Reduction of the metal centre in aqueous permanganate ion involves

The value of \(n\) in the molecular formula \(\mathrm{Be}_n \mathrm{Al}_2 \mathrm{Si}_6 \mathrm{O}_{18}\) is

Different possible thermal decomposition pathways for peroxyesters are shown below. Match each pathway from List I with an appropriate structure from List II and select the correct answer using the code given below the lists.

	List – I		List – II
A	Pathway P	P.
B.	Pathway Q	Q.
C.	Pathway R	R.
D.	Pathway S	S.

The major product of the following reaction is

The total number of lone pairs of electrons in ${\mathrm{N}}_2{\mathrm{O}}_3$ is

Match the reaction in Column I with appropriate options in Column II

Match each of the reactions given in Column I with the corresponding products(s) given in Column II.

The circuit shown in the figure contains an inductor \(L\), a capacitor \(C_0\), a resistor \(R_0\) and an ideal battery. The circuit also contains two keys \(\mathrm{K}_1\) and \(\mathrm{K}_2\). Initially, both the keys are open and there is no charge on the capacitor. At an instant, key \(K_1\) is closed and immediately after this the current in \(R_0\) is found to be \(I_1\). After a long time, the current attains a steady state value \(I_2\). Thereafter, \(\mathrm{K}_2\) is closed and simultaneously \(\mathrm{K}_1\) is opened and the voltage across \(C_0\) oscillates with amplitude \(V_0\) and angular frequency \(\omega_0\).

Match the quantities mentioned in List-I with their values in List-II and choose the correct option.

Let \(H_1, H_2, \ldots, H_n\) be mutually exclusive events with \(P\left(H_i\right)>0, i=1,2, \ldots, n\). Let \(E\) be any other event with \(0 < P(E) < 1\).
Statement I \(P\left(H_i / E\right)>P\left(E / H_i\right) P\left(H_i\right)\) for \(i=1,2, \ldots, n\).
Statement II \(\sum_{i=1}^n P\left(H_i\right)=1\)

A ray of light, travelling in the direction $\frac{1}{2}\left(\hat{\mathrm{i}}+\sqrt{3} \hat{\mathrm{j}}\right)$, is incident on a plane mirror. After reflection, it travels along the direction $\frac{1}{2}\left(\hat{\mathrm{i}}-\sqrt{3}\widehat{ \mathrm{j}}\right)$. The angle of incidence is

A monoatomic ideal gas undergoes a process in which the ratio of \(P\) to \(V\) at any instant is constant and equals to 1 . What is the molar heat capacity of the gas?

Statement I Molecules that are not superimposable on their mirror images are chiral.
Statement II All chiral molecules have chiral centres.

In an insulated vessel, \(0.05 \mathrm{~kg}\) steam at \(373 \mathrm{~K}\) and \(0.45 \mathrm{~kg}\) of ice at \(253 \mathrm{~K}\) are mixed. Find the final temperature of the mixture (in kelvin).
\( \text { Given, }L_{\text {fusion }} =80 \mathrm{cal} / \mathrm{g}=336 \mathrm{~J} / \mathrm{g},\) \( L_{\text {vaporization }}=540 \mathrm{cal} / \mathrm{g}=2268 \mathrm{~J} / \mathrm{g}, \)
\( S_{\text {ice }} =2100 \mathrm{~J} / \mathrm{kg}, K=0.5 \mathrm{cal} / \mathrm{gK} \text { and } S_{\text {water }}=\) \(4200 \mathrm{~J} / \mathrm{kg}, K=1 \mathrm{cal} / \mathrm{gK} .\)