In an aluminum (Al) bar of square cross section, a square hole is drilled and is filled with iron (Fe) as shown in the figure. The electrical resistivities of Al and Fe are $2.7\times {10}^{-8}\mathrm{\Omega }\mathrm{}\mathrm{m}$ and $1.0\times {10}^{-7}\mathrm{\Omega }\mathrm{}\mathrm{m}$ , respectively. The electrical resistance between the two faces P and Q of the composite bar is

Match List I of the nuclear processes with List II containing parent nucleus and one of the end products of each process and then select the correct answer using the codes given below the lists :

List - i	List - II
A. Alpha decay	P. \({ }_8^{15} O \rightarrow{ }_7^{15} N+\ldots \ldots \ldots\).
B. \(\beta^{+}\)decay	Q. \({ }_{92}^{238} U \rightarrow{ }_{90}^{234} Th +\ldots \ldots \ldots\).
C. Fission	R. \({ }_{83}^{185} Bi \rightarrow{ }_{82}^{184} Pb+\ldots \ldots \ldots\).
D. Proton emission	S. \({ }_{94}^{239} Pu \rightarrow{ }_{57}^{140} La +\ldots \ldots \ldots\).

Two large circular discs separated by a distance of $0.01\mathrm{m}$ are connected to a battery via a switch as shown in the figure. Charged oil drops of density $900\mathrm{kg}{\mathrm{m}}^{-3}$ are released through a tiny hole at the centre of the top disc. Once some oil drops achieve terminal velocity, the switch is closed to apply a voltage of $200\mathrm{V}$ across the discs. As a result, an oil drop of radius $8\times {10}^{-7}\mathrm{m}$ stops moving vertically and floats between the discs. The number of electrons present in this oil drop is __________. (neglect the buoyancy force, take acceleration due to gravity $=10\mathrm{m}{\mathrm{s}}^{-2}$ and charge on an electron (e) $=1.6\times {10}^{-19}\mathrm{C}$)

Statement 1 For practical purposes, the earth is used as a reference at zero potential in electrical circuits.
and Statement 2 The electrical potential of a sphere of radius \(R\) with charge \(Q\) uniformly distributed on the surface is given by \(\frac{Q}{4 \pi \varepsilon_0 R}\).

A circular coil of radius $R$ and $N$ turns has negligible resistance. As shown in the schematic figure, its two ends are connected to two wires and it is hanging by those wires with its plane being vertical. The wires are connected to a capacitor with charge $Q$ through a switch. The coil is in a horizontal uniform magnetic field $B_o$ parallel to the plane of the coil. When the switch is closed, the capacitor gets discharged through the coil in a very short time. By the time the capacitor is discharged fully, magnitude of the angular momentum gained by the coil will be (assume that the discharge time is so short that the coil has hardly rotated during this time)

Positive and negative point charges of equal magnitude are kept at \(\left(0,0, \frac{a}{2}\right)\) and \(\left(0,0, \frac{-a}{2}\right)\), respectively. The work done by the electric field when another positive point charge is moved from \((-a, 0,0)\) to \((0, a, 0)\) is

An accident in a nuclear laboratory resulted in deposition of a certain amount of radioactive material of half-life 18 days inside the laboratory. Tests revealed that the radiation was 64 times more than the permissible level required for safe operation of the laboratory. What is the minimum number of days after which the laboratory can be considered safe for use?

Paragraph I: An organic compound \(\mathrm{P}\) with molecular formula \(\mathrm{C}_5 \mathrm{H}_{18} \mathrm{O}_2\) decolorizes bromine water and also shows positive iodoform test. \(\mathrm{P}\) on ozonolysis followed by treatment with \(\mathrm{H}_2 \mathrm{O}_2\) gives \(\mathrm{Q}\) and \(\mathrm{R}\). While compound \(\mathrm{Q}\) shows positive iodoform test, compound \(\mathrm{R}\) does not give positive iodoform test. \(\mathrm{Q}\) and \(\mathrm{R}\) on oxidation with pyridinium chlorochromate (PCC) followed by heating give \(\mathrm{S}\) and \(\mathrm{T}\), respectively. Both \(\mathrm{S}\) and \(\mathrm{T}\) show positive iodoform test.
Complete copolymerization of 500 moles of \(\mathrm{Q}\) and 500 moles of \(\mathrm{R}\) gives one mole of a single acyclic copolymer \(\mathrm{U}\).
[Given, atomic mass: \(\mathrm{H}=1, \mathrm{C}=12, \mathrm{O}=16\)]
Question: The molecular weight of \(\mathbf{U}\) is

One mole of an ideal gas expands adiabatically from an initial state $(T_A, V_0)$ to final state $(T_f, 5V_0)$. Another mole of the same gas expands isothermally from a different initial state $(T_B, V_0)$ to the same final state $(T_f, 5V_0)$. The ratio of the specific heats at constant pressure and constant volume of this ideal gas is $\gamma$. What is the ratio $\frac{T_A}{T_B}$?

A real gas behaves like an ideal gas if its

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 \(n=100, M=3 \mathrm{~kg}, m=0.01 \mathrm{~kg}\); \(b=2 \mathrm{~m} ; a=1 \mathrm{~m} ; g=10 \mathrm{~m} / \mathrm{s}^2\).

Four solid spheres each of diameter \(\sqrt{5} \mathrm{~cm}\) and mass \(0.5 \mathrm{~kg}\) ar placed with their centres at the corners of a square of side \(4 \mathrm{~cm}\). The moment of inertia of the system about the diagonal of the square is \(N \times 10^{-4} \mathrm{~kg}-\mathrm{m}^2\), then \(N\) is

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