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
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
2024 session1_27jan_shift2

30 maths questions

Q61 Sequences and series, recurrence and convergence Multiple-choice on sequence properties View
If $\alpha , \beta$ are the roots of the equation, $\mathrm { x } ^ { 2 } - \mathrm { x } - 1 = 0$ and $\mathrm { S } _ { \mathrm { n } } = 2023 \alpha ^ { \mathrm { n } } + 2024 \beta ^ { \mathrm { n } }$, then
(1) $2 \quad \mathrm {~S} _ { 12 } = \mathrm { S } _ { 11 } + \mathrm { S } _ { 10 }$
(2) $\mathrm { S } _ { 12 } = \mathrm { S } _ { 11 } + \mathrm { S } _ { 10 }$
(3) $2 \mathrm {~S} _ { 11 } = \mathrm { S } _ { 12 } + \mathrm { S } _ { 10 }$
(4) $\mathrm { S } _ { 11 } = \mathrm { S } _ { 10 } + \mathrm { S } _ { 12 }$
Q62 Indices and Surds Number-Theoretic Reasoning with Indices View
Let $\alpha = \frac { ( 4 ! ) ! } { ( 4 ! ) ^ { 3 ! } }$ and $\beta = \frac { ( 5 ! ) ! } { ( 5 ! ) ^ { 4 ! } }$. Then :
(1) $\alpha \in \mathrm { N }$ and $\beta \notin \mathrm { N }$
(2) $\alpha \notin \mathrm { N }$ and $\beta \in \mathrm { N }$
(3) $\alpha \in \mathrm { N }$ and $\beta \in \mathrm { N }$
(4) $\alpha \notin \mathrm { N }$ and $\beta \notin \mathrm { N }$
Q63 Arithmetic Sequences and Series Find Specific Term from Given Conditions View
The $20 ^ { \text {th} }$ term from the end of the progression $20,19 \frac { 1 } { 4 } , 18 \frac { 1 } { 2 } , 17 \frac { 3 } { 4 } , \ldots , - 129 \frac { 1 } { 4 }$ is :-
(1) - 118
(2) - 110
(3) - 115
(4) - 100
Q64 Standard trigonometric equations Solve trigonometric equation for solutions in an interval View
If $2 \tan ^ { 2 } \theta - 5 \sec \theta = 1$ has exactly 7 solutions in the interval $0 , \frac { \mathrm { n } \pi } { 2 }$, for the least value of $\mathrm { n } \in \mathrm { N }$ then $\sum _ { \mathrm { k } = 1 } ^ { \mathrm { n } } \frac { \mathrm { k } } { 2 ^ { \mathrm { k } } }$ is equal to :
(1) $\frac { 1 } { 2 ^ { 15 } } 2 ^ { 14 } - 14$
(2) $\frac { 1 } { 2 _ { 1 } ^ { 14 } } 2 ^ { 15 } - 15$
(3) $1 - \frac { 15 } { 2 ^ { 13 } }$
(4) $\frac { 1 } { 2 ^ { 13 } } 2 ^ { 14 } - 15$
Q65 Combinations & Selection Subset Counting with Set-Theoretic Conditions View
Let $A$ and $B$ be two finite sets with $m$ and $n$ elements respectively. The total number of subsets of the set $A$ is 56 more than the total number of subsets of $B$. Then the distance of the point $\mathrm { P } ( \mathrm { m } , \mathrm { n } )$ from the point $\mathrm { Q } ( - 2 , - 3 )$ is
(1) 10
(2) 6
(3) 4
(4) 8
Q66 Inequalities Solve Polynomial/Rational Inequality for Solution Set View
Let $R$ be the interior region between the lines $3 x - y + 1 = 0$ and $x + 2 y - 5 = 0$ containing the origin. The set of all values of $a$, for which the points $\mathrm { a } ^ { 2 } , \mathrm { a } + 1$ lie in R , is :
(1) $( - 3 , - 1 ) \cup - \frac { 1 } { 3 } , 1$
(2) $( - 3,0 ) \cup \frac { 1 } { 3 } , 1$
(3) $( - 3,0 ) \cup \frac { 2 } { 3 } , 1$
(4) $( - 3 , - 1 ) \cup \frac { 1 } { 3 } , 1$
Q67 Conic sections Chord Properties and Midpoint Problems View
Let $e _ { 1 }$ be the eccentricity of the hyperbola $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 9 } = 1$ and $e _ { 2 }$ be the eccentricity of the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 , a > b$, which passes through the foci of the hyperbola. If $e _ { 1 } e _ { 2 } = 1$, then the length of the chord of the ellipse parallel to the x -axis and passing through $( 0,2 )$ is:
(1) $4 \sqrt { 5 }$
(2) $\frac { 8 \sqrt { 5 } } { 3 }$
(3) $\frac { 10 \sqrt { 5 } } { 3 }$
(4) $3 \sqrt { 5 }$
Q68 Chain Rule Limit Evaluation Involving Composition or Substitution View
If $\lim _ { x \rightarrow 0 } \frac { 3 + \alpha \sin x + \beta \cos x + \log _ { e } ( 1 - x ) } { 3 \tan ^ { 2 } x } = \frac { 1 } { 3 }$, then $2 \alpha - \beta$ is equal to :
(1) 2
(2) 7
(3) 5
(4) 1
Q69 Matrices Determinant and Rank Computation View
The values of $\alpha$, for which $$2 \alpha + 3 \quad 3 \alpha + 1 \quad 0$$ (1) ( - 2, 1)
(2) ( - 3, 0)
(3) $- \frac { 3 } { 2 } , \frac { 3 } { 2 }$
(4) $( 0,3 )$
Q70 Standard trigonometric equations Inverse trigonometric equation View
Considering only the principal values of inverse trigonometric functions, the number of positive real values of $x$ satisfying $\tan ^ { - 1 } ( \mathrm { x } ) + \tan ^ { - 1 } ( 2 \mathrm { x } ) = \frac { \pi } { 4 }$ is :
(1) More than 2
(2) 1
(3) 2
(4) 0
Q71 Composite & Inverse Functions Determine Domain or Range of a Composite Function View
Let $\mathrm { f } : \mathrm { R } - \frac { - 1 } { 2 } \rightarrow \mathrm { R }$ and $\mathrm { g } : \mathrm { R } - \frac { - 5 } { 2 } \rightarrow \mathrm { R }$ be defined as $\mathrm { fx } = \frac { 2 \mathrm { x } + 3 } { 2 \mathrm { x } + 1 }$ and $\mathrm { gx } = \frac { | \mathrm { x } | + 1 } { 2 \mathrm { x } + 5 }$. Then the domain of the function fog is :
(1) $\mathrm { R } - - \frac { 5 } { 2 }$
(2) $R$
(3) $R - \frac { 1 } { 4 }$
(4) $\mathrm { R } - - \frac { 5 } { 2 } , - \frac { 7 } { 4 }$
Q72 Curve Sketching Continuity and Discontinuity Analysis of Piecewise Functions View
Consider the function $\mathrm { f } : ( 0,2 ) \rightarrow \mathrm { R }$ defined by $\mathrm { f } ( \mathrm { x } ) = \frac { \mathrm { x } } { 2 } + \frac { 2 } { \mathrm { x } }$ and the function $\mathrm { g } ( \mathrm { x } )$ defined by $\mathrm { gx } = \begin{array} { c c } \min \{ \mathrm { f } ( \mathrm { t } ) \} , & 0 < \mathrm { t } \leq \mathrm { x } \text { and } 0 < \mathrm { x } \leq 1 \\ \frac { 3 } { 2 } + \mathrm { x } , & 1 < \mathrm { x } < 2 \end{array}$. Then
(1) g is continuous but not differentiable at $\mathrm { x } = 1$
(2) g is not continuous for all $\mathrm { x } \in ( 0,2 )$
(3) g is neither continuous nor differentiable at $\mathrm { x } = 1$
(4) $g$ is continuous and differentiable for all $x \in ( 0,2 )$
Q73 Stationary points and optimisation Determine intervals of increase/decrease or monotonicity conditions View
Let $\mathrm { g } ( \mathrm { x } ) = 3 \mathrm { f } ^ { \mathrm { x } } + \mathrm { f } ( 3 - \mathrm { x } )$ and $\mathrm { f } ^ { \prime \prime } ( \mathrm { x } ) > 0$ for all $\mathrm { x } \in ( 0,3 )$. If g is decreasing in ( $0 , \alpha$ ) and increasing in $( \alpha , 3 )$, then $8 \alpha$ is
(1) 24
(2) 0
(3) 18
(4) 20
Q74 Integration by Substitution Substitution to Transform Integral Form (Show Transformed Expression) View
The integral $\int \frac { x ^ { 8 } - x ^ { 2 } d x } { x ^ { 12 } + 3 x ^ { 6 } + 1 \tan ^ { - 1 } x ^ { 3 } + \frac { 1 } { x ^ { 3 } } }$ is equal to :
(1) $\log \tan ^ { - 1 } x ^ { 3 } + { \frac { 1 } { x ^ { 3 } } } ^ { \frac { 1 } { 3 } } + C$
(2) $\log _ { e } \tan ^ { - 1 } x ^ { 3 } + { \frac { 1 } { x ^ { 3 } } } ^ { \frac { 1 } { 2 } } + C$
(3) $\log _ { e } \tan ^ { - 1 } x ^ { 3 } + \frac { 1 } { x ^ { 3 } } + C$
(4) $\log _ { e } \tan ^ { - 1 } x ^ { 3 } + { \frac { 1 } { x ^ { 3 } } } ^ { 3 } + C$
Q75 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
For $0 < \mathrm { a } < 1$, the value of the integral $\int _ { 0 } ^ { \pi } \frac { d x } { 1 - 2 a \cos x + a ^ { 2 } }$ is :
(1) $\frac { \pi ^ { 2 } } { \pi + a ^ { 2 } }$
(2) $\frac { \pi ^ { 2 } } { \pi - a ^ { 2 } }$
(3) $\frac { \pi } { 1 - a ^ { 2 } }$
(4) $\frac { \pi } { 1 + a ^ { 2 } }$
Q76 Differential equations Solving Separable DEs with Initial Conditions View
If $y = y ( x )$ is the solution curve of the differential equation $x ^ { 2 } - 4 d y - y ^ { 2 } - 3 y d x = 0 , x > 2 , y ( 4 ) = \frac { 3 } { 2 }$ and the slope of the curve is never zero, then the value of $\mathrm { y } ( 10 )$ equals :
(1) $\frac { 3 } { 1 + ( 8 ) ^ { \frac { 1 } { 4 } } }$
(2) $\frac { 3 } { 1 + 2 \sqrt { z } }$
(3) $\frac { 3 } { 1 - 2 \sqrt { 2 } }$
(4) $\frac { 3 } { 1 - ( 8 ) ^ { \frac { 1 } { 4 } } }$
Q77 Vectors 3D & Lines Section Division and Coordinate Computation View
The position vectors of the vertices $A , B$ and $C$ of a triangle are $2 \hat { i } - 3 \hat { j } + 3 \hat { k } , \quad 2 \hat { i } + 2 \hat { j } + 3 \hat { k }$ and $- \hat { i } + \hat { j } + 3 \hat { k }$ respectively. Let $l$ denotes the length of the angle bisector AD of $\angle \mathrm { BAC }$ where D is on the line segment BC , then $2 l ^ { 2 }$ equals :
(1) 49
(2) 42
(3) 50
(4) 45
Q78 Vectors 3D & Lines Distance from a Point to a Line (Show/Compute) View
Let the position vectors of the vertices $A , B$ and $C$ of a triangle be $2 \hat { i } + 2 \hat { j } + \hat { k } , \hat { i } + 2 \hat { j } + 2 \hat { k }$ and $2 \hat { i } + \hat { j } + 2 \hat { k }$ respectively. Let $l _ { 1 } , l _ { 2 }$ and $l _ { 3 }$ be the lengths of perpendiculars drawn from the ortho centre of the triangle on the sides $A B , B C$ and $C A$ respectively, then $l _ { 1 } ^ { 2 } + l _ { 2 } ^ { 2 } + l _ { 3 } ^ { 2 }$ equals :
(1) $\frac { 1 } { 5 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 1 } { 3 }$
Q79 Vectors: Lines & Planes Perpendicular/Orthogonal Projection onto a Plane View
Let the image of the point $( 1,0,7 )$ in the line $\frac { x } { 1 } = \frac { y - 1 } { 2 } = \frac { z - 2 } { 3 }$ be the point $( \alpha , \beta , \gamma )$. Then which one of the following points lies on the line passing through $( \alpha , \beta , \gamma )$ and making angles $\frac { 2 \pi } { 3 }$ and $\frac { 3 \pi } { 4 }$ with y - axis and z axis respectively and an acute angle with x -axis?
(1) $( 1 , - 2,1 + \sqrt { 2 } )$
(2) $( 1,2,1 - \sqrt { 2 } )$
(3) $( 3,4,3 - 2 \sqrt { 2 } )$
(4) $( 3 , - 4,3 + 2 \sqrt { 2 } )$
Q80 Probability Definitions Conditional Probability and Bayes' Theorem View
An urn contains 6 white and 9 black balls. Two successive draws of 4 balls are made without replacement. The probability, that the first draw gives all white balls and the second draw gives all black balls, is :
(1) $\frac { 5 } { 256 }$
(2) $\frac { 5 } { 715 }$
(3) $\frac { 3 } { 715 }$
(4) $\frac { 3 } { 256 }$
Q81 Complex Numbers Arithmetic Modulus Computation View
Let the complex numbers $\alpha$ and $\frac { 1 } { \alpha }$ lie on the circles $\mathrm { z } - \mathrm { z } _ { 0 } { } ^ { 2 } = 4$ and $\mathrm { z } - \mathrm { z } _ { 0 } { } ^ { 2 } = 16$ respectively, where $\mathrm { z } _ { 0 } = 1 + \mathrm { i }$. Then, the value of $100 | \alpha | ^ { 2 }$ is $\_\_\_\_$ .
Q82 Binomial Theorem (positive integer n) Find a Specific Coefficient in a Product of Binomial/Polynomial Expressions View
The coefficient of $x ^ { 2012 }$ in the expansion of $1 - x ^ { 2008 } 1 + x + x ^ { 2007 }$ is equal to $\_\_\_\_$ .
Q83 Straight Lines & Coordinate Geometry Collinearity and Concurrency View
If the sum of squares of all real values of $\alpha$, for which the lines $2 x - y + 3 = 0,6 x + 3 y + 1 = 0$ and $\alpha x + 2 y - 2 = 0$ do not form a triangle is $p$, then the greatest integer less than or equal to $p$ is $\_\_\_\_$ .
Q84 Circles Circles Tangent to Each Other or to Axes View
Consider a circle $x - \alpha ^ { 2 } + y - \beta ^ { 2 } = 50$, where $\alpha , \beta > 0$. If the circle touches the line $y + x = 0$ at the point P , whose distance from the origin is $4 \sqrt { 2 }$, then $( \alpha + \beta ) ^ { 2 }$ is equal to $\_\_\_\_$ .
Q85 Measures of Location and Spread View
The mean and standard deviation of 15 observations were found to be 12 and 3 respectively. On rechecking it was found that an observation was read as 10 in place of 12 . If $\mu$ and $\sigma ^ { 2 }$ denote the mean and variance of the correct observations respectively, then $15 \mu + \mu ^ { 2 } + \sigma ^ { 2 }$ is equal to $\_\_\_\_$ .
Q86 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Let $A$ be a $2 \times 2$ real matrix and $I$ be the identity matrix of order 2 . If the roots of the equation $| A - x I | = 0$ be - 1 and 3 , then the sum of the diagonal elements of the matrix $A ^ { 2 }$ is $\_\_\_\_$ .
Q87 Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution View
Let $\mathrm { fx } = \int _ { 0 } ^ { \mathrm { x } } \mathrm { gt } \log _ { \mathrm { e } } \frac { 1 - \mathrm { t } } { 1 + \mathrm { t } } \mathrm { dt }$, where g is a continuous odd function. If $\int _ { - \frac { \pi } { 2 } } ^ { \frac { \pi } { 2 } } \mathrm { fx } + \frac { \mathrm { x } ^ { 2 } \cos \mathrm { x } } { 1 + \mathrm { e } ^ { \mathrm { x } } } \mathrm { dx } = \frac { \pi ^ { 2 } } { \alpha } - \alpha$, then $\alpha$ is equal to $\_\_\_\_$.
Q88 Areas by integration View
If the area of the region $( x , y ) : 0 \leq y \leq \min 2 x , 6 x - x ^ { 2 }$ is $A$, then $12 A$ is equal to $\_\_\_\_$ .
Q89 Differential equations Solving Separable DEs with Initial Conditions View
If the solution curve, of the differential equation $\frac { d y } { d x } = \frac { x + y - 2 } { x - y }$ passing through the point $( 2,1 )$ is $\tan ^ { - 1 } \frac { y - 1 } { x - 1 } - \frac { 1 } { \beta } \log _ { e } \alpha + \frac { y - 1 } { x - 1 } ^ { 2 } = \log _ { e } x - 1$, then $5 \beta + \alpha$ is equal to $\_\_\_\_$.
Q90 Vectors: Lines & Planes Distance Computation (Point-to-Plane or Line-to-Line) View
The lines $\frac { x - 2 } { 2 } = \frac { y } { - 2 } = \frac { z - 7 } { 16 }$ and $\frac { x + 3 } { 4 } = \frac { y + 2 } { 3 } = \frac { z + 2 } { 1 }$ intersect at the point $P$. If the distance of $P$ from the line $\frac { \mathrm { x } + 1 } { 2 } = \frac { \mathrm { y } - 1 } { 3 } = \frac { \mathrm { z } - 1 } { 1 }$ is $l$, then $14 l ^ { 2 }$ is equal to $\_\_\_\_$ .