jee-advanced

Q1 3 marks Differential equations Finding a DE from a Limit or Implicit Condition View

Let $f ( x )$ be a continuously differentiable function on the interval $( 0 , \infty )$ such that $f ( 1 ) = 2$ and
$$\lim _ { t \rightarrow x } \frac { t ^ { 10 } f ( x ) - x ^ { 10 } f ( t ) } { t ^ { 9 } - x ^ { 9 } } = 1$$
for each $x > 0$. Then, for all $x > 0 , f ( x )$ is equal to
(A) $\frac { 31 } { 11 x } - \frac { 9 } { 11 } x ^ { 10 }$
(B) $\frac { 9 } { 11 x } + \frac { 13 } { 11 } x ^ { 10 }$
(C) $\frac { - 9 } { 11 x } + \frac { 31 } { 11 } x ^ { 10 }$
(D) $\frac { 13 } { 11 x } + \frac { 9 } { 11 } x ^ { 10 }$

Q2 3 marks Conditional Probability Bayes' Theorem with Production/Source Identification View

A student appears for a quiz consisting of only true-false type questions and answers all the questions. The student knows the answers of some questions and guesses the answers for the remaining questions. Whenever the student knows the answer of a question, he gives the correct answer. Assume that the probability of the student giving the correct answer for a question, given that he has guessed it, is $\frac { 1 } { 2 }$. Also assume that the probability of the answer for a question being guessed, given that the student's answer is correct, is $\frac { 1 } { 6 }$. Then the probability that the student knows the answer of a randomly chosen question is
(A) $\frac { 1 } { 12 }$
(B) $\frac { 1 } { 7 }$
(C) $\frac { 5 } { 7 }$
(D) $\frac { 5 } { 12 }$

Q3 3 marks Addition & Double Angle Formulae Multi-Step Composite Problem Using Identities View

Let $\frac { \pi } { 2 } < x < \pi$ be such that $\cot x = \frac { - 5 } { \sqrt { 11 } }$. Then
$$\left( \sin \frac { 11 x } { 2 } \right) ( \sin 6 x - \cos 6 x ) + \left( \cos \frac { 11 x } { 2 } \right) ( \sin 6 x + \cos 6 x )$$
is equal to
(A) $\frac { \sqrt { 11 } - 1 } { 2 \sqrt { 3 } }$
(B) $\frac { \sqrt { 11 } + 1 } { 2 \sqrt { 3 } }$
(C) $\frac { \sqrt { 11 } + 1 } { 3 \sqrt { 2 } }$
(D) $\frac { \sqrt { 11 } - 1 } { 3 \sqrt { 2 } }$

Q4 3 marks Conic sections Tangent and Normal Line Problems View

Consider the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$. Let $S ( p , q )$ be a point in the first quadrant such that $\frac { p ^ { 2 } } { 9 } + \frac { q ^ { 2 } } { 4 } > 1$. Two tangents are drawn from $S$ to the ellipse, of which one meets the ellipse at one end point of the minor axis and the other meets the ellipse at a point $T$ in the fourth quadrant. Let $R$ be the vertex of the ellipse with positive $x$-coordinate and $O$ be the center of the ellipse. If the area of the triangle $\triangle O R T$ is $\frac { 3 } { 2 }$, then which of the following options is correct?
(A) $q = 2 , p = 3 \sqrt { 3 }$
(B) $q = 2 , p = 4 \sqrt { 3 }$
(C) $q = 1 , p = 5 \sqrt { 3 }$
(D) $q = 1 , p = 6 \sqrt { 3 }$

Q5 4 marks Complex Numbers Arithmetic True/False or Property Verification Statements View

Let $S = \{ a + b \sqrt { 2 } : a , b \in \mathbb { Z } \} , T _ { 1 } = \left\{ ( - 1 + \sqrt { 2 } ) ^ { n } : n \in \mathbb { N } \right\}$, and $T _ { 2 } = \left\{ ( 1 + \sqrt { 2 } ) ^ { n } : n \in \mathbb { N } \right\}$.
Then which of the following statements is (are) TRUE?
(A) $\mathbb { Z } \bigcup T _ { 1 } \bigcup T _ { 2 } \subset S$
(B) $T _ { 1 } \cap \left( 0 , \frac { 1 } { 2024 } \right) = \phi$, where $\phi$ denotes the empty set.
(C) $T _ { 2 } \cap ( 2024 , \infty ) \neq \phi$
(D) For any given $a , b \in \mathbb { Z } , \cos ( \pi ( a + b \sqrt { 2 } ) ) + i \sin ( \pi ( a + b \sqrt { 2 } ) ) \in \mathbb { Z }$ if and only if $b = 0$, where $i = \sqrt { - 1 }$.

Q6 4 marks Discriminant and conditions for roots Parameter range for no real roots (positive definite) View

Let $\mathbb { R } ^ { 2 }$ denote $\mathbb { R } \times \mathbb { R }$. Let
$$S = \left\{ ( a , b , c ) : a , b , c \in \mathbb { R } \text { and } a x ^ { 2 } + 2 b x y + c y ^ { 2 } > 0 \text { for all } ( x , y ) \in \mathbb { R } ^ { 2 } - \{ ( 0,0 ) \} \right\}$$
Then which of the following statements is (are) TRUE?
(A) $\left( 2 , \frac { 7 } { 2 } , 6 \right) \in S$
(B) If $\left( 3 , b , \frac { 1 } { 12 } \right) \in S$, then $| 2 b | < 1$.
(C) For any given $( a , b , c ) \in S$, the system of linear equations
$$\begin{aligned} & a x + b y = 1 \\ & b x + c y = - 1 \end{aligned}$$
has a unique solution.
(D) For any given $( a , b , c ) \in S$, the system of linear equations
$$\begin{aligned} & ( a + 1 ) x + b y = 0 \\ & b x + ( c + 1 ) y = 0 \end{aligned}$$
has a unique solution.

Q7 4 marks Straight Lines & Coordinate Geometry Locus Determination View

Let $\mathbb { R } ^ { 3 }$ denote the three-dimensional space. Take two points $P = ( 1,2,3 )$ and $Q = ( 4,2,7 )$. Let $\operatorname { dist } ( X , Y )$ denote the distance between two points $X$ and $Y$ in $\mathbb { R } ^ { 3 }$. Let
$$\begin{gathered} S = \left\{ X \in \mathbb { R } ^ { 3 } : ( \operatorname { dist } ( X , P ) ) ^ { 2 } - ( \operatorname { dist } ( X , Q ) ) ^ { 2 } = 50 \right\} \text { and } \\ T = \left\{ Y \in \mathbb { R } ^ { 3 } : ( \operatorname { dist } ( Y , Q ) ) ^ { 2 } - ( \operatorname { dist } ( Y , P ) ) ^ { 2 } = 50 \right\} \end{gathered}$$
Then which of the following statements is (are) TRUE?
(A) There is a triangle whose area is 1 and all of whose vertices are from $S$.
(B) There are two distinct points $L$ and $M$ in $T$ such that each point on the line segment $L M$ is also in $T$.
(C) There are infinitely many rectangles of perimeter 48, two of whose vertices are from $S$ and the other two vertices are from $T$.
(D) There is a square of perimeter 48, two of whose vertices are from $S$ and the other two vertices are from $T$.

Q8 4 marks Laws of Logarithms Solve a Logarithmic Equation View

Let $a = 3 \sqrt { 2 }$ and $b = \frac { 1 } { 5 ^ { 1 / 6 } \sqrt { 6 } }$. If $x , y \in \mathbb { R }$ are such that
$$\begin{aligned} & 3 x + 2 y = \log _ { a } ( 18 ) ^ { \frac { 5 } { 4 } } \\ & 2 x - y = \log _ { b } ( \sqrt { 1080 } ) \end{aligned}$$
then $4 x + 5 y$ is equal to $\_\_\_\_$ .

Q9 4 marks Roots of polynomials Vieta's formulas: compute symmetric functions of roots View

Let $f ( x ) = x ^ { 4 } + a x ^ { 3 } + b x ^ { 2 } + c$ be a polynomial with real coefficients such that $f ( 1 ) = - 9$. Suppose that $i \sqrt { 3 }$ is a root of the equation $4 x ^ { 3 } + 3 a x ^ { 2 } + 2 b x = 0$, where $i = \sqrt { - 1 }$. If $\alpha _ { 1 } , \alpha _ { 2 } , \alpha _ { 3 }$, and $\alpha _ { 4 }$ are all the roots of the equation $f ( x ) = 0$, then $\left| \alpha _ { 1 } \right| ^ { 2 } + \left| \alpha _ { 2 } \right| ^ { 2 } + \left| \alpha _ { 3 } \right| ^ { 2 } + \left| \alpha _ { 4 } \right| ^ { 2 }$ is equal to $\_\_\_\_$ .

Q10 4 marks Matrices Determinant and Rank Computation View

Let $S = \left\{ A = \left( \begin{array} { l l l } 0 & 1 & c \\ 1 & a & d \\ 1 & b & e \end{array} \right) : a , b , c , d , e \in \{ 0,1 \} \right.$ and $\left. | A | \in \{ - 1,1 \} \right\}$, where $| A |$ denotes the determinant of $A$. Then the number of elements in $S$ is $\_\_\_\_$ .

Q11 4 marks Combinations & Selection Partitioning into Teams or Groups View

A group of 9 students, $s _ { 1 } , s _ { 2 } , \ldots , s _ { 9 }$, is to be divided to form three teams $X , Y$, and $Z$ of sizes 2,3 , and 4 , respectively. Suppose that $s _ { 1 }$ cannot be selected for the team $X$, and $s _ { 2 }$ cannot be selected for the team $Y$. Then the number of ways to form such teams, is $\_\_\_\_$ .

Q12 4 marks Vectors 3D & Lines Vector Algebra and Triple Product Computation View

Let $\overrightarrow { O P } = \frac { \alpha - 1 } { \alpha } \hat { i } + \hat { j } + \hat { k } , \overrightarrow { O Q } = \hat { i } + \frac { \beta - 1 } { \beta } \hat { j } + \hat { k }$ and $\overrightarrow { O R } = \hat { i } + \hat { j } + \frac { 1 } { 2 } \hat { k }$ be three vectors, where $\alpha , \beta \in \mathbb { R } - \{ 0 \}$ and $O$ denotes the origin. If $( \overrightarrow { O P } \times \overrightarrow { O Q } ) \cdot \overrightarrow { O R } = 0$ and the point $( \alpha , \beta , 2 )$ lies on the plane $3 x + 3 y - z + l = 0$, then the value of $l$ is $\_\_\_\_$ .

Q13 4 marks Discrete Probability Distributions Probability Distribution Table Completion and Expectation Calculation View

Let $X$ be a random variable, and let $P ( X = x )$ denote the probability that $X$ takes the value $x$. Suppose that the points $( x , P ( X = x ) ) , x = 0,1,2,3,4$, lie on a fixed straight line in the $x y$-plane, and $P ( X = x ) = 0$ for all $x \in \mathbb { R } - \{ 0,1,2,3,4 \}$. If the mean of $X$ is $\frac { 5 } { 2 }$, and the variance of $X$ is $\alpha$, then the value of $24 \alpha$ is $\_\_\_\_$ .

Q14 3 marks Matrices Matrix Entry and Coefficient Identities View

Let $\alpha$ and $\beta$ be the distinct roots of the equation $x ^ { 2 } + x - 1 = 0$. Consider the set $T = \{ 1 , \alpha , \beta \}$. For a $3 \times 3$ matrix $M = \left( a _ { i j } \right) _ { 3 \times 3 }$, define $R _ { i } = a _ { i 1 } + a _ { i 2 } + a _ { i 3 }$ and $C _ { j } = a _ { 1 j } + a _ { 2 j } + a _ { 3 j }$ for $i = 1,2,3$ and $j = 1,2,3$.
Match each entry in List-I to the correct entry in List-II.
List-I
(P) The number of matrices $M = \left( a _ { i j } \right) _ { 3 \times 3 }$ with all entries in $T$ such that $R _ { i } = C _ { j } = 0$ for all $i , j$, is
(Q) The number of symmetric matrices $M = \left( a _ { i j } \right) _ { 3 \times 3 }$ with all entries in $T$ such that $C _ { j } = 0$ for all $j$, is
(R) Let $M = \left( a _ { i j } \right) _ { 3 \times 3 }$ be a skew symmetric matrix such that $a _ { i j } \in T$ for $i > j$. Then the number of elements in the set $\left\{ \left( \begin{array} { l } x \\ y \\ z \end{array} \right) : x , y , z \in \mathbb { R } , M \left( \begin{array} { l } x \\ y \\ z \end{array} \right) = \left( \begin{array} { c } a _ { 12 } \\ 0 \\ - a _ { 23 } \end{array} \right) \right\}$ is
(S) Let $M = \left( a _ { i j } \right) _ { 3 \times 3 }$ be a matrix with all entries in $T$ such that $R _ { i } = 0$ for all $i$. Then the absolute value of the determinant of $M$ is
List-II
(1) 1
(2) 12
(3) infinite
(4) 6
(5) 0
The correct option is
(A) $( \mathrm { P } ) \rightarrow ( 4 )$, $( \mathrm { Q } ) \rightarrow ( 2 )$, $( \mathrm { R } ) \rightarrow ( 5 )$, $( \mathrm { S } ) \rightarrow ( 1 )$
(B) $( \mathrm { P } ) \rightarrow ( 2 )$, $( \mathrm { Q } ) \rightarrow ( 4 )$, $( \mathrm { R } ) \rightarrow ( 1 )$, $( \mathrm { S } ) \rightarrow ( 5 )$
(C) $( \mathrm { P } ) \rightarrow ( 2 )$, $( \mathrm { Q } ) \rightarrow ( 4 )$, $( \mathrm { R } ) \rightarrow ( 3 )$, $( \mathrm { S } ) \rightarrow ( 5 )$
(D) $( \mathrm { P } ) \rightarrow ( 1 )$, $( \mathrm { Q } ) \rightarrow ( 5 )$, $( \mathrm { R } ) \rightarrow ( 3 )$, $( \mathrm { S } ) \rightarrow ( 4 )$

Q15 3 marks Circles Circles Tangent to Each Other or to Axes View

Let the straight line $y = 2 x$ touch a circle with center $( 0 , \alpha ) , \alpha > 0$, and radius $r$ at a point $A _ { 1 }$. Let $B _ { 1 }$ be the point on the circle such that the line segment $A _ { 1 } B _ { 1 }$ is a diameter of the circle. Let $\alpha + r = 5 + \sqrt { 5 }$.
Match each entry in List-I to the correct entry in List-II.
List-I
(P) $\alpha$ equals
(Q) $r$ equals
(R) $A _ { 1 }$ equals
(S) $B _ { 1 }$ equals
List-II
(1) $( - 2,4 )$
(2) $\sqrt { 5 }$
(3) $( - 2,6 )$
(4) 5
(5) $( 2,4 )$
The correct option is
(A) $( \mathrm { P } ) \rightarrow ( 4 )$, $( \mathrm { Q } ) \rightarrow ( 2 )$, $( \mathrm { R } ) \rightarrow ( 1 )$, $( \mathrm { S } ) \rightarrow ( 3 )$
(B) $( \mathrm { P } ) \rightarrow ( 2 )$, $( \mathrm { Q } ) \rightarrow ( 4 )$, $( \mathrm { R } ) \rightarrow ( 1 )$, $( \mathrm { S } ) \rightarrow ( 3 )$
(C) $( \mathrm { P } ) \rightarrow ( 4 )$, $( \mathrm { Q } ) \rightarrow ( 2 )$, $( \mathrm { R } ) \rightarrow ( 5 )$, $( \mathrm { S } ) \rightarrow ( 3 )$
(D) $( \mathrm { P } ) \rightarrow ( 2 )$, $( \mathrm { Q } ) \rightarrow ( 4 )$, $( \mathrm { R } ) \rightarrow ( 3 )$, $( \mathrm { S } ) \rightarrow ( 5 )$

Q16 3 marks Vectors: Lines & Planes MCQ: Identify Correct Equation or Representation View

Let $\gamma \in \mathbb { R }$ be such that the lines $L _ { 1 } : \frac { x + 11 } { 1 } = \frac { y + 21 } { 2 } = \frac { z + 29 } { 3 }$ and $L _ { 2 } : \frac { x + 16 } { 3 } = \frac { y + 11 } { 2 } = \frac { z + 4 } { \gamma }$ intersect. Let $R _ { 1 }$ be the point of intersection of $L _ { 1 }$ and $L _ { 2 }$. Let $O = ( 0,0,0 )$, and $\hat { n }$ denote a unit normal vector to the plane containing both the lines $L _ { 1 }$ and $L _ { 2 }$.
Match each entry in List-I to the correct entry in List-II.
List-I
(P) $\gamma$ equals
(Q) A possible choice for $\hat { n }$ is
(R) $\overrightarrow { O R _ { 1 } }$ equals
(S) A possible value of $\overrightarrow { O R _ { 1 } } \cdot \hat { n }$ is
List-II
(1) $- \hat { i } - \hat { j } + \hat { k }$
(2) $\sqrt { \frac { 3 } { 2 } }$
(3) 1
(4) $\frac { 1 } { \sqrt { 6 } } \hat { i } - \frac { 2 } { \sqrt { 6 } } \hat { j } + \frac { 1 } { \sqrt { 6 } } \hat { k }$
(5) $\sqrt { \frac { 2 } { 3 } }$
The correct option is
(A) $(\mathrm{P}) \rightarrow (3)$, $(\mathrm{Q}) \rightarrow (4)$, $(\mathrm{R}) \rightarrow (1)$, $(\mathrm{S}) \rightarrow (2)$
(B) $(\mathrm{P}) \rightarrow (5)$, $(\mathrm{Q}) \rightarrow (4)$, $(\mathrm{R}) \rightarrow (1)$, $(\mathrm{S}) \rightarrow (2)$
(C) $(\mathrm{P}) \rightarrow (3)$, $(\mathrm{Q}) \rightarrow (4)$, $(\mathrm{R}) \rightarrow (1)$, $(\mathrm{S}) \rightarrow (5)$
(D) $(\mathrm{P}) \rightarrow (3)$, $(\mathrm{Q}) \rightarrow (1)$, $(\mathrm{R}) \rightarrow (4)$, $(\mathrm{S}) \rightarrow (5)$

Q17 3 marks Curve Sketching Continuity and Differentiability of Special Functions View

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ and $g : \mathbb { R } \rightarrow \mathbb { R }$ be functions defined by
$$f ( x ) = \left\{ \begin{array} { l l } x | x | \sin \left( \frac { 1 } { x } \right) , & x \neq 0 , \\ 0 , & x = 0 , \end{array} \quad \text { and } \quad g ( x ) = \begin{cases} 1 - 2 x , & 0 \leq x \leq \frac { 1 } { 2 } \\ 0 , & \text { otherwise } \end{cases} \right.$$
Let $a , b , c , d \in \mathbb { R }$. Define the function $h : \mathbb { R } \rightarrow \mathbb { R }$ by
$$h ( x ) = a f ( x ) + b \left( g ( x ) + g \left( \frac { 1 } { 2 } - x \right) \right) + c ( x - g ( x ) ) + d g ( x ) , x \in \mathbb { R }$$
Match each entry in List-I to the correct entry in List-II.
List-I
(P) If $a = 0 , b = 1 , c = 0$, and $d = 0$, then
(Q) If $a = 1 , b = 0 , c = 0$, and $d = 0$, then
(R) If $a = 0 , b = 0 , c = 1$, and $d = 0$, then
(S) If $a = 0 , b = 0 , c = 0$, and $d = 1$, then
List-II
(1) $h$ is one-one.
(2) $h$ is onto.
(3) $h$ is differentiable on $\mathbb { R }$.
(4) the range of $h$ is $[ 0,1 ]$.
(5) the range of $h$ is $\{ 0,1 \}$.
The correct option is
(A) $(\mathrm{P}) \rightarrow (4)$, $(\mathrm{Q}) \rightarrow (3)$, $(\mathrm{R}) \rightarrow (1)$, $(\mathrm{S}) \rightarrow (2)$
(B) $(\mathrm{P}) \rightarrow (5)$, $(\mathrm{Q}) \rightarrow (2)$, $(\mathrm{R}) \rightarrow (4)$, $(\mathrm{S}) \rightarrow (3)$
(C) $(\mathrm{P}) \rightarrow (5)$, $(\mathrm{Q}) \rightarrow (3)$, $(\mathrm{R}) \rightarrow (2)$, $(\mathrm{S}) \rightarrow (4)$
(D) $(\mathrm{P}) \rightarrow (4)$, $(\mathrm{Q}) \rightarrow (2)$, $(\mathrm{R}) \rightarrow (1)$, $(\mathrm{S}) \rightarrow (3)$

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