Let $\mathrm{g}\mathrm{ }:\mathbb{R}\to \mathbb{R}$ be a differentiable function with $\mathrm{g}\left(0\right)=0,\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }{\mathrm{g}}^{\mathrm{\prime }}\left(0\right)=0$ and ${\mathrm{g}}^{\mathrm{\prime }}\left(1\right)\mathrm{ }\ne 0.\mathrm{ }$ Let $\mathrm{f}\left(\mathrm{x}\right)=\mathrm{ }\left\{\begin{array}{cc}\frac{\mathrm{x}}{\left|\mathrm{x}\right|}\mathrm{ }\mathrm{g}\left(\mathrm{x}\right),& \mathrm{x}\mathrm{ }\ne 0\\ 0,& \mathrm{x}=0\end{array}\right.$ and $\mathrm{h}\left(\mathrm{x}\right)={\mathrm{e}}^{\left|\mathrm{x}\right|}$ for all $\mathrm{x}\in \mathbb{R}.$ Let $\left(\mathrm{foh}\right)\mathrm{ }\left(\mathrm{x}\right)$ denote $\mathrm{f}\left(\mathrm{h}\left(\mathrm{x}\right)\right)\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{ }\left(\mathrm{hof}\right)\mathrm{ }\left(\mathrm{x}\right)\mathrm{ }$denote $\mathrm{h}\left(\mathrm{f}\left(\mathrm{x}\right)\right)$ .Then which of the following is (are) true?

Paragraph:
Consider the functions defined implicitly by the equation \(y^3-3 y+x=0\) on various intervals in the real line. If \(x \in(-\infty,-2) \cup(2, \infty)\), the equation implicitly defines a unique real valued differentiable function \(y=f(x)\). If \(x \in(-2,2)\), the equation implicitly defines a unique real valued differentiable function \(y=g(x)\), satisfying \(g(0)=0\).Question:
\(\int_{-1}^1 g^{\prime}(x) d x\) is equal to

For any positive integer $n$, define $f_n:\left(0, \infty \right)\to R$ as $f_n\left(x\right)={\sum }_{i=1}^n{\tan }^{-1}\left(\frac{1}{1+\left(x+j\right)\left(x+j-1\right)}\right)$ for all $x\in \left(0,\infty \right)$ . $($Here, the inverse trigonometric function ${\tan }^{-1}x$ assumes values in $\left(-\frac{\pi }{2}, \frac{\pi }{2}\right)$$)$ Then, which of the following statement(s) is (are) TRUE?

The common tangents to the circle ${\mathrm{x}}^2+{\mathrm{y}}^2=2$ and the parabola ${\mathrm{y}}^2=8\mathrm{x}$ touch the circle at the points P, and Q the parabola at the points R, S. Then the area of the quadrilateral PQRS is

The coefficient of $x^9$ in the expansion of $\left(1+x\right) \left(1+x^2\right) \left(1+x^3\right)\dots \dots \left(1+x^{100}\right)$ is

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 ____.

Paragraph:
Let \(M=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: x^{2}+y^{2} \leq r^{2}\right\}\),
where \(r>0 .\) Consider the geometric progression \(a_{n}=\frac{1}{2^{n-1}}, n=1,2,3, \ldots .\) Let \(S_{0}=0\) and, for \(n \geq 1\), let \(S_{n}\) denote the sum of the first \(n\) terms of this progression. For \(n \geq 1\), let \(C_{n}\) denote the circle with center \(\left(S_{n-1}, 0\right)\) and radius \(a_{n}\), and \(D_{n}\) denote the circle with center \(\left(S_{n-1}, S_{n-1}\right)\) and radius \(a_{n}\).

Question:
Consider \(M\) with \(r=\frac{\left(2^{199}-1\right) \sqrt{2}}{2^{198}} .\) The number of all those circles \(D_{n}\) that are inside \(M\) is

Among the following, the correct statement(s) is(are)

Total number of hydroxyl groups present in a molecule of the major product $\mathrm{P}$ is ______

A series LCR circuit is connected to a $45 \sin \left(\omega t\right)$ Volt source. The resonant angular frequency of the circuit is ${10}^5 \mathrm{rad} {\mathrm{s}}^{-1}$ and current amplitude at resonance is $I_0$. When the angular frequency of the source is $\omega =8\times {10}^4 \mathrm{rad} {\mathrm{s}}^{-1}$, the current amplitude in the circuit is $0.05 I_0$. If $L=50 \mathrm{mH}$, match each entry in List-$\mathrm{I}$ with an appropriate value from List-$\mathrm{II}$ and choose the correct option.

	List-$\mathrm{I}$		List-$\mathrm{II}$
$\mathrm{P}$	$I_0$ in $\mathrm{mA}$	$1$	$44.4$
$\mathrm{Q}$	The quality factor of the circuit	$2$	$18$
$\mathrm{R}$	The bandwidth of the circuit in $\mathrm{rad} {\mathrm{s}}^{-1}$	$3$	$400$
$\mathrm{S}$	The peak power dissipated at resonance in Watt	$4$	$2250$
		$5$	$500$

An ideal gas is undergoing a cyclic thermodynamic process in different ways as shown in the corresponding P-V diagrams in column 3 of the table. Consider only the path from state 1 to state 2. W denotes the corresponding work done on the system. The equations and plots in the table have standard notations as used in thermodynamic process. Here $\gamma$ is the ratio of heat capacities at constant pressure and constant volume. The number of moles in the gas is n.

Column – 1	Column – 2	Column – 3
(I) \(W_{1 \rightarrow 2}=\frac{1}{\gamma-1}\) \(\left(P_2 V_2-P_1 V_1\right)\)	(i) Isothermal	(P)
(II) \(W_{1 \rightarrow 2}=\) \(-P V_2+P V_1\)	(ii) Isochoric	(Q)
(III) \(W_{1 \rightarrow 2}=0\)	(iii) Isobaric	(R)
(IV) \(W_{1 \rightarrow 2}=\) \(-n R T \ln \left(\frac{V_2}{V_1}\right)\)	(iv) Adiabatic	(S)

Which of the following options is the only correct representation of the process $∆U=∆Q-P∆V$?

An electric dipole with dipole moment $\frac{{\mathrm{p}}_0}{\sqrt{2}}\left(\widehat{\mathrm{i}}+\widehat{\mathrm{j}}\right)$ is held fixed at the origin $\mathrm{O}$ in the presence of an uniform electric field of magnitude ${\mathrm{E}}_0.$ If the potential is constant on a circle of radius $\mathrm{R}$ centered at the origin as shown in figure, then the correct statement(s) is/are:
( ${\mathrm{\epsilon }}_0$ is permittivity of free space, $\mathrm{R}>>$ dipole size)

In the chemical reaction between stoichiometric quantities of ${\mathrm{KMnO}}_4$ $\mathrm{KI}$ in weakly basic solution, what is the number of moles of ${\mathrm{I}}_2$ released for $4$ ${\mathrm{KMnO}}_4$ consumed?