jee-main

Papers (169)
2025
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2024
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2023
session1_01feb_shift1 24 session1_01feb_shift2 3 session1_24jan_shift1 13 session1_24jan_shift2 12 session1_25jan_shift1 28 session1_25jan_shift2 27 session1_29jan_shift1 29 session1_29jan_shift2 28 session1_30jan_shift1 2 session1_30jan_shift2 29 session1_31jan_shift1 28 session1_31jan_shift2 17 session2_06apr_shift1 5 session2_06apr_shift2 17 session2_08apr_shift1 29 session2_08apr_shift2 14 session2_10apr_shift1 29 session2_10apr_shift2 15 session2_11apr_shift1 5 session2_11apr_shift2 4 session2_12apr_shift1 26 session2_13apr_shift1 25 session2_13apr_shift2 20 session2_15apr_shift1 20
2022
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2021
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2020
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2019
session1_09jan_shift1 6 session1_09jan_shift2 29 session1_10jan_shift1 30 session1_10jan_shift2 12 session1_11jan_shift1 6 session1_11jan_shift2 5 session1_12jan_shift1 10 session1_12jan_shift2 20 session2_08apr_shift1 29 session2_08apr_shift2 29 session2_09apr_shift1 29 session2_09apr_shift2 29 session2_10apr_shift1 2 session2_10apr_shift2 3 session2_12apr_shift1 3 session2_12apr_shift2 9
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
2022 session1_26jun_shift1

26 maths questions

Q61 Complex Numbers Argand & Loci Intersection of Loci and Simultaneous Geometric Conditions View
Let $A = \left\{ z \in C : \left| \frac { z + 1 } { z - 1 } \right| < 1 \right\}$ and $B = \left\{ z \in C : \arg \left( \frac { z - 1 } { z + 1 } \right) = \frac { 2 \pi } { 3 } \right\}$. Then $A \cap B$ is
(1) a portion of a circle centred at $\left( 0 , - \frac { 1 } { \sqrt { 3 } } \right)$ that lies in the second and third quadrants only
(2) a portion of a circle centred at $\left( 0 , - \frac { 1 } { \sqrt { 3 } } \right)$ that lies in the second quadrant only
(3) an empty set
(4) a portion of a circle of radius $\frac { 2 } { \sqrt { 3 } }$ that lies in the third quadrant only
Q63 Straight Lines & Coordinate Geometry Area Computation in Coordinate Geometry View
Let $R$ be the point $( 3,7 )$ and let $P$ and $Q$ be two points on the line $x + y = 5$ such that $PQR$ is an equilateral triangle. Then the area of $\triangle PQR$ is
(1) $\frac { 25 } { 4 \sqrt { 3 } }$
(2) $\frac { 25 \sqrt { 3 } } { 2 }$
(3) $\frac { 25 } { \sqrt { 3 } }$
(4) $\frac { 25 } { 2 \sqrt { 3 } }$
Q64 Circles Circle Equation Derivation View
Let $C$ be a circle passing through the points $A ( 2 , - 1 )$ and $B ( 3,4 )$. The line segment $AB$ is not a diameter of $C$. If $r$ is the radius of $C$ and its centre lies on the circle $( x - 5 ) ^ { 2 } + ( y - 1 ) ^ { 2 } = \frac { 13 } { 2 }$, then $r ^ { 2 }$ is equal to
(1) 32
(2) $\frac { 65 } { 2 }$
(3) $\frac { 61 } { 2 }$
(4) 30
Q65 Circles Tangent Lines and Tangent Lengths View
Let the normal at the point $P$ on the parabola $y ^ { 2 } = 6 x$ pass through the point $( 5 , - 8 )$. If the tangent at $P$ to the parabola intersects its directrix at the point $Q$, then the ordinate of the point $Q$ is
(1) $\frac { - 9 } { 4 }$
(2) $\frac { 9 } { 4 }$
(3) $\frac { - 5 } { 2 }$
(4) $- 3$
Q66 Standard trigonometric equations Inverse trigonometric equation View
$\lim _ { x \rightarrow \frac { 1 } { \sqrt { 2 } } } \frac { \sin \left( \cos ^ { - 1 } x \right) - x } { 1 - \tan \left( \cos ^ { - 1 } x \right) }$ is equal to
(1) $\frac { 1 } { \sqrt { 2 } }$
(2) $\frac { - 1 } { \sqrt { 2 } }$
(3) $\sqrt { 2 }$
(4) $- 1$
Q68 Measures of Location and Spread View
The mean of the numbers $a , b , 8 , 5 , 10$ is 6 and their variance is 6.8. If $M$ is the mean deviation of the numbers about the mean, then $25M$ is equal to
(1) 60
(2) 55
(3) 50
(4) 75
Q69 3x3 Matrices Determinant of Parametric or Structured Matrix View
Let $A$ be a $3 \times 3$ invertible matrix. If $| \operatorname { adj } ( 24 A ) | = | \operatorname { adj } ( 3 \operatorname { adj } ( 2A ) ) |$, then $| A | ^ { 2 }$ is equal to
(1) $2 ^ { 6 }$
(2) $2 ^ { 12 }$
(3) 512
(4) $6 ^ { 6 }$
Q70 Simultaneous equations View
The ordered pair $( a , b )$, for which the system of linear equations $3 x - 2 y + z = b$ $5 x - 8 y + 9 z = 3$ $2 x + y + a z = - 1$ has no solution, is
(1) $\left( 3 , \frac { 1 } { 3 } \right)$
(2) $\left( - 3 , \frac { 1 } { 3 } \right)$
(3) $\left( - 3 , - \frac { 1 } { 3 } \right)$
(4) $\left( 3 , - \frac { 1 } { 3 } \right)$
Q71 Composite & Inverse Functions Evaluate Composition from Algebraic Definitions View
Let $f ( x ) = \frac { x - 1 } { x + 1 } , x \in R - \{ 0 , - 1 , 1 \}$. If $f ^ { n + 1 } ( x ) = f \left( f ^ { n } ( x ) \right)$ for all $n \in N$, then $f ^ { 6 } ( 6 ) + f ^ { 7 } ( 7 )$ is equal to
(1) $\frac { 7 } { 6 }$
(2) $- \frac { 3 } { 2 }$
(3) $\frac { 7 } { 12 }$
(4) $- \frac { 11 } { 12 }$
Q72 Chain Rule Piecewise Function Differentiability Analysis View
$f , g : R \rightarrow R$ be two real valued function defined as $f ( x ) = \left\{ \begin{array} { c l } - | x + 3 | & , x < 0 \\ e ^ { x } & , x \geq 0 \end{array} \right.$ and $g ( x ) = \left\{ \begin{array} { l l } x ^ { 2 } + k _ { 1 } x , & x < 0 \\ 4 x + k _ { 2 } , & x \geq 0 \end{array} \right.$, where $k _ { 1 }$ and $k _ { 2 }$ are real constants. If $gof$ is differentiable at $x = 0$, then $gof ( - 4 ) + gof ( 4 )$ is equal to
(1) $4 \left( e ^ { 4 } + 1 \right)$
(2) $2 \left( 2 e ^ { 4 } + 1 \right)$
(3) $4 e ^ { 4 }$
(4) $2 \left( 2 e ^ { 4 } - 1 \right)$
Q73 Stationary points and optimisation Find absolute extrema on a closed interval or domain View
The sum of the absolute minimum and the absolute maximum values of the function $f ( x ) = \left| 3 x - x ^ { 2 } + 2 \right| - x$ in the interval $[ - 1 , 2 ]$ is
(1) $\frac { \sqrt { 17 } + 3 } { 2 }$
(2) $\frac { \sqrt { 17 } + 5 } { 2 }$
(3) 5
(4) $\frac { 9 - \sqrt { 17 } } { 2 }$
Q74 Tangents, normals and gradients Prove a given line is tangent to a curve View
Let $S$ be the set of all the natural numbers, for which the line $\frac { x } { a } + \frac { y } { b } = 2$ is a tangent to the curve $\left( \frac { x } { a } \right) ^ { n } + \left( \frac { y } { b } \right) ^ { n } = 2$ at the point $( a , b ) , ab \neq 0$. Then
(1) $S = \phi$
(2) $n ( S ) = 1$
(3) $S = \{ 2k : k \in N \}$
(4) $S = N$
Q75 Stationary points and optimisation Find absolute extrema on a closed interval or domain View
Let $f ( x ) = 2 \cos ^ { - 1 } x + 4 \cot ^ { - 1 } x - 3 x ^ { 2 } - 2 x + 10 , x \in [ - 1 , 1 ]$. If $[ a , b ]$ is the range of the function, then $4a - b$ is equal to
(1) 11
(2) $11 - \pi$
(3) $11 + \pi$
(4) $15 - \pi$
Q76 Areas by integration View
The area bounded by the curve $y = \left| x ^ { 2 } - 9 \right|$ and the line $y = 3$ is
(1) $8 \sqrt { 6 } - 16 \sqrt { 12 } - 72$
(2) $8 \sqrt { 6 } + 8 \sqrt { 12 } - 72$
(3) $16 \sqrt { 6 } + 16 \sqrt { 12 } - 72$
(4) $16 \sqrt { 6 } - 16 \sqrt { 12 } - 64$
Q77 Vectors Introduction & 2D Dot Product Computation View
If $\vec { a } \cdot \vec { b } = 1 , \vec { b } \cdot \vec { c } = 2$ and $\vec { c } \cdot \vec { a } = 3$, then the value of $[ \vec { a } \times ( \vec { b } \times \vec { c } ) \quad \vec { b } \times ( \vec { c } \times \vec { a } ) \quad \vec { c } \times ( \vec { b } \times \vec { a } ) ]$ is
(1) 0
(2) $- 6 \vec { a } \cdot ( \vec { b } \times \vec { c } )$
(3) $12 \vec { c } \cdot ( \vec { a } \times \vec { b } )$
(4) $- 12 \vec { b } \cdot ( \vec { c } \times \vec { a } )$
Q78 Vectors 3D & Lines MCQ: Relationship Between Two Lines View
If the two lines $l _ { 1 } : \frac { x - 2 } { 3 } = \frac { y + 1 } { - 2 } , z = 2$ and $l _ { 2 } : \frac { x - 1 } { 1 } = \frac { 2 y + 3 } { \alpha } = \frac { z + 5 } { 2 }$ are perpendicular, then an angle between the lines $l _ { 2 }$ and $l _ { 3 } : \frac { 1 - x } { 3 } = \frac { 2 y - 1 } { - 4 } = \frac { z } { 4 }$ is
(1) $\cos ^ { - 1 } \left( \frac { 29 } { 4 } \right)$
(2) $\sec ^ { - 1 } \left( \frac { 29 } { 4 } \right)$
(3) $\cos ^ { - 1 } \left( \frac { 2 } { 29 } \right)$
(4) $\cos ^ { - 1 } \left( \frac { 2 } { \sqrt { 29 } } \right)$
Q79 Vectors 3D & Lines Plane Rotation About a Line View
Let the plane $2 x + 3 y + z + 20 = 0$ be rotated through a right angle about its line of intersection with the plane $x - 3 y + 5 z = 8$. If the mirror image of the point $\left( 2 , - \frac { 1 } { 2 } , 2 \right)$ in the rotated plane is $B ( a , b , c )$, then
(1) $\frac { a } { 8 } = \frac { b } { 5 } = \frac { c } { - 4 }$
(2) $\frac { a } { 4 } = \frac { b } { 5 } = \frac { c } { - 2 }$
(3) $\frac { a } { 8 } = \frac { b } { - 5 } = \frac { c } { 4 }$
(4) $\frac { a } { 4 } = \frac { b } { 5 } = \frac { c } { 2 }$
Q80 Binomial Distribution Compute Cumulative or Complement Binomial Probability View
Let a biased coin be tossed 5 times. If the probability of getting 4 heads is equal to the probability of getting 5 heads, then the probability of getting atmost two heads is
(1) $\frac { 46 } { 6 ^ { 4 } }$
(2) $\frac { 275 } { 6 ^ { 5 } }$
(3) $\frac { 41 } { 5 ^ { 5 } }$
(4) $\frac { 36 } { 5 ^ { 4 } }$
Q81 Roots of polynomials Vieta's formulas: compute symmetric functions of roots View
The sum of the cubes of all the roots of the equation $x ^ { 4 } - 3 x ^ { 3 } - 2 x ^ { 2 } + 3 x + 1 = 0$ is $\_\_\_\_$.
Q82 Combinations & Selection Selection with Group/Category Constraints View
There are ten boys $B _ { 1 } , B _ { 2 } , \ldots , B _ { 10 }$ and five girls $G _ { 1 } , G _ { 2 } , \ldots G _ { 5 }$ in a class. Then the number of ways of forming a group consisting of three boys and three girls, if both $B _ { 1 }$ and $B _ { 2 }$ together should not be the members of a group, is $\_\_\_\_$.
Q84 Trig Proofs Trigonometric Identity Simplification View
If $\sin ^ { 2 } \left( 10 ^ { \circ } \right) \sin \left( 20 ^ { \circ } \right) \sin \left( 40 ^ { \circ } \right) \sin \left( 50 ^ { \circ } \right) \sin \left( 70 ^ { \circ } \right) = \alpha - \frac { 1 } { 16 } \sin \left( 10 ^ { \circ } \right)$, then $16 + \alpha ^ { - 1 }$ is equal to $\_\_\_\_$.
Q85 Conic sections Tangent and Normal Line Problems View
Let the common tangents to the curves $4 \left( x ^ { 2 } + y ^ { 2 } \right) = 9$ and $y ^ { 2 } = 4 x$ intersect at the point $Q$. Let an ellipse, centered at the origin $O$, has lengths of semi-minor and semi-major axes equal to $OQ$ and 6, respectively. If $e$ and $l$ respectively denote the eccentricity and the length of the latus rectum of this ellipse, then $\frac { l } { e ^ { 2 } }$ is equal to $\_\_\_\_$.
Q87 Areas by integration View
Let $f ( x ) = \max \{ | x + 1 | , | x + 2 | , \ldots , | x + 5 | \}$. Then $\int _ { - 6 } ^ { 0 } f ( x ) \, dx$ is equal to $\_\_\_\_$.
Q88 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
The value of the integral $\frac { 48 } { \pi ^ { 4 } } \int _ { 0 } ^ { \pi } \left( \frac { 3 \pi x ^ { 2 } } { 2 } - x ^ { 3 } \right) \frac { \sin x } { 1 + \cos ^ { 2 } x } \, dx$ is equal to $\_\_\_\_$.
Q89 Differential equations First-Order Linear DE: General Solution View
Let the solution curve $y = y ( x )$ of the differential equation $\left( 4 + x ^ { 2 } \right) dy - 2 x \left( x ^ { 2 } + 3 y + 4 \right) dx = 0$ pass through the origin. Then $y ( 2 )$ is equal to $\_\_\_\_$.
Q90 Differential equations Solving Separable DEs with Initial Conditions View
Let $S = ( 0 , 2 \pi ) - \left\{ \frac { \pi } { 2 } , \frac { 3 \pi } { 4 } , \frac { 3 \pi } { 2 } , \frac { 7 \pi } { 4 } \right\}$. Let $y = y ( x ) , x \in S$, be the solution curve of the differential equation $\frac { dy } { dx } = \frac { 1 } { 1 + \sin 2 x } , y \left( \frac { \pi } { 4 } \right) = \frac { 1 } { 2 }$. If the sum of abscissas of all the points of intersection of the curve $y = y ( x )$ with the curve $y = \sqrt { 2 } \sin x$ is $\frac { k \pi } { 12 }$, then $k$ is equal to $\_\_\_\_$.