One mole of an ideal gas at $900 \mathrm{K}$, undergoes two reversible processes, $\mathrm{I}$ followed by $\mathrm{II}$, as shown below. If the work done by the gas in the two processes are same, the value of $\ln \frac{{\mathrm{V}}_3}{{\mathrm{V}}_2}$ is ($\mathrm{U}$ : internal energy, $\mathrm{S}:$ entropy, $\mathrm{p}:$ pressure, $\mathrm{V}:$ volume, $\mathrm{R}$ : gas constant)  (Given: molar heat capacity at constant volume, ${\mathrm{C}}_{\mathrm{V},\mathrm{m}}$ of the gas is $\frac{5}{2}\mathrm{R}$)

The number of $-{\mathrm{CH}}_{2-}$ (methylene) groups in the product formed from the following reaction sequence is _____ .

The thermal dissociation equilibrium of ${\mathrm{CaCO}}_3\left(\mathrm{s}\right)$ is studied under different conditions.
${\text{CaCO}}_3\left(s\right)\rightleftharpoons \text{CaO}\left(s\right)+{\text{CO}}_2\left(\text{g}\right)$
For this equilibrium, the correct statement(s) is/are

Among ${\mathrm{B}}_2{\mathrm{H}}_6,\mathrm{ }{\mathrm{B}}_3{\mathrm{N}}_3{\mathrm{H}}_6,\mathrm{ }{\mathrm{N}}_2\mathrm{O},\mathrm{ }{\mathrm{N}}_2{\mathrm{O}}_4,\mathrm{ }{\mathrm{H}}_2{\mathrm{S}}_2{\mathrm{O}}_3$ and ${\mathrm{H}}_2{\mathrm{S}}_2{\mathrm{O}}_8,$ the number of molecules containing covalent bond between two atoms of the same kind is __________

Paragraph:
Tollen's reagent is used for the detection of aldehyde when a solution of \(\mathrm{AgNO}_3\) is added to glucose with \(\mathrm{NH}_4 \mathrm{OH}\) then gluconic acid is formed
\(\mathrm{Ag}^{+}+e^{-} \rightarrow \mathrm{Ag} E_{\text {red }}^{\circ}=0.8 \mathrm{~V} \)
\( \mathrm{C}_6 \mathrm{H}_{12} \mathrm{O}_6+\mathrm{H}_2 \mathrm{O} \rightarrow \text { Gluconic acid }\left(\mathrm{C}_6 \mathrm{H}_{12} \mathrm{O}_7\right)\) \(+~2 \mathrm{H}^{+}+2 e^{-} ; -E_{\mathrm{red}}^{\circ}=-0.05 \mathrm{~V} \)
\( \mathrm{Ag}\left(\mathrm{NH}_3\right)_2^{+}+e^{-} \rightarrow \mathrm{Ag}(s)+2 \mathrm{NH}_3 ; E_{\mathrm{red}}^{\circ}\) \(=0.337 \mathrm{~V}\)
[Use \(2.303 \times \frac{R T}{F}=0.0592\) and \(\frac{F}{R T}=38.92\) at \(298 \mathrm{~K}\)]
Question:
\(2 \mathrm{Ag}^{+}+\mathrm{C}_6 \mathrm{H}_{12} \mathrm{O}_6+\mathrm{H}_2 \mathrm{O} \rightarrow 2 \mathrm{Ag}(\mathrm{s})\) \(+~\mathrm{C}_6 \mathrm{H}_{12} \mathrm{O}_4+2 \mathrm{H}^{+}\). Find \(\ln \mathrm{K}\) of this reaction.

Match the specified conditions in solids listed in Column I with their properties in Column II. Indicate your answer by darkening the appropriate bubbles of the \(4 \times 4\) matrix given in the ORS.

Column I	Column II
A. Simple cubic and face-centered cubic	p. Have these cell parameters \(a=b=c \text { and } \alpha=\beta=\gamma\)
B. Cubic and rhombohedral	q. Are two crystal systems
C. Cubic and tetragonal	r. Have only two crystallographic angles of \(90^{\circ}\)
D. Hexagonal and monoclinic	s. Belong to same crystal system

Match the following

Column I	Column II
(A) \(CH _3- CHBr - CD _3\) on treatment with alc. KOH gives \(CH _2= CH - CD _3\) as a major product.	(P) E1 reaction
(B) \(Ph - CHBr - CH _3\) reacts faster than \(Ph - CHBr - CD _3\)	(Q) E2 reaction
(C) \(Ph - CH _2- CH _2 Br\) on treatment with \(C _2 H _5 OD / C _2 H _5 O ^{-}\)gives \(Ph - CD = CH _2\) as the major product	(R) El cb reaction
(D) \(PhCH _2 CH _2 Br\) and \(PhCH _2 CH _2 Br\) react with same rate	(S) First order reaction

Paragraph:
Let \(M=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: x^{2}+y^{2} \leq r^{2}\right\}\),
where \(r>0 .\) Consider the geometric progression \(a_{n}=\frac{1}{2^{n-1}}, n=1,2,3, \ldots .\) Let \(S_{0}=0\) and, for \(n \geq 1\), let \(S_{n}\) denote the sum of the first \(n\) terms of this progression. For \(n \geq 1\), let \(C_{n}\) denote the circle with center \(\left(S_{n-1}, 0\right)\) and radius \(a_{n}\), and \(D_{n}\) denote the circle with center \(\left(S_{n-1}, S_{n-1}\right)\) and radius \(a_{n}\).

Question:
Consider \(M\) with \(r=\frac{\left(2^{199}-1\right) \sqrt{2}}{2^{198}} .\) The number of all those circles \(D_{n}\) that are inside \(M\) is

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)

Mixture(s) showing positive deviation from Raoult's law at ${35}^{\mathrm{o}}\mathrm{C}$ is (are)

Let $f :R\to R$ and $g :R\to R$ be respectively given by $f\left(x\right)=\left|x\right|+1$ and $g\left(x\right)=x^2+1.$ Define $h :R\to R$ by $h\left(x\right)= \left\{\begin{array}{ccc}\max \left\{f\left(x\right), g\left(x\right)\right\}& if& x \le 0\\ \min \left\{f\left(x\right), g\left(x\right)\right\}& if& x>0\end{array}\right.$
Then number of points at which $h\left(x\right)$ is not differentiable is_________

Suppose $a, b$ denote the distinct real roots of the quadratic polynomial $x^2+20x-2020$ and suppose $c, d$ denote the distinct complex roots of the quadratic polynomial $x^2-20x+2020$. Then the value of $ac\left(a-c\right)+ad\left(a-d\right)+bc\left(b-c\right)+bd\left(b-d\right)$

Let \(z_k=\cos \left(\frac{2 k \pi}{10}\right)+i \sin \left(\frac{2 k \pi}{10}\right) ; k=1,2, \ldots, 9\)

List - I	List - II
(A) For each \(z_k\) there exists a \(z_j\) such \(z_k \cdot z_j=1\)	(P) True
(B) There exists a \(k \in\{1,2, \ldots, 9\}\) such that \(z_1 \cdot z=z_k\) has no solution z in the set of complex numbers	(Q) False
(C) \(\frac{\left|1-z_1\right|\left|1-z_2\right|\ldots\left|1-z_9\right|}{10}\) equals	(R) 1
(D) \(1-\sum_{k=1}^9 \cos \left(\frac{2 k \pi}{10}\right)\) equals	(S) 2