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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:
Energy of the state \(S_1\) in units of the hydrogen atom ground state energy is

The acidic hydrolysis of ether (X) shown below is fastest when

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Copper is the most noble of the first row transition metals and occurs in small deposits in several countries. Ores of copper include chalcanthite \(\left(\mathrm{CuSO}_4 \cdot 5 \mathrm{H}_2 \mathrm{O}\right)\), atacamite \(\left(\mathrm{Cu}_2 \mathrm{Cl}(\mathrm{OH})_3\right)\), cuprite \(\left(\mathrm{Cu}_2 \mathrm{O}\right)\), copper glance \(\left(\mathrm{Cu}_2 \mathrm{~S}\right)\) and malachite \(\left(\mathrm{Cu}_2(\mathrm{OH})_2 \mathrm{CO}_3\right)\). However, 80\% of the world copper production comes from the ore chalcopyrite \(\left(\mathrm{CuFeS}_2\right)\). The extraction of copper from chalcopyrite involves partial roasting, removal of iron and self-reduction.
Question:
In self-reduction, the reducing species is

The boiling point of water in a \(0.1\) molal silver nitrate solution (solution \(\mathbf{A}\) ) is \(\mathbf{x}^{\circ} \mathrm{C}\). To this solution \(\mathbf{A}\), an equal volume of \(0.1\) molal aqueous barium chloride solution is added to make a new solution B. The difference in the boiling points of water in the two solutions \(\mathbf{A}\) and \(\mathbf{B}\) is \(\mathbf{y} \times 10^{-2}{ }^{\circ} \mathrm{C}\).
(Assume: Densities of the solutions \(\mathbf{A}\) and \(\mathbf{B}\) are the same as that of water and the soluble salts dissociate completely.
Use: Molal elevation constant (Ebullioscopic Constant), \(K_{b}=0.5 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}\); Boiling point of pure water as \(100^{\circ} \mathrm{C}\).)
The value of $\mathrm{x}$ is ___.

Aqueous solution of \(\text{Na} _2\text S_2\text O _3\) on reaction with \(\text{Cl} _2\) gives

Considering ideal gas behavior, the expansion work done (in kJ) when 144 g of water is electrolyzed completely under constant pressure at 300 K is _____
Use: Universal gas constant \((\mathrm{R})=8.3 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\); Atomic mass (in amu): \(\mathrm{H}=1, \mathrm{O}=16\) (mark absolute value as answer)

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An aqueous solution of metal ion \(\mathbf{M} 1\) reacts separately with reagents \(\mathbf{Q}\) and \(\mathbf{R}\) in excess to give tetrahedral and square planar complexes, respectively. An aqueous solution of another metal ion M2 always forms tetrahedral complexes with these reagents. Aqueous solution of \(\mathbf{M} 2\) on reaction with reagent \(\mathbf{S}\) gives white precipitate which dissolves in excess of \(\mathbf{S}\). The reactions are summarized in the scheme given below:


Question:
\(\mathbf{M} 1, \mathbf{Q}\) and \(\mathbf{R}\), respectively are

Two uniform strings of mass per unit length \(\mu\) and \(4 \mu\), and length \(L\) and \(2 L\), respectively, are joined at point \(\mathrm{O}\), and tied at two fixed ends \(\mathrm{P}\) and \(\mathrm{Q}\), as shown in the figure. The strings are under a uniform tension \(T\). If we define the frequency \(v_0=\frac{1}{2 L} \sqrt{\frac{T}{\mu}}\), which of the following statement(s) is(are) correct?

The number of points in $\left(-\infty ,\infty \right)$, for which $x^2-x\sin x-\cos x=0$, is:

Total number of cis $\mathrm{N}-\mathrm{M}\mathrm{n}-\mathrm{C}\mathrm{l}$ bond angles (that is, $\mathrm{M}\mathrm{n}-\mathrm{N}$ and $\mathrm{M}\mathrm{n}-\mathrm{C}\mathrm{l}$ bonds in cis positions) present in a molecule of cis- $\left[\mathrm{M}\mathrm{n}{\left(\mathrm{e}\mathrm{n}\right)}_2{\mathrm{C}\mathrm{l}}_2\right]$ complex is _________ $\left(en={\mathrm{N}\mathrm{H}}_2{\mathrm{C}\mathrm{H}}_2{\mathrm{C}\mathrm{H}}_2{\mathrm{N}\mathrm{H}}_2\right)$

In the circuit shown below, the switch \(\mathrm{S}\) is connected to position \(\mathrm{P}\) for a long time so that the charge on the capacitor becomes \(q_{1} \mu \mathrm{C}\). Then \(\mathrm{S}\) is switched to position \(\mathrm{Q}\). After a long time, the charge on the capacitor is \(q_{2} \mu \mathrm{C}\).

The magnitude of $q_1$ is _______.

Suppose $\det \left[\begin{array}{cc}\sum _{k=0}^{n}k& \sum _{k=0}^n{}_{\mathrm{ }}{}^n{C}_k{k}^2\\ \sum _{k=0}^n{}_{\mathrm{ }}{}^n{C}_{k}k& \sum _{k=0}^n{}_{\mathrm{ }}{}^n{C}_k{3}^k\end{array}\right]=0,$ holds for some positive integer $n.$ Then $\sum _{k=0}^n\frac{{}_{ }{}^n{C}_k}{k+1}$ equals

Match each of the diatomic molecules in Column I with its property/properties in Column II.

Column I	Column II
A. \(B _2\)	p. Paramagnetic
B. \(N_2\)	q. Undergoes oxidation
C. \(O _2^{-}\)	r. Undergoes reduction
D. \(O _2\)	s. Bond order
	t. Mixing of 's' and 'p' orbitals