$2 \mathrm{~mol} \,\mathrm{of}\, \mathrm{Hg}(\mathrm{g})$ is combusted in a fixed volume bomb calorimeter with excess of $\mathrm{O}_{2}$ at $298 \mathrm{~K}$ and 1 atm into $\mathrm{HgO}(s)$. During the reaction, temperature increases from $298.0 \mathrm{~K}$ to $312.8 \mathrm{~K}$. If heat capacity of the bomb calorimeter and enthalpy of formation of $\mathrm{Hg}(g)$ are $20.00 \mathrm{~kJ} \mathrm{~K}^{-1}$ and $61.32 \mathrm{~kJ}$ $\mathrm{mol}^{-1}$ at $298 \mathrm{~K}$, respectively, the calculated standard molar enthalpy of formation of $\mathrm{HgO}(s)$ at 298 $\mathrm{K}$ is $\mathrm{X}\, \mathrm{kJ}\, \mathrm{mol}^{-1}$. The value of $|\mathrm{X}|$ is _________ .

[Given: Gas constant $\mathrm{R}=8.3 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$ ]

The reduction potential $\left(E^{0}\right.$, in $\left.\mathrm{V}\right)$ of $\mathrm{MnO}_{4}^{-}(\mathrm{aq}) / \mathrm{Mn}(\mathrm{s})$ is __________.

[Given: $E_{\left(\mathrm{MnO}_{4}^{-}(\mathrm{aq}) / \mathrm{MnO}_{2}(\mathrm{~s})\right)}^{0}=1.68 \mathrm{~V} ; E_{\left(\mathrm{MnO}_{2}(\mathrm{~s}) / \mathrm{Mn}^{2+}(\mathrm{aq})\right)}^{0}=1.21 \mathrm{~V} ; E_{\left(\mathrm{Mn}^{2+}(\mathrm{aq}) / \mathrm{Mn}(\mathrm{s})\right)}^{0}=-1.03 \mathrm{~V}$ ]

A solution is prepared by mixing $0.01 \mathrm{~mol}$ each of $\mathrm{H}_{2} \mathrm{CO}_{3}, \mathrm{NaHCO}_{3}, \mathrm{Na}_{2} \mathrm{CO}_{3}$, and $\mathrm{NaOH}$ in $100 \mathrm{~mL}$ of water. $p \mathrm{H}$ of the resulting solution is _________.

[Given: $p \mathrm{~K}_{\mathrm{a} 1}$ and $p \mathrm{~K}_{\mathrm{a} 2}$ of $\mathrm{H}_{2} \mathrm{CO}_{3}$ are $6.37$ and 10.32, respectively; $\log 2=0.30$ ]

The treatment of an aqueous solution of $3.74 \mathrm{~g}$ of $\mathrm{Cu}\left(\mathrm{NO}_{3}\right)_{2}$ with excess KI results in a brown solution along with the formation of a precipitate. Passing $\mathrm{H}_{2} \mathrm{~S}$ through this brown solution gives another precipitate $\mathbf{X}$. The amount of $\mathbf{X}$ (in $g$ ) is ___________.

[Given: Atomic mass of $\mathrm{H}=1, \mathrm{~N}=14, \mathrm{O}=16, \mathrm{~S}=32, \mathrm{~K}=39, \mathrm{Cu}=63, \mathrm{I}=127$ ]

Dissolving $1.24 \mathrm{~g}$ of white phosphorous in boiling $\mathrm{NaOH}$ solution in an inert atmosphere gives a gas $\mathbf{Q}$. The amount of $\mathrm{CuSO}_{4}$ (in g) required to completely consume the gas $\mathbf{Q}$ is _________.

[Given: Atomic mass of $\mathrm{H}=1, \mathrm{O}=16, \mathrm{Na}=23, \mathrm{P}=31, \mathrm{~S}=32, \mathrm{Cu}=63$ ]

Consider the following reaction.

On estimation of bromine in $1.00 \mathrm{~g}$ of $\mathbf{R}$ using Carius method, the amount of $\mathrm{AgBr}$ formed (in $\mathrm{g}$ ) is ___________.

[Given: Atomic mass of $\mathrm{H}=1, \mathrm{C}=12, \mathrm{O}=16, \mathrm{P}=31, \mathrm{Br}=80, \mathrm{Ag}=108]$

The weight percentage of hydrogen in $\mathbf{Q}$, formed in the following reaction sequence, is ________.

[Given: Atomic mass of $\mathrm{H}=1, \mathrm{C}=12, \mathrm{~N}=14, \mathrm{O}=16, \mathrm{~S}=32, \mathrm{Cl}=35$ ]

If the reaction sequence given below is carried out with 15 moles of acetylene, the amount of the product $\mathbf{D}$ formed (in $\mathrm{g}$ ) is ___________ .

The yields of $\mathbf{A}, \mathbf{B}, \mathbf{C}$ and $\mathbf{D}$ are given in parentheses.

[Given: Atomic mass of $\mathrm{H}=1, \mathrm{C}=12, \mathrm{O}=16, \mathrm{Cl}=35$ ]

Considering the reaction sequence given below, the correct statement(s) is(are)

Considering the following reaction sequence,

the correct option(s) is(are)

Match the rate expressions in LIST-I for the decomposition of $X$ with the corresponding profiles provided in LIST-II. $X_{\mathrm{s}}$ and $\mathrm{k}$ are constants having appropriate units.

List-I	List-II
(I) rate $=\frac{\mathrm{k}[\mathrm{X}]}{\mathrm{X}_{\mathrm{s}}+[\mathrm{X}]}$ under all possible initial concentrations of $\mathrm{X}$	(P)
(II) rate $=\frac{k[X]}{X_{s}+[X]}$ where initial concentrations of $X$ are much less than $X_{s}$	(Q)
(III) rate $=\frac{k[X]}{X_{s}+[X]}$ where initial concentrations of $\mathrm{X}$ are much higher than $X_{s}$	(R)
(IV) rate $=\frac{k[X]^{2}}{X_{s}+[X]}$ where initial concentration of $X$ is much higher than $\mathrm{X}_{\mathrm{s}}$	(S)
	(T)

LIST-I contains compounds and LIST-II contains reactions

Match each compound in LIST-I with its formation reaction(s) in LIST-II, and choose the correct option

LIST-I contains metal species and LIST-II contains their properties.

[Given: Atomic number of $\mathrm{Cr}=24, \mathrm{Ru}=44, \mathrm{Fe}=26$ ]

Match each metal species in LIST-I with their properties in LIST-II, and choose the correct option

Match the compounds in LIST-I with the observations in LIST-II, and choose the correct option.

Consider the equation

$$ \int_{1}^{e} \frac{\left(\log _{\mathrm{e}} x\right)^{1 / 2}}{x\left(a-\left(\log _{\mathrm{e}} x\right)^{3 / 2}\right)^{2}} d x=1, \quad a \in(-\infty, 0) \cup(1, \infty) $$

Which of the following statements is/are TRUE?

Let $$a_{1}, a_{2}, a_{3}, \ldots$$ be an arithmetic progression with $$a_{1}=7$$ and common difference 8. Let $$T_{1}, T_{2}, T_{3}, \ldots$$ be such that $$T_{1}=3$$ and $$T_{n+1}-T_{n}=a_{n}$$ for $$n \geq 1$$. Then, which of the following is/are TRUE ?

Consider the parabola $$y^{2}=4 x$$. Let $$S$$ be the focus of the parabola. A pair of tangents drawn to the parabola from the point $$P=(-2,1)$$ meet the parabola at $$P_{1}$$ and $$P_{2}$$. Let $$Q_{1}$$ and $$Q_{2}$$ be points on the lines $$S P_{1}$$ and $$S P_{2}$$ respectively such that $$P Q_{1}$$ is perpendicular to $$S P_{1}$$ and $$P Q_{2}$$ is perpendicular to $$S P_{2}$$. Then, which of the following is/are TRUE?

Let $$|M|$$ denote the determinant of a square matrix $$M$$. Let $$g:\left[0, \frac{\pi}{2}\right] \rightarrow \mathbb{R}$$ be the function defined by

$$ g(\theta)=\sqrt{f(\theta)-1}+\sqrt{f\left(\frac{\pi}{2}-\theta\right)-1} $$

where

$$ f(\theta)=\frac{1}{2}\left|\begin{array}{ccc} 1 & \sin \theta & 1 \\ -\sin \theta & 1 & \sin \theta \\ -1 & -\sin \theta & 1 \end{array}\right|+\left|\begin{array}{ccc} \sin \pi & \cos \left(\theta+\frac{\pi}{4}\right) & \tan \left(\theta-\frac{\pi}{4}\right) \\ \sin \left(\theta-\frac{\pi}{4}\right) & -\cos \frac{\pi}{2} & \log _{e}\left(\frac{4}{\pi}\right) \\ \cot \left(\theta+\frac{\pi}{4}\right) & \log _{e}\left(\frac{\pi}{4}\right) & \tan \pi \end{array}\right| . $$

Let $$p(x)$$ be a quadratic polynomial whose roots are the maximum and minimum values of the function $$g(\theta)$$, and $$p(2)=2-\sqrt{2}$$. Then, which of the following is/are TRUE ?

Consider the following lists :

The correct option is:

Two players, $$P_{1}$$ and $$P_{2}$$, play a game against each other. In every round of the game, each player rolls a fair die once, where the six faces of the die have six distinct numbers. Let $$x$$ and $$y$$ denote the readings on the die rolled by $$P_{1}$$ and $$P_{2}$$, respectively. If $$x>y$$, then $$P_{1}$$ scores 5 points and $$P_{2}$$ scores 0 point. If $$x=y$$, then each player scores 2 points. If $$x < y$$, then $$P_{1}$$ scores 0 point and $$P_{2}$$ scores 5 points. Let $$X_{i}$$ and $$Y_{i}$$ be the total scores of $$P_{1}$$ and $$P_{2}$$, respectively, after playing the $$i^{\text {th }}$$ round.

The correct option is:

Let $$p, q, r$$ be nonzero real numbers that are, respectively, the $$10^{\text {th }}, 100^{\text {th }}$$ and $$1000^{\text {th }}$$ terms of a harmonic progression. Consider the system of linear equations

$$$ \begin{gathered} x+y+z=1 \\ 10 x+100 y+1000 z=0 \\ q r x+p r y+p q z=0 \end{gathered} $$$

The correct option is:

Consider the ellipse

$$$ \frac{x^{2}}{4}+\frac{y^{2}}{3}=1 $$$

Let $H(\alpha, 0), 0<\alpha<2$, be a point. A straight line drawn through $H$ parallel to the $y$-axis crosses the ellipse and its auxiliary circle at points $E$ and $F$ respectively, in the first quadrant. The tangent to the ellipse at the point $E$ intersects the positive $x$-axis at a point $G$. Suppose the straight line joining $F$ and the origin makes an angle $\phi$ with the positive $x$-axis.

The correct option is:

Two spherical stars $A$ and $B$ have densities $\rho_{A}$ and $\rho_{B}$, respectively. $A$ and $B$ have the same radius, and their masses $M_{A}$ and $M_{B}$ are related by $M_{B}=2 M_{A}$. Due to an interaction process, star $A$ loses some of its mass, so that its radius is halved, while its spherical shape is retained, and its density remains $\rho_{A}$. The entire mass lost by $A$ is deposited as a thick spherical shell on $B$ with the density of the shell being $\rho_{A}$. If $v_{A}$ and $v_{B}$ are the escape velocities from $A$ and $B$ after the interaction process, the ratio $\frac{v_{B}}{v_{A}}=\sqrt{\frac{10 n}{15^{1 / 3}}}$. The value of $n$ is __________ .

In the following circuit $C_{1}=12 \mu F, C_{2}=C_{3}=4 \mu F$ and $C_{4}=C_{5}=2 \mu F$. The charge stored in $C_{3}$ is ____________ $\mu C$.

A rod of length $2 \mathrm{~cm}$ makes an angle $\frac{2 \pi}{3} \mathrm{rad}$ with the principal axis of a thin convex lens. The lens has a focal length of $10 \mathrm{~cm}$ and is placed at a distance of $\frac{40}{3} \mathrm{~cm}$ from the object as shown in the figure. The height of the image is $\frac{30 \sqrt{3}}{13} \mathrm{~cm}$ and the angle made by it with respect to the principal axis is $\alpha$ rad. The value of $\alpha$ is $\frac{\pi}{n} r a d$, where $n$ is __________ .

A solid sphere of mass $1 \mathrm{~kg}$ and radius $1 \mathrm{~m}$ rolls without slipping on a fixed inclined plane with an angle of inclination $\theta=30^{\circ}$ from the horizontal. Two forces of magnitude $1 \mathrm{~N}$ each, parallel to the incline, act on the sphere, both at distance $r=0.5 \mathrm{~m}$ from the center of the sphere, as shown in the figure. The acceleration of the sphere down the plane is _________ $m \,s^{-2} .\left(\right.$ Take $g=10\, m s^{-2}$)

A projectile is fired from horizontal ground with speed $v$ and projection angle $\theta$. When the acceleration due to gravity is $g$, the range of the projectile is $d$. If at the highest point in its trajectory, the projectile enters a different region where the effective acceleration due to gravity is $g^{\prime}=\frac{g}{0.81}$, then the new range is $d^{\prime}=n d$. The value of $n$ is ___________ .

A medium having dielectric constant $K>1$ fills the space between the plates of a parallel plate capacitor. The plates have large area, and the distance between them is $d$. The capacitor is connected to a battery of voltage $V$, as shown in Figure (a). Now, both the plates are moved by a distance of $\frac{d}{2}$ from their original positions, as shown in Figure (b).

In the process of going from the configuration depicted in Figure (a) to that in Figure (b), which of the following statement(s) is(are) correct?

The figure shows a circuit having eight resistances of $1 \Omega$ each, labelled $R_{1}$ to $R_{8}$, and two ideal batteries with voltages $\varepsilon_{1}=12 \mathrm{~V}$ and $\varepsilon_{2}=6 \mathrm{~V}$.

Which of the following statement(s) is(are) correct?

An ideal gas of density $\rho=0.2 \mathrm{~kg} \mathrm{~m}^{-3}$ enters a chimney of height $h$ at the rate of $\alpha=$ $0.8 \mathrm{~kg} \mathrm{~s}^{-1}$ from its lower end, and escapes through the upper end as shown in the figure. The cross-sectional area of the lower end is $A_{1}=0.1 \mathrm{~m}^{2}$ and the upper end is $A_{2}=0.4 \mathrm{~m}^{2}$. The pressure and the temperature of the gas at the lower end are $600 \mathrm{~Pa}$ and $300 \mathrm{~K}$, respectively, while its temperature at the upper end is $150 \mathrm{~K}$. The chimney is heat insulated so that the gas undergoes adiabatic expansion. Take $g=10 \mathrm{~m} \mathrm{~s}^{-2}$ and the ratio of specific heats of the gas $\gamma=2$. Ignore atmospheric pressure.

Which of the following statement(s) is(are) correct?

Three plane mirrors form an equilateral triangle with each side of length $L$. There is a small hole at a distance $l>0$ from one of the corners as shown in the figure. A ray of light is passed through the hole at an angle $\theta$ and can only come out through the same hole. The cross section of the mirror configuration and the ray of light lie on the same plane.

Which of the following statement(s) is(are) correct?

Six charges are placed around a regular hexagon of side length $a$ as shown in the figure. Five of them have charge $q$, and the remaining one has charge $x$. The perpendicular from each charge to the nearest hexagon side passes through the center 0 of the hexagon and is bisected by the side.

Which of the following statement(s) is(are) correct in SI units?

The binding energy of nucleons in a nucleus can be affected by the pairwise Coulomb repulsion. Assume that all nucleons are uniformly distributed inside the nucleus. Let the binding energy of a proton be $E_{b}^{p}$ and the binding energy of a neutron be $E_{b}^{n}$ in the nucleus.

Which of the following statement(s) is(are) correct?

A small circular loop of area $A$ and resistance $R$ is fixed on a horizontal $x y$-plane with the center of the loop always on the axis $\hat{n}$ of a long solenoid. The solenoid has $m$ turns per unit length and carries current $I$ counterclockwise as shown in the figure. The magnetic field due to the solenoid is in $\hat{n}$ direction. List-I gives time dependences of $\hat{n}$ in terms of a constant angular frequency $\omega$. List-II gives the torques experienced by the circular loop at time $t=\frac{\pi}{6 \omega}$. Let $\alpha=\frac{A^{2} \mu_{0}^{2} m^{2} I^{2} \omega}{2 R}$.

Which one of the following options is correct?

List I describes four systems, each with two particles $A$ and $B$ in relative motion as shown in figures. List II gives possible magnitudes of their relative velocities (in $m s^{-1}$ ) at time $t=\frac{\pi}{3} s$.

List-I	List-II
(I) $A$ and $B$ are moving on a horizontal circle of radius $1 \mathrm{~m}$ with uniform angular speed $\omega=1 \mathrm{rad} \mathrm{s}^{-1}$. The initial angular positions of $A$ and $B$ at time $t=0$ are $\theta=0$ and $\theta=\frac{\pi}{2}$, respectively.	(P) $\frac{\sqrt{3}+1}{2}$
(II) Projectiles $A$ and $B$ are fired (in the same vertical plane) at $t=0$ and $t=0.1 \mathrm{~s}$ respectively, with the same speed $v=\frac{5 \pi}{\sqrt{2}} \mathrm{~m} \mathrm{~s}^{-1}$ and at $45^{\circ}$ from the horizontal plane. The initial separation between $A$ and $B$ is large enough so that they do not collide. $\left(g=10 \mathrm{~ms}^{-2}\right)$.	(Q) $\frac{\sqrt{3}-1}{\sqrt{2}}$
(III) Two harmonic oscillators $A$ and $B$ moving in the $x$ direction according to $x_{A}=x_{0} \sin \frac{t}{t_{0}}$ and $x_{B}=x_{0} \sin \left(\frac{t}{t_{0}}+\frac{\pi}{2}\right)$ respectively, starting from $t=0$. Take $x_{0}=1 \mathrm{~m}, t_{0}=1 \mathrm{~s}$.	(R) $\sqrt{10}$
(IV) Particle $A$ is rotating in a horizontal circular path of radius $1 \mathrm{~m}$ on the $x y$ plane, with constant angular speed $\omega=1 \mathrm{rad} \mathrm{s}^{-1}$. Particle $B$ is moving up at a constant speed $3 \mathrm{~m} \mathrm{~s}^{-1}$ in the vertical direction as shown in the figure. (Ignore gravity.)	(S) $\sqrt{2}$
	(T) $\sqrt{25\pi^{2}+1}$

Which one of the following options is correct?

List I describes thermodynamic processes in four different systems. List II gives the magnitudes (either exactly or as a close approximation) of possible changes in the internal energy of the system due to the process.

Which one of the following options is correct?

List I contains four combinations of two lenses (1 and 2) whose focal lengths (in $\mathrm{cm}$ ) are indicated in the figures. In all cases, the object is placed $20 \mathrm{~cm}$ from the first lens on the left, and the distance between the two lenses is $5 \mathrm{~cm}$. List II contains the positions of the final images.

List-I	List-II
(I)	(P) Final image is formed at $7.5 \mathrm{~cm}$ on the right side of lens 2 .
(II)	(Q) Final image is formed at $60.0 \mathrm{~cm}$ on the right side of lens 2 .
(III)	(R) Final image is formed at $30.0 \mathrm{~cm}$ on the left side of lens $2 .$
(IV)	(S) Final image is formed at $6.0 \mathrm{~cm}$ on the right side of lens 2 .
	(T) Final image is formed at $30.0 \mathrm{~cm}$ on the right side of lens 2 .

Which one of the following options is correct?