In the following reaction sequence in aqueous solution, the species $\text{X,}\text{Y}$ and $\text{Z}$ respectively, are
\(S _2 O _3^{2-} \xrightarrow{ Ag ^{+}} \underset{\substack{\text { Clear } \\ \text { Solution }}}{ X } \xrightarrow{ Ag ^{+}} \underset{\substack{\text { White } \\ \text { Precipitate }}}{ Y }\) \(\xrightarrow{\text { With time }} \underset{\substack{\text { Black } \\ \text { Precipitate }}}{ Z }\)

A sample \((5.6 \mathrm{~g})\) containing iron is completely dissolved in cold dilute \(\mathrm{HCl}\) to prepare a \(250 \mathrm{~mL}\) of solution. Titration of \(25.0 \mathrm{~mL}\) of this solution requires \(12.5 \mathrm{~mL}\) of \(0.03 \mathrm{M} \mathrm{KMnO}_{4}\) solution to reach the end point. Number of moles of \(\mathrm{Fe}^{2+}\) present in \(250 \mathrm{~mL}\) solution is \(\mathbf{x} \times 10^{-2}\) (consider complete dissolution of \(\mathrm{FeCl}_{2}\) ). The amount of iron present in the sample is \(\mathbf{y} \%\) by weight.
(Assume: \(\mathrm{KMnO}_{4}\) reacts only with \(\mathrm{Fe}^{2+}\) in the solution Use: Molar mass of iron as \(56 \mathrm{~g} \mathrm{~mol}^{-1}\) )
The value of $\mathrm{x}$ is________ .

Complete reaction of acetaldehyde with excess formaldehyde, upon heating with conc. \(\mathrm{NaOH}\) solution, gives \(\mathbf{P}\) and \(\mathbf{Q}\). Compound \(\mathbf{P}\) does not give Tollens' test, whereas \(\mathbf{Q}\) on acidification gives positive Tollens' test. Treatment of \(\mathbf{P}\) with excess cyclohexanone in the presence of catalytic amount of \(p\)-toluenesulfonic acid (PTSA) gives product \(\mathbf{R}\).
Sum of the number of methylene groups \(\left(-\mathrm{CH}_2-\right)\) and oxygen atoms in \(\mathbf{R}\) is ________.

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Carbon-14 is used to determine the age of organic material. The procedure is based on the formation of \({ }^{14} \mathrm{C}\) by neutron capture in the upper atmosphere.
\(
{ }_7^{14} \mathrm{~N}+{ }_0^1 n \longrightarrow{ }_6^{14} \mathrm{C}+{ }_1 n^1
\)
\({ }^{14} \mathrm{C}\) is absorbed by living organisms during photosynthesis. The \({ }^{14} \mathrm{C}\) content is constant in living organism once the plant or animal dies, the uptake of carbon dioxide by it ceases and the level of \({ }^{14} \mathrm{C}\) in the dead being, falls to the decay which \({ }^{14} \mathrm{C}\) undergoes.
\(
{ }_6^{14} \mathrm{C} \longrightarrow{ }_7^{14} \mathrm{~N}+\beta^{-}
\)
The half life period of \({ }^{14} \mathrm{C}\) is 5770 years. The decay constant \((\lambda)\) can be calculated by using the following formula \(\lambda=\frac{0.693}{t_{1 / 2}}\).
The comparison of the \(\beta^{-}\)activity of the dead matter with that of the carbon still in circulation enables measurement of the period of the isolation of the material from the living cycle. The method however, ceases to be accurate over periods longer than 30,000 years. The proportion of \({ }^{14} \mathrm{C}\) to \({ }^{12} \mathrm{C}\) in living matter is \(1: 10^{12}\).
Question:
Which of the following option is correct?

(I) 1, 2-dihydroxy benzene
(II) 1, 3-dihydroxy benzene
(III) 1, 4-dihydroxy benzene
(IV) Hydroxy benzene
The increasing order of boiling points of above mentioned alcohols is

The freezing point (in \({ }^{\circ} \mathrm{C}\) ) of solution containing \(0.1 \mathrm{~g}\) of \(\mathrm{K}_3\left[\mathrm{Fe}(\mathrm{CN})_6\right]\) (molecular weight 329\()\) in \(100 \mathrm{~g}\) of water \(\left(K_f=1.86 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}\right)\) is

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An acylic hydrocarbon \(P\), having molecular formula \(\mathrm{C}_6 \mathrm{H}_{10}\), gave acetone as the only organic product through the following sequence of reactions, in which \(Q\) is an intermediate organic compounds.

Question:
The structure of compound P is

Let $\overrightarrow{a}, \overrightarrow{b}$ and $\overrightarrow{c}$ be three non - coplanar unit vectors such that the angle between every pair of them is $\frac{\pi }{3}.$ If $\overrightarrow{a} \times \overrightarrow{b}+ \overrightarrow{b} \times \overrightarrow{c}=p\overrightarrow{a}+q\overrightarrow{b}+r\overrightarrow{c}$ , where p, q and r are scalars, then the value of $\frac{p^2+2q^2+r^2}{q^2}$ is _______

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

If the function \(f(x)=x^3+e^{\frac{x}{2}}\) and \(g(x)=f^{-1}(x)\), then the value of \(g^{\prime}(1)\) is

A small object of uniform density rolls up a curved surface with an initial velocity \(v\). It reaches up to a maximum height of \(\frac{3 v^2}{4 g}\) with respect to the initial\(
\text { position. The object is }
\)

At \(298 \mathrm{~K}\), the limiting molar conductivity of a weak monobasic acid is \(4 \times 10^{2} \mathrm{~S} \mathrm{~cm}^{2} \mathrm{~mol}^{-1}\). At \(298 \mathrm{~K}\), for an aqueous solution of the acid the degree of dissociation is \(\boldsymbol{\alpha}\) and the molar conductivity is \(\mathbf{y} \times 10^{2} \mathrm{~S} \mathrm{~cm}^{2} \mathrm{~mol}^{-1}\). At \(298 \mathrm{~K}\), upon 20 times dilution with water, the molar conductivity of the solution becomes \(3 \mathbf{y} \times 10^{2} \mathrm{~S} \mathrm{~cm}^{2} \mathrm{~mol}^{-1}\)
The value of $\mathrm{\alpha }$ is _______ .

Let \(S=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: x \geq 0, y \geq 0, y^2 \leq 4 x, y^2 \leq 12-2 x\right.\) and \(\left.3 y+\sqrt{8} x \leq 5 \sqrt{8}\right\}\). If the area of the region \(S\) is \(\alpha \sqrt{2}\), then \(\alpha\) is equal to