A colorless aqueous solution contains nitrates of two metals, $\mathrm{X}$ and $\mathrm{Y}$. When it was added to an aqueous solution of $\mathrm{NaCl}$, a white precipitate was formed. This precipitate was found to be partly soluble in hot water to give a residue $\mathrm{P}$ and a solution $\mathrm{Q}$. The residue $\mathrm{P}$ was soluble in aq. ${\mathrm{NH}}_3$ and also in excess sodium thiosulfate. The hot solution $\mathrm{Q}$ gave a yellow precipitate with $\mathrm{KI}$. The metals $\mathrm{X}$ and $\mathrm{Y}$, respectively, are

If the distribution of molecular speeds of a gas is as per the figure shown below, then the ratio of the most probable, the average, and the root mean square speeds, respectively, is

When 100 mL of 1.0 M HCl was mixed with 100 mL of 1.0 M NaOH in an insulated beaker at constant pressure, a temperature increase of ${5.7}^{\mathrm{o}}\mathrm{C}$ was measured for the beaker and its contents (Expt.1). Because the enthalpy of neutralization of a strong acid with a strong base is a constant $(-57.0\mathrm{ }\mathrm{k}\mathrm{J}\mathrm{ }\mathrm{m}\mathrm{o}{\mathrm{l}}^{-1})$ , this experiment could be used to measure the calorimeter constant. In a second experiment (Expt. 2), 100 mL of 2.0 M acetic acid $({\mathrm{K}}_{\mathrm{a}}=2.0\times {10}^{-5})$ was mixed with 100 mL of 1.0 M NaOH (under identical conditions to Expt. 1) where a temperature rise of ${5.6}^{\mathrm{o}}\mathrm{C}$ was measured. (Consider heat capacity of all solutions as $4.2\mathrm{ }\mathrm{J}\mathrm{ }{\mathrm{g}}^{-1}\mathrm{ }{\mathrm{K}}^{-1}$ and density of all solutions as $1.0\mathrm{ }\mathrm{g}\mathrm{ }\mathrm{m}{\mathrm{L}}^{-1}$ )
Enthalpy of dissociation $\left(\mathrm{i}\mathrm{n}\mathrm{ }\mathrm{k}\mathrm{J}\mathrm{ }\mathrm{m}\mathrm{o}{\mathrm{l}}^{-1}\right)$ of acetic acid obtained from the Expt. 2 is

The electrochemical extraction of aluminium from bauxite ore involves

The wave function, ${\mathrm{\psi }}_{\mathrm{n},\mathrm{}\mathrm{l},\mathrm{}{\mathrm{m}}_{\mathrm{l}}}$ is a mathematical function whose value depends upon spherical polar coordinates $\left(\mathrm{r},\mathrm{}\mathrm{\theta },\mathrm{}\mathrm{\varphi }\right)$ of the electron and characterized by the quantum numbers $\mathrm{n},\mathrm{}\mathrm{l}$ and ${\mathrm{m}}_{\mathrm{l}}$ . Here $\mathrm{r}$ is the distance from the nucleus, $\mathrm{\theta }$ is colatitude and $\mathrm{\varphi }$ is azimuth. In the mathematical functions given in the Table, $\mathrm{Z}$is the atomic number and ${\mathrm{a}}_{\mathrm{o}}$ is Bohr radius

Column 1	Column 2	Column 3
(I) 1s orbital	\(\begin{array}{l}\text { (i) } \psi_{ n , 1, m_1} \\ \propto\left(\frac{ Z }{ a _{ o }}\right)^{\frac{3}{2}} e ^{-\left(\frac{ Zr }{ e _{ o }}\right)}\end{array}\)	(P)
(II) 2s orbital	(ii) One radial node	(Q) Probability density at the nucleus \(\propto \frac{1}{ a _{ o }^3}\).
(III) \(2 p _{ z }\) orbital	\(\begin{array}{l}\text { (iii) } \psi_{ n }, 1, m_{ l } \\ \propto\left(\frac{ Z }{ a _{ o }}\right)^{\frac{5}{2}} re ^{-\left(\frac{ Zr }{2 a _{ o }}\right)} \cos \theta\end{array}\)	(R) Probability density is maximum at the nucleus.
(IV) \(3 d_{ z }^2\) orbital	(iv) \(x y\)-plane is a nodal plane	(S) Energy needed to excite an electron from \(n =2\) state to \(n =4\) state is \(\frac{27}{32}\) times the energy needed to excite are electron from \(n =2\) state to \(n =6\) state.

For the given orbital in column 1, the only correct combination for any hydrogen-like species is

In the scheme given below, \(\text X\) and \(\text Y\), respectively, are
Metal halide \(\quad \xrightarrow{\text { aq. NaOH} } \quad\) White precipitate \((\text P )+\) Filtrate \((\text Q )\)
\(P \xrightarrow[\text { heat }]{\substack{\text { aq. } H _2 SO _4 \\ PbO _2 \text { (excess) }}}\text X\) (a coloured species in solution)
\(Q \xrightarrow[\text { warm }]{ MnO ( OH )_2, \text { Conc. } H _2 SO _4}\text Y\) (gives blue-coloration with KI-starch paper)

Paragraph:
An aqueous solution of a mixture of two inorganic salts, when treated with dilute \(H C\) ?, gave a precipitate \((\boldsymbol{P})\) and a filtrate \((\boldsymbol{Q})\). The precipitate \(\boldsymbol{P}\) was found to dissolve in hot water. The filtrate \((\boldsymbol{Q})\) remained unchanged, when treated with \(\mathrm{H}_{2} \mathrm{~S}\) in a dilute mineral acid medium. However, it gave a precipitate \((\boldsymbol{R})\) with \(H_{2} S\) in an ammoniacal medium. The precipitate \(\boldsymbol{R}\) gave a coloured solution \((\boldsymbol{S})\), when treated with \(\mathrm{H}_{2} \mathrm{O}_{2}\) in an aqueous \(\mathrm{NaOH}\) medium.

Question:
The precipitate \(\boldsymbol{P}\) contains

Reaction of \(\mathrm{Br}_2\) with \(\mathrm{Na}_2 \mathrm{CO}_3\) in aqueous solution gives sodium bromide and sodium bromate with evolution of \(\mathrm{CO}_2\) gas. The number of sodium bromide molecules involved in the balanced chemical equation is

In the circuit shown, $L=1\text{μ}H,C=1\text{μ}FandR=1k\mathrm{\Omega }.$ They are connected in series with an a.c. source $V=V_0\sin \text{ω}t$ as shown. Which of the following options is/are correct?

Let $n \ge 2$ be an integer. Take n distinct points on a circle and join each pair of points by a line segment. Colour the line segment joining every pair of adjacent points by blue and the rest by red. If the number of red and blue line segments are equal, then the value of n is ________

Consider the Bohr's model of a one-electron atom where the electron moves around the nucleus. In the following List-I contains some quantities for the $n^{\mathrm{t}\mathrm{h}}$ orbit of the atom and List-II contains options showing how they depend on $n$ .
 

List - I	List - II
(I) Radius of the $n^{\mathrm{t}\mathrm{h}}$ orbit	$\left(\mathrm{P}\right)$ $\propto n^{-2}$
(II) Angular momentum of the electron in the $n^{th}$ orbit	$\left(\mathrm{Q}\right)$ $\propto n^{-1}$
(III) Kinetic energy of the electron in the $n^{\mathrm{t}\mathrm{h}}$ orbit	$\left(\mathrm{R}\right)$ $\propto n^0$
(IV) Potential energy of the electron in the $n^{\mathrm{t}\mathrm{h}}$ orbit	$\left(\mathrm{S}\right)$ $\propto n^1$
	$\left(\mathrm{T}\right)$ $\propto n^2$
	$\left(\mathrm{U}\right)$ $\propto n^{1/2}$

Which of the following options has the correct combination considering List-I and List-II?

Extraction of copper from copper pyrite $\left(\mathrm{C}\mathrm{u}\mathrm{F}\mathrm{e}{\mathrm{S}}_2\right)$ involves

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 state \(S_1\) is