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 $n_1 and n_2$ be the number of red and black balls, respectively, in box I. Let $n_{3}and n_4$ be the number of red and black balls, respectively, in box II.
A ball is drawn at random from box I and transferred to box II. If the probability of drawing a red ball from box I, after this transfer, is $\frac{1}{3}$ , then the correct options(s) with the possible values of $n_{1}and n_2$ is(are)

For any $3\times 3$ matrix $M$, let $\left|M\right|$ denote the determinant of $M$. Let $E=\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 4\\ 8& 13& 18\end{array}\right], P=\left[\begin{array}{lll}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right] \& F=\left[\begin{array}{ccc}1& 3& 2\\ 8& 18& 13\\ 2& 4& 3\end{array}\right]$ If $Q$ is a nonsingular matrix of order $3\times 3$, then which of the following statements is (are) TRUE ?

Let $\mathrm{F}\left(\mathrm{x}\right)=\mathrm{ }\underset{\mathrm{x}}{\overset{{\mathrm{x}}^2+\frac{\mathrm{\pi }}{6}}{\int }}2{\cos }^2\mathrm{t}\mathrm{ }\mathrm{d}\mathrm{t}$ for all $\mathrm{x}\mathrm{ }\in \mathbb{R}\mathrm{ }\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{ }\mathrm{f}\mathrm{ }:\mathrm{ }\left[0,\frac{1}{2}\right]\to \left[0,\mathrm{ }\mathrm{\infty }\right)$ be a continuous function. For $\mathrm{a}\in \left[0,\frac{1}{2}\right],$ if ${\mathrm{F}}^{\mathrm{\prime }}\left(\mathrm{a}\right)+2$ is the area of the region bounded by $\mathrm{x}=0,\mathrm{ }\mathrm{y}=0,\mathrm{ }\mathrm{y}=\mathrm{f}\left(\mathrm{x}\right)\mathrm{ }\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{ }\mathrm{x}=\mathrm{a},$ then $\mathrm{f}\left(0\right)$ is

Let \(\theta \in\left(0, \frac{\pi}{4}\right)\) and \(t_1=(\tan \theta)^{\tan \theta},\) \(t_2=(\tan \theta)^{\cot \theta}\) and \(t_4=(\cot \theta)^{\tan \theta}\), then

Match the following.
 

	List – I		List – II
(A)	The number of polynomials f(x) with non – negative integer coefficients of degree $\le 2,$ satisfying f(0) =0 and $\underset{0}{\overset{1}{\int }}f\left(x\right)dx=1, is$	(P)	8
(B)	The number of point in the interval $\left[- \sqrt{13}, \sqrt{13}\right]$ at which $f\left(x\right)=\sin \left(x^2\right)+\cos \left(x^2\right)$ attains its maximum value, is	(Q)	2
(C)	$\underset{-2}{\overset{2}{\int }}\frac{3x^2}{\left(1+e^x\right)}dx$ equals	(R)	4
(D)	$\frac{\left(\underset{-\frac{1}{2}}{\overset{\frac{1}{2}}{\int }}\cos 2x .\log \left(\frac{1+x}{1-x}\right)dx\right)}{\left(\underset{0}{\overset{\frac{1}{2}}{\int }}\cos 2x .\log \left(\frac{1+x}{1-x}\right)dx\right)}$	(S)	0

Let \(a=3 \sqrt{2}\) and \(b=\frac{1}{5^{1 / 6} \sqrt{6}}\). If \(x, y \in \mathbb{R}\) are such that
\(\begin{gathered}3 x+2 y=\log _a(18)^{\frac{5}{4}} \\2 x-y=\log _b(\sqrt{1080})\end{gathered}\)
and then \(4 x+5 y\) is equal to . ________.

One mole of an ideal gas is taken from \(a\) to \(b\) along two paths denoted by the solid and the dashed lines as shown in the graph below. If the work done along the solid line path is \(W_s\) and that along the dotted line path is \(W_d\), then the integer closest to the ratio \(\frac{W_d}{W_s}\) is

Paragraph
The key feature of Bohr's theory of spectrum of hydrogen atom is the quantization of angular momentum when an electron is revolving around a proton. We will extend this to a general rotational motion to find quantized rotational energy of a diatomic molecule assuming it to be rigid. The rule to be applied is Bohr's quantization condition.
Question:
It is found that the excitation frequency from ground to the first excited state of rotation for the \(\mathrm{CO}\) molecule is close to \(\frac{4}{\pi} \times 10^{11} \mathrm{~Hz}\). Then the moment of inertia of CO molecule about its centre of mass is close to (Take \(h=2 \pi \times 10^{-34} J-s\) )

An open-ended U-tube of uniform cross-sectional area contains water (density ${10}^3\mathrm{kg}{\mathrm{m}}^{-3}$ ). Initially the water level stands at $0.29\mathrm{m}$ from the bottom in each arm. Kerosene oil (a water-immiscible liquid) of density $800\mathrm{kg}{\mathrm{m}}^{-3}$ is added to the left arm until its length is $0.1\mathrm{m}$, as shown in the schematic figure below. The ratio $\left(\frac{h_1}{h_2}\right)$ of the heights of the liquid in the two arms 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____

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 one of the following options is the correct combination?

Two non-conducting spheres of radii $R_1$ and $R_2$ and carrying uniform volume charge densities $+\rho$ and $-\rho$, respectively, are placed such that they partially overlap, as shown in the figure. At all points in the overlapping region,