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
08apr 29 15apr 28 15apr_shift1 28 15apr_shift2 2 16apr 15
2017
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2016
03apr 30 09apr 30 10apr 28
2015
04apr 29 10apr 30
2014
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2013
07apr 29 09apr 14 22apr 5 23apr 14 25apr 13
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
2017 08apr

29 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 Curve Sketching 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) $\sqrt { 3 }$
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 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 Simultaneous equations 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 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 Standard trigonometric equations 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 Applied differentiation 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 Indefinite & Definite Integrals 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 Introduction & 2D 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) $\frac { 15 } { 64 }$