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Q1 3 marks Addition & Double Angle Formulae Addition/Subtraction Formula Evaluation View

Considering only the principal values of the inverse trigonometric functions, the value of
$$\tan \left( \sin ^ { - 1 } \left( \frac { 3 } { 5 } \right) - 2 \cos ^ { - 1 } \left( \frac { 2 } { \sqrt { 5 } } \right) \right)$$
is
(A) $\frac { 7 } { 24 }$
(B) $\frac { - 7 } { 24 }$
(C) $\frac { - 5 } { 24 }$
(D) $\frac { 5 } { 24 }$

Q2 3 marks Areas Between Curves Area Between Curves with Parametric or Implicit Region Definition View

Let $S = \left\{ ( x , y ) \in \mathbb { R } \times \mathbb { R } : x \geq 0 , y \geq 0 , y ^ { 2 } \leq 4 x , y ^ { 2 } \leq 12 - 2 x \right.$ and $\left. 3 y + \sqrt { 8 } x \leq 5 \sqrt { 8 } \right\}$. If the area of the region $S$ is $\alpha \sqrt { 2 }$, then $\alpha$ is equal to
(A) $\frac { 17 } { 2 }$
(B) $\frac { 17 } { 3 }$
(C) $\frac { 17 } { 4 }$
(D) $\frac { 17 } { 5 }$

Q3 3 marks Chain Rule Limit Evaluation Involving Composition or Substitution View

Let $k \in \mathbb { R }$. If $\lim _ { x \rightarrow 0 + } ( \sin ( \sin k x ) + \cos x + x ) ^ { \frac { 2 } { x } } = e ^ { 6 }$, then the value of $k$ is
(A) 1
(B) 2
(C) 3
(D) 4

Q4 3 marks Curve Sketching Number of Solutions / Roots via Curve Analysis View

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a function defined by
$$f ( x ) = \left\{ \begin{array} { c l } x ^ { 2 } \sin \left( \frac { \pi } { x ^ { 2 } } \right) , & \text { if } x \neq 0 \\ 0 , & \text { if } x = 0 \end{array} \right.$$
Then which of the following statements is TRUE?
(A) $f ( x ) = 0$ has infinitely many solutions in the interval $\left[ \frac { 1 } { 10 ^ { 10 } } , \infty \right)$.
(B) $f ( x ) = 0$ has no solutions in the interval $\left[ \frac { 1 } { \pi } , \infty \right)$.
(C) The set of solutions of $f ( x ) = 0$ in the interval $\left( 0 , \frac { 1 } { 10 ^ { 10 } } \right)$ is finite.
(D) $f ( x ) = 0$ has more than 25 solutions in the interval $\left( \frac { 1 } { \pi ^ { 2 } } , \frac { 1 } { \pi } \right)$.

Q5 4 marks Sequences and series, recurrence and convergence Convergence proof and limit determination View

Let $S$ be the set of all $( \alpha , \beta ) \in \mathbb { R } \times \mathbb { R }$ such that
$$\lim _ { x \rightarrow \infty } \frac { \sin \left( x ^ { 2 } \right) \left( \log _ { e } x \right) ^ { \alpha } \sin \left( \frac { 1 } { x ^ { 2 } } \right) } { x ^ { \alpha \beta } \left( \log _ { e } ( 1 + x ) \right) ^ { \beta } } = 0 .$$
Then which of the following is (are) correct?
(A) $( - 1,3 ) \in S$
(B) $( - 1,1 ) \in S$
(C) $( 1 , - 1 ) \in S$
(D) $( 1 , - 2 ) \in S$

Q6 4 marks Vectors 3D & Lines Line-Plane Intersection View

A straight line drawn from the point $P ( 1,3,2 )$, parallel to the line $\frac { x - 2 } { 1 } = \frac { y - 4 } { 2 } = \frac { z - 6 } { 1 }$, intersects the plane $L _ { 1 } : x - y + 3 z = 6$ at the point $Q$. Another straight line which passes through $Q$ and is perpendicular to the plane $L _ { 1 }$ intersects the plane $L _ { 2 } : 2 x - y + z = - 4$ at the point $R$. Then which of the following statements is (are) TRUE?
(A) The length of the line segment $PQ$ is $\sqrt { 6 }$
(B) The coordinates of $R$ are $( 1,6,3 )$
(C) The centroid of the triangle $PQR$ is $\left( \frac { 4 } { 3 } , \frac { 14 } { 3 } , \frac { 5 } { 3 } \right)$
(D) The perimeter of the triangle $PQR$ is $\sqrt { 2 } + \sqrt { 6 } + \sqrt { 11 }$

Q7 4 marks Circles Tangent Lines and Tangent Lengths View

Let $A _ { 1 } , B _ { 1 } , C _ { 1 }$ be three points in the $xy$-plane. Suppose that the lines $A _ { 1 } C _ { 1 }$ and $B _ { 1 } C _ { 1 }$ are tangents to the curve $y ^ { 2 } = 8 x$ at $A _ { 1 }$ and $B _ { 1 }$, respectively. If $O = ( 0,0 )$ and $C _ { 1 } = ( - 4,0 )$, then which of the following statements is (are) TRUE?
(A) The length of the line segment $OA _ { 1 }$ is $4 \sqrt { 3 }$
(B) The length of the line segment $A _ { 1 } B _ { 1 }$ is 16
(C) The orthocenter of the triangle $A _ { 1 } B _ { 1 } C _ { 1 }$ is $( 0,0 )$
(D) The orthocenter of the triangle $A _ { 1 } B _ { 1 } C _ { 1 }$ is $( 1,0 )$

Q8 4 marks Composite & Inverse Functions Custom Operation or Property Verification View

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a function such that $f ( x + y ) = f ( x ) + f ( y )$ for all $x , y \in \mathbb { R }$, and $g : \mathbb { R } \rightarrow ( 0 , \infty )$ be a function such that $g ( x + y ) = g ( x ) g ( y )$ for all $x , y \in \mathbb { R }$. If $f \left( \frac { - 3 } { 5 } \right) = 12$ and $g \left( \frac { - 1 } { 3 } \right) = 2$, then the value of $\left( f \left( \frac { 1 } { 4 } \right) + g ( - 2 ) - 8 \right) g ( 0 )$ is $\_\_\_\_$ .

Q9 4 marks Conditional Probability Sequential/Multi-Stage Conditional Probability View

A bag contains $N$ balls out of which 3 balls are white, 6 balls are green, and the remaining balls are blue. Assume that the balls are identical otherwise. Three balls are drawn randomly one after the other without replacement. For $i = 1,2,3$, let $W _ { i } , G _ { i }$, and $B _ { i }$ denote the events that the ball drawn in the $i ^ { \text {th } }$ draw is a white ball, green ball, and blue ball, respectively. If the probability $P \left( W _ { 1 } \cap G _ { 2 } \cap B _ { 3 } \right) = \frac { 2 } { 5 N }$ and the conditional probability $P \left( B _ { 3 } \mid W _ { 1 } \cap G _ { 2 } \right) = \frac { 2 } { 9 }$, then $N$ equals $\_\_\_\_$ .

Q10 4 marks Standard trigonometric equations Count zeros or intersection points involving trigonometric curves View

Let the function $f : \mathbb { R } \rightarrow \mathbb { R }$ be defined by
$$f ( x ) = \frac { \sin x } { e ^ { \pi x } } \frac { \left( x ^ { 2023 } + 2024 x + 2025 \right) } { \left( x ^ { 2 } - x + 3 \right) } + \frac { 2 } { e ^ { \pi x } } \frac { \left( x ^ { 2023 } + 2024 x + 2025 \right) } { \left( x ^ { 2 } - x + 3 \right) }$$
Then the number of solutions of $f ( x ) = 0$ in $\mathbb { R }$ is $\_\_\_\_$ .

Q11 4 marks Vectors 3D & Lines Vector Algebra and Triple Product Computation View

Let $\vec { p } = 2 \hat { i } + \hat { j } + 3 \hat { k }$ and $\vec { q } = \hat { i } - \hat { j } + \hat { k }$. If for some real numbers $\alpha , \beta$, and $\gamma$, we have
$$15 \hat { i } + 10 \hat { j } + 6 \hat { k } = \alpha ( 2 \vec { p } + \vec { q } ) + \beta ( \vec { p } - 2 \vec { q } ) + \gamma ( \vec { p } \times \vec { q } )$$
then the value of $\gamma$ is $\_\_\_\_$ .

Q12 4 marks Circles Chord Length and Chord Properties View

A normal with slope $\frac { 1 } { \sqrt { 6 } }$ is drawn from the point $( 0 , - \alpha )$ to the parabola $x ^ { 2 } = - 4 a y$, where $a > 0$. Let $L$ be the line passing through $( 0 , - \alpha )$ and parallel to the directrix of the parabola. Suppose that $L$ intersects the parabola at two points $A$ and $B$. Let $r$ denote the length of the latus rectum and $s$ denote the square of the length of the line segment $AB$. If $r : s = 1 : 16$, then the value of $24 a$ is $\_\_\_\_$ .

Q13 4 marks Indefinite & Definite Integrals Accumulation Function Analysis View

Let the function $f : [ 1 , \infty ) \rightarrow \mathbb { R }$ be defined by $$f ( t ) = \left\{ \begin{array} { c c } ( - 1 ) ^ { n + 1 } 2 , & \text { if } t = 2 n - 1 , n \in \mathbb { N } , \\ \frac { ( 2 n + 1 - t ) } { 2 } f ( 2 n - 1 ) + \frac { ( t - ( 2 n - 1 ) ) } { 2 } f ( 2 n + 1 ) , & \text { if } 2 n - 1 < t < 2 n + 1 , n \in \mathbb { N } . \end{array} \right.$$ Define $g ( x ) = \int _ { 1 } ^ { x } f ( t ) d t , x \in ( 1 , \infty )$. Let $\alpha$ denote the number of solutions of the equation $g ( x ) = 0$ in the interval $( 1,8 ]$ and $\beta = \lim _ { x \rightarrow 1 + } \frac { g ( x ) } { x - 1 }$. Then the value of $\alpha + \beta$ is equal to $\_\_\_\_$ .

Q14 3 marks Combinations & Selection Subset Counting with Set-Theoretic Conditions View

Let $S = \{ 1,2,3,4,5,6 \}$ and $X$ be the set of all relations $R$ from $S$ to $S$ that satisfy both the following properties: i. $R$ has exactly 6 elements. ii. For each $( a , b ) \in R$, we have $| a - b | \geq 2$. Let $Y = \{ R \in X$ : The range of $R$ has exactly one element $\}$ and $Z = \{ R \in X : R$ is a function from $S$ to $S \}$. Let $n ( A )$ denote the number of elements in a set $A$. If $n ( X ) = {}^{ m } C _ { 6 }$, then the value of $m$ is $\_\_\_\_$ .

Q15 3 marks Combinations & Selection Subset Counting with Set-Theoretic Conditions View

Let $S = \{ 1,2,3,4,5,6 \}$ and $X$ be the set of all relations $R$ from $S$ to $S$ that satisfy both the following properties: i. $R$ has exactly 6 elements. ii. For each $( a , b ) \in R$, we have $| a - b | \geq 2$. Let $Y = \{ R \in X$ : The range of $R$ has exactly one element $\}$ and $Z = \{ R \in X : R$ is a function from $S$ to $S \}$. Let $n ( A )$ denote the number of elements in a set $A$. If the value of $n ( Y ) + n ( Z )$ is $k ^ { 2 }$, then $| k |$ is $\_\_\_\_$ .

Q16 3 marks Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View

Let $f : \left[ 0 , \frac { \pi } { 2 } \right] \rightarrow [ 0,1 ]$ be the function defined by $f ( x ) = \sin ^ { 2 } x$ and let $g : \left[ 0 , \frac { \pi } { 2 } \right] \rightarrow [ 0 , \infty )$ be the function defined by $g ( x ) = \sqrt { \frac { \pi x } { 2 } - x ^ { 2 } }$. The value of $2 \int _ { 0 } ^ { \frac { \pi } { 2 } } f ( x ) g ( x ) d x - \int _ { 0 } ^ { \frac { \pi } { 2 } } g ( x ) d x$ is $\_\_\_\_$ .

Q17 3 marks Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View

Let $f : \left[ 0 , \frac { \pi } { 2 } \right] \rightarrow [ 0,1 ]$ be the function defined by $f ( x ) = \sin ^ { 2 } x$ and let $g : \left[ 0 , \frac { \pi } { 2 } \right] \rightarrow [ 0 , \infty )$ be the function defined by $g ( x ) = \sqrt { \frac { \pi x } { 2 } - x ^ { 2 } }$. The value of $\frac { 16 } { \pi ^ { 3 } } \int _ { 0 } ^ { \frac { \pi } { 2 } } f ( x ) g ( x ) d x$ is $\_\_\_\_$ .

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