Let $\alpha$ and $\beta$ be real numbers such that $-\frac{\pi}{4}<\beta<0<\alpha<\frac{\pi}{4}$.

If $\sin (\alpha+\beta)=\frac{1}{3}$ and $\cos (\alpha-\beta)=\frac{2}{3}$, then the greatest integer less than or equal to

$$ \left(\frac{\sin \alpha}{\cos \beta}+\frac{\cos \beta}{\sin \alpha}+\frac{\cos \alpha}{\sin \beta}+\frac{\sin \beta}{\cos \alpha}\right)^{2} $$ is

If $y(x)$ is the solution of the differential equation

$$ x d y-\left(y^{2}-4 y\right) d x=0 \text { for } x > 0, y(1)=2, $$

and the slope of the curve $y=y(x)$ is never zero, then the value of $10 y(\sqrt{2})$ is

The greatest integer less than or equal to

$$ \int_{1}^{2} \log _{2}\left(x^{3}+1\right) d x+\int_{1}^{\log _{2} 9}\left(2^{x}-1\right)^{\frac{1}{3}} d x $$

is ___________.

The product of all positive real values of $x$ satisfying the equation

$$ x^{\left(16\left(\log _{5} x\right)^{3}-68 \log _{5} x\right)}=5^{-16} $$

is __________.

If

$$ \beta=\lim \limits_{x \to 0} \frac{e^{x^{3}}-\left(1-x^{3}\right)^{\frac{1}{3}}+\left(\left(1-x^{2}\right)^{\frac{1}{2}}-1\right) \sin x}{x \sin ^{2} x}, $$

then the value of $6 \beta$ is ___________.

Let $\beta$ be a real number. Consider the matrix

$$ A=\left(\begin{array}{ccc} \beta & 0 & 1 \\ 2 & 1 & -2 \\ 3 & 1 & -2 \end{array}\right) $$

If $A^{7}-(\beta-1) A^{6}-\beta A^{5}$ is a singular matrix, then the value of $9 \beta$ is _________.

Consider the hyperbola

$$ \frac{x^{2}}{100}-\frac{y^{2}}{64}=1 $$

with foci at $S$ and $S_{1}$, where $S$ lies on the positive $x$-axis. Let $P$ be a point on the hyperbola, in the first quadrant. Let $\angle S P S_{1}=\alpha$, with $\alpha<\frac{\pi}{2}$. The straight line passing through the point $S$ and having the same slope as that of the tangent at $P$ to the hyperbola, intersects the straight line $S_{1} P$ at $P_{1}$. Let $\delta$ be the distance of $P$ from the straight line $S P_{1}$, and $\beta=S_{1} P$. Then the greatest integer less than or equal to $\frac{\beta \delta}{9} \sin \frac{\alpha}{2}$ is ________.

Consider the functions $f, g: \mathbb{R} \rightarrow \mathbb{R}$ defined by

$$ f(x)=x^{2}+\frac{5}{12} \quad \text { and } \quad g(x)= \begin{cases}2\left(1-\frac{4|x|}{3}\right), & |x| \leq \frac{3}{4} \\ 0, & |x|>\frac{3}{4}\end{cases} $$

If $\alpha$ is the area of the region

$$ \left\{(x, y) \in \mathbb{R} \times \mathbb{R}:|x| \leq \frac{3}{4}, 0 \leq y \leq \min \{f(x), g(x)\}\right\}, $$

then the value of $9 \alpha$ is

Let $P Q R S$ be a quadrilateral in a plane, where

$Q R=1, \angle P Q R=\angle Q R S=70^{\circ}, \angle P Q S=15^{\circ}$ and $\angle P R S=40^{\circ}$.

If $\angle R P S=\theta^{\circ}, P Q=\alpha$ and $P S=\beta$, then the interval(s) that contain(s) the value of

$4 \alpha \beta \sin \theta^{\circ}$ is/are

Let

$$ \alpha=\sum\limits_{k = 1}^\infty {{{\sin }^{2k}}\left( {{\pi \over 6}} \right)} $$

Let $g:[0,1] \rightarrow \mathbb{R}$ be the function defined by

$$ g(x)=2^{\alpha x}+2^{\alpha(1-x)} . $$

Then, which of the following statements is/are TRUE ?

Let $\bar{z}$ denote the complex conjugate of a complex number $z$. If $z$ is a non-zero complex number for which both real and imaginary parts of $$ (\bar{z})^{2}+\frac{1}{z^{2}} $$ are integers, then which of the following is/are possible value(s) of $|z|$ ?

Let $G$ be a circle of radius $R>0$. Let $G_{1}, G_{2}, \ldots, G_{n}$ be $n$ circles of equal radius $r>0$. Suppose each of the $n$ circles $G_{1}, G_{2}, \ldots, G_{n}$ touches the circle $G$ externally. Also, for $i=1,2, \ldots, n-1$, the circle $G_{i}$ touches $G_{i+1}$ externally, and $G_{n}$ touches $G_{1}$ externally. Then, which of the following statements is/are TRUE?

Let $\hat{\imath}, \hat{\jmath}$ and $\hat{k}$ be the unit vectors along the three positive coordinate axes. Let

$$ \begin{aligned} & \vec{a}=3 \hat{\imath}+\hat{\jmath}-\hat{k} \text {, } \\ & \vec{b}=\hat{\imath}+b_{2} \hat{\jmath}+b_{3} \hat{k}, \quad b_{2}, b_{3} \in \mathbb{R} \text {, } \\ & \vec{c}=c_{1} \hat{\imath}+c_{2} \hat{\jmath}+c_{3} \hat{k}, \quad c_{1}, c_{2}, c_{3} \in \mathbb{R} \end{aligned} $$

be three vectors such that $b_{2} b_{3}>0, \vec{a} \cdot \vec{b}=0$ and

$$ \left(\begin{array}{ccc} 0 & -c_{3} & c_{2} \\ c_{3} & 0 & -c_{1} \\ -c_{2} & c_{1} & 0 \end{array}\right)\left(\begin{array}{l} 1 \\ b_{2} \\ b_{3} \end{array}\right)=\left(\begin{array}{r} 3-c_{1} \\ 1-c_{2} \\ -1-c_{3} \end{array}\right) . $$

Then, which of the following is/are TRUE?

For $x \in \mathbb{R}$, let the function $y(x)$ be the solution of the differential equation

$$ \frac{d y}{d x}+12 y=\cos \left(\frac{\pi}{12} x\right), \quad y(0)=0 $$

Then, which of the following statements is/are TRUE ?

Consider 4 boxes, where each box contains 3 red balls and 2 blue balls. Assume that all 20 balls are distinct. In how many different ways can 10 balls be chosen from these 4 boxes so that from each box at least one red ball and one blue ball are chosen ?

If $M=\left(\begin{array}{rr}\frac{5}{2} & \frac{3}{2} \\ -\frac{3}{2} & -\frac{1}{2}\end{array}\right)$, then which of the

following matrices is equal to $M^{2022} ?$

Suppose that

Box-I contains 8 red, 3 blue and 5 green balls,

Box-II contains 24 red, 9 blue and 15 green balls,

Box-III contains 1 blue, 12 green and 3 yellow balls,

Box-IV contains 10 green, 16 orange and 6 white balls.

A ball is chosen randomly from Box-I; call this ball $b$. If $b$ is red then a ball is chosen randomly from Box-II, if $b$ is blue then a ball is chosen randomly from Box-III, and if $b$ is green then a ball is chosen randomly from Box-IV. The conditional probability of the event 'one of the chosen balls is white' given that the event 'at least one of the chosen balls is green' has happened, is equal to

For positive integer $n$, define

$$ f(n)=n+\frac{16+5 n-3 n^{2}}{4 n+3 n^{2}}+\frac{32+n-3 n^{2}}{8 n+3 n^{2}}+\frac{48-3 n-3 n^{2}}{12 n+3 n^{2}}+\cdots+\frac{25 n-7 n^{2}}{7 n^{2}} . $$

Then, the value of $$\mathop {\lim }\limits_{n \to \infty } f\left( n \right)$$ is equal to :