To find the distance d over which a signal can be seen clearly in foggy conditions, a railways engineer uses dimensional analysis and assumes that the distance depends on the mass density $\mathrm{\rho }$ of the fog, intensity (power/area) S of the light from the signal and its frequency f. The engineer finds that d is proportional to ${\mathrm{S}}^{\frac{1}{\mathrm{n}}}$ . The value of n is

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A small spherical monoatomic ideal gas bubble \(\left(\gamma=\frac{5}{3}\right)\) is trapped inside a liquid of density \(\rho_1\) (see figure). Assume that the bubble does not exchange any heat with the liquid. The bubble contains \(n\) moles of gas. The temperature of the gas when the bubble is at the bottom is \(T_0\), the height of the liquid is \(H\) and the atmospheric pressure is \(p_0\) (Neglect surface tension).

Question:
When the gas bubble is at height \(y\) from the bottom, its temperature is

The focal length of a thin biconvex lens is \(20 \mathrm{~cm}\). When an object is moved from a distance of \(25 \mathrm{~cm}\) in front of it to \(50 \mathrm{~cm}\), the magnification of its image changes from \(m_{25}\) to \(m_{50}\). The ratio \(\frac{m_{25}}{m_{50}}\) is

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One twirls a circular ring (of mass \(M\) and radius \(R\) ) near the tip of one's finger as shown in Figure 1 . In the process the finger never loses contact with the inner rim of the ring. The finger traces out the surface of a cone, shown by the dotted line. The radius of the path traced out by the point where the ring and the finger is in contact is \(r\). The finger rotates with an angular velocity \(\omega_{0}\). The rotating ring rolls without slipping on the outside of a smaller circle described by the point where the ring and the finger is in contact (Figure 2). The coefficient of friction between the ring and the finger is \(\mu\) and the acceleration due to gravity is \(g\).








Question:

The minimum value of \(\omega_{0}\) below which the ring will drop down is

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.

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Phase space diagrams are useful tools in analyzing all kinds of dynamical problems. They are especially useful in studying the changes in motion as initial position and momentum are changed. Here we consider some simple dynamical systems in one-dimension. For such systems, phase space is a plane in which position is plotted along horizontal axis and momentum is plotted along vertical axis. The phase space diagram is \(x(t) v s p(t)\) curve in this plane. The arrow on the
curve indicates the time flow. For example, the phase space diagram for a particle moving with constant velocity is a straight line as shown in the figure. We use the sign convention in which position or momentum upwards (or to right) is positive and downwards (or to left) is negative.

Question :
The phase space diagram for simple harmonic motion is a circle centered at the origin. In the figure, the two circles represent the same oscillator but for different initial conditions, and \(E_1\) and \(E_2\) are the total mechanical energies respectively. Then,

In an $X$-ray tube, electrons emitted from a filament (cathode) carrying current $I$ hit a target (anode) at a distance $d$ from the cathode. The target is kept at a potential $V$ higher than the cathode resulting in emission of continuous and characteristic X-rays. If the filament current $I$ is decreased to $\frac{I}{2},$ the potential difference $V$ is increased to $2V,$ and the separation distance $d$ is reduced to $\frac{d}{2},$ then

If $y\left(x\right)$ is the solution of the differential equation $xdy-\left(y^2-4y\right)dx=0$ for $x>0, y\left(1\right)=2$, and the slope of the curve $y=y\left(x\right)$ is never zero, then the value of $10y\left(\sqrt{2}\right)$ is _____ .

In the following reaction, the product(s) formed is/are:

Let \(A_n=\left(\frac{3}{4}\right)-\left(\frac{3}{4}\right)^2+\left(\frac{3}{4}\right)^3+\ldots+(-1)^{n-1}\)\(\left(\frac{3}{4}\right)^n\) and \(B_n=1-A_n\). Find a least odd natural number \(n_0\), so that \(B_n>A_n, \forall n \geq n_0\).

Consider the $6\times 6$ square in the figure. Let $A_1,A_2,\dots ,A_{49}$ be the points of intersections (dots in the picture) in some order. We say that $A_i$ and $A_j$ are friends if they are adjacent along a row or along a column. Assume that each point $A_i$ has an equal chance of being chosen.

(There are two questions based on $\mathrm{PARAGRAPH} "\mathrm{II}"$, the question given below is one of them)
Two distinct points are chosen randomly out of the points $A_1,A_2,\dots \dots ,A_{49}$. Let $p$ be the probability that they are friends. Then the value of $7 p$ is

A hyperbola, having the transverse axis of length \(2 \sin \theta\), is confocal with the ellipse \(3 x^2+4 y^2=12\). Then, its equation is

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A point charge \(Q\) is moving in a circular orbit of radius \(R\) in the \(x-y\) plane with an angular velocity \(\omega .\) This can be considered as equivalent to a loop carrying a steady current \(\frac{Q \omega}{2 \pi}\). A uniform magnetic field along the positive \(z\)-axis is now switched on, which increases at a constant rate from 0 to \(B\) in one second. Assume that the radius of the orbit remains constant. The application of the magnetic field induces an emf in the orbit. The induced emf is defined as the work done by an induced electric field in moving a unit positive charge around a closed loop. It is known that, for an orbiting charge, the magnetic dipole moment is proportional to the angular momentum with a proportionality constant \(\gamma\).

Question:

The change in the magnetic dipole moment associated with the orbit, at the end of the time interval of the magnetic field change, is