jee-main

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
session1_24feb_shift1 9 session1_24feb_shift2 4 session1_25feb_shift1 29 session1_25feb_shift2 29 session1_26feb_shift2 15 session2_16mar_shift1 29 session2_16mar_shift2 18 session2_17mar_shift1 21 session2_17mar_shift2 27 session2_18mar_shift1 18 session2_18mar_shift2 9 session3_20jul_shift1 29 session3_20jul_shift2 29 session3_22jul_shift1 9 session3_25jul_shift1 8 session3_25jul_shift2 14 session3_27jul_shift1 4 session3_27jul_shift2 7 session4_01sep_shift2 14 session4_26aug_shift1 5 session4_26aug_shift2 2 session4_27aug_shift1 3 session4_27aug_shift2 29 session4_31aug_shift1 28 session4_31aug_shift2 4
2020
session1_07jan_shift1 28 session1_07jan_shift2 20 session1_08jan_shift1 5 session1_08jan_shift2 11 session1_09jan_shift1 26 session1_09jan_shift2 16 session2_02sep_shift1 18 session2_02sep_shift2 16 session2_03sep_shift1 23 session2_03sep_shift2 8 session2_04sep_shift1 14 session2_04sep_shift2 27 session2_05sep_shift1 22 session2_05sep_shift2 29 session2_06sep_shift1 11 session2_06sep_shift2 10
2019
session1_09jan_shift1 6 session1_09jan_shift2 29 session1_10jan_shift1 29 session1_10jan_shift2 14 session1_11jan_shift1 6 session1_11jan_shift2 5 session1_12jan_shift1 10 session1_12jan_shift2 29 session2_08apr_shift1 29 session2_08apr_shift2 29 session2_09apr_shift1 29 session2_09apr_shift2 29 session2_10apr_shift1 2 session2_10apr_shift2 5 session2_12apr_shift1 3 session2_12apr_shift2 9
2018
08apr 30 15apr 28 15apr_shift1 28 15apr_shift2 6 16apr 19
2017
02apr 30 08apr 30 09apr 34
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
07may 17 12may 21 19may 14 26may 17 offline 30
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
2017 08apr

30 maths questions

Q61 Factor & Remainder Theorem Polynomial Construction from Root/Value Conditions View
Let $p ( x )$ be a quadratic polynomial such that $p ( 0 ) = 1$. If $p ( x )$ leaves remainder 4 when divided by $x - 1$ and it leaves remainder 6 when divided by $x + 1$ then:
(1) $p ( - 2 ) = 19$
(2) $p ( 2 ) = 19$
(3) $p ( - 2 ) = 11$
(4) $p ( 2 ) = 11$
Q62 Complex Numbers Argand & Loci Circle Equation and Properties via Complex Number Manipulation View
Let $z \in C$, the set of complex numbers. Then the equation, $2 | z + 3 i | - | z - i | = 0$ represents:
(1) A circle with radius $\frac { 8 } { 3 }$
(2) An ellipse with length of minor axis $\frac { 16 } { 9 }$
(3) An ellipse with length of major axis $\frac { 16 } { 3 }$
(4) A circle with diameter $\frac { 10 } { 3 }$
Q63 Permutations & Arrangements Dictionary Order / Rank of a Permutation View
If all the words, with or without meaning, are written using the letters of the word QUEEN and are arranged as in English dictionary, then the position of the word QUEEN is:
(1) $47 ^ { t h }$
(2) $45 ^ { t h }$
(3) $46 ^ { t h }$
(4) $44 ^ { \text {th } }$
Q64 Arithmetic Sequences and Series Arithmetic-Geometric Hybrid Problem View
If the arithmetic mean of two numbers $a$ and $b , a > b > 0$, is five times their geometric mean, then $\frac { a + b } { a - b }$ is equal to:
(1) $\frac { 7 \sqrt { 3 } } { 12 }$
(2) $\frac { 3 \sqrt { 2 } } { 4 }$
(3) $\frac { \sqrt { 6 } } { 2 }$
(4) $\frac { 5 \sqrt { 6 } } { 12 }$
Q65 Arithmetic Sequences and Series Compute Partial Sum of an Arithmetic Sequence View
If the sum of the first $n$ terms of the series $\sqrt { 3 } + \sqrt { 75 } + \sqrt { 243 } + \sqrt { 507 } + \ldots$ is $435 \sqrt { 3 }$, then $n$ equals:
(1) 13
(2) 15
(3) 29
(4) 18
Q66 Number Theory Modular Arithmetic Computation View
If $(27)^{999}$ is divided by 7, then the remainder is
(1) 3
(2) 1
(3) 6
(4) 2
Q67 Circles Circle-Related Locus Problems View
The locus of the point of intersection of the straight lines, $t x - 2 y - 3 t = 0$ and $x - 2 t y + 3 = 0 ( t \in R )$, is:
(1) A hyperbola with the length of conjugate axis 3
(2) A hyperbola with eccentricity $\sqrt { 5 }$
(3) An ellipse with the length of major axis 6
(4) An ellipse with eccentricity $\frac { 2 } { \sqrt { 5 } }$
Q68 Circles Chord Length and Chord Properties View
If two parallel chords of a circle, having diameter 4 units, lie on the opposite sides of the center and subtend angles $\cos ^ { - 1 } \left( \frac { 1 } { 7 } \right)$ and $\sec ^ { - 1 } ( 7 )$ at the center respectively, then the distance between these chords is:
(1) $\frac { 8 } { \sqrt { 7 } }$
(2) $\frac { 16 } { 7 }$
(3) $\frac { 4 } { \sqrt { 7 } }$
(4) $\frac { 8 } { 7 }$
Q69 Circles Tangent Lines and Tangent Lengths View
If the common tangents to the parabola, $x ^ { 2 } = 4 y$ and the circle, $x ^ { 2 } + y ^ { 2 } = 4$ intersect at the point $P$, then the distance of $P$ from the origin (units), is:
(1) $2 ( 3 + 2 \sqrt { 2 } )$
(2) $3 + 2 \sqrt { 2 }$
(3) $\sqrt { 2 } + 1$
(4) $2 ( \sqrt { 2 } + 1 )$
Q70 Circles Optimization on a Circle View
If a point $P ( 0 , - 2 )$ and $Q$ is any point on the circle, $x ^ { 2 } + y ^ { 2 } - 5 x - y + 5 = 0$, then the maximum value of $( P Q ) ^ { 2 }$ is
(1) $8 + 5 \sqrt { 3 }$
(2) $\frac { 47 + 10 \sqrt { 6 } } { 2 }$
(3) $14 + 5 \sqrt { 3 }$
(4) $\frac { 25 + \sqrt { 6 } } { 2 }$
Q71 Circles Area and Geometric Measurement Involving Circles View
Consider an ellipse, whose center is at the origin and its major axis is along the $x$-axis. If its eccentricity is $\frac { 3 } { 5 }$ and the distance between its foci is 6, then the area (in sq. units) of the quadrilateral inscribed in the ellipse, with the vertices as the vertices of the ellipse, is:
(1) 32
(2) 80
(3) 40
(4) 8
Q72 Chain Rule Limit Computation from Algebraic Expressions View
$\lim _ { x \rightarrow 3 } \frac { \sqrt { 3 x } - 3 } { \sqrt { 2 x - 4 } - \sqrt { 2 } }$ is equal to
(1) $\frac { 1 } { \sqrt { 2 } }$
(2) $\frac { 1 } { 2 \sqrt { 2 } }$
(3) $\frac { \sqrt { 3 } } { 2 }$
(4) $\sqrt { 3 }$
Q73 Proof Proof of Equivalence or Logical Relationship Between Conditions View
The proposition $( \sim p ) \vee ( p \wedge \sim q )$ is equivalent to
(1) $p \rightarrow \sim q$
(2) $p \wedge \sim q$
(3) $q \rightarrow p$
(4) none
Q74 Measures of Location and Spread View
The mean age of 25 teachers in a school is 40 years. A teacher retires at the age of 60 years and a new teacher is appointed in his place. If the mean age of the teachers in this school now is 39 years, then the age (in years) of the newly appointed teacher is
(1) 35
(2) 40
(3) 25
(4) 30
Q75 3x3 Matrices True/False or Multiple-Select Conceptual Reasoning View
Let $A$ be any $3 \times 3$ invertible matrix. Then which one of the following is not always true?
(1) $\operatorname { adj } ( \operatorname { adj } ( \mathrm { A } ) ) = | A | ^ { 2 } \cdot ( \operatorname { adj } ( \mathrm {~A} ) ) ^ { - 1 }$
(2) $\operatorname { adj } ( \operatorname { adj } ( \mathrm { A } ) ) = | A | \cdot ( \operatorname { adj } ( \mathrm { A } ) ) ^ { - 1 }$
(3) $\operatorname { adj } ( \operatorname { adj } ( \mathrm { A } ) ) = | A | \cdot A$
(4) $\operatorname { adj } ( \mathrm { A } ) = | A | \cdot A ^ { - 1 }$
Q76 3x3 Matrices Linear System with Parameter — Infinite Solutions View
The number of real values of $\lambda$ for which the system of linear equations, $2 x + 4 y - \lambda z = 0, 4 x + \lambda y + 2 z = 0$ and $\lambda x + 2 y + 2 z = 0$, has infinitely many solutions, is:
(1) 3
(2) 1
(3) 2
(4) 0
Q77 3x3 Matrices Determinant and Rank Computation View
If $S = \left\{ x \in [ 0,2 \pi ] : \left| \begin{array} { c c c } 0 & \cos x & - \sin x \\ \sin x & 0 & \cos x \\ \cos x & \sin x & 0 \end{array} \right| = 0 \right\}$, then $\sum _ { x \in S } \tan \left( \frac { \pi } { 3 } + x \right)$ is equal to:
(1) $4 + 2 \sqrt { 3 }$
(2) $- 4 - 2 \sqrt { 3 }$
(3) $- 2 + \sqrt { 3 }$
(4) $- 2 - \sqrt { 3 }$
Q78 Composite & Inverse Functions Inverse trigonometric equation View
The value of $\tan ^ { - 1 } \left[ \frac { \sqrt { 1 + x ^ { 2 } } + \sqrt { 1 - x ^ { 2 } } } { \sqrt { 1 + x ^ { 2 } } - \sqrt { 1 - x ^ { 2 } } } \right] , | x | < \frac { 1 } { 2 } , x \neq 0$, is equal to:
(1) $\frac { \pi } { 4 } + \frac { 1 } { 2 } \cos ^ { - 1 } x ^ { 2 }$
(2) $\frac { \pi } { 4 } - \cos ^ { - 1 } x ^ { 2 }$
(3) $\frac { \pi } { 4 } - \frac { 1 } { 2 } \cos ^ { - 1 } x ^ { 2 }$
(4) $\frac { \pi } { 4 } + \cos ^ { - 1 } x ^ { 2 }$
Q79 Composite & Inverse Functions Recover a Function from a Composition or Functional Equation View
Let $f ( x ) = 2 ^ { 10 } x + 1$ and $g ( x ) = 3 ^ { 10 } x - 1$. If $( f \circ g ) ( x ) = x$, then $x$ is equal to:
(1) $\frac { 2 ^ { 10 } - 1 } { 2 ^ { 10 } - 3 ^ { - 10 } }$
(2) $\frac { 1 - 2 ^ { - 10 } } { 3 ^ { 10 } - 2 ^ { - 10 } }$
(3) $\frac { 3 ^ { 10 } - 1 } { 3 ^ { 10 } - 2 ^ { - 10 } }$
(4) $\frac { 1 - 3 ^ { - 10 } } { 2 ^ { 10 } - 3 ^ { - 10 } }$
Q80 Differentiating Transcendental Functions Finding parameter values from differentiability or equation constraints View
If $y = \left[ x + \sqrt { x ^ { 2 } - 1 } \right] ^ { 15 } + \left[ x - \sqrt { x ^ { 2 } - 1 } \right] ^ { 15 }$, then $\left( x ^ { 2 } - 1 \right) \frac { d ^ { 2 } y } { d x ^ { 2 } } + x \frac { d y } { d x }$ is equal to
(1) $224 y ^ { 2 }$
(2) $125 y$
(3) $225 y$
(4) $225 y ^ { 2 }$
Q81 Tangents, normals and gradients Geometric properties of tangent lines (intersections, lengths, areas) View
The tangent at the point $( 2 , - 2 )$ to the curve, $x ^ { 2 } y ^ { 2 } - 2 x = 4 ( 1 - y )$ does not pass through the point:
(1) $( - 2 , - 7 )$
(2) $( 8,5 )$
(3) $( - 4 , - 9 )$
(4) $\left( 4 , \frac { 1 } { 3 } \right)$
Q82 Standard Integrals and Reverse Chain Rule Definite Integral Evaluation via Substitution or Standard Forms View
The integral $\int \sqrt { 1 + 2 \cot x ( \operatorname { cosec } x + \cot x ) } d x , \left( 0 < x < \frac { \pi } { 2 } \right)$ is equal to
(1) $2 \log \left| \sin \frac { x } { 2 } \right| + c$
(2) $4 \log \left| \sin \frac { x } { 2 } \right| + c$
(3) $4 \log \left| \cos \frac { x } { 2 } \right| + c$
(4) $2 \log \left| \cos \frac { x } { 2 } \right| + c$
Q83 Standard Integrals and Reverse Chain Rule Definite Integral Evaluation (Computational) View
The integral $\int _ { \frac { \pi } { 12 } } ^ { \frac { \pi } { 4 } } \frac { 8 \cos 2 x } { ( \tan x + \cot x ) ^ { 3 } } d x$ equals
(1) $\frac { 13 } { 256 }$
(2) $\frac { 15 } { 64 }$
(3) $\frac { 13 } { 32 }$
(4) $\frac { 15 } { 128 }$
Q84 Areas by integration View
The area (in sq. units) of the smaller portion enclosed between the curves, $x ^ { 2 } + y ^ { 2 } = 4$ and $y ^ { 2 } = 3 x$, is:
(1) $\frac { 1 } { \sqrt { 3 } } + \frac { 4 \pi } { 3 }$
(2) $\frac { 1 } { \sqrt { 3 } } + \frac { 2 \pi } { 3 }$
(3) $\frac { 1 } { 2 \sqrt { 3 } } + \frac { \pi } { 3 }$
(4) $\frac { 1 } { 2 \sqrt { 3 } } + \frac { 2 \pi } { 3 }$
Q85 Differential equations First-Order Linear DE: General Solution View
The curve satisfying the differential equation, $y d x - \left( x + 3 y ^ { 2 } \right) d y = 0$ and passing through the point $( 1,1 )$ also passes through the point
(1) $\left( \frac { 1 } { 4 } , - \frac { 1 } { 2 } \right)$
(2) $\left( - \frac { 1 } { 3 } , \frac { 1 } { 3 } \right)$
(3) $\left( \frac { 1 } { 4 } , \frac { 1 } { 2 } \right)$
(4) $\left( \frac { 1 } { 3 } , - \frac { 1 } { 3 } \right)$
Q86 Vectors 3D & Lines Area Computation Using Vectors View
The area (in sq. units) of the parallelogram whose diagonals are along the vectors $8 \hat { \mathrm { i } } - 6 \hat { \mathrm { j } }$ and $3 \hat { \mathrm { i } } + 4 \hat { \mathrm { j } } - 12 \widehat { \mathrm { k } }$, is:
(1) 20
(2) 65
(3) 52
(4) 26
Q87 Vectors: Lines & Planes Perpendicular/Orthogonal Projection onto a Plane View
The coordinates of the foot of the perpendicular from the point $( 1 , - 2,1 )$ on the plane containing the lines $\frac { x + 1 } { 6 } = \frac { y - 1 } { 7 } = \frac { z - 3 } { 8 }$ and $\frac { x - 1 } { 3 } = \frac { y - 2 } { 5 } = \frac { z - 3 } { 7 }$, is:
(1) $( 2 , - 4,2 )$
(2) $( 1,1,1 )$
(3) $( 0,0,0 )$
(4) $( - 1,2 , - 1 )$
Q88 Vectors: Lines & Planes Find Parametric Representation of a Line View
The line of intersection of the planes $\vec { r } \cdot ( 3 \hat { i } - \hat { j } + \widehat { k } ) = 1$ and $\vec { r } \cdot ( \hat { i } + 4 \hat { j } - 2 \widehat { k } ) = 2$, is,
(1) $\frac { x - \frac { 6 } { 13 } } { 2 } = \frac { y - \frac { 5 } { 13 } } { 7 } = \frac { z } { - 13 }$
(2) $\frac { x - \frac { 4 } { 7 } } { 2 } = \frac { y } { - 7 } = \frac { z + \frac { 5 } { 7 } } { 13 }$
(3) $\frac { x - \frac { 6 } { 13 } } { 2 } = \frac { y - \frac { 5 } { 13 } } { - 7 } = \frac { z } { - 13 }$
(4) $\frac { x - \frac { 4 } { 7 } } { - 2 } = \frac { y } { 7 } = \frac { z - \frac { 5 } { 7 } } { 13 }$
Q89 Probability Definitions Probability Using Set/Event Algebra View
An unbiased coin is tossed eight times. The probability of obtaining at least one head and at least one tail is:
(1) $\frac { 127 } { 128 }$
(2) $\frac { 63 } { 64 }$
(3) $\frac { 255 } { 256 }$
(4) $\frac { 1 } { 2 }$
Q90 Independent Events View
Three persons $\mathrm { P } , \mathrm { Q }$ and R independently try to hit a target. If the probabilities of their hitting the target are $\frac { 3 } { 4 } , \frac { 1 } { 2 }$ and $\frac { 5 } { 8 }$ respectively, then the probability that the target is hit by P or Q but not by R is:
(1) $\frac { 39 } { 64 }$
(2) $\frac { 21 } { 64 }$
(3) $\frac { 9 } { 64 }$
(4) $\frac { 15 } { 64 }$