Let $E_1=${$x\in \mathrm{ℝ}: x\ne 1$ and $\frac{x}{x-1}>0$} and $E_2=${$x\in E_1: {\sin }^{-1}\left({\log }_e\left(\frac{x}{x-1}\right)\right)$ is a real number} (Here the inverse trigonometric function ${\sin }^{-1}\left(x\right)$ assumes values in $\left[-\frac{\mathrm{\pi }}{2}. \frac{\mathrm{\pi }}{2}\right]$.)
Let $f:E_1\to \mathrm{ℝ}$ be the function defined by $f\left(x\right)={\log }_e\left(\frac{x}{x-1}\right)$
And $g:E_2\to \mathbb{R}$ be the function defined by $g\left(x\right)={\sin }^{-1}\left({\log }_e\left(\frac{x}{x-1}\right)\right)$ .

LIST-I	LIST-II
A. The range of $f$ is	P. $\left(-\infty ,\frac{1}{1-e}\right]\cup \left[\frac{e}{e-1}, \infty \right)$
B. The range of $g$ contains	Q. (0, 1)
C. The domain of $f$ contains	R. $\left[-\frac{1}{2},\frac{1}{2}\right]$
D. The domain of $g$ is	S. $\left(-\infty ,0\right)\cup \left(0, \infty \right)$
	T. $\left(-\infty ,\frac{e}{e-1}\right]$
	U. $\left(-\infty ,0\right)\cup \left(\frac{1}{2},\frac{e}{e-1}\right]$

The correct option is:

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

Consider the ellipse \(\frac{x^2}{9}+\frac{y^2}{4}=1\). Let \(S(p, q)\) be a point in the first quadrant such that \(\frac{p^2}{9}+\frac{q^2}{4}>1\). Two tangents are drawn from \(S\) to the ellipse, of which one meets the ellipse at one end point of the minor axis and the other meets the ellipse at a point \(T\) in the fourth quadrant. Let \(R\) be the vertex of the ellipse with positive \(x\)-coordinate and \(O\) be the center of the ellipse. If the area of the triangle \(\triangle O R T\) is \(\frac{3}{2}\), then which of the following options is correct?

Paragraph: 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 \(g(x)=\sqrt{\frac{\pi x}{2}-x^2}\).
Question: 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

Consider the ellipse $\frac{x^2}{4}+\frac{y^2}{3}=1$. Let $H\left(\alpha ,0\right),0<\alpha <2$, be a point. A straight line drawn through $H$ parallel to the $y$-axis crosses the ellipse and its auxiliary circle at points $E$ and $F$ respectively, in the first quadrant. The tangent to the ellipse at the point $E$ intersects the positive $x$-axis at a point $G$. Suppose the straight line joining $F$ and the origin makes an angle $\varphi$ with the positive $x$-axis.

	List-$\mathrm{I}$		List-$\mathrm{II}$
$\left(\mathrm{I}\right)$	If $\varphi =\frac{\pi }{4}$, then the area of the triangle $FGH$ is	$\left(\mathrm{P}\right)$	$\frac{{\left(\sqrt{3}-1\right)}^4}{8}$
$\left(\mathrm{II}\right)$	If $\varphi =\frac{\pi }{3}$, then the area of the triangle $FGH$ is	$\left(\mathrm{Q}\right)$	$1$
$\left(\mathrm{III}\right)$	If $\varphi =\frac{\pi }{6}$, then the area of the triangle $FGH$ is	$\left(\mathrm{R}\right)$	$\frac{3}{4}$
$\left(\mathrm{IV}\right)$	If $\varphi =\frac{\pi }{12}$, then the area of the triangle $FGH$ is	$\left(\mathrm{S}\right)$	$\frac{1}{2\sqrt{3}}$
		$\left(\mathrm{T}\right)$	$\frac{3\sqrt{3}}{2}$

The correct option is:

If $\mathrm{I}=\frac{2}{\mathrm{\pi }}\underset{-\frac{\mathrm{\pi }}{4}}{\overset{\frac{\mathrm{\pi }}{4}}{\int }}\frac{\mathrm{d}\mathrm{x}}{\left(1+{\mathrm{e}}^{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{x}}\right)\left(2-\mathrm{c}\mathrm{o}\mathrm{s}2\mathrm{x}\right)}$ then find $27{\mathrm{I}}^2$ equals____

The set of values of \(\theta\) satisfying the inequation \(2 \sin ^2 \theta-5 \sin \theta+2>0\), where \(0 < \theta < 2 \pi\), is

A point mass is subjected to two simultaneous sinusoidal displacements in \(x\)-direction, \(x_1(t)=A \sin \omega t\) and \(x_2(t)=A \sin \left(\omega t+\frac{2 \pi}{3}\right)\). Adding a third sinusoidal displacement \(x_3(t)=B \sin (\omega t+\phi)\) bring the mass to a complete rest. The values of \(B\) and \(\phi\) are

A thin uniform rod of length \(L\) and certain mass is kept on a frictionless horizontal table with a massless string of length \(L\) fixed to one end (top view is shown in the figure). The other end of the string is pivoted to a point \(\mathrm{O}\). If a horizontal impulse \(P\) is imparted to the rod at a distance \(x=L / n\) from the mid-point of the rod (see figure), then the rod and string revolve together around the point \(\mathrm{O}\), with the rod remaining aligned with the string. In such a case, the value of \(n\) is _______ .

In an experiment to measure the speed of sound by a resonating air column, a tuning fork of frequency $500 Hz$ is used. The length of the air column is varied by changing the level of water in the resonance tube. Two successive resonances are heard at air columns of length $50.7 cm and 83.9 cm$ . Which of the following statements is (are) true?

Among ${\mathrm{B}}_2{\mathrm{H}}_6,\mathrm{ }{\mathrm{B}}_3{\mathrm{N}}_3{\mathrm{H}}_6,\mathrm{ }{\mathrm{N}}_2\mathrm{O},\mathrm{ }{\mathrm{N}}_2{\mathrm{O}}_4,\mathrm{ }{\mathrm{H}}_2{\mathrm{S}}_2{\mathrm{O}}_3$ and ${\mathrm{H}}_2{\mathrm{S}}_2{\mathrm{O}}_8,$ the number of molecules containing covalent bond between two atoms of the same kind is __________

In a radioactive decay process, the activity is defined as $A=-\frac{dN}{dt}$, where $N\left(t\right)$ is the number of radioactive nuclei at time $\left(t\right)$. Two radioactive sources, $S_1$ and $S_2$ have same activity at time $t=0$. At a later time, the activities of $S_1$ and $S_2$ are $A_1$ and $A_2$, respectively. When $S_1$ and $S_2$ have just completed their $3^{\mathrm{rd}}$ and $7^{\mathrm{th}}$ half-lives, respectively, the ratio $\frac{A_1}{A_2}$ is _____.

The surface of copper gets tarnished by the formation of copper oxide. $N_2$ gas was passed to prevent the oxide formation during heating of copper at 1250 K. However, the $N_2$ gas contains 1 mole % of water vapour as impurity. The water vapour oxidises copper as per the reaction given below:

$2Cu\left(s\right)+ H_{2}O\left(g\right)\to C{u}_{2}O\left(s\right) + H_2\left(g\right)$

$P_{H_2}$ Is the minimum partial pressure of $H_2$ (in bar) needed to prevent the oxidation at 1250 K. The value of $\ln \left(p{H}_2\right)$ is ____.

(Given: total pressure \(=1\) bar, R (universal gas constant) \(=8 J K ^{-1} mol^{-1}, \ln (10) =2.3 \cdot Cu (s)\) and \(Cu _2 O (s)\) are mutually immiscible.

At \(1250 K: 2 Cu ( s )+\frac{1}{2} O _2(g) \rightarrow Cu _2 O ( s ) ;\) \(\Delta G^{\ominus}=-78,000 J mol ^{-1}\)

\(H _2(g)+\frac{1}{2} O _2(g) \rightarrow H _2 O ( g ) ; \quad \Delta^{\ominus}=\)\(-1,78,000 J mol ^{-1} ; G\) is the Gibbs energy)/mi> \(\Theta=\) \(-78,000 J mol -1\)

\(H _2(g)+\frac{1}{2} O _2(g) \rightarrow H _2 O ( g ) ; \quad\)\(\Delta^{\ominus}=-1,78,000 J mol ^{-1} ; G\) is the Gibbs energy)