Two equilateral-triangular prisms \(\mathrm{P}_1\) and \(\mathrm{P}_2\) are kept with their sides parallel to each other, in vacuum, as shown in the figure. A light ray enters prism \(P_1\) at an angle of incidence \(\theta\) such that the outgoing ray undergoes minimum deviation in prism \(P_2\). If the respective refractive indices of \(\mathrm{P}_1\) and \(\mathrm{P}_2\) are \(\sqrt{\frac{3}{2}}\) and \(\sqrt{3}\), then \(\theta=\sin ^{-1}\left[\sqrt{\frac{3}{2}} \sin \left(\frac{\pi}{\beta}\right)\right]\), where the value of \(\beta\) is _______ .

A uniform circular disc of mass 1.5 kg and radius 0.5 m is initially at rest on a horizontal frictionless surface. Three forces of equal magnitude $\mathrm{F}\mathrm{}=\mathrm{}0.5\mathrm{}\mathrm{N}$ are applied simultaneously along the three sides of an equilateral triangle XYZ with its vertices on the perimeter of the disc (see figure). One second after applying the forces, the angular speed of the disc in rad ${\mathrm{s}}^{-1}$ is

A long straight wire carries a current, $I=2$ ampere. A semi-circular conducting rod is placed beside it on two conducting parallel rails of negligible resistance. Both the rails are parallel to the wire. The wire, the rod and the rails lie in the same horizontal plane, as shown in the figure. Two ends of the semi-circular rod are at distances $1 \mathrm{cm}$ and $4 \mathrm{cm}$ from the wire.  At time $t=0$, the rod starts moving on the rails with a speed $v=3.0 \mathrm{m} {\mathrm{s}}^{-1}$ (see the figure). A resistor $R=1.4 \mathrm{\Omega }$ and a capacitor $C_0=5.0 \mathrm{\mu F}$ are connected in series between the rails. At time $t=0$, $C_0$ is uncharged. Which of the following statement(s) is (are) correct? [${\mu }_0=4\pi \times {10}^{-7}$ SI units. Take $\ln 2=0.7$]

The circular scale of a screw gauge has 50 divisions and pitch of \(0.5 \mathrm{~mm}\). Find the diameter of sphere. Main scale reading is 2 .

In the following circuit, the current through the resistor $R(=2\mathrm{\Omega })$ is $\mathrm{I}$ Amperes. The value of $\mathrm{I}\mathrm{}$ is

Column I gives certain situations in which a straight metallic wire of resistance \(R\) is used and Column II gives some resulting effects. Match the statements in Column I with the statements in Column II and indicate your answer by darkening appropriate bubbles in the \(4 \times 4\) matrix given in the ORS.

Paragraph :

A thermal power plant produces electric power of \(600 \mathrm{~kW}\) at \(4000 \mathrm{~V}\), which is to be transported to a place \(20 \mathrm{~km}\) away from the power plant for consumers' usage. It can be transported either directly with a cable of large current carrying capacity or by using a combination of step-up and step-down transformers at the two ends. The drawback of the direct transmission is the large energy dissipation. In the method using transformers, the dissipation is much smaller. In this method, a step-up transformer is used at the plant side so that the current is reduced to a smaller value. At the consumers' end, a step-down transformer is used to supply power to the consumers at the specified lower voltage. It is reasonable to assume that the power cable is purely resistive and the transformers are ideal with a power factor unity. All the currents and voltages mentioned are rms values.

Question :

In the method using the transformers, assume that the ratio of the number of turns in the primary to that in the secondary in the step-up transformer is \(1: 10\). If the power to the consumers has to be supplied at \(200 \mathrm{~V}\), the ratio of the number of turns in the primary to that in the secondary in the step-down transformer is

Statement I Micelles are formed by surfactant molecules above the critical micellar concentration (CMC).
Statement II The conductivity of a solution having surfactant molecules deceases sharply at the CMC.

There is a rectangular plate of mass \(M \mathrm{~kg}\) of dimensions \((a \times b)\). The plate is held in horizontal position by striking \(n\) small balls each of mass \(m\) per unit area per unit time. These are striking in the shaded half region of the plate. The balls are colliding elastically with velocity \(v\). What is \(v\) ? It is given \(n=100, M=3 \mathrm{~kg}, m=0.01 \mathrm{~kg}\); \(b=2 \mathrm{~m} ; a=1 \mathrm{~m} ; g=10 \mathrm{~m} / \mathrm{s}^2\).

A right-angled prism of refractive index ${\mu }_1$ is placed in a rectangular block of refractive index ${\mu }_2$, which is surrounded by a medium of refractive index ${\mu }_3$, as shown in the figure. A ray of light $e$ enters the rectangular block at normal incidence. Depending upon the relationships between ${\mu }_1\text{, }{\mu }_2$ and ${\mu }_3$, it takes one of the four possible paths $ef, eg, eh$ or $ei$.

Match the paths in List $\mathrm{I}$ with conditions of refractive indices in List $\mathrm{II}$ and select the correct answer using the codes given below the lists:

	List I		List II
A.	$e\to f$	P.	${\mu }_1>\sqrt{2}{\mu }_2$
B.	$e\to g$	Q.	${\mu }_2>{\mu }_1\text{and}{\mu }_2>{\mu }_3$
C.	$e\to h$	R.	${\mu }_1={\mu }_2$
D.	$e\to i$	S.	${\mu }_2<{\mu }_1<\sqrt{2}{\mu }_2\text{and}{\mu }_2>{\mu }_3$

Let $X$ be the set consisting of the first $2018$ terms of the arithmetic progression $\mathrm{1,6},11,\dots ,$ and $Y$ be the set consisting of the first $2018$ terms of the arithmetic progression $\mathrm{9,16,23},\dots$ . Then, the number of elements in the set $X\cup Y$ is___.

Column II shows five systems in which two objects are labelled as \(X\) and \(Y\). Also in each case a point \(P\) is shown. Column-I gives some statements about \(X\) and/or \(Y\). Match these statements to the appropriate system(s) from Column-II.

Consider an electrochemical cell: $\text{A}\left(\text{s}\right)\left|{\text{ A}}^{\text{n+}}\left(\text{aq, 2 M}\right)\right|\left|{\text{ B}}^{\text{2n+}}\left(\text{aq, 1 M}\right)\right|\text{ B}\left(\text{s}\right)$. The value of ${\text{ΔH}}^{\text{o}}$ for the cell reaction is twice that of ${\text{ΔG}}^{\text{o}}$ at $\text{300 K}\text{.}$ If the emf of the cell is zero, the magnitude of $\mathrm{\Delta }{\text{S}}^o$ $\left(\text{in} {\text{J}}^{-1}{\text{K}}^{-1} {\text{mol}}^{-1}\right)$ of the cell reaction per mole of $\text{B}$ formed at $\text{300}\text{K}$ is ____
(Given: R $\text{=}\text{8}\text{.3}{\text{JK}}^{-\text{1}}{\text{mol}}^{-1}.$ $\text{H,}\text{S}$ and $\text{G}$ are enthalpy, entropy and Gibbs energy, respectively.)