Let $\left|X\right|$ denote the number of elements in set $X.$ Let $S=\left\{1, 2, 3, 4, 5, 6\right\}$ be a sample space, where each element is equally likely to occur. If $A$ and $B$ are independent events associated with $S,$ then the number of ordered pairs $\left(A, B\right)$ such that $1\le \left|B\right|<\left|A\right|,$ equals

The value of ${\left({\left({\log }_{2}9\right)}^2\right)}^{\frac{1}{{\log }_2\left({\log }_{2}9\right)}}\times {\left(\sqrt{7}\right)}^{\frac{1}{{\log }_{4}7}}$ is_____.

For every pair of continuous functions $f, g :\left[0, 1\right]\to R$ such that $\max \left\{f\left(x\right):x \in \left[0, 1\right]\right\}=\max \{g\left(x\right) :x \in \left[0, 1\right]\},$then the correct statement (s) is (are)

Let $f :\left(0, \infty \right)\to R$ be given by $f\left(x\right)= \underset{\frac{1}{x}}{\overset{x}{\int }}{e}^{-\left(t+\frac{1}{t}\right)}\frac{dt}{t},$ then

Let \(f:(0,1) \rightarrow R\) be defined by \(f(x)=\frac{b-x}{1-b x}\), where \(b\) is a constant such that \(0 < b < 1\). Then,

Let $\mathrm{A}\mathrm{P}\mathrm{ }\left(\mathrm{a};\mathrm{d}\right)$ denote the set of all the terms of an infinite arithmetic progression with first term a and common difference $\mathrm{d}>0.$ If $\mathrm{A}\mathrm{P}\left(1;3\right)\cap \mathrm{A}\mathrm{P}\left(2;5\right)\cap \mathrm{A}\mathrm{P}\left(3;7\right)=\mathrm{A}\mathrm{P}\left(\mathrm{a};\mathrm{d}\right)$ then $\mathrm{a}+\mathrm{d}$ equals ____

Let ${\text{S}}_{\text{n}}=\underset{\text{k}=1}{\overset{4\text{n}}{\Sigma }}{\left(-1\right)}^{\frac{\text{k}\left(\text{k}+1\right)}{2}}{\text{k}}^2$. Then S_n can take value(s)

Let $C_1$ be the circle of radius $1$ with center at the origin. Let $C_2$ be the circle of radius $r$ with center at the point $A=(4,1)$, where $1<r<3$. Two distinct common tangents $PQ$ and $ST$ of $C_1$ and $C_2$ are drawn. The tangent $PQ$ touches $C_1$ at $P$ and $C_2$ at $Q$. The tangent $ST$ touches $C_1$ at $S$ and $C_2$ at $T$. Midpoints of the line segments $PQ$ and $ST$ are joined to form a line which meets the $x$-axis at a point $B$. If $AB=\sqrt{5}$, then the value of $r^2$ is

Two identical non-conducting solid spheres of same mass and charge are suspended in air from a common point by two non-conducting, massless strings of same length. At equilibrium, the angle between the strings is $\alpha$. The spheres are now immersed in a dielectric liquid of density $800 \mathrm{kg} {\mathrm{m}}^{-3}$ and dielectric constant $21$. If the angle between the strings remains the same after the immersion, then

The colour of the $X_2$ molecules of group 17 elements changes gradually from yellow to violet down the group. This is due to -

Paragraph :
A dense collection of equal number of electrons and positive ions is called neutral plasma. Certain solids containing fixed positive ions surrounded by free electrons can be treated as neutral plasma. Let \(N\) be the number density of free electrons, each of mass \(m\). When the electrons are subjected to an electric field, they are displaced relatively away from the heavy positive ions.If the electric field becomes zero, the electrons being to oscillate about the positive ions with a natural angular frequency \(\omega_p\), which is called the plasma frequency. To sustain the oscillations, a time varying electric field needs to be applied that has an angular frequency \(\omega\), where a part of the energy is absorbed and a part of it is reflected. As \(\omega\) approaches \(\omega_p\), all the free electrons are set to resonance together and all the energy is reflected. This is the explanation of high reflectivity of metals.
Question :
Taking the electronic charge as \(e\) and the permittivity as \(\varepsilon_0\), use dimensional analysis to determine the correct expression for \(\omega_p\).

Let $f\left(x\right)=\sin \left(\pi \cos x\right)$ and $g\left(x\right)=\cos \left(2\pi \sin x\right)$ be two functions defined for $x>0.$ Define the following sets whose elements are written in the increasing order:
$X=\left\{x:f\left(x\right)=0\right\}, Y=\left\{x:f^{\text{'}}\left(x\right)=0\right\}$
$Z=\left\{x:g\left(x\right)=0\right\}, W=\left\{x:g^{\text{'}}\left(x\right)=0\right\}$
List- $\mathrm{I}$ contains the sets $X, Y, Z$ and $W.$ List- $\mathrm{I}\mathrm{I}$ contains some information regarding these sets.
 

	List-  I		List-  II
$\left(I\right)$	$X$	$\left(P\right)$	$\supseteq \left\{\frac{\pi }{2},\frac{3\pi }{2}, 4\pi , 7\pi \right\}$
$\left(II\right)$	$Y$	$\left(Q\right)$	an arithmetic progression
$\left(III\right)$	$Z$	$\left(R\right)$	NOT an arithmetic progression
$\left(IV\right)$	$W$	$\left(S\right)$	$\supseteq \left\{\frac{\pi }{6},\frac{7\pi }{6},\frac{13\pi }{6}\right\}$
		$\left(T\right)$	$\supseteq \left\{\frac{\pi }{3},\frac{2\pi }{3}, \pi \right\}$
		$\left(U\right)$	$\supseteq \left\{\frac{\pi }{6},\frac{3\pi }{4}\right\}$

Which of the following is the only correct combination?

Let $f :\mathbb{R}\to \mathbb{R}\mathbb{ }$ be a continuous odd function, which vanishes exactly at one point and $f\left(1\right)=\frac{1}{2}.$ Suppose that $F\left(x\right)= \underset{-1}{\overset{x}{\int }}f\left(t\right)dt$ for all $x\in \left[-1, 2\right]$ and $G\left(x\right)= \underset{-1}{\overset{x}{\int }}t\left|f\left(f\left(t\right)\right)\right|dt$ for all $x\in \left[-1, 2\right].$ If $\underset{x\to 1}{\lim }\frac{F\left(x\right)}{G\left(x\right)}=\frac{1}{14},$ then the value of $f\left(\frac{1}{2}\right)$ is