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Q1 3 marks Differentiating Transcendental Functions Limit involving transcendental functions View

Let $x _ { 0 }$ be the real number such that $e ^ { x _ { 0 } } + x _ { 0 } = 0$. For a given real number $\alpha$, define
$$g ( x ) = \frac { 3 x e ^ { x } + 3 x - \alpha e ^ { x } - \alpha x } { 3 \left( e ^ { x } + 1 \right) }$$
for all real numbers $x$.
Then which one of the following statements is TRUE?

(A)	For $\alpha = 2 , \lim _ { x \rightarrow x _ { 0 } } \left\| \frac { g ( x ) + e ^ { x _ { 0 } } } { x - x _ { 0 } } \right\| = 0$
(B)	For $\alpha = 2 , \lim _ { x \rightarrow x _ { 0 } } \left\| \frac { g ( x ) + e ^ { x _ { 0 } } } { x - x _ { 0 } } \right\| = 1$
(C)	For $\alpha = 3 , ~ \lim _ { x \rightarrow x _ { 0 } } \left\| \frac { g ( x ) + e ^ { x _ { 0 } } } { x - x _ { 0 } } \right\| = 0$
(D)	For $\alpha = 3 , ~ \lim _ { x \rightarrow x _ { 0 } } \left\| \frac { g ( x ) + e ^ { x _ { 0 } } } { x - x _ { 0 } } \right\| = \frac { 2 } { 3 }$

Q2 3 marks Areas by integration View

Let $\mathbb { R }$ denote the set of all real numbers. Then the area of the region
$$\left\{ ( x , y ) \in \mathbb { R } \times \mathbb { R } : x > 0 , y > \frac { 1 } { x } , 5 x - 4 y - 1 > 0,4 x + 4 y - 17 < 0 \right\}$$
is

(A)	$\frac { 17 } { 16 } - \log _ { e } 4$	(B)	$\frac { 33 } { 8 } - \log _ { e } 4$
(C)	$\frac { 57 } { 8 } - \log _ { e } 4$	(D)	$\frac { 17 } { 2 } - \log _ { e } 4$

Q3 3 marks Trigonometric equations in context View

The total number of real solutions of the equation
$$\theta = \tan ^ { - 1 } ( 2 \tan \theta ) - \frac { 1 } { 2 } \sin ^ { - 1 } \left( \frac { 6 \tan \theta } { 9 + \tan ^ { 2 } \theta } \right)$$
is (Here, the inverse trigonometric functions $\sin ^ { - 1 } x$ and $\tan ^ { - 1 } x$ assume values in $\left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]$ and ( $- \frac { \pi } { 2 } , \frac { \pi } { 2 }$ ), respectively.)

(A)

1

(B)

2

(C)

3

(D)

5

Q4 3 marks Circles Tangent Lines and Tangent Lengths View

Let $S$ denote the locus of the point of intersection of the pair of lines
$$\begin{gathered} 4 x - 3 y = 12 \alpha \\ 4 \alpha x + 3 \alpha y = 12 \end{gathered}$$
where $\alpha$ varies over the set of non-zero real numbers. Let $T$ be the tangent to $S$ passing through the points $( p , 0 )$ and $( 0 , q ) , q > 0$, and parallel to the line $4 x - \frac { 3 } { \sqrt { 2 } } y = 0$.
Then the value of $p q$ is

(A)

$- 6 \sqrt { 2 }$

(B)

$- 3 \sqrt { 2 }$

(C)

$- 9 \sqrt { 2 }$

(D)

$- 12 \sqrt { 2 }$

Q5 4 marks Matrices Determinant and Rank Computation View

Let $I = \left( \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right)$ and $P = \left( \begin{array} { l l } 2 & 0 \\ 0 & 3 \end{array} \right)$. Let $Q = \left( \begin{array} { l l } x & y \\ z & 4 \end{array} \right)$ for some non-zero real numbers $x , y$, and $z$, for which there is a $2 \times 2$ matrix $R$ with all entries being non-zero real numbers, such that $Q R = R P$.
Then which of the following statements is (are) TRUE?

(A)	The determinant of $Q - 2 I$ is zero
(B)	The determinant of $Q - 6 I$ is 12
(C)	The determinant of $Q - 3 I$ is 15
(D)	$y z = 2$

Q6 4 marks Areas Between Curves Area Involving Conic Sections or Circles View

Let $S$ denote the locus of the mid-points of those chords of the parabola $y ^ { 2 } = x$, such that the area of the region enclosed between the parabola and the chord is $\frac { 4 } { 3 }$. Let $\mathcal { R }$ denote the region lying in the first quadrant, enclosed by the parabola $y ^ { 2 } = x$, the curve $S$, and the lines $x = 1$ and $x = 4$.
Then which of the following statements is (are) TRUE?

(A)	$( 4 , \sqrt { 3 } ) \in S$
(B)	$( 5 , \sqrt { 2 } ) \in S$
(C)	Area of $\mathcal { R }$ is $\frac { 14 } { 3 } - 2 \sqrt { 3 }$
(D)	Area of $\mathcal { R }$ is $\frac { 14 } { 3 } - \sqrt { 3 }$

Q7 4 marks Straight Lines & Coordinate Geometry Line Equation and Parametric Representation View

Let $P \left( x _ { 1 } , y _ { 1 } \right)$ and $Q \left( x _ { 2 } , y _ { 2 } \right)$ be two distinct points on the ellipse
$$\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$$
such that $y _ { 1 } > 0$, and $y _ { 2 } > 0$. Let $C$ denote the circle $x ^ { 2 } + y ^ { 2 } = 9$, and $M$ be the point $( 3,0 )$.
Suppose the line $x = x _ { 1 }$ intersects $C$ at $R$, and the line $x = x _ { 2 }$ intersects C at $S$, such that the $y$-coordinates of $R$ and $S$ are positive. Let $\angle R O M = \frac { \pi } { 6 }$ and $\angle S O M = \frac { \pi } { 3 }$, where $O$ denotes the origin $( 0,0 )$. Let $| X Y |$ denote the length of the line segment $X Y$.
Then which of the following statements is (are) TRUE?

(A)	The equation of the line joining $P$ and $Q$ is $2 x + 3 y = 3 ( 1 + \sqrt { 3 } )$
(B)	The equation of the line joining $P$ and $Q$ is $2 x + y = 3 ( 1 + \sqrt { 3 } )$
(C)	If $N _ { 2 } = \left( x _ { 2 } , 0 \right)$, then $3 \left\| N _ { 2 } Q \right\| = 2 \left\| N _ { 2 } S \right\|$
(D)	If $N _ { 1 } = \left( x _ { 1 } , 0 \right)$, then $9 \left\| N _ { 1 } P \right\| = 4 \left\| N _ { 1 } R \right\|$

Q8 4 marks Stationary points and optimisation Find critical points and classify extrema of a given function View

Let $\mathbb { R }$ denote the set of all real numbers. Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be defined by
$$f ( x ) = \begin{cases} \frac { 6 x + \sin x } { 2 x + \sin x } & \text { if } x \neq 0 \\ \frac { 7 } { 3 } & \text { if } x = 0 \end{cases}$$
Then which of the following statements is (are) TRUE?

(A)	The point $x = 0$ is a point of local maxima of $f$
(B)	The point $x = 0$ is a point of local minima of $f$
(C)	Number of points of local maxima of $f$ in the interval $[ \pi , 6 \pi ]$ is 3
(D)	Number of points of local minima of $f$ in the interval $[ 2 \pi , 4 \pi ]$ is 1

Q9 4 marks First order differential equations (integrating factor) View

Let $y ( x )$ be the solution of the differential equation
$$x ^ { 2 } \frac { d y } { d x } + x y = x ^ { 2 } + y ^ { 2 } , \quad x > \frac { 1 } { e }$$
satisfying $y ( 1 ) = 0$. Then the value of $2 \frac { ( y ( e ) ) ^ { 2 } } { y \left( e ^ { 2 } \right) }$ is $\_\_\_\_$.

Q10 4 marks Binomial Theorem (positive integer n) Find the Largest Term or Coefficient in a Binomial Expansion View

Let $a _ { 0 } , a _ { 1 } , \ldots , a _ { 23 }$ be real numbers such that
$$\left( 1 + \frac { 2 } { 5 } x \right) ^ { 23 } = \sum _ { i = 0 } ^ { 23 } a _ { i } x ^ { i }$$
for every real number $x$. Let $a _ { r }$ be the largest among the numbers $a _ { j }$ for $0 \leq j \leq 23$. Then the value of $r$ is $\_\_\_\_$.

Q11 4 marks Conditional Probability Bayes' Theorem with Production/Source Identification View

A factory has a total of three manufacturing units, $M _ { 1 } , M _ { 2 }$, and $M _ { 3 }$, which produce bulbs independent of each other. The units $M _ { 1 } , M _ { 2 }$, and $M _ { 3 }$ produce bulbs in the proportions of $2 : 2 : 1$, respectively. It is known that $20 \%$ of the bulbs produced in the factory are defective. It is also known that, of all the bulbs produced by $M _ { 1 } , 15 \%$ are defective. Suppose that, if a randomly chosen bulb produced in the factory is found to be defective, the probability that it was produced by $M _ { 2 }$ is $\frac { 2 } { 5 }$.
If a bulb is chosen randomly from the bulbs produced by $M _ { 3 }$, then the probability that it is defective is $\_\_\_\_$.

Q12 4 marks Vector Product and Surfaces View

Consider the vectors
$$\vec { x } = \hat { \imath } + 2 \hat { \jmath } + 3 \hat { k } , \quad \vec { y } = 2 \hat { \imath } + 3 \hat { \jmath } + \hat { k } , \quad \text { and } \quad \vec { z } = 3 \hat { \imath } + \hat { \jmath } + 2 \hat { k }$$
For two distinct positive real numbers $\alpha$ and $\beta$, define
$$\vec { X } = \alpha \vec { x } + \beta \vec { y } - \vec { z } , \quad \vec { Y } = \alpha \vec { y } + \beta \vec { z } - \vec { x } , \quad \text { and } \quad \vec { Z } = \alpha \vec { z } + \beta \vec { x } - \vec { y }$$
If the vectors $\vec { X } , \vec { Y }$, and $\vec { Z }$ lie in a plane, then the value of $\alpha + \beta - 3$ is $\_\_\_\_$.

Q13 4 marks Complex numbers 2 Modulus and Argument Computation View

For a non-zero complex number $z$, let $\arg ( z )$ denote the principal argument of $z$, with $- \pi < \arg ( z ) \leq \pi$. Let $\omega$ be the cube root of unity for which $0 < \arg ( \omega ) < \pi$. Let
$$\alpha = \arg \left( \sum _ { n = 1 } ^ { 2025 } ( - \omega ) ^ { n } \right) .$$
Then the value of $\frac { 3 \alpha } { \pi }$ is $\_\_\_\_$.

Q14 4 marks Composite & Inverse Functions Derivative of an Inverse Function View

Let $\mathbb { R }$ denote the set of all real numbers. Let $f : \mathbb { R } \rightarrow \mathbb { R }$ and $g : \mathbb { R } \rightarrow ( 0,4 )$ be functions defined by
$$f ( x ) = \log _ { e } \left( x ^ { 2 } + 2 x + 4 \right) , \text { and } g ( x ) = \frac { 4 } { 1 + e ^ { - 2 x } }$$
Define the composite function $f \circ g ^ { - 1 }$ by $\left( f \circ g ^ { - 1 } \right) ( x ) = f \left( g ^ { - 1 } ( x ) \right)$, where $g ^ { - 1 }$ is the inverse of the function $g$.
Then the value of the derivative of the composite function $f \circ g ^ { - 1 }$ at $x = 2$ is $\_\_\_\_$.

Q15 4 marks Addition & Double Angle Formulae Telescoping Sum of Trigonometric Terms View

Let
$$\alpha = \frac { 1 } { \sin 60 ^ { \circ } \sin 61 ^ { \circ } } + \frac { 1 } { \sin 62 ^ { \circ } \sin 63 ^ { \circ } } + \cdots + \frac { 1 } { \sin 118 ^ { \circ } \sin 119 ^ { \circ } } .$$
Then the value of
$$\left( \frac { \operatorname { cosec } 1 ^ { \circ } } { \alpha } \right) ^ { 2 }$$
is $\_\_\_\_$.

Q16 4 marks Integration by Substitution Substitution Combined with Symmetry or Companion Integral View

If
$$\alpha = \int _ { \frac { 1 } { 2 } } ^ { 2 } \frac { \tan ^ { - 1 } x } { 2 x ^ { 2 } - 3 x + 2 } d x$$
then the value of $\sqrt { 7 } \tan \left( \frac { 2 \alpha \sqrt { 7 } } { \pi } \right)$ is $\_\_\_\_$. (Here, the inverse trigonometric function $\tan ^ { - 1 } x$ assumes values in $\left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$.)

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