If $\alpha =3{\sin }^{-1}\left(\frac{6}{11}\right) and \beta =3{\cos }^{-1}\left(\frac{4}{9}\right),$ where the inverse trigonometric functions take only the principal values, then the correct options(s) is(are)

If \(|z|=1\) and \(z \neq \pm 1\), then all the values of \(\frac{z}{1-z^2}\) lie on

Let \(\alpha, \beta\) be the roots of the equation \(x^2-p x+r=0\) and \(\frac{\alpha}{2}, 2 \beta\) be the roots of the equation \(x^2-q x+r=0\). Then, the value of \(r\) is

Let \(S_1=\{(i, j, k): i, j, k \in\{1,2, \ldots, 10\}\}\)
\(S_2=\{(i, j): 1 \leq i < j+2 \leq 10, i, j\) \(\in\{1,2, \ldots, 10\}\} \)
\(S_3=\{(i, j, k, l): 1 \leq i < j < k < l, i,\) \(j, k, l \in\{1,2, \ldots . ., 10\}\}\)
and \(S_4=\{(i, j, k, l): i, j, k\) and \(l\) are distint elements in \(\{1,2, \ldots, 10\}\}\) If the total number of elements in the set \(S_r\) is \(n_r, r=1,2,3,4\) then which of the following statements is (are) TRUE?

The number of $5$ digit numbers which are divisible by $4$, with digits from the set $\{\mathrm{1,2},\mathrm{3,4},5\}$ and the repetition of digits is allowed, is _____.

If \(\vec{a}\) and \(\vec{b}\) are vectors such that \(|\vec{a}+\vec{b}|=\sqrt{29}\) and \(\vec{a} \times(2 \hat{i}+3 \hat{j}+4 \hat{k})=(2 \hat{i}+3 \hat{j}+4 \hat{k}) \times \vec{b}\), then a possible value of \((\vec{a}+\vec{b}) \cdot(-7 \hat{i}+2 \hat{j}+3 \hat{k})\) is

The number of distinct solutions of the equation $\frac{5}{4}{\cos }^2\mathrm{ }2\mathrm{x}+{\cos }^4\mathrm{x}+{\sin }^4\mathrm{x}+{\cos }^6\mathrm{x}+{\sin }^6\mathrm{x}=2$ in the interval $[0,\mathrm{ }2\mathrm{\pi }]$ is

Answer the following by appropriately matching the lists based on the information given in the paragraph.
In a thermodynamics process on an ideal monatomic gas, the infinitesimal heat absorbed by the gas is given by $\mathrm{T}\mathrm{\Delta }\mathrm{X},$ where $\mathrm{T}$ is temperature of the system and $\mathrm{\Delta }\mathrm{X}$ is the infinitesimal change in a thermodynamic quantity $\mathrm{X}$ of the system. For a mole of monatomic ideal gas $\mathrm{X}=\frac{3}{2}\mathrm{R}\mathrm{l}\mathrm{n}\left(\frac{\mathrm{T}}{{\mathrm{T}}_{\mathrm{A}}}\right)+\mathrm{R}\mathrm{l}\mathrm{n}\left(\frac{\mathrm{V}}{{\mathrm{V}}_{\mathrm{A}}}\right).$ Here, $\mathrm{R}$ is gas constant, $\mathrm{V}$ is volume of gas, ${\mathrm{T}}_{\mathrm{A}}$ and ${\mathrm{V}}_{\mathrm{A}}$ are constants.
The List-I below gives some quantities involved in a process and List-II gives some possible values of these quantities.

	List I		List II
$\left(\mathrm{a}\right)$	Work done by the system in process $1\to 2\to 3$	$\left(\mathrm{p}\right)$	$\frac{1}{3}\mathrm{R}{\mathrm{T}}_0\mathrm{l}\mathrm{n}2$
$\left(\mathrm{b}\right)$	Change in internal energy in process $1\to 2\to 3$	$\left(\mathrm{q}\right)$	$\frac{1}{3}\mathrm{R}{\mathrm{T}}_0$
$\left(\mathrm{c}\right)$	Heat absorbed by the system in process $1\to 2\to 3$	$\left(\mathrm{r}\right)$	$\mathrm{R}{\mathrm{T}}_0$
$\left(\mathrm{d}\right)$	Heat absorbed by the system in process $1\to 2$	$\left(\mathrm{s}\right)$	$\frac{4}{3}\mathrm{R}{\mathrm{T}}_0$
		$\left(\mathrm{t}\right)$	$\frac{1}{3}\mathrm{R}{\mathrm{T}}_0\left(3+\mathrm{l}\mathrm{n}2\right)$
		$\left(\mathrm{u}\right)$	$\frac{5}{6}\mathrm{R}{\mathrm{T}}_0$

If the process carried out on one mole of monatomic ideal gas is as shown in figure in the $\mathrm{T}$ - $\mathrm{V}$ -diagram with ${\mathrm{P}}_0{\mathrm{V}}_0=\frac{1}{3}\mathrm{R}{\mathrm{T}}_0,$ the correct match is,

There are three bags ${\mathrm{B}}_1,{\mathrm{B}}_2$ and ${\mathrm{B}}_3.$ The bag ${\mathrm{B}}_1$ contains $5$ red and $5$ green balls, ${\mathrm{B}}_2$ contains $3$ red and $5$ green balls, and ${\mathrm{B}}_3$ contains $5$ red and $3$ green balls, Bags ${\mathrm{B}}_1,{\mathrm{B}}_2$ and ${\mathrm{B}}_3$ have probabilities $\frac{3}{10},\frac{3}{10}$ and $\frac{4}{10}$ respectively of being chosen. A bag is selected at random and a ball is chosen at random from the bag. Then which of the following options is/are correct?

Consider an expanding sphere of instantaneous radius $R$ whose total mass remains constant. The expansion is such that the instantaneous density $\rho$ remains uniform throughout the volume. The rate of fractional change in density $\left(\frac{1}{\rho }\frac{d\rho }{dt}\right)$ is constant. The velocity $v$ of any point on the surface of the expanding sphere is proportional to

Under hydrolytic conditions, the compounds used for the preparation of linear polymer and for chain termination, respectively, are

Answer the following by appropriately matching the lists based on the information given in the paragraph.
A musical instrument is made using four different metal strings, $1,\mathrm{ }2,\mathrm{ }3$ and $4$ with mass per unit length $\mathrm{\mu },2\mathrm{\mu },3\mathrm{\mu }$ and $4\mathrm{\mu }$ respectively. The instrument is played by vibrating the strings by varying the free length in between the range ${\mathrm{L}}_0$ and $2{\mathrm{L}}_0.$ It is found that in string- $1\left(\mathrm{\mu }\right)$ at free length ${\mathrm{L}}_0$ and tension ${\mathrm{T}}_0$ the fundamental mode frequency is ${\mathrm{f}}_0.$
List-I gives the above four strings while list-II lists the magnitude of some quantity.

	List I		List II
$\left(\mathrm{a}\right)$	String $-1\left(\mathrm{\mu }\right)$	$\left(\mathrm{p}\right)$	$1$
$\left(\mathrm{b}\right)$	String $-2\left(2\mathrm{\mu }\right)$	$\left(\mathrm{q}\right)$	$\frac{1}{2}$
$\left(\mathrm{c}\right)$	String $-3\left(3\mathrm{\mu }\right)$	$\left(\mathrm{r}\right)$	$\frac{1}{\sqrt{2}}$
$\left(\mathrm{d}\right)$	String $-4\left(4\mathrm{\mu }\right)$	$\left(\mathrm{s}\right)$	$\frac{1}{\sqrt{3}}$
		$\left(\mathrm{t}\right)$	$\frac{3}{16}$
		$\left(\mathrm{u}\right)$	$\frac{1}{16}$

The length of the strings $1,\mathrm{ }2,\mathrm{ }3$ and $4$ are kept fixed at ${\mathrm{L}}_0,\frac{3{\mathrm{L}}_0}{2},\frac{5{\mathrm{L}}_0}{4}$ and $\frac{7{\mathrm{L}}_0}{4},$ respectively. Strings $1,\mathrm{ }2,\mathrm{ }3$ and $4$ are vibrated at their $1^{\mathrm{s}\mathrm{t}},\mathrm{ }{3}^{\mathrm{r}\mathrm{d}},\mathrm{ }{5}^{\mathrm{t}\mathrm{h}}$ and ${14}^{\mathrm{t}\mathrm{h}}$ harmonics, respectively such that all the strings have same frequency. The correct match for the tension in the four strings in the units of ${\mathrm{T}}_0$ will be.

Paragraph:
The circle \(x^2+y^2-8 x=0\) and hyperbola \(\frac{x^2}{9}-\frac{y^2}{4}=1\) intersect at the points \(A\) and \(B\).Question:
Equation of a common tangent with positive slope to the circle as well as to the hyperbola is