Paragraph :
A wooden cylinder of diameter \(4 r\), height \(H\) and density \(\rho / 3\) is kept on a hole of diameter \(2 r\) of a tank, filled with liquid of density \(\rho\) as shown in the figure.

Question :
The block in the above question is maintained at the position by external means and the level of liquid is lowered. The height \(h_2\) when this external force reduces to zero is

Paragraph :
Two plane harmonic sound waves are expressed by the equations.
\(
\begin{aligned}
& y_1(x, t)=A \cos (0.5 \pi x-100 \pi t) \\
& y_2(x, t)=A \cos (0.46 \pi x-92 \pi t)
\end{aligned}
\)
(All parameters are in MKS)
Question :
What is the speed of the sound?

Column I shows four situations of standard Young's double slit arrangement with the screen placed far away from the slits \(S_1\) and \(S_2\). In each of these cases \(S_1 P_0=S_2 P_0\), \(S_1 P_1-S_2 P_1=\frac{\lambda}{4}\) and \(S_1 P_2-S_2 P_2=\frac{\lambda}{3}\), where \(\lambda\) is the wavelength of the light used. In the cases \(\mathrm{B}, \mathrm{C}\) and \(\mathrm{D}\), a transparent sheet of refractive index \(\mu\) and thickness \(t\) is pasted on slit \(S_2\). The thickness of the sheets are different in different cases. The phase difference between the light waves reaching a point \(P\) on the screen from the two slits is denoted by \(\delta(P)\) and the intensity by \(I(P)\). Match each situation given in Column-I with the statement(s) in Column-II valid for that situation.

A cube of unit volume contains \(35 \times 10^7\) photons of frequency \(10^{15} \mathrm{~Hz}\). If the energy of all the photons is viewed as the average energy being contained in the electromagnetic waves within the same volume, then the amplitude of the magnetic field is \(\alpha \times 10^{-9} \mathrm{~T}\). Taking permeability of free space \(\mu_0=4 \pi \times 10^{-7} \mathrm{Tm} / \mathrm{A}\), Planck's constant \(h=6 \times 10^{-34} \mathrm{Js}\) and \(\pi=\frac{22}{7}\), the value of \(\alpha\) is_______

Paragraph:
The figure shows a surface \(X Y\) separating two transparent media, medium-1 and medium-2. The lines \(a b\) and \(c d\) represent wavefronts of a light wave travelling in medium-1 and incident on \(X Y\). The lines ef and \(g h\) represent wavefronts of the light wave in medium-2 after refraction.
Question:
Light travels as a

Two solid cylinders \(P\) and \(Q\) of same mass and same radius start rolling down a fixed inclined plane from the same height at the same time. Cylinder \(P\) has most of its mass concentrated near its surface, while \(Q\) has most of its mass concentrated near the axis. Which statement(s) is(are) correct?

Two non-reactive monoatomic ideal gases have their atomic masses in the ratio 2 : 3. The ratio of their partial pressures, when enclosed in a vessel kept at a constant temperature, is 4 : 3. The ratio of their densities is 

The probability that a missile hits a target successfully is $0.75.$ In order to destroy the target completely, at least three successful hits are required. Then the minimum number of missiles that have to be fired so that the probability of completely destroying the target is not less than $0.95,$ is ___________

Paragraph: Let $S$ be the circle in the $xy$ – plane defined by the equation $x^2 + y^2 = 4.$

Question : Let $P$ be a point on the circle $S$ with both coordinates being positive. Let the tangent to $S$ at $P$ intersect the coordinate axes at the points $M$ and $MN$ Then, the mid-point of the line segment must lie on the curve

Statement I p-hydroxybenzoic acid has a lower boiling point than o-hydroxybenzoic acid.
Statement II o-hydroxybenzoic acid has intramolecular hydrogen bonding.

Choose the reaction(s) from the following options, for which the standard enthalpy of reaction is equal to the standard enthalpy of formation.

An optical arrangement consists of two concave mirrors $M_1$ and $M_2$, and a convex lens $L$ with a common principal axis, as shown in the figure. The focal length of $L$ is $10 \mathrm{cm}$. The radii of curvature of $M_1$ and $M_2$ are $20 \mathrm{cm}$ and $24 \mathrm{cm}$, respectively. The distance between $L$ and $M_2$ is $20 \mathrm{cm}$. A point object $S$ is placed at the mid-point between $L$ and $M_2$ on the axis. When the distance between $L$ and $M_1$ is $\frac{n}{7} \mathrm{cm}$, one of the images coincides with $S$. The value of $n$ is _____(Consider the case which gives minimum value).

Paragraph:

The figure shows a circular loop of radius \(a\) with two long parallel wires (numbered \(1\) and \(2\) ) all in the plane of the paper. The distance of each wire from the centre of the loop is \(d\). The loop and the wires are carrying the same current \(I\). The current in the loop is in the counterclockwise direction if seen from above.



Question:

When \(d \approx a\) but wires are not touching the loop, it is found that the net magnetic field on the axis of the loop is zero at a height \(h\) above the loop. In that case