For the identification of $\mathrm{\beta }$ -naphthol using dye test, it is necessary to use

The number of resonance structures for N is

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The reaction of \(\mathrm{K}_{3}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]\) with freshly prepared \(\mathrm{FeSO}_{4}\) solution produces a dark blue precipitate called Turnbull's blue. Reaction of \(\mathrm{K}_{4}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]\) with the \(\mathrm{FeSO}_{4}\) solution in complete absence of air produces a white precipitate \(\mathbf{X}\), which turns blue in air. Mixing the \(\mathrm{FeSO}_{4}\) solution with \(\mathrm{NaNO}_{3}\), followed by a slow addition of concentrated \(\mathrm{H}_{2} \mathrm{SO}_{4}\) through the side of the test tube produces a brown ring.
Question:
Among the following, the brown ring is due to the formation of

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

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\(RCONH _2\) is converted into \(RNH _2\) by means of Hofmann bromamide degradation.

In this reaction, \(R \mathrm{CONHBr}\) is formed from which this reaction has derived its name. Electron donating group at phenyl activates the reaction. Hofmann degradation reaction is an intramolecular reaction.
Question:
Which is the rate determining step in Hofmann bromamide degradation?

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

The mole fraction of a solute in a solution is $0.1$. At $298 \mathrm{K}$, the molarity of this solution is the same as its molality. The density of this solution at $298 \mathrm{K}$ is $2.0\mathrm{ }\mathrm{g}\mathrm{ }\mathrm{c}{\mathrm{m}}^{–3}$. The ratio of the molecular weights of the solute and solvent, $\left(\frac{\mathrm{M}{\mathrm{W}}_{\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{e}}}{\mathrm{M}{\mathrm{W}}_{\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{v}\mathrm{e}\mathrm{n}\mathrm{t}}}\right)$, is:

The Fischer projection of $\mathrm{D}$-erythrose is shown below.
$\mathrm{D}$-Erythrose and its isomers are listed as $\mathrm{P},\mathrm{Q},\mathrm{R},$ and $\mathrm{S}$ in Column I. Choose the correct relationship of $\mathrm{P},\mathrm{Q},\mathrm{R},$ and $\mathrm{S}$ with $\mathrm{D}$-erythrose from Column II.

	Column - I		Column -II
P.		1.	Diastereomer
Q.		2.	Identical
R.		3.	Enantiomer
S.

List-I shows various functional dependencies of energy \((E)\) on the atomic number \((Z)\). Energies associated with certain phenomena are given in List-II. Choose the option that describes the correct match between the entries in List-I to those in List-II.

List - I	List - II
(P) \(E\text { propto } Z^2\)	(1) energy of characteristic x-rays
(Q) \(E\text { propto } (Z-1)^2\)	(2) electrostatic part of the nuclear binding energy for stable nuclei with mass numbers in the range 30 to 170
(R) \(E \text { propto } Z(Z-1)\)	(3) energy of continuous x-rays
(S) E is practically independent of Z	(4) average nuclear binding energy per nucleon for stable nuclei with mass number in the range 30 to 170
	(5) energy of radiation due to electronic transitions from hydrogen-like atoms

In an electrochemicalcell, dichromate ions in aqueous acidic medium are reduced to \(\mathrm{Cr}^{3+}\). The current (in amperes) that flows through the cell for 48.25 minutes to produce 1 mole of \(\mathrm{Cr}^{3+}\) is _____ .
Use: 1 Faraday \(=96500 \mathrm{C} \mathrm{mol}^{-1}\)

A ring and a disc are initially at rest, side by side, at the top of an inclined plane which makes an angle $60°$ with the horizontal. They start to roll without slipping at the same instant of time along the shortest path. If the time difference between their reaching the ground is $\left(2-\sqrt{3}\right)/\sqrt{10} \mathrm{s}$, then the height of the top of the inclined plane (in meters) is __________. Take $g=10\mathrm{ }\mathrm{m}\mathrm{ }{\mathrm{s}}^{-2}$.

Let \(f: R \rightarrow R\) be a function such that \(f(x+y)=f(x)+f(y), \forall x, y \in R\). If \(f(x)\) is differentiable at \(x=0\), then

Function \(x=A \sin ^2 \omega t+B \cos ^2 \omega t+C\) \(\sin \omega t \cos \omega t\) represents SHM.