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.

A length - scale $\left(\mathcal{l}\right)$ depends on the permittivity $\left(\mathrm{\epsilon }\right)$ of a dielectric material, Boltzmann constant $k_B$ , the absolute temperature T, the number per unit volume (n) of certain charged particles, and the charge (q) carried by each of the particles, Which of the following expressions (s) for $\mathcal{l}$ is (are) dimensionally correct?

Consider a thin square plate floating on a viscous liquid in a large tank. The height $h$ of the liquid in the tank is much less than the width of the tank. The floating plate is pulled horizontally with a constant velocity $u_0$ . Which of the following statements is (are) true?

An annular disk of mass $M$, inner radius $a$ and outer radius $b$ is placed on a horizontal surface with coefficient of friction $\mu$, as shown in the figure. At some time, an impulse $J_0\stackrel{\wedge }{x}$ is applied at a height $h$ above the center of the disk. If $h=h_m$ then the disk rolls without slipping along the $x$-axis. Which of the following statement(s) is(are) correct?

A moving coil galvanometer has 50 turns and each turn has an area $2\times {10}^{-4} {\mathrm{m}}^2$ . The magnetic field produced by the magnet inside the galvanometer is $0.02 \mathrm{T}$ . The torsional constant of the suspension wire is ${10}^{-4} \mathrm{N}\mathrm{ }\mathrm{m}\mathrm{ }{\mathrm{rad}}^{-1}$ . When a current flows through the galvanometer, a full scale deflection occurs if the coil rotates by $0.2 \mathrm{rad}$ . The resistance of the coil of the galvanometer is $50 \mathrm{\Omega }$ . This galvanometer is to be converted into an ammeter capable of measuring current in the range $0-1.0 \mathrm{A}$ . For this purpose, a shunt resistance is to be added in parallel to the galvanometer. The value of this shunt resistance, in ohms, is __________.

A bob of mass $m$, suspended by a string of length $l_1$, is given a minimum velocity required to complete a full circle in the vertical plane. At the highest point, it collides elastically with another bob of mass $m$, suspended by a string of length $l_2$, which is initially at rest. Both the strings are mass-less and inextensible. If the second bob, after collision acquires the minimum speed required to complete a full circle in the vertical plane, the ratio $\frac{l_1}{l_2}$ is

Column-II gives certain systems undergoing a process. Column-I suggests changes in some of the parameters related to the system. Match the statements in Column-I to the appropriate process (es) from Column-II.

Airplanes A and B are flying with constant velocity in the same vertical plane at angles ${30}^{\mathrm{o}}$ and ${60}^{\mathrm{o}}$ with respect to the horizontal respectively as shown in the figure. The speed of A is $100\sqrt{3}\mathrm{}\mathrm{m}{\mathrm{s}}^{-1}$ . At time t = 0 s, an observer in A finds B at a distance of 500 m. This observer sees B moving with a constant velocity perpendicular to the line of motion of A. If at $\mathrm{t}={\mathrm{t}}_0$ , A just escapes being hit by B, t₀ in seconds is

Paragraph :
When liquid medicine of density \(\rho\) is to be put in the eye, it is done with the help of a dropper. As the bulb on the top of the dropper is pressed, a drop forms at the opening of the dropper. We wish to estimate the size of the drop.
We first assume that the drop formed at the opening is spherical because that requires a minimum increase in its surface energy. To determine the size, we calculate the net vertical force due to the surface tension \(T\) when the radius of the drop is \(R\). When this force becomes smaller than the weight of the drop, the drop gets detached from the dropper.
Question :
If \(r=5 \times 10^{-4} \mathrm{~m}, \rho=10^3 \mathrm{~kg} \mathrm{~m}^{-3}\), \(g=10 \mathrm{~ms}^{-2}, T=0.11 \mathrm{Nm}^{-1}\), the radius of the drop when it detaches from the dropper is approximately

Let $\alpha$ and $\beta$ be non zero real numbers such that $2\left(\cos \beta -\cos \alpha \right)+\cos \alpha \cos \beta =1$. Then which of the following is/are true?

Let $n_1 and n_2$ be the number of red and black balls, respectively, in box I. Let $n_{3}and n_4$ be the number of red and black balls, respectively, in box II.
A ball is drawn at random from box I and transferred to box II. If the probability of drawing a red ball from box I, after this transfer, is $\frac{1}{3}$ , then the correct options(s) with the possible values of $n_{1}and n_2$ is(are)

Match the nuclear processes given in column I with the appropriate option(s) in column II.

	Column I		Column II
A.	Nuclear fusion	P	Absorption of thermal neutrons by ${}_{92}{}^{235}\mathrm{U}$
B.	Fission in a nuclear reactor	Q	${}_{27}{}^{60}\mathrm{C}\mathrm{o}$ nucleus
C.	$\mathrm{\beta }$ -decay	R	Energy production in stars via hydrogen conversion to helium
D.	$\mathrm{\gamma }$ -ray emission	S	Heavy water
		T	Neutrino emission

One of the products formed from the reaction of permanganate ion with iodide ion in neutral aqueous medium is