Three plane mirrors form an equilateral triangle with each side of length $L$. There is a small hole at a distance $l>0$ from one of the corners as shown in the figure. A ray of light is passed through the hole at an angle $\theta$ and can only come out through the same hole. The cross section of the mirror configuration and the ray of light lie on the same plane.

Which of the following statement(s) is(are) correct?

A flat surface of a thin uniform disk $A$of radius$R$ is glued to a horizontal table. Another thin uniform disk$B$ of mass $M$and with the same radius $R$rolls without slipping on the circumference of $A$, as shown in the figure. A flat surface of $B$ also lies on the plane of the table. The center of mass of$B$ has fixed angular speed $\omega$about the vertical axis passing through the center of $A$. The angular momentum of $B$ is $nM\omega R^2$with respect to the center of $A$. Which of the following is the value of $n$?

In a radioactive sample, ${\mathrm{}}_{19}^{40}\mathrm{K}$ nuclei either decay into stable ${\mathrm{}}_{20}^{40}\mathrm{C}\mathrm{a}$ nuclei with decay constant $4.5\times {10}^{-10}$ per year or into stable ${\mathrm{}}_{18}^{40}\mathrm{A}\mathrm{r}$ nuclei with decay constant $0.5\times {10}^{-10}$ per year. Given that in this sample all the stable ${\mathrm{}}_{20}^{40}\mathrm{C}\mathrm{a}$ and ${\mathrm{}}_{18}^{40}\mathrm{A}\mathrm{r}$ nuclei are produced by the ${\mathrm{}}_{19}^{40}\mathrm{K}$ nuclei only. In time $\mathrm{t}\times {10}^9$ years, if the ratio of the sum of stable ${\mathrm{}}_{20}^{40}\mathrm{C}\mathrm{a}$ and ${\mathrm{}}_{18}^{40}\mathrm{A}\mathrm{r}$ nuclei to the radioactive ${\mathrm{}}_{19}^{40}\mathrm{K}$ nuclei is $99,$ the value of t will be : [Given $\mathrm{l}\mathrm{n}\mathrm{}10=2.3$ ]

In a circuit, a metal filament lamp is connected in series with a capacitor of capacitance \(\mathrm{C} \mu F\) across a \(200 \mathrm{~V}, 50 \mathrm{~Hz}\) supply. The power consumed by the lamp is \(500 \mathrm{~W}\) while the voltage drop across it is \(100 \mathrm{~V}\). Assume that there is no inductive load in the circuit. Take \(r m s\) values of the voltages. The magnitude of the phaseangle (in degrees) between the current and the supply voltage is \(\varphi\). Assume, \(\pi \sqrt{3} \approx 5\).
The value of $\varphi$ is _____.

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The nuclear charge \((\mathrm{Ze})\) is non-uniformly distributed within a nucleus of radius \(R\). The charge density \(\rho(r)\) [charge per unit volume] is dependent only on the radial distance \(r\) from the centre of the nucleus as shown in figure. The electric field is only along the radial direction.

Question:
For \(a=0\), the value of \(d\) (maximum value of \(\rho\) as shown in the figure) is

The circuit shown in the figure contains an inductor \(L\), a capacitor \(C_0\), a resistor \(R_0\) and an ideal battery. The circuit also contains two keys \(\mathrm{K}_1\) and \(\mathrm{K}_2\). Initially, both the keys are open and there is no charge on the capacitor. At an instant, key \(K_1\) is closed and immediately after this the current in \(R_0\) is found to be \(I_1\). After a long time, the current attains a steady state value \(I_2\). Thereafter, \(\mathrm{K}_2\) is closed and simultaneously \(\mathrm{K}_1\) is opened and the voltage across \(C_0\) oscillates with amplitude \(V_0\) and angular frequency \(\omega_0\).

Match the quantities mentioned in List-I with their values in List-II and choose the correct option.

A real gas behaves like an ideal gas if its

A normal with slope \(\frac{1}{\sqrt{6}}\) is drawn from the point \((0,-\alpha)\) to the parabola \(x^2=-4 a y\), where \(a>0\). Let \(L\) be the line passing through \((0,-\alpha)\) and parallel to the directrix of the parabola. Suppose that \(L\) intersects the parabola at two points \(A\) and \(B\). Let \(r\) denote the length of the latus rectum and \(s\) denote the square of the length of the line segment \(A B\). If \(r: s=1: 16\), then the value of 24ais ________.

In a radioactive decay chain reaction, ${}_{90}^{230}\mathrm{Th}$ nucleus decays into ${}_{84}{}^{214}\mathrm{Po}$ nucleus. The ratio of the number of $\alpha$ to number of ${\beta }^{-}$ particles emitted in this process is _______.

For the reaction sequence given below, the correct statement(s) is(are)

One mole of an ideal gas at $900 \mathrm{K}$, undergoes two reversible processes, $\mathrm{I}$ followed by $\mathrm{II}$, as shown below. If the work done by the gas in the two processes are same, the value of $\ln \frac{{\mathrm{V}}_3}{{\mathrm{V}}_2}$ is ($\mathrm{U}$ : internal energy, $\mathrm{S}:$ entropy, $\mathrm{p}:$ pressure, $\mathrm{V}:$ volume, $\mathrm{R}$ : gas constant)  (Given: molar heat capacity at constant volume, ${\mathrm{C}}_{\mathrm{V},\mathrm{m}}$ of the gas is $\frac{5}{2}\mathrm{R}$)

Suppose a ${}_{88}{}^{226}Ra$ nucleus at rest and in ground state undergoes $\mathrm{\alpha }$ -decay to a ${}_{86}{}^{222}Rn$ nucleus in its excited state. The kinetic energy of the emitted $\mathrm{\alpha }$ particle is found to be $4.44\mathrm{ }\mathrm{M}\mathrm{e}\mathrm{V}.{}_{86}{}^{222}Rn$ nucleus then goes to its ground state by $\mathrm{\gamma }$ -decay. The energy of the emitted $\mathrm{\gamma }$ -photon is _______ $\mathrm{k}\mathrm{e}\mathrm{V},$
[Given: atomic mass of ${\mathrm{ }}_{88}^{226}\mathrm{R}\mathrm{a}=226.005\mathrm{u},$ atomic mass of ${\mathrm{ }}_{86}^{222}\mathrm{R}\mathrm{n}=222.000\mathrm{u},$ atomic mass of $\mathrm{\alpha }$ particle $=4.000\mathrm{u},1\mathrm{u}=931\mathrm{M}\mathrm{e}\mathrm{V}/{\mathrm{c}}^2,\mathrm{c}$ is speed of the light]

Let $P=\left[\begin{array}{ccc}1& 0& 0\\ 4& 1& 0\\ 16& 4& 1\end{array}\right]$ and $I$ be the identity matrix of order $3$. If $Q=\left[q_{ij}\right]$ is a matrix such that $P^{50}-Q=I,$ then the value of $\frac{q_{31}+q_{32}}{q_{21}}$ is equal to