The sum of the number of lone pairs of electrons on each central atom in the following species is \(\left[ TeBr _6\right]^{2-},\left[ Br F _2\right]^{+}, S N F_3\) and \(\left[ XeF _3\right]^{-}\)[Atomic number : \(N=7, F=9, S =16, B r=35,\) \(T e=52, X e=54]\)

Match the compounds in LIST-$\mathrm{I}$ with the observations in LIST-$\mathrm{II}$, and choose the correct option.

	LIST-$\mathrm{I}$		LIST-$\mathrm{II}$
$\left(\mathrm{I}\right)$	Aniline	$\left(\mathrm{P}\right)$	Sodium fusion extract of the compound on boiling with ${\mathrm{FeSO}}_4$, followed by acidification with conc. ${\mathrm{H}}_2{\mathrm{SO}}_4$, gives Prussian blue color.
$\left(\mathrm{II}\right)$	$\mathrm{o}-$Cresol	$\left(\mathrm{Q}\right)$	Sodium fusion extract of the compound on treatment with sodium nitroprusside gives blood red color.
$\left(\mathrm{III}\right)$	Cysteine	$\left(\mathrm{R}\right)$	Addition of the compound to a saturated solution of ${\mathrm{NaHCO}}_3$ results in effervescence.
$\left(\mathrm{IV}\right)$	Caprolactam	$\left(\mathrm{S}\right)$	The compound reacts with bromine water to give a white precipitate.
		$\left(\mathrm{T}\right)$	Treating the compound with neutral ${\mathrm{FeCl}}_3$ solution produces violet color.

Which of the following liberates ${\mathrm{O}}_2$ upon hydrolysis?

Among the following, the correct statement(s) about polymers is(are)

Paragraph:
The hydrogen-like species \(\mathrm{Li}^{2+}\) is in a spherically symmetric state \(S_1\) with one radial node. Upon absorbing light the ion undergoes transition to a state \(S_2\). The state \(S_2\) has one radial node and its energy is equal to the ground state energy of the hydrogen atom.
Question:
The orbtial angular momentum quantum number of the state \(S_2\) is

Consider the following volume-temperature \((\mathrm{V}-\mathrm{T})\) diagram for the expansion of 5 moles of an ideal monoatomic gas.

Considering only P-V work is involved, the total change in enthalpy (in Joule) for the transformation of state in the sequence \(\mathbf{X} \rightarrow \mathbf{Y} \rightarrow \mathbf{Z}\) is ________.
[Use the given data: Molar heat capacity of the gas for the given temperature range, \(\mathrm{C}_{\mathrm{V}, \mathrm{m}}=12 \mathrm{~J} \mathrm{~K}^{-1}\) \(\mathrm{mol}^{-1}\) and gas constant, \(\left.\mathrm{R}=8.3 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right]\)

Based on VSEPR model, match the xenon compounds given in List-I with the corresponding geometries and the number of lone pairs on xenon given in List-II and choose the correct option.

List - I	List - II
(P) \(XeF _2\)	(1) Trigonal bipyramidal and two lone pair of electrons
(Q) \(XeF _4\)	(2) Tetrahedral and one lone pair of electrons
(R) \(XeO _3\)	(3) Octahedral and two lone pair of electrons
(S) \(XeO _3 F_2\)	(4) Trigonal bipyramidal and no lone pair of electrons
	(5) Trigonal bipyramidal and three lone pair of electrons

A particle of mass $m$ is projected from the ground with an initial speed $u_{\circ }$ at an angle $\alpha$ with the horizontal. At the highest point of its trajectory, it makes a completely inelastic collision with another identical particle, which was thrown vertically upward from the ground with the same initial speed $u_{\circ }$. The angle that the composite system makes with the horizontal immediately after the collision is

To determine the half-life of a radioactive element, a student plots a graph of \(\ln \left|\frac{d N(t)}{d t}\right|\) versust. Here \(\frac{d N(t)}{d t}\) is the rate of radioactive decay at time \(t\). If the number of radioactive nuclei of this element decreases by a factor of \(p\) after \(4.16 \mathrm{yr}\), the value of \(p\) is

Let $f\left(x\right)=\frac{1-x(1+\left|1-x\right|)}{|1-x|}\cos \left(\frac{1}{1-x}\right) \mathrm{for} x\ne 1,$ then

Answer the following by appropriately matching the lists based on the information given in the paragraph.
In a thermodynamics process on an ideal monatomic gas, the infinitesimal heat absorbed by the gas is given by $\mathrm{T}\mathrm{\Delta }\mathrm{X},$ where $\mathrm{T}$ is temperature of the system and $\mathrm{\Delta }\mathrm{X}$ is the infinitesimal change in a thermodynamic quantity $\mathrm{X}$ of the system. For a mole of monatomic ideal gas $\mathrm{X}=\frac{3}{2}\mathrm{R}\mathrm{l}\mathrm{n}\left(\frac{\mathrm{T}}{{\mathrm{T}}_{\mathrm{A}}}\right)+\mathrm{R}\mathrm{l}\mathrm{n}\left(\frac{\mathrm{V}}{{\mathrm{V}}_{\mathrm{A}}}\right).$ Here, $\mathrm{R}$ is gas constant, $\mathrm{V}$ is volume of gas, ${\mathrm{T}}_{\mathrm{A}}$ and ${\mathrm{V}}_{\mathrm{A}}$ are constants.
The List-I below gives some quantities involved in a process and List-II gives some possible values of these quantities.

	List I		List II
$\left(\mathrm{a}\right)$	Work done by the system in process $1\to 2\to 3$	$\left(\mathrm{p}\right)$	$\frac{1}{3}\mathrm{R}{\mathrm{T}}_0\mathrm{l}\mathrm{n}2$
$\left(\mathrm{b}\right)$	Change in internal energy in process $1\to 2\to 3$	$\left(\mathrm{q}\right)$	$\frac{1}{3}\mathrm{R}{\mathrm{T}}_0$
$\left(\mathrm{c}\right)$	Heat absorbed by the system in process $1\to 2\to 3$	$\left(\mathrm{r}\right)$	$\mathrm{R}{\mathrm{T}}_0$
$\left(\mathrm{d}\right)$	Heat absorbed by the system in process $1\to 2$	$\left(\mathrm{s}\right)$	$\frac{4}{3}\mathrm{R}{\mathrm{T}}_0$
		$\left(\mathrm{t}\right)$	$\frac{1}{3}\mathrm{R}{\mathrm{T}}_0\left(3+\mathrm{l}\mathrm{n}2\right)$
		$\left(\mathrm{u}\right)$	$\frac{5}{6}\mathrm{R}{\mathrm{T}}_0$

If the process carried out on one mole of monatomic ideal gas is as shown in figure in the $\mathrm{T}$ - $\mathrm{V}$ -diagram with ${\mathrm{P}}_0{\mathrm{V}}_0=\frac{1}{3}\mathrm{R}{\mathrm{T}}_0,$ the correct match is,

The correct option(s) related to adsorption processes is(are)

Let \(z=\cos \theta+i \sin \theta\). Then, the value of \(\sum_{m=1}^{15} \operatorname{Im}\left(z^{2 m-1}\right)\) at \(\theta=2^{\circ}\) is