Sometimes it is convenient to construct a system of units so that all quantities can be expressed in terms of only one physical quantity. In one such system, dimensions of different quantities are given in terms of a quantity $X$ as follows: $\left[\mathrm{position}\right]=\left[X^{\alpha }\right];$ $\left[\mathrm{speed}\right]=\left[X^{\beta }\right]$; $\left[\mathrm{acceleration}\right]=\left[X^p\right]$;$\left[\mathrm{linear}\mathrm{momentum}\right]=\left[X^q\right]$; $\left[\mathrm{force}\right]=\left[X^r\right]$. Then

In Circuit-$1$ and Circuit-$2$ shown in the figures, $R_1=1\mathrm{\Omega },R_2=2\mathrm{\Omega }$ and $R_3=3\mathrm{\Omega }$.
$P_1$ and $P_2$ are the power dissipations in Circuit-$1$ and Circuit-$2$ when the switches $S_1$ and $S_2$ are in open conditions, respectively.
$Q_1$ and $Q_2$ are the power dissipations in Circuit-$1$ and Circuit-$2$ when the switches $S_1$ and $S_2$ are in closed conditions, respectively.

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

One mole of an ideal gas undergoes two different cyclic processes $\mathrm{I}$ and $\mathrm{II}$, as shown in the $P-V$ diagrams below. In cycle $\mathrm{I}$, processes $a, b, c$ and $d$ are isobaric, isothermal, isobaric and isochoric, respectively. In cycle $\mathrm{II}$, processes $a\text{'}, b\text{'}, c\text{'}$ and $d\text{'}$ are isothermal, isochoric, isobaric and isochoric, respectively. The total work done during cycle $\mathrm{I}$ is $W_{\mathrm{I}}$ and that during cycle $\mathrm{II}$ is $W_{\mathrm{II}}$. The ratio $\frac{W_{\mathrm{I}}}{W_{\mathrm{II}}}$ is _____.

A ray of light, travelling in the direction $\frac{1}{2}\left(\hat{\mathrm{i}}+\sqrt{3} \hat{\mathrm{j}}\right)$, is incident on a plane mirror. After reflection, it travels along the direction $\frac{1}{2}\left(\hat{\mathrm{i}}-\sqrt{3}\widehat{ \mathrm{j}}\right)$. The angle of incidence is

Two identical uniform discs roll without slipping on two different surfaces AB and CD (see figure) starting at A and C with linear speeds \(v_1\) and \(v_2\), respectively, and always remain in contact with the surfaces. If they reach B and D with the same linear speed and \(v_1=3 m / s\), then \(v_2\) in \(m / s\) is \(\left( g =10 \frac{m}{ s ^2}\right)\)

A light disc made of aluminum (a nonmagnetic material) is kept horizontally and is free to rotate about its axis as shown in the figure. A strong magnet is held vertically at a point above the disc away from its axis. On revolving the magnet about the axis of the disc, the disc will (figure is schematic and not drawn to scale)

A planet of mass M, has two natural satellites with masses $m_1\mathrm{a}\mathrm{n}\mathrm{d}{m}_2$ . The radii of their circular orbits are $R_1 \mathrm{a}\mathrm{n}\mathrm{d} R_2$ respectively. Ignore the gravitational force between the satellites. Define $v_1,L_1,K_1 \mathrm{a}\mathrm{n}\mathrm{d} T_1$ to be, respectively, the orbital speed, angular momentum, kinetic energy and time period of revolution of satellite $1; \mathrm{a}\mathrm{n}\mathrm{d} v_2,L_2,K_2\mathrm{a}\mathrm{n}\mathrm{d} T_2$ and to be the corresponding quantities of satellite 2. Given $m_1/m_2=2 \mathrm{a}\mathrm{n}\mathrm{d} R_1/R_2=\frac{1}{4}$ , match the ratios in List-I to the numbers in List-II.

	LIST -I		LIST-II
A)	$\frac{v_1}{v_2}$	P)	$\frac{1}{8}$
B)	$\frac{L_1}{L_2}$	Q)	1
C)	$\frac{K_1}{K_2}$	R)	2
D)	$\frac{T_1}{T_2}$	S)	8

The ellipse \(E_{1}: \frac{x^{2}}{9}+\frac{y^{2}}{4}=1\) is inscribed in a rectangle \(R\) whose sides are parallel to the coordinate axes. Another ellipse \(E_{2}\) passing through the point \((0,4)\) circumscribes the rectangle \(R\). The eccentricity of the ellipse \(E_{2}\) is

Bleaching powder and bleach solution are produced on a large scale and used in several household products. The effectiveness of bleach solution is often measured by iodometry.
Question:
\(25 \mathrm{~mL}\) of household solution was mixed with \(30 \mathrm{~mL}\) of \(0.50\) M KI and \(10 \mathrm{~mL}\) of \(4 \mathrm{~N}\) acetic acid. In the titration of the liberated iodine, \(48 \mathrm{~mL}\) of \(0.25 \mathrm{~N~} \mathrm{Na}_{2} \mathrm{~S}_{2} \mathrm{O}_{3}\) was used to reach the end point. The molarity of the household bleach solution is

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A special metal \(S\) conducts electricity without any resistance. A closed wire loop, made of \(S\), does not allow any change in flux through itself by inducing a suitable current to generate a compensating flux. The induced current in the loop cannot decay due to its zero resistance. This current gives rise to a magnetic moment which in turn repels the source of magnetic field or flux. Consider such a loop, of radius \(a\), with its center at the origin. A magnetic dipole of moment \(m\) is brought along the axis of this loop from infinity to a point at distance \(r(\gg a)\) from the center of the loop with its north pole always facing the loop, as shown in the figure below.

The magnitude of magnetic field of a dipole \(m\), at a point on its axis at distance \(r\), is \(\frac{\mu_{0}}{2 \pi} \frac{m}{r^{3}}\), where \(\mu_{0}\) is the permeability of free space. The magnitude of the force between two magnetic dipoles with moments, \(m_{1}\) and \(m_{2}\), separated by a distance \(r\) on the common axis, with their north poles facing each other, is \(\frac{k m_{1} m_{2}}{r^{4}}\), where \(k\) is a constant of appropriate dimensions. The direction of this force is along the line joining the two dipoles.

Question :
The work done in bringing the dipole from infinity to a distance$r$from the centre of the loop by the given process is proportional to:

Most materials have the refractive index, \(n>1\). So, when a light ray from air enters a naturally occurring material, then by Snell's

law, \(\frac{\sin \theta_{1}}{\sin \theta_{2}}=\frac{n_{2}}{n_{1}}\), it is understood that the refracted ray bends towards the normal. But it never emerges on the same side of the normal as the incident ray. According to electromagnetism, the refractive index of the medium is given by the relation, \(\mathrm{n}=c / v=\pm \sqrt{\varepsilon_{r} \mu_{r}}\), where \(c\) is the speed of electromagnetic waves in vacuum, \(v\) its speed in the medium, \(\varepsilon_{r}\) and \(\mu_{r}\) are the relative permittivity and permeability of the medium respectively. In normal materials, both \(\varepsilon_{r}\) and \(\mu_{r}\), are positive, implying positive \(n\) for the medium. When both \(\varepsilon_{r}\) and \(\mu_{r}\) are negative, one must choose the negative root of \(n\). Such negative refractive index materials can now be artificially prepared and are called metamaterials. They exhibit significantly different optical behavior, without violating any physical laws. Since \(n\) is negative, it results in a change in the direction of propagation of the refracted light. However, similar to normal materials, the frequency of light remains unchanged upon refraction even in meta-materials.
Question : Choose the correct statement.

Let $G$ be a circle of radius $R>0$. Let $G_1,G_2,\dots ,G_n$ be $n$ circles of equal radius $r>0$. Suppose each of the $n$ circles $G_1,G_2,\dots ,G_n$ touches the circle $G$ externally. Also, for $i=1,2,\dots ,n-1$, the circle $G_i$ touches $G_{i+1}$ externally, and $G_n$ touches $G_1$ externally. Then, which of the following statements is/are TRUE?

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Let \(F_{1}\left(x_{1}, 0\right)\) and \(F_{2}\left(x_{2}, 0\right)\), for \(x_{1}<0\) and \(x_{2}>0\), be the foci of the ellipse \(\frac{x^{2}}{9}+\frac{y^{2}}{8}=1 .\) Suppose a parabola having vertex at the origin and focus at \(F_{2}\) intersects the ellipse at point \(M\) in the first quadrant and at point \(N\) in the fourth quadrant.


Question:

If the tangents to the ellipse at \(M\) and \(N\) meet at \(R\) and the normal to the parabola at \(M\) meets the \(x\)-axis at \(Q\), then the ratio of area of the triangle \(M Q R\) to area of the quadrilateral \(M F_{1} N F_{2}\) is