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
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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
2020 session2_04sep_shift2

24 maths questions

Q21 Travel graphs View
The speed verses time graph for a particle is shown in the figure. The distance travelled (in $m$) by the particle during the time interval $\mathrm { t } = 0$ to $\mathrm { t } = 5$ s will be $\_\_\_\_$
Q51 Solving quadratics and applications Finding roots or coefficients of a quadratic using Vieta's relations View
Let $\lambda \neq 0$ be in $R$. If $\alpha$ and $\beta$ are the roots of the equation, $x ^ { 2 } - x + 2 \lambda = 0$ and $\alpha$ and $\gamma$ are the roots of the equation, $3 x ^ { 2 } - 10 x + 27 \lambda = 0$, then $\frac { \beta \gamma } { \lambda }$ is equal to:
(1) 27
(2) 18
(3) 9
(4) 36
Q52 Complex Numbers Arithmetic Roots of Unity and Cyclotomic Expressions View
If $a$ and $b$ are real numbers such that $( 2 + \alpha ) ^ { 4 } = a + b \alpha$, where $\alpha = \frac { - 1 + i \sqrt { 3 } } { 2 }$, then $a + b$ is equal to:
(1) 9
(2) 24
(3) 33
(4) 57
Q53 Arithmetic Sequences and Series Find Specific Term from Given Conditions View
Let $a _ { 1 } , a _ { 2 } , \ldots , a _ { n }$ be a given A.P. whose common difference is an integer and $S _ { n } = a _ { 1 } + a _ { 2 } + \ldots + a _ { n }$. If $a _ { 1 } = 1 , a _ { n } = 300$ and $15 \leq n \leq 50$, then the ordered pair $\left( \mathrm { S } _ { n - 4 } , a _ { n - 4 } \right)$ is equal to:
(1) $( 2490,249 )$
(2) $( 2480,249 )$
(3) $( 2480,248 )$
(4) $( 2490,248 )$
Q54 Binomial Theorem (positive integer n) Determine Parameters from Conditions on Coefficients or Terms View
If for some positive integer $n$, the coefficients of three consecutive terms in the binomial expansion of $( 1 + x ) ^ { n + 5 }$ are in the ratio $5 : 10 : 14$, then the largest coefficient in the expansion is:
(1) 462
(2) 330
(3) 792
(4) 252
Q55 Straight Lines & Coordinate Geometry Line Equation and Parametric Representation View
If the perpendicular bisector of the line segment joining the points $P ( 1,4 )$ and $Q ( k , 3 )$ has $y$-intercept equal to $-4$, then a value of $k$ is:
(1) $-2$
(2) $-4$
(3) $\sqrt { 14 }$
(4) $\sqrt { 15 }$
Q56 Circles Circle Equation Derivation View
The circle passing through the intersection of the circles, $x ^ { 2 } + y ^ { 2 } - 6 x = 0$ and $x ^ { 2 } + y ^ { 2 } - 4 y = 0$ having its centre on the line, $2 x - 3 y + 12 = 0$, also passes through the point:
(1) $( - 1,3 )$
(2) $( - 3,6 )$
(3) $( - 3,1 )$
(4) $( 1 , - 3 )$
Q57 Conic sections Tangent and Normal Line Problems View
Let $x = 4$ be a directrix to an ellipse whose centre is at the origin and its eccentricity is $\frac { 1 } { 2 }$. If $P ( 1 , \beta ) , \beta > 0$ is a point on this ellipse, then the equation of the normal to it at $P$ is
(1) $4 x - 3 y = 2$
(2) $8 x - 2 y = 5$
(3) $7 x - 4 y = 1$
(4) $4 x - 2 y = 1$
Q58 Proof Direct Proof of a Stated Identity or Equality View
Contrapositive of the statement: 'If a function $f$ is differentiable at $a$, then it is also continuous at $a$', is
(1) If a function $f$ is continuous at $a$, then it is not differentiable at $a$.
(2) If a function $f$ is not continuous at $a$, then it is not differentiable at $a$.
(3) If a function $f$ is not continuous at $a$, then it is differentiable at $a$.
(4) If a function $f$ is continuous at $a$, then it is differentiable at $a$.
Q59 Sine and Cosine Rules Heights and distances / angle of elevation problem View
The angle of elevation of a cloud $C$ from a point $P$, $200$ m above a still lake is $30 ^ { \circ }$. If the angle of depression of the image of $C$ in the lake from the point $P$ is $60 ^ { \circ }$, then $PC$ (in m) is equal to
(1) 100
(2) $200 \sqrt { 3 }$
(3) 400
(4) $400 \sqrt { 3 }$
Q60 Principle of Inclusion/Exclusion View
Let $\cup _ { i = 1 } ^ { 50 } X _ { i } = \cup _ { i = 1 } ^ { n } Y _ { i } = T$, where each $X _ { i }$ contains 10 elements and each $Y _ { i }$ contains 5 elements. If each element of the set $T$ is an element of exactly 20 of sets $X _ { i }$'s and exactly 6 of sets $Y _ { i }$'s then $n$ is equal to:
(1) 15
(2) 50
(3) 45
(4) 30
Q61 Matrices Linear System and Inverse Existence View
If the system of equations $x + y + z = 2$ $2 x + 4 y - z = 6$ $3 x + 2 y + \lambda z = \mu$ has infinitely many solutions, then:
(1) $\lambda + 2 \mu = 14$
(2) $2 \lambda - \mu = 5$
(3) $\lambda - 2 \mu = - 5$
(4) $2 \lambda + \mu = 14$
Q62 Matrices Determinant and Rank Computation View
Suppose the vectors $x _ { 1 } , x _ { 2 }$ and $x _ { 3 }$ are the solutions of the system of linear equations, $A x = b$ when the vector $b$ on the right side is equal to $b _ { 1 } , b _ { 2 }$ and $b _ { 3 }$ respectively. If $x _ { 1 } = \left[ \begin{array} { l } 1 \\ 1 \\ 1 \end{array} \right] , x _ { 2 } = \left[ \begin{array} { l } 0 \\ 2 \\ 1 \end{array} \right] , x _ { 3 } = \left[ \begin{array} { l } 0 \\ 0 \\ 1 \end{array} \right] ; b _ { 1 } = \left[ \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right] , b _ { 2 } = \left[ \begin{array} { l } 0 \\ 2 \\ 0 \end{array} \right] , b _ { 3 } = \left[ \begin{array} { l } 0 \\ 0 \\ 2 \end{array} \right]$, then the determinant of $A$ is equal to
(1) 4
(2) 2
(3) $\frac { 1 } { 2 }$
(4) $\frac { 3 } { 2 }$
Q63 Stationary points and optimisation Find absolute extrema on a closed interval or domain View
The minimum value of $2 ^ { \sin x } + 2 ^ { \cos x }$ is:
(1) $2 ^ { - 1 + \frac { 1 } { \sqrt { 2 } } }$
(2) $2 ^ { - 1 + \sqrt { 2 } }$
(3) $2 ^ { 1 - \sqrt { 2 } }$
(4) $2 ^ { 1 - \frac { 1 } { \sqrt { 2 } } }$
Q64 Applied differentiation Finding parameter values from differentiability or equation constraints View
The function $f ( x ) = \left\{ \begin{array} { l l } \frac { \pi } { 4 } + \tan ^ { - 1 } x , & | x | \leq 1 \\ \frac { 1 } { 2 } ( | x | - 1 ) , & | x | > 1 \end{array} \right.$ is:
(1) continuous on $R - \{ 1 \}$ and differentiable on $R - \{ - 1,1 \}$.
(2) both continuous and differentiable on $R - \{ 1 \}$
(3) continuous on $R - \{ - 1 \}$ and differentiable on $R - \{ - 1,1 \}$
(4) both continuous and differentiable on $R - \{ - 1 \}$
Q65 Applied differentiation Properties of differentiable functions (abstract/theoretical) View
Let $f : ( 0 , \infty ) \rightarrow ( 0 , \infty )$ be a differentiable function such that $f ( 1 ) = e$ and $\lim _ { t \rightarrow x } \frac { t ^ { 2 } f ^ { 2 } ( x ) - x ^ { 2 } f ^ { 2 } ( t ) } { t - x } = 0$. If $f ( x ) = 1$, then $x$ is equal to:
(1) $\frac { 1 } { e }$
(2) $2 e$
(3) $\frac { 1 } { 2 e }$
(4) $e$
Q66 Stationary points and optimisation Geometric or applied optimisation problem View
The area (in sq. units) of the largest rectangle $ABCD$ whose vertices $A$ and $B$ lie on the $x$-axis and vertices $C$ and $D$ lie on the parabola, $y = x ^ { 2 } - 1$ below the $x$-axis, is:
(1) $\frac { 2 } { 3 \sqrt { 3 } }$
(2) $\frac { 1 } { 3 \sqrt { 3 } }$
(3) $\frac { 4 } { 3 }$
(4) $\frac { 4 } { 3 \sqrt { 3 } }$
Q67 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
The integral $\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 3 } } \tan ^ { 3 } x \cdot \sin ^ { 2 } 3 x \left( 2 \sec ^ { 2 } x \cdot \sin ^ { 2 } 3 x + 3 \tan x \cdot \sin 6 x \right) d x$ is equal to:
(1) $\frac { 7 } { 18 }$
(2) $- \frac { 1 } { 9 }$
(3) $- \frac { 1 } { 18 }$
(4) $\frac { 9 } { 2 }$
Q68 Differential equations Solving Separable DEs with Initial Conditions View
The solution of the differential equation $\frac { d y } { d x } - \frac { y + 3 x } { \log _ { e } ( y + 3 x ) } + 3 = 0$ is (where $C$ is a constant of integration)
(1) $x - \frac { 1 } { 2 } \left( \log _ { e } ( y + 3 x ) \right) ^ { 2 } = C$
(2) $x - \log _ { e } ( y + 3 x ) = C$
(3) $y + 3 x - \frac { 1 } { 2 } \left( \log _ { e } x \right) ^ { 2 } = C$
(4) $x - 2 \log _ { e } ( y + 3 x ) = C$
Q69 Vectors: Lines & Planes Distance Computation (Point-to-Plane or Line-to-Line) View
The distance of the point $( 1 , - 2,3 )$ from the plane $x - y + z = 5$ measured parallel to the line $\frac { x } { 2 } = \frac { y } { 3 } = \frac { z } { - 6 }$ is:
(1) $\frac { 7 } { 5 }$
(2) 1
(3) $\frac { 1 } { 7 }$
(4) 7
Q70 Discrete Probability Distributions Recurrence Relations and Sequences Involving Probabilities View
In a game two players $A$ and $B$ take turns in throwing a pair of fair dice starting with player $A$ and total of scores on the two dice, in each throw is noted. $A$ wins the game if he throws a total of 6 before $B$ throws a total of 7 and $B$ wins the game if he throws a total of 7 before $A$ throws a total of six. The game stops as soon as either of the players wins. The probability of $A$ winning the game is:
(1) $\frac { 5 } { 31 }$
(2) $\frac { 31 } { 61 }$
(3) $\frac { 5 } { 6 }$
(4) $\frac { 30 } { 61 }$
Q71 Combinations & Selection Selection with Group/Category Constraints View
A test consists of 6 multiple choice questions, each having 4 alternative answers of which only one is correct. The number of ways, in which a candidate answers all six questions such that exactly four of the answers are correct, is $\_\_\_\_$
Q72 Circles Optimization on a Circle View
Let $PQ$ be a diameter of the circle $x ^ { 2 } + y ^ { 2 } = 9$. If $\alpha$ and $\beta$ are the lengths of the perpendiculars from $P$ and $Q$ on the straight line, $x + y = 2$ respectively, then the maximum value of $\alpha \beta$ is $\_\_\_\_$
Q73 Measures of Location and Spread View
If the variance of the following frequency distribution:
Class:$10 - 20$$20 - 30$$30 - 40$
Frequency:2$x$2

is 50, then $x$ is equal to $\_\_\_\_$