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
session1_24feb_shift1 10 session1_24feb_shift2 7 session1_25feb_shift1 29 session1_25feb_shift2 29 session1_26feb_shift2 17 session2_16mar_shift1 29 session2_16mar_shift2 15 session2_17mar_shift1 20 session2_17mar_shift2 24 session2_18mar_shift1 12 session2_18mar_shift2 11 session3_20jul_shift1 30 session3_20jul_shift2 29 session3_22jul_shift1 7 session3_25jul_shift1 2 session3_25jul_shift2 15 session3_27jul_shift1 3 session3_27jul_shift2 4 session4_01sep_shift2 11 session4_26aug_shift1 5 session4_26aug_shift2 2 session4_27aug_shift1 3 session4_27aug_shift2 28 session4_31aug_shift1 28 session4_31aug_shift2 4
2020
session1_07jan_shift1 26 session1_07jan_shift2 17 session1_08jan_shift1 5 session1_08jan_shift2 12 session1_09jan_shift1 22 session1_09jan_shift2 18 session2_02sep_shift1 19 session2_02sep_shift2 17 session2_03sep_shift1 21 session2_03sep_shift2 9 session2_04sep_shift1 10 session2_04sep_shift2 24 session2_05sep_shift1 23 session2_05sep_shift2 27 session2_06sep_shift1 13 session2_06sep_shift2 10
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
08apr 29 15apr 28 15apr_shift1 28 15apr_shift2 2 16apr 15
2017
02apr 28 08apr 29 09apr 30
2016
03apr 30 09apr 30 10apr 28
2015
04apr 29 10apr 30
2014
06apr 28 09apr 28 11apr 4 12apr 5 19apr 29
2013
07apr 29 09apr 14 22apr 5 23apr 14 25apr 13
2012
07may 18 12may 22 19may 13 26may 17 offline 30
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
2019 session2_09apr_shift1

29 maths questions

Q61 Discriminant and conditions for roots Root relationships and Vieta's formulas View
Let $p , q \in Q$. If $2 - \sqrt { 3 }$ is a root of the quadratic equation $x ^ { 2 } + p x + q = 0$, then
(1) $p ^ { 2 } - 4 q + 12 = 0$
(2) $q ^ { 2 } + 4 p + 14 = 0$
(3) $p ^ { 2 } - 4 q - 12 = 0$
(4) $q ^ { 2 } - 4 p - 16 = 0$
Q62 Complex Numbers Argand & Loci Circle Equation and Properties via Complex Number Manipulation View
All the points in the set $S = \left\{ \frac { \alpha + i } { \alpha - i } , \alpha \in R \right\} , i = \sqrt { - 1 }$ lie on a
(1) straight line whose slope is - 1
(2) circle whose radius is $\sqrt { 2 }$
(3) circle whose radius is 1
(4) straight line whose slope is 1
Q63 Combinations & Selection Selection with Group/Category Constraints View
A committee of 11 member is to be formed from 8 males and 5 females. If $m$ is the number of ways the committee is formed with at least 6 males and $n$ is the number of ways the committee is formed with at least 3 females, then:
(1) $m = n = 68$
(2) $n = m - 8$
(3) $m = n = 78$
(4) $m + n = 68$
Q64 Arithmetic Sequences and Series Find General Term Formula View
Let the sum of the first $n$ terms of a non-constant A.P., $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots , a _ { n }$ be $50 n + \frac { n ( n - 7 ) } { 2 } A$, where $A$ is a constant. If $d$ is the common difference of this A.P., then the ordered pair $\left( d , a _ { 50 } \right)$ is equal to
(1) $( 50,50 + 46 A )$
(2) $( A , 50 + 45 A )$
(3) $( 50,50 + 45 A )$
(4) $( A , 50 + 46 A )$
Q65 Binomial Theorem (positive integer n) Determine Parameters from Conditions on Coefficients or Terms View
If the fourth term in the Binomial expansion of $\left( \frac { 2 } { x } + x ^ { \log _ { 8 } x } \right) ^ { 6 } , ( x > 0 )$ is $20 \times 8 ^ { 7 }$, then a value of $x$ is
(1) $8 ^ { - 2 }$
(2) 8
(3) $8 ^ { 3 }$
(4) $8 ^ { 2 }$
Q66 Trig Proofs Trigonometric Identity Simplification View
The value of $\cos ^ { 2 } 10 ^ { \circ } - \cos 10 ^ { \circ } \cos 50 ^ { \circ } + \cos ^ { 2 } 50 ^ { \circ }$ is
(1) $\frac { 3 } { 4 }$
(2) $\frac { 3 } { 4 } + \cos 20 ^ { \circ }$
(3) $\frac { 3 } { 2 }$
(4) $\frac { 3 } { 2 } \left( 1 + \cos 20 ^ { \circ } \right)$
Q67 Standard trigonometric equations Solve trigonometric equation for solutions in an interval View
Let $S = \left\{ \theta \in [ - 2 \pi , 2 \pi ] : 2 \cos ^ { 2 } \theta + 3 \sin \theta = 0 \right\}$. Then the sum of the elements of $S$ is:
(1) $\pi$
(2) $\frac { 13 \pi } { 6 }$
(3) $\frac { 5 \pi } { 3 }$
(4) $2 \pi$
Q68 Straight Lines & Coordinate Geometry Slope and Angle Between Lines View
Slope of a line passing through $P ( 2,3 )$ and intersecting the line $x + y = 7$ at a distance of 4 units from $P$, is
(1) $\frac { \sqrt { 7 } - 1 } { \sqrt { 7 } + 1 }$
(2) $\frac { 1 - \sqrt { 7 } } { 1 + \sqrt { 7 } }$
(3) $\frac { \sqrt { 5 } - 1 } { \sqrt { 5 } + 1 }$
(4) $\frac { 1 - \sqrt { 5 } } { 1 + \sqrt { 5 } }$
Q69 Circles Circle-Related Locus Problems View
If a tangent to the circle $x ^ { 2 } + y ^ { 2 } = 1$ intersects the coordinate axes at distinct points $P$ and $Q$, then the locus of the mid-point of $PQ$ is:
(1) $x ^ { 2 } + y ^ { 2 } - 16 x ^ { 2 } y ^ { 2 } = 0$
(2) $x ^ { 2 } + y ^ { 2 } - 4 x ^ { 2 } y ^ { 2 } = 0$
(3) $x ^ { 2 } + y ^ { 2 } - 2 x y = 0$
(4) $x ^ { 2 } + y ^ { 2 } - 2 x ^ { 2 } y ^ { 2 } = 0$
Q70 Conic sections Focal Chord and Parabola Segment Relations View
If one end of a focal chord of the parabola, $y ^ { 2 } = 16 x$ is at $( 1,4 )$, then the length of this focal chord is
(1) 24
(2) 25
(3) 22
(4) 20
Q71 Conic sections Tangent and Normal Line Problems View
If the line $y = m x + 7 \sqrt { 3 }$ is normal to the hyperbola $\frac { x ^ { 2 } } { 24 } - \frac { y ^ { 2 } } { 18 } = 1$, then a value of $m$ is:
(1) $\frac { \sqrt { 5 } } { 2 }$
(2) $\frac { 3 } { \sqrt { 5 } }$
(3) $\frac { \sqrt { 15 } } { 2 }$
(4) $\frac { 2 } { \sqrt { 5 } }$
Q73 Measures of Location and Spread View
If the standard deviation of the numbers $- 1,0,1 , k$ is $\sqrt { 5 }$ where $k > 0$, then $k$ is equal to
(1) $\sqrt { 6 }$
(2) $4 \sqrt { \frac { 5 } { 3 } }$
(3) $2 \sqrt { \frac { 10 } { 3 } }$
(4) $2 \sqrt { 6 }$
Q74 Matrices Linear System and Inverse Existence View
If $\left[ \begin{array} { l l } 1 & 1 \\ 0 & 1 \end{array} \right] \left[ \begin{array} { l l } 1 & 2 \\ 0 & 1 \end{array} \right] \left[ \begin{array} { l l } 1 & 3 \\ 0 & 1 \end{array} \right] \ldots \left[ \begin{array} { c c } 1 & n - 1 \\ 0 & 1 \end{array} \right] = \left[ \begin{array} { c c } 1 & 78 \\ 0 & 1 \end{array} \right]$, then the inverse of $\left[ \begin{array} { l l } 1 & n \\ 0 & 1 \end{array} \right]$ is:
(1) $\left[ \begin{array} { c c } 1 & - 12 \\ 0 & 1 \end{array} \right]$
(2) $\left[ \begin{array} { c c } 1 & 0 \\ 12 & 1 \end{array} \right]$
(3) $\left[ \begin{array} { c c } 1 & 0 \\ 13 & 1 \end{array} \right]$
(4) $\left[ \begin{array} { c c } 1 & - 13 \\ 0 & 1 \end{array} \right]$
Q75 3x3 Matrices Determinant of Parametric or Structured Matrix View
Let $\alpha$ and $\beta$ be the roots of the equation $x ^ { 2 } + x + 1 = 0$. Then for $y \neq 0$ in $R , \left| \begin{array} { c c c } y + 1 & \alpha & \beta \\ \alpha & y + \beta & 1 \\ \beta & 1 & y + \alpha \end{array} \right|$ is equal to
(1) $y ^ { 3 }$
(2) $y \left( y ^ { 2 } - 1 \right)$
(3) $y ^ { 3 } - 1$
(4) $y \left( y ^ { 2 } - 3 \right)$
Q76 Composite & Inverse Functions Injectivity, Surjectivity, or Bijectivity Classification View
If the function $f : R - \{ 1 , - 1 \} \rightarrow A$ defined by $f ( x ) = \frac { x ^ { 2 } } { 1 - x ^ { 2 } }$, is surjective, then $A$ is equal to
(1) $[ 0 , \infty )$
(2) $R - \{ - 1 \}$
(3) $R - [ - 1,0 )$
(4) $R - ( - 1,0 )$
Q77 Modulus function Differentiability of functions involving modulus View
Let $f ( x ) = 15 - | x - 10 | ; x \in R$. Then the set of all values of $x$, at which the function $g ( x ) = f ( f ( x ) )$ is not differentiable, is:
(1) $\{ 5,10,15 \}$
(2) $\{ 10 \}$
(3) $\{ 10,15 \}$
(4) $\{ 5,10,15,20 \}$
Q78 Composite & Inverse Functions Find or Apply an Inverse Function Formula View
If the function $f$ defined on $\left( \frac { \pi } { 6 } , \frac { \pi } { 3 } \right)$ by $f ( x ) = \left\{ \begin{array} { l l } \frac { \sqrt { 2 } \cos x - 1 } { \cot x - 1 } , & x \neq \frac { \pi } { 4 } \\ k , & x = \frac { \pi } { 4 } \end{array} \right.$ is continuous, then $k$ is equal to
(1) $\frac { 1 } { 2 }$
(2) 1
(3) 2
(4) $\frac { 1 } { \sqrt { 2 } }$
Q79 Exponential Functions Functional Equation with Exponentials View
Let $\sum _ { k = 1 } ^ { 10 } f ( a + k ) = 16 \left( 2 ^ { 10 } - 1 \right)$, where the function $f$ satisfies $f ( x + y ) = f ( x ) f ( y )$ for all natural numbers $x , y$ and $f ( 1 ) = 2$. Then the natural number ' $a$ ' is:
(1) 3
(2) 16
(3) 4
(4) 2
Q80 Stationary points and optimisation Find critical points and classify extrema of a given function View
If $f ( x )$ is a non-zero polynomial of degree four, having local extreme points at $x = - 1,0,1$; then the set $S = \{ x \in R : f ( x ) = f ( 0 ) \}$ contains exactly
(1) Two irrational and two rational numbers
(2) Four rational numbers
(3) Two irrational and one rational number
(4) Four irrational numbers
Q81 Tangents, normals and gradients Determine unknown parameters from tangent conditions View
If the tangent to the curve, $y = x ^ { 3 } + a x - b$ at the point $( 1 , - 5 )$ is perpendicular to the line, $- x + y + 4 = 0$, then which one of the following points lies on the curve?
(1) $( 2 , - 2 )$
(2) $( 2 , - 1 )$
(3) $( - 2,1 )$
(4) $( - 2,2 )$
Q82 Applied differentiation Tangent line computation and geometric consequences View
Let $S$ be the set of all values of $x$ for which the tangent to the curve $y = f ( x ) = x ^ { 3 } - x ^ { 2 } - 2 x$ at ( $x , y$ ) is parallel to the line segment joining the points $( 1 , f ( 1 ) )$ and $( - 1 , f ( - 1 ) )$, then $S$ is equal to
(1) $\left\{ - \frac { 1 } { 3 } , - 1 \right\}$
(2) $\left\{ - \frac { 1 } { 3 } , 1 \right\}$
(3) $\left\{ \frac { 1 } { 3 } , 1 \right\}$
(4) $\left\{ \frac { 1 } { 3 } , - 1 \right\}$
Q83 Integration by Substitution Substitution to Transform Integral Form (Show Transformed Expression) View
$\int \sec ^ { 2 } x \cdot \cot ^ { \frac { 4 } { 3 } } x \, d x$ is equal to
(1) $3 \tan ^ { - \frac { 1 } { 3 } } x + C$
(2) $- \frac { 3 } { 4 } \tan ^ { - \frac { 4 } { 3 } } x + C$
(3) $- 3 \tan ^ { - \frac { 1 } { 3 } } x + C$
(4) $- 3 \cot ^ { - \frac { 1 } { 3 } } x + C$
Q84 Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution View
The value of $\int _ { 0 } ^ { \pi / 2 } \frac { \sin ^ { 3 } x } { \sin x + \cos x } d x$ is:
(1) $\frac { \pi - 1 } { 2 }$
(2) $\frac { \pi - 2 } { 8 }$
(3) $\frac { \pi - 1 } { 4 }$
(4) $\frac { \pi - 2 } { 4 }$
Q85 Areas by integration View
The area (in sq. units) of the region $A = \left\{ ( x , y ) : x ^ { 2 } \leq y \leq x + 2 \right\}$ is
(1) $\frac { 13 } { 6 }$
(2) $\frac { 31 } { 6 }$
(3) $\frac { 9 } { 2 }$
(4) $\frac { 10 } { 3 }$
Q86 Differential equations First-Order Linear DE: General Solution View
The solution of the differential equation $x \frac { d y } { d x } + 2 y = x ^ { 2 } , ( x \neq 0 )$ with $y ( 1 ) = 1$, is
(1) $y = \frac { x ^ { 3 } } { 5 } + \frac { 1 } { 5 x ^ { 2 } }$
(2) $y = \frac { 3 } { 4 } x ^ { 2 } + \frac { 1 } { 4 x ^ { 2 } }$
(3) $y = \frac { x ^ { 2 } } { 4 } + \frac { 3 } { 4 x ^ { 2 } }$
(4) $y = \frac { 4 } { 5 } x ^ { 3 } + \frac { 1 } { 5 x ^ { 2 } }$
Q87 Vectors: Cross Product & Distances View
Let $\vec { \alpha } = 3 \hat { i } + \hat { j }$ and $\vec { \beta } = 2 \hat { i } - \hat { j } + 3 \hat { k }$. If $\vec { \beta } = \overrightarrow { \beta _ { 1 } } - \overrightarrow { \beta _ { 2 } }$, where $\overrightarrow { \beta _ { 1 } }$ is parallel to $\vec { \alpha }$ and $\overrightarrow { \beta _ { 2 } }$ is perpendicular to $\vec { \alpha }$, then $\overrightarrow { \beta _ { 1 } } \times \overrightarrow { \beta _ { 2 } }$ is equal to:
(1) $\frac { 1 } { 2 } ( - 3 \hat { i } + 9 \hat { j } + 5 \widehat { k } )$
(2) $3 \hat { i } - 9 \hat { j } - 5 \widehat { k }$
(3) $- 3 \hat { i } + 9 \hat { j } + 5 \widehat { k }$
(4) $\frac { 1 } { 2 } ( 3 \hat { i } - 9 \hat { j } + 5 \hat { k } )$
Q88 Vectors: Lines & Planes Find Cartesian Equation of a Plane View
A plane passing though the points $( 0 , - 1,0 )$ and $( 0,0,1 )$ and making an angle $\frac { \pi } { 4 }$ with the plane $y - z + 5 = 0$, also passes through the point
(1) $( \sqrt { 2 } , - 1,4 )$
(2) $( \sqrt { 2 } , 1,4 )$
(3) $( - \sqrt { 2 } , - 1 , - 4 )$
(4) $( - \sqrt { 2 } , 1 , - 4 )$
Q89 Vectors: Lines & Planes Find Intersection of a Line and a Plane View
If the line, $\frac { x - 1 } { 2 } = \frac { y + 1 } { 3 } = \frac { z - 2 } { 4 }$ meets the plane, $x + 2 y + 3 z = 15$ at a point $P$, then the distance of $P$ from the origin is,
(1) $2 \sqrt { 5 }$
(2) $\frac { 9 } { 2 }$
(3) $\frac { \sqrt { 5 } } { 2 }$
(4) $\frac { 7 } { 2 }$
Q90 Probability Definitions Probability Using Set/Event Algebra View
Four persons can hit a target correctly with probabilities $\frac { 1 } { 2 } , \frac { 1 } { 3 } , \frac { 1 } { 4 }$ and $\frac { 1 } { 8 }$ respectively. If all hit at the target independently, then the probability that the target would be hit, is
(1) $\frac { 25 } { 192 }$
(2) $\frac { 7 } { 32 }$
(3) $\frac { 1 } { 192 }$
(4) $\frac { 25 } { 32 }$