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- $I$ contains the sets $X, Y, Z$ and $W.$ List- $\mathrm{I}\mathrm{I}$ contains some information regarding these sets.
 

List- $\mathrm{I}$	List- $\mathrm{I}\mathrm{I}$
$\left(I\right)$	$X$	$\left(P\right)$	$\supseteq \left\{\frac{\pi }{2},\frac{3\pi }{2}, 4\pi , 7\pi \right\}$
$\left(II\right)$	$Y$	$\left(Q\right)$	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 \(\alpha, \beta\) and \(\gamma\) be real numbers such that the system of linear equations
\[
\begin{array}{c}
x+2 y+3 z=\alpha \\
4 x+5 y+6 z=\beta \\
7 x+8 y+9 z=\gamma-1
\end{array}
\]
is consistent. Let \(|M|\) represent the determinant of the matrix
\[
M=\left[\begin{array}{ccc}
\alpha & 2 & \gamma \\
\beta & 1 & 0 \\
-1 & 0 & 1
\end{array}\right]
\]
Let \(P\) be the plane containing all those \((\alpha, \beta, \gamma)\) for which the above system of linear equations is consistent, and \(D\) be the square of the distance of the point \((0,1,0)\) from the plane \(P\).
The value of $\left|M\right|$ is

Let the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) be defined by
\(f(x)=\frac{\sin x}{e^{\pi x}} \frac{\left(x^{2023}+2024 x+2025\right)}{\left(x^2-x+3\right)}+\frac{2}{e^{\pi x}} \frac{\left(x^{2023}+2024 x+2025\right)}{\left(x^2-x+3\right)}\).
Then the number of solutions of \(f(x)=0\) in \(\mathbb{R}\) is

The centres of two circles \(C_1\) and \(C_2\) each of unit radius are at a distance of 6 units from each other. Let \(P\) be the mid-point of the line segment joining the centres of \(C_1\) and \(C_2\) and \(C\) be a circle touching circles \(C_1\) and \(C_2\) externally. If a common tangents to \(C_1\) and \(C\) passing through \(P\) is also a common tangent to \(C_2\) and \(C\), then the radius of the circle \(C\) is

Five persons $\mathrm{A},\mathrm{ }\mathrm{B},\mathrm{ }\mathrm{C},\mathrm{ }\mathrm{D}$ and $\mathrm{E}$ are seated in a circular arrangement. If each of them is given a hat of one of the three colours red, blue and green, then the number of ways of distributing the hats such that the persons seated in adjacent seats get different coloured hats is

Let $P$ be the plane $\sqrt{3}x+2y+3z=16$ and let
$S=\left\{\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}: {\alpha }^2+{\beta }^2+{\gamma }^2=1\right.$ and the distance of $\left(\alpha , \beta , \gamma \right)$ from the plane $P$ is $\frac{7}{2}\}$.
Let $\overrightarrow{u}, \overrightarrow{v}$ and $\overrightarrow{w}$ be three distinct vectors in $S$ such that $\left|\overrightarrow{u}-\overrightarrow{v}\right|=\left|\overrightarrow{v}-\overrightarrow{w}\right|=\left|\overrightarrow{w}-\overrightarrow{u}\right|$. Let $V$ be the volume of the parallelepiped determined by vectors $\overrightarrow{u}, \overrightarrow{v}$ and $\overrightarrow{w}$. Then the value of $\frac{80}{\sqrt{3}}V$ is

Let \(a\) and \(b\) be two nonzero real numbers. If the coefficient of \(x^5\) in the expansion of \(\left(a x^2+\frac{70}{27 b x}\right)^4\) is equal to the coefficient of \(x^{-5}\) in the expansion of \(\left(a x-\frac{1}{b x^2}\right)^7\), then the value of \(2 b\) is

Paragraph:
The reactions of \(\mathrm{Cl}_{2}\) gas with cold-dilute and hot-concentrated \(\mathrm{NaOH}\) in water give sodium salts of two (different) oxoacids of chlorine, \(\boldsymbol{P}\) and \(\boldsymbol{Q}\), respectively. The \(\mathrm{Cl}_{2}\) gas reacts with \(\mathrm{SO}_{2}\) gas, in presence of charcoal, to give a product \(\boldsymbol{R}\). \(\boldsymbol{R}\) reacts with white phosphorus to give a compound \(\boldsymbol{S}\). On hydrolysis, \(\boldsymbol{S}\) gives an oxoacid of phosphorus, \(\boldsymbol{T}\).
Question:
\(P\) and \(Q\), respectively, are the sodium salts of

$S_1$ and $S_2$ are two identical sound sources of frequency $656 \mathrm{Hz}$. The source $S_1$ is located at $O$ and $S_2$ moves anti-clockwise with a uniform speed $4\sqrt{2} \mathrm{m} {\mathrm{s}}^{-1}$ on a circular path around $O$, as shown in the figure. There are three points $P, Q$ and $R$ on this path such that $P$ and $R$ are diametrically opposite while $Q$ is equidistant from them. A sound detector is placed at point $P$. The source $S_1$ can move along direction $OP$.
[Given: The speed of sound in air is $324 \mathrm{m} {\mathrm{s}}^{-1}$]

When only $S_2$ is emitting sound and it is at $Q$, the frequency of sound measured by the detector in $\mathrm{Hz}$ is _____.

Let $f\left(x\right)= \underset{n\to \infty }{\lim }{\left(\frac{n^n\left(x+n\right)\left(x+\frac{n}{2}\right)\dots ..\left(x+\frac{n}{n}\right)}{n!\left(x^2+n^2\right) \left(x^2+\frac{n^2}{4}\right)\dots ..\left(x^2+\frac{n^2}{n^2}\right)}\right)}^{\frac{x}{n}}$ , for all $x>0$ . Then

A hydrogen atom, initially at rest in its ground state, absorbs a photon of frequency \(v_1\) and ejects the electron with a kinetic energy of 10 eV . The electron then combines with a positron at rest to form a positronium atom in its ground state and simultaneously emits a photon of frequency \(v_2\). The center of mass of the resulting positronium atom moves with a kinetic energy of 5 eV . It is given that positron has the same mass as that of electron and the positronium atom can be considered as a Bohr atom, in which the electron and the positron orbit around their center of mass. Considering no other energy loss during the whole process, the difference between the two photon energies (in eV ) is \(\qquad\)

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

List$-\mathrm{I}$ includes starting materials and reagents of selected chemical reactions. List$-\mathrm{II}$gives structures of compounds that may be formed as intermediate products and/or final products from the reactions of List$-\mathrm{I}$.

List-I	List-II
(I)	$\left(\mathrm{P}\right)$
(II)	$\left(\mathrm{Q}\right)$
(III)	$\left(\mathrm{R}\right)$
(IV)	$\left(\mathrm{S}\right)$
	$\left(\mathrm{T}\right)$
	$\left(\mathrm{U}\right)$

Which of the following options has the correct combination considering List$-\mathrm{I}$ and List$-\mathrm{II}$?