 Trig Proofs Trigonometric Equation Constraint Deduction View
Q63. Suppose $\theta \epsilon \left[ 0 , \frac { \pi } { 4 } \right]$ is a solution of $4 \cos \theta - 3 \sin \theta = 1$. Then $\cos \theta$ is equal to :
(1) $\frac { 4 } { ( 3 \sqrt { 6 } + 2 ) }$
(2) $\frac { 6 + \sqrt { 6 } } { ( 3 \sqrt { 6 } + 2 ) }$
(3) $\frac { 4 } { ( 3 \sqrt { 6 } - 2 ) }$
(4) $\frac { 6 - \sqrt { 6 } } { ( 3 \sqrt { 6 } - 2 ) }$
Q64 Straight Lines & Coordinate Geometry Triangle Properties and Special Points View
Q64. Let two straight lines drawn from the origin O intersect the line $3 x + 4 y = 12$ at the points P and Q such that $\triangle \mathrm { OPQ }$ is an isosceles triangle and $\angle \mathrm { POQ } = 90 ^ { \circ }$. If $l = \mathrm { OP } ^ { 2 } + \mathrm { PQ } ^ { 2 } + \mathrm { QO } ^ { 2 }$, then the greatest integer less than or equal to $l$ is :
(1) 42
(2) 46
(3) 44
(4) 48
Q65 Vectors 3D & Lines Vector Algebra and Triple Product Computation View
Q65. If $\mathrm { A } ( 1 , - 1,2 ) , \mathrm { B } ( 5,7 , - 6 ) , \mathrm { C } ( 3,4 , - 10 )$ and $\mathrm { D } ( - 1 , - 4 , - 2 )$ are the vertices of a quadrilateral $A B C D$, then its area is :
(1) $48 \sqrt { 7 }$
(2) $12 \sqrt { 29 }$
(3) $24 \sqrt { 7 }$
(4) $24 \sqrt { 29 }$
Q66 Circles Circle Equation Derivation View
Q66. Let a circle $C$ of radius 1 and closer to the origin be such that the lines passing through the point $( 3,2 )$ and parallel to the coordinate axes touch it. Then the shortest distance of the circle C from the point $( 5,5 )$ is :
(1) $2 \sqrt { 2 }$
(2) $4 \sqrt { 2 }$
(3) 4
(4) 5
Q67 Circles Circle Equation Derivation View
Q67. Let the line $2 x + 3 y - \mathrm { k } = 0 , \mathrm { k } > 0$, intersect the $x$-axis and $y$-axis at the points A and B , respectively. If the equation of the circle having the line segment AB as a diameter is $x ^ { 2 } + y ^ { 2 } - 3 x - 2 y = 0$ and the length of the latus rectum of the ellipse $x ^ { 2 } + 9 y ^ { 2 } = k ^ { 2 }$ is $\frac { m } { n }$, where $m$ and $n$ are coprime, then $2 \mathrm {~m} + \mathrm { n }$ is equal to
(1) 11
(2) 10
(3) 12
(4) 13
Q68 3x3 Matrices Determinant of Parametric or Structured Matrix View
Q68. Let $A$ and $B$ be two square matrices of order 3 such that $| A | = 3$ and $| B | = 2$. Then $\left| \mathrm { A } ^ { \mathrm { T } } \mathrm { A } ( \operatorname { adj } ( 2 \mathrm {~A} ) ) ^ { - 1 } ( \operatorname { adj } ( 4 \mathrm {~B} ) ) ( \operatorname { adj } ( \mathrm { AB } ) ) ^ { - 1 } \mathrm { AA } ^ { \mathrm { T } } \right|$ is equal to :
(1) 108
(2) 32
(3) 81
(4) 64
Q69 Simultaneous equations View
Q69.
$$11 x + y + \lambda z = - 5$$
If the system of equations $2 x + 3 y + 5 z = 3$ has infinitely many solutions, then $\lambda ^ { 4 } - \mu$ is equal to :
$$8 x - 19 y - 39 z = \mu$$
(1) 51
(2) 45
(3) 47
(4) 49
Q70 Combinations & Selection Counting Functions or Mappings with Constraints View
Q70. Let $A = \{ 1,3,7,9,11 \}$ and $B = \{ 2,4,5,7,8,10,12 \}$. Then the total number of one-one maps $f : \mathrm { A } \rightarrow \mathrm { B }$, such that $f ( 1 ) + f ( 3 ) = 14$, is :
(1) 480
(2) 240
(3) 120
(4) 180
Q71 Composite & Inverse Functions Derivative of an Inverse Function View
Q71. Let $f ( x ) = x ^ { 5 } + 2 x ^ { 3 } + 3 x + 1 , x \in \mathbf { R }$, and $g ( x )$ be a function such that $g ( f ( x ) ) = x$ for all $x \in \mathbf { R }$. Then $\frac { g ( 7 ) } { g ^ { \prime } ( 7 ) }$ is equal to :
(1) 14
(2) 42
(3) 7
(4) 1
Q72 Chain Rule Continuity Conditions via Composition View
Q72. If the function $f ( x ) = \frac { \sin 3 x + \alpha \sin x - \beta \cos 3 x } { x ^ { 3 } } , x \in \mathbf { R }$, is continuous at $x = 0$, then $f ( 0 )$ is equal to :
(1) 2
(2) - 2
(3) 4
(4) - 4
Q73 Stationary points and optimisation Geometric or applied optimisation problem View
Q73. Let a rectangle $A B C D$ of sides 2 and 4 be inscribed in another rectangle $P Q R S$ such that the vertices of the rectangle $A B C D$ lie on the sides of the rectangle $P Q R S$. Let $a$ and $b$ be the sides of the rectangle $P Q R S$ when its area is maximum. Then $( a + b ) ^ { 2 }$ is equal to :
(1) 72
(2) 60
(3) 64
(4) 80
Q74 Stationary points and optimisation Determine intervals of increase/decrease or monotonicity conditions View
Q74. For the function $f ( x ) = \sin x + 3 x - \frac { 2 } { \pi } \left( x ^ { 2 } + x \right)$, where $x \in \left[ 0 , \frac { \pi } { 2 } \right]$, consider the following two statements : (I) f is increasing in ( $0 , \frac { \pi } { 2 }$ ). (II) $f ^ { \prime }$ is decreasing in ( $0 , \frac { \pi } { 2 }$ ).
Between the above two statements,
(1) only (II) is true.
(2) only (I) is true.
(3) neither (I) nor (II) is true.
(4) both (I) and (II) are true
Q75 Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution View
Q75. The value of $\int _ { - \pi } ^ { \pi } \frac { 2 y ( 1 + \sin y ) } { 1 + \cos ^ { 2 } y } d y$ is :
(1) $2 \pi ^ { 2 }$
(2) $\frac { \pi ^ { 2 } } { 2 }$
(3) $\frac { \pi } { 2 }$
(4) $\pi ^ { 2 }$
Q76 Integration by Substitution Substitution to Evaluate a Definite Integral (Numerical Answer) View
Q76. The integral $\int _ { 0 } ^ { \pi / 4 } \frac { 136 \sin x } { 3 \sin x + 5 \cos x } d x$ is equal to :
(1) $3 \pi - 50 \log _ { e } 2 + 20 \log _ { e } 5$
(2) $3 \pi - 25 \log _ { e } 2 + 10 \log _ { e } 5$
(3) $3 \pi - 10 \log _ { e } ( 2 \sqrt { 2 } ) + 10 \log _ { e } 5$
(4) $3 \pi - 30 \log _ { e } 2 + 20 \log _ { e } 5$
Q77 First order differential equations (integrating factor) View
Q77. If $y = y ( x )$ is the solution of the differential equation $\frac { \mathrm { d } y } { \mathrm {~d} x } + 2 y = \sin ( 2 x ) , y ( 0 ) = \frac { 3 } { 4 }$, then $y \left( \frac { \pi } { 8 } \right)$ is equal to:
(1) $\mathrm { e } ^ { \pi / 8 }$
(2) $e ^ { \pi / 4 }$
(3) $e ^ { - \pi / 4 }$
(4) $e ^ { - \pi / 8 }$
Q78 Vectors 3D & Lines MCQ: Relationship Between Two Lines View
Q78. If the line $\frac { 2 - x } { 3 } = \frac { 3 y - 2 } { 4 \lambda + 1 } = 4 - z$ makes a right angle with the line $\frac { x + 3 } { 3 \mu } = \frac { 1 - 2 y } { 6 } = \frac { 5 - z } { 7 }$, then $4 \lambda + 9 \mu$ is equal to :
(1) 4
(2) 13
(3) 5
(4) 6
Q79 Vectors 3D & Lines Line-Plane Intersection View
Q79. Let d be the distance of the point of intersection of the lines $\frac { x + 6 } { 3 } = \frac { y } { 2 } = \frac { z + 1 } { 1 }$ and $\frac { x - 7 } { 4 } = \frac { y - 9 } { 3 } = \frac { z - 4 } { 2 }$ from the point $( 7,8,9 )$. Then $\mathrm { d } ^ { 2 } + 6$ is equal to :
(1) 69
(2) 78
(3) 72
(4) 75
Q80 Discriminant and conditions for roots Probability involving discriminant conditions View
Q80. The coefficients $a , b , c$ in the quadratic equation $a x ^ { 2 } + b x + c = 0$ are chosen from the set $\{ 1,2,3,4,5,6,7,8 \}$ . The probability of this equation having repeated roots is :
(1) $\frac { 1 } { 128 }$
(2) $\frac { 1 } { 64 }$
(3) $\frac { 3 } { 256 }$
(4) $\frac { 3 } { 128 }$
Q81 Permutations & Arrangements Distribution of Objects into Bins/Groups View
Q81. The number of ways of getting a sum 16 on throwing a dice four times is $\_\_\_\_$
Q82 Arithmetic Sequences and Series Multi-Part Structured Problem on AP View
Q82. Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ be in an arithmetic progression of positive terms. Let $\mathrm { A } _ { \mathrm { k } } = \mathrm { a } _ { 1 } ^ { 2 } - \mathrm { a } _ { 2 } ^ { 2 } + \mathrm { a } _ { 3 } ^ { 2 } - \mathrm { a } _ { 4 } ^ { 2 } + \ldots + \mathrm { a } _ { 2 \mathrm { k } - 1 } ^ { 2 } - \mathrm { a } _ { 2 \mathrm { k } } ^ { 2 }$. If $\mathrm { A } _ { 3 } = - 153 , \mathrm {~A} _ { 5 } = - 435$ and $\mathrm { a } _ { 1 } ^ { 2 } + \mathrm { a } _ { 2 } ^ { 2 } + \mathrm { a } _ { 3 } ^ { 2 } = 66$, then $\mathrm { a } _ { 17 } - \mathrm { A } _ { 7 }$ is equal to $\_\_\_\_$
Q83 Generalised Binomial Theorem View
Q83. If the constant term in the expansion of $\left( 1 + 2 x - 3 x ^ { 3 } \right) \left( \frac { 3 } { 2 } x ^ { 2 } - \frac { 1 } { 3 x } \right) ^ { 9 }$ is p , then 108 p is equal to
Q84 Circles Optimization on a Circle View
Q84. Suppose $A B$ is a focal chord of the parabola $y ^ { 2 } = 12 x$ of length $l$ and slope $\mathrm { m } < \sqrt { 3 }$. If the distance of the chord AB from the origin is d , then $l \mathrm {~d} ^ { 2 }$ is equal to $\_\_\_\_$
Q85 Chain Rule Limit Involving Derivative Definition of Composed Functions View
Q85. Let $f$ be a differentiable function in the interval $( 0 , \infty )$ such that $f ( 1 ) = 1$ and $\lim _ { t \rightarrow x } \frac { t ^ { 2 } f ( x ) - x ^ { 2 } f ( t ) } { t - x } = 1$ for each $x > 0$. Then $2 f ( 2 ) + 3 f ( 3 )$ is equal to $\_\_\_\_$
Q86 Hypergeometric Distribution View
Q86. From a lot of 10 items, which include 3 defective items, a sample of 5 items is drawn at random. Let the random variable X denote the number of defective items in the sample. If the variance of X is $\sigma ^ { 2 }$, then $96 \sigma ^ { 2 }$ is equal to $\_\_\_\_$
Q87 Modulus function Counting solutions satisfying modulus conditions View
Q87. The number of distinct real roots of the equation $| x | | x + 2 | - 5 | x + 1 | - 1 = 0$ is $\_\_\_\_$
Q88 Sign Change & Interval Methods View
Q88. If $S = \{ a \in \mathbf { R } : | 2 a - 1 | = 3 [ a ] + 2 \{ a \} \}$, where $[ t ]$ denotes the greatest integer less than or equal to $t$ and $\{ t \}$ represents the fractional part of $t$, then $72 \sum _ { a \in S } a$ is equal to $\_\_\_\_$
Q89 Areas Between Curves Compute Area Directly (Numerical Answer) View
Q89. The area of the region enclosed by the parabolas $y = x ^ { 2 } - 5 x$ and $y = 7 x - x ^ { 2 }$ is
Q90 Vectors: Cross Product & Distances View
Q90. Let $\overrightarrow { \mathrm { a } } = \hat { i } - 3 \hat { j } + 7 \hat { k } , \overrightarrow { \mathrm {~b} } = 2 \hat { i } - \hat { j } + \hat { k }$ and $\overrightarrow { \mathrm { c } }$ be a vector such that $\overrightarrow { \mathrm { a } } + 2 \overrightarrow { \mathrm {~b} } ) \times \overrightarrow { \mathrm { c } } = 3 ( \overrightarrow { \mathrm { c } } \times \overrightarrow { \mathrm { a } } )$. If $\vec { a } \cdot \vec { c } = 130$, then $\vec { b } \cdot \vec { c }$ is equal to $\_\_\_\_$
ANSWER KEYS

1. (2)2. (3)
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17. (2)18. (4)
25. (4)26. (5)
33. (4)34. (3)
41. (1)42. (2)
49. (4)50. (2)
57. (5)58. (2)
65. (2)66. (3)
73. (1)74. (4)
81. (125)82. (910)
89. (72)90. (30)

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