jee-advanced

Q1 4 marks Composite & Inverse Functions Existence or Properties of Functions and Inverses (Proof-Based) View

Let $S = ( 0,1 ) \cup ( 1,2 ) \cup ( 3,4 )$ and $T = \{ 0,1,2,3 \}$. Then which of the following statements is(are) true?
(A) There are infinitely many functions from $S$ to $T$
(B) There are infinitely many strictly increasing functions from $S$ to $T$
(C) The number of continuous functions from $S$ to $T$ is at most 120
(D) Every continuous function from $S$ to $T$ is differentiable

Q2 4 marks Circles Tangent Lines and Tangent Lengths View

Let $T _ { 1 }$ and $T _ { 2 }$ be two distinct common tangents to the ellipse $E : \frac { x ^ { 2 } } { 6 } + \frac { y ^ { 2 } } { 3 } = 1$ and the parabola $P : y ^ { 2 } = 12 x$. Suppose that the tangent $T _ { 1 }$ touches $P$ and $E$ at the points $A _ { 1 }$ and $A _ { 2 }$, respectively and the tangent $T _ { 2 }$ touches $P$ and $E$ at the points $A _ { 4 }$ and $A _ { 3 }$, respectively. Then which of the following statements is(are) true?
(A) The area of the quadrilateral $A _ { 1 } A _ { 2 } A _ { 3 } A _ { 4 }$ is 35 square units
(B) The area of the quadrilateral $A _ { 1 } A _ { 2 } A _ { 3 } A _ { 4 }$ is 36 square units
(C) The tangents $T _ { 1 }$ and $T _ { 2 }$ meet the $x$-axis at the point $( - 3,0 )$
(D) The tangents $T _ { 1 }$ and $T _ { 2 }$ meet the $x$-axis at the point $( - 6,0 )$

Q3 4 marks Areas by integration View

Let $f : [ 0,1 ] \rightarrow [ 0,1 ]$ be the function defined by $f ( x ) = \frac { x ^ { 3 } } { 3 } - x ^ { 2 } + \frac { 5 } { 9 } x + \frac { 17 } { 36 }$. Consider the square region $S = [ 0,1 ] \times [ 0,1 ]$. Let $G = \{ ( x , y ) \in S : y > f ( x ) \}$ be called the green region and $R = \{ ( x , y ) \in S : y < f ( x ) \}$ be called the red region. Let $L _ { h } = \{ ( x , h ) \in S : x \in [ 0,1 ] \}$ be the horizontal line drawn at a height $h \in [ 0,1 ]$. Then which of the following statements is(are) true?
(A) There exists an $h \in \left[ \frac { 1 } { 4 } , \frac { 2 } { 3 } \right]$ such that the area of the green region above the line $L _ { h }$ equals the area of the green region below the line $L _ { h }$
(B) There exists an $h \in \left[ \frac { 1 } { 4 } , \frac { 2 } { 3 } \right]$ such that the area of the red region above the line $L _ { h }$ equals the area of the red region below the line $L _ { h }$
(C) There exists an $h \in \left[ \frac { 1 } { 4 } , \frac { 2 } { 3 } \right]$ such that the area of the green region above the line $L _ { h }$ equals the area of the red region below the line $L _ { h }$
(D) There exists an $h \in \left[ \frac { 1 } { 4 } , \frac { 2 } { 3 } \right]$ such that the area of the red region above the line $L _ { h }$ equals the area of the green region below the line $L _ { h }$

Q4 3 marks Connected Rates of Change Pointwise Limit of a Difference Quotient View

Let $f : ( 0,1 ) \rightarrow \mathbb { R }$ be the function defined as $f ( x ) = \sqrt { n }$ if $x \in \left[ \frac { 1 } { n + 1 } , \frac { 1 } { n } \right)$ where $n \in \mathbb { N }$. Let $g : ( 0,1 ) \rightarrow \mathbb { R }$ be a function such that $\int _ { x ^ { 2 } } ^ { x } \sqrt { \frac { 1 - t } { t } } d t < g ( x ) < 2 \sqrt { x }$ for all $x \in ( 0,1 )$. Then $\lim _ { x \rightarrow 0 } f ( x ) g ( x )$
(A) does NOT exist
(B) is equal to 1
(C) is equal to 2
(D) is equal to 3

Q5 3 marks Vectors 3D & Lines Shortest Distance Between Two Lines View

Let $Q$ be the cube with the set of vertices $\left\{ \left( x _ { 1 } , x _ { 2 } , x _ { 3 } \right) \in \mathbb { R } ^ { 3 } : x _ { 1 } , x _ { 2 } , x _ { 3 } \in \{ 0,1 \} \right\}$. Let $F$ be the set of all twelve lines containing the diagonals of the six faces of the cube $Q$. Let $S$ be the set of all four lines containing the main diagonals of the cube $Q$; for instance, the line passing through the vertices $( 0,0,0 )$ and $( 1,1,1 )$ is in $S$. For lines $\ell _ { 1 }$ and $\ell _ { 2 }$, let $d \left( \ell _ { 1 } , \ell _ { 2 } \right)$ denote the shortest distance between them. Then the maximum value of $d \left( \ell _ { 1 } , \ell _ { 2 } \right)$, as $\ell _ { 1 }$ varies over $F$ and $\ell _ { 2 }$ varies over $S$, is
(A) $\frac { 1 } { \sqrt { 6 } }$
(B) $\frac { 1 } { \sqrt { 8 } }$
(C) $\frac { 1 } { \sqrt { 3 } }$
(D) $\frac { 1 } { \sqrt { 12 } }$

Q6 3 marks Geometric Probability View

Let $X = \left\{ ( x , y ) \in \mathbb { Z } \times \mathbb { Z } : \frac { x ^ { 2 } } { 8 } + \frac { y ^ { 2 } } { 20 } < 1 \right.$ and $\left. y ^ { 2 } < 5 x \right\}$. Three distinct points $P , Q$ and $R$ are randomly chosen from $X$. Then the probability that $P , Q$ and $R$ form a triangle whose area is a positive integer, is
(A) $\frac { 71 } { 220 }$
(B) $\frac { 73 } { 220 }$
(C) $\frac { 79 } { 220 }$
(D) $\frac { 83 } { 220 }$

Q7 3 marks Conic sections Tangent and Normal Line Problems View

Let $P$ be a point on the parabola $y ^ { 2 } = 4 a x$, where $a > 0$. The normal to the parabola at $P$ meets the $x$-axis at a point $Q$. The area of the triangle $P F Q$, where $F$ is the focus of the parabola, is 120. If the slope $m$ of the normal and $a$ are both positive integers, then the pair $( a , m )$ is
(A) $( 2,3 )$
(B) $( 1,3 )$
(C) $( 2,4 )$
(D) $( 3,4 )$

Q8 4 marks Standard trigonometric equations Inverse trigonometric equation View

Let $\tan ^ { - 1 } ( x ) \in \left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$, for $x \in \mathbb { R }$. Then the number of real solutions of the equation $\sqrt { 1 + \cos ( 2 x ) } = \sqrt { 2 } \tan ^ { - 1 } ( \tan x )$ in the set $\left( - \frac { 3 \pi } { 2 } , - \frac { \pi } { 2 } \right) \cup \left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right) \cup \left( \frac { \pi } { 2 } , \frac { 3 \pi } { 2 } \right)$ is equal to

Q9 4 marks Areas by integration View

Let $n \geq 2$ be a natural number and $f : [ 0,1 ] \rightarrow \mathbb { R }$ be the function defined by
$$f ( x ) = \begin{cases} n ( 1 - 2 n x ) & \text { if } 0 \leq x \leq \frac { 1 } { 2 n } \\ 2 n ( 2 n x - 1 ) & \text { if } \frac { 1 } { 2 n } \leq x \leq \frac { 3 } { 4 n } \\ 4 n ( 1 - n x ) & \text { if } \frac { 3 } { 4 n } \leq x \leq \frac { 1 } { n } \\ \frac { n } { n - 1 } ( n x - 1 ) & \text { if } \frac { 1 } { n } \leq x \leq 1 \end{cases}$$
If $n$ is such that the area of the region bounded by the curves $x = 0 , x = 1 , y = 0$ and $y = f ( x )$ is 4, then the maximum value of the function $f$ is

Q10 4 marks Arithmetic Sequences and Series Telescoping or Non-Standard Summation Involving an AP View

Let $7 \overbrace { 5 \cdots 5 } ^ { r } 7$ denote the $( r + 2 )$ digit number where the first and the last digits are 7 and the remaining $r$ digits are 5. Consider the sum $S = 77 + 757 + 7557 + \cdots + 7 \overbrace { 5 \cdots 5 } ^ { 98 } 7$. If $S = \frac { 7 \overbrace { 5 \cdots 5 } ^ { 99 } 7 + m } { n }$, where $m$ and $n$ are natural numbers less than 3000, then the value of $m + n$ is

Q11 4 marks Complex Numbers Argand & Loci Algebraic Conditions for Geometric Properties (Real, Imaginary, Collinear) View

Let $A = \left\{ \frac { 1967 + 1686 i \sin \theta } { 7 - 3 i \cos \theta } : \theta \in \mathbb { R } \right\}$. If $A$ contains exactly one positive integer $n$, then the value of $n$ is

Q12 4 marks Vectors: Lines & Planes Volume of Pyramid/Tetrahedron Using Planes and Lines View

Let $P$ be the plane $\sqrt { 3 } x + 2 y + 3 z = 16$ and let $S = \left\{ \alpha \hat { i } + \beta \hat { j } + \gamma \hat { k } : \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 1 \right.$ and the distance of $( \alpha , \beta , \gamma )$ from the plane $P$ is $\left. \frac { 7 } { 2 } \right\}$. Let $\vec { u } , \vec { v }$ and $\vec { w }$ be three distinct vectors in $S$ such that $| \vec { u } - \vec { v } | = | \vec { v } - \vec { w } | = | \vec { w } - \vec { u } |$. Let $V$ be the volume of the parallelepiped determined by vectors $\vec { u } , \vec { v }$ and $\vec { w }$. Then the value of $\frac { 80 } { \sqrt { 3 } } V$ is

Q13 4 marks Binomial Theorem (positive integer n) Determine Parameters from Conditions on Coefficients or Terms View

Let $a$ and $b$ be two nonzero real numbers. If the coefficient of $x ^ { 5 }$ in the expansion of $\left( a x ^ { 2 } + \frac { 70 } { 27 b x } \right) ^ { 4 }$ is equal to the coefficient of $x ^ { - 5 }$ in the expansion of $\left( a x - \frac { 1 } { b x ^ { 2 } } \right) ^ { 7 }$, then the value of $2 b$ is

Q14 3 marks Matrices Linear System and Inverse Existence View

Let $\alpha , \beta$ and $\gamma$ be real numbers. Consider the following system of linear equations
$x + 2 y + z = 7$
$x + \alpha z = 11$
$2 x - 3 y + \beta z = \gamma$
Match each entry in List-I to the correct entries in List-II.
List-I
(P) If $\beta = \frac { 1 } { 2 } ( 7 \alpha - 3 )$ and $\gamma = 28$, then the system has
(Q) If $\beta = \frac { 1 } { 2 } ( 7 \alpha - 3 )$ and $\gamma \neq 28$, then the system has
(R) If $\beta \neq \frac { 1 } { 2 } ( 7 \alpha - 3 )$ where $\alpha = 1$ and $\gamma \neq 28$, then the system has
(S) If $\beta \neq \frac { 1 } { 2 } ( 7 \alpha - 3 )$ where $\alpha = 1$ and $\gamma = 28$, then the system has
List-II
(1) a unique solution
(2) no solution
(3) infinitely many solutions
(4) $x = 11 , y = - 2$ and $z = 0$ as a solution
(5) $x = - 15 , y = 4$ and $z = 0$ as a solution
The correct option is:
(A) $( P ) \rightarrow ( 3 ) \quad ( Q ) \rightarrow ( 2 ) \quad ( R ) \rightarrow ( 1 ) \quad ( S ) \rightarrow ( 4 )$
(B) $( P ) \rightarrow ( 3 ) \quad ( Q ) \rightarrow ( 2 ) \quad ( R ) \rightarrow ( 5 ) \quad ( S ) \rightarrow ( 4 )$
(C) $( P ) \rightarrow ( 2 ) \quad ( Q ) \rightarrow ( 1 ) \quad ( R ) \rightarrow ( 4 ) \quad ( S ) \rightarrow ( 5 )$
(D) $( P ) \rightarrow ( 2 ) \quad ( Q ) \rightarrow ( 1 ) \quad ( R ) \rightarrow ( 1 ) \quad ( S ) \rightarrow ( 3 )$

Q15 3 marks Measures of Location and Spread View

Consider the given data with frequency distribution

$x _ { i }$	3	8	11	10	5	4
$f _ { i }$	5	2	3	2	4	4

Match each entry in List-I to the correct entries in List-II.
List-I
(P) The mean of the above data is
(Q) The median of the above data is
(R) The mean deviation about the mean of the above data is
(S) The mean deviation about the median of the above data is
List-II
(1) 2.5
(2) 5
(3) 6
(4) 2.7
(5) 2.4
The correct option is:
(A) $( P ) \rightarrow ( 3 )$ $( Q ) \rightarrow ( 2 )$ $( R ) \rightarrow ( 4 )$ $( S ) \rightarrow ( 5 )$
(B) $( P ) \rightarrow ( 3 )$ $( Q ) \rightarrow ( 2 )$ $( R ) \rightarrow ( 1 )$ $( S ) \rightarrow ( 5 )$
(C) $( P ) \rightarrow ( 2 )$ $( Q ) \rightarrow ( 3 )$ $( R ) \rightarrow ( 4 )$ $( S ) \rightarrow ( 1 )$
(D) $( P ) \rightarrow ( 3 )$ $( Q ) \rightarrow ( 3 )$ $( R ) \rightarrow ( 5 )$ $( S ) \rightarrow ( 5 )$

Q16 3 marks Vectors: Lines & Planes Distance Computation (Point-to-Plane or Line-to-Line) View

Let $\ell _ { 1 }$ and $\ell _ { 2 }$ be the lines $\vec { r } _ { 1 } = \lambda ( \hat { i } + \hat { j } + \hat { k } )$ and $\vec { r } _ { 2 } = ( \hat { j } - \hat { k } ) + \mu ( \hat { i } + \hat { k } )$, respectively. Let $X$ be the set of all the planes $H$ that contain the line $\ell _ { 1 }$. For a plane $H$, let $d ( H )$ denote the smallest possible distance between the points of $\ell _ { 2 }$ and $H$. Let $H _ { 0 }$ be a plane in $X$ for which $d \left( H _ { 0 } \right)$ is the maximum value of $d ( H )$ as $H$ varies over all planes in $X$.
Match each entry in List-I to the correct entries in List-II.
List-I
(P) The value of $d \left( H _ { 0 } \right)$ is
(Q) The distance of the point $( 0,1,2 )$ from $H _ { 0 }$ is
(R) The distance of origin from $H _ { 0 }$ is
(S) The distance of origin from the point of intersection of planes $y = z , x = 1$ and $H _ { 0 }$ is
List-II
(1) $\sqrt { 3 }$
(2) $\frac { 1 } { \sqrt { 3 } }$
(3) 0
(4) $\sqrt { 2 }$
(5) $\frac { 1 } { \sqrt { 2 } }$
The correct option is:
(A) $( P ) \rightarrow ( 2 )$ $( Q ) \rightarrow ( 4 )$ $( R ) \rightarrow ( 5 )$ $( S ) \rightarrow ( 1 )$
(B) $( P ) \rightarrow ( 5 )$ $( Q ) \rightarrow ( 4 )$ $( R ) \rightarrow ( 3 )$ $( S ) \rightarrow ( 1 )$
(C) $( P ) \rightarrow ( 2 )$ $( Q ) \rightarrow ( 1 )$ $( R ) \rightarrow ( 3 )$ $( S ) \rightarrow ( 2 )$
(D) $( P ) \rightarrow ( 5 )$ $( Q ) \rightarrow ( 1 )$ $( R ) \rightarrow ( 4 )$ $( S ) \rightarrow ( 2 )$

Q17 3 marks Complex Numbers Argand & Loci Solving Complex Equations with Geometric Interpretation View

Let $z$ be a complex number satisfying $| z | ^ { 3 } + 2 z ^ { 2 } + 4 \bar { z } - 8 = 0$, where $\bar { z }$ denotes the complex conjugate of $z$. Let the imaginary part of $z$ be nonzero.
Match each entry in List-I to the correct entries in List-II.
List-I
(P) $| z | ^ { 2 }$ is equal to
(Q) $| z - \bar { z } | ^ { 2 }$ is equal to
(R) $| z | ^ { 2 } + | z + \bar { z } | ^ { 2 }$ is equal to
(S) $| z + 1 | ^ { 2 }$ is equal to
List-II
(1) 12
(2) 4
(3) 8
(4) 10
(5) 7
The correct option is:
(A) $( P ) \rightarrow ( 1 ) \quad ( Q ) \rightarrow ( 3 ) \quad ( R ) \rightarrow ( 5 ) \quad ( S ) \rightarrow ( 4 )$
(B) $( P ) \rightarrow ( 2 ) \quad ( Q ) \rightarrow ( 1 ) \quad ( R ) \rightarrow ( 3 ) \quad ( S ) \rightarrow ( 5 )$
(C) $( P ) \rightarrow ( 2 ) \quad ( Q ) \rightarrow ( 4 ) \quad ( R ) \rightarrow ( 5 ) \quad ( S ) \rightarrow ( 1 )$
(D) $( P ) \rightarrow ( 2 ) \quad ( Q ) \rightarrow ( 3 ) \quad ( R ) \rightarrow ( 5 ) \quad ( S ) \rightarrow ( 4 )$

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