Three moles of ${\mathrm{B}}_2{\mathrm{H}}_6$ are completely reacted with methanol. The number of moles of boron containing product formed is

The carboxyl functional group \((-\mathrm{COOH})\) is present in

Reaction of iso-propylbenzene with \(\mathrm{O}_2\) followed by the treatment with \(\mathrm{H}_3 \mathrm{O}^{+}\)forms phenol and a by-product \(\mathbf{P}\). Reaction of \(\mathbf{P}\) with 3 equivalents of \(\mathrm{Cl}_2\) gives compound \(\mathbf{Q}\). Treatment of \(\mathbf{Q}\) with \(\mathrm{Ca}(\mathrm{OH})_2\) produces compound \(\mathbf{R}\) and calcium salt \(\mathbf{S}\).
The correct statement(s) regarding \(\mathbf{P}, \mathbf{Q}, \mathbf{R}\) and \(\mathbf{S}\) is(are)

In the following reactions, P, Q, R and S are the major products.

The correct statement(s) about P, Q, R and S is(are)

Dissolving $1.24 \mathrm{g}$ of white phosphorous in boiling $\mathrm{NaOH}$ solution in an inert atmosphere gives a gas $\mathrm{Q}$. The amount of ${\mathrm{CuSO}}_4$ (in $\mathrm{g}$) required to completely consume the gas $\mathrm{Q}$ is____
[Given: Atomic mass of
\(\text H =1,\text O =16, \text{Na} =23,\text P =31,\text S=32,\) \(\text{Cu} =63]\)

Choose the correct reason(s) for the stability of the lyophobic colloidal particles.

A trinitro compound, $1,3,5$-tris-($4$-nitrophenyl) benzene, on complete reaction with an excess of $\mathrm{Sn}/\mathrm{HCl}$ gives a major product, which on treatment with an excess of ${\mathrm{NaNO}}_2/\mathrm{HCl}$ at $0°\mathrm{C}$ provides $\mathbf{P}$ as the product. $\mathbf{P}$, upon treatment with excess of ${\mathrm{H}}_2\mathrm{O}$ at room temperature, gives the product $\mathbf{Q}$. Bromination of $\mathbf{Q}$ in aqueous medium furnishes the product $\mathbf{R}$. The compound $\mathbf{P}$ upon treatment with an excess of phenol under basic conditions gives the product $\mathbf{S}$. The molar mass difference between compounds $\mathbf{Q}$ and $\mathbf{R}$ is $474 \mathrm{g} {\mathrm{mol}}^{-1}$ and between compounds $\mathbf{P}$ and $\mathbf{S}$ is $172.5 \mathrm{g} {\mathrm{mol}}^{-1}$.
The total number of carbon atoms and heteroatoms present in one molecule of \(S\) is
[Use: Molar mass (in \(\text{gmol} ^{-1}\) ): \(\text H =1,\text C =12,\text N=14,\text O =16,\) \(\text{Br} =80, \text{Cl} =35.5\) Atoms other than C and H are considered as heteroatoms]

Paragraph : If the measurement errors in all the independent quantities are known, then it is possible to determine the error in any dependent quantity. This is done by the use of series expansion and truncating the expansion at the first power of the error. For example, consider the relation $z =\frac{x}{y}$. If the errors in $x, y \mathrm{a}\mathrm{n}\mathrm{d} z \mathrm{a}\mathrm{r}\mathrm{e} ∆x, ∆y \mathrm{a}\mathrm{n}\mathrm{d} ∆z$ , respectively, then

$z\pm ∆z=\frac{x\pm ∆x}{y\pm ∆y}=\frac{x}{y}\left(1\pm \frac{∆x}{x}\right){\left(1\pm \frac{∆y}{y}\right)}^{-1}$

The series expansion for ${\left(1\pm \frac{∆y}{y}\right)}^{-1}$, to first power in $∆y/y, \mathrm{i}\mathrm{s} 1\mp \left(∆y/y\right)$. The relative errors in independent variables are always added. So the error in $z$ will be

$\text{Δ}z=z\left(\frac{\text{Δ}x}{x}+\frac{\text{Δ}y}{y}\right).$
The above derivation makes the assumption that \(\Delta r / x \ll 1, \Delta y / y < 1\). Therefore, the higher powers of these quantities are neglected.
Question : Consider the ratio \(r=\frac{(1-a)}{(1+a)}\) to be determined by measuring a dimensionless quantity a If the error in the measurement of a is \(\Delta a(\frac{\Delta a}{a}<<1)\), then what is the error \(\Delta r\) in determining \(r\) ?

During an experiment with a metre bridge, the galvanometer shows a null point when the jockey is pressed at $40.0\mathrm{cm}$using a standard resistance of $90\mathrm{}\mathrm{\Omega }$ ,as shown in the figure. The least count of the scale used in the metre bridge is $1\mathrm{mm}$. The unknown resistance is

Match the following columns

A solid sphere of radius R and density ρ is attached to one end of a mass-less spring of force constant k. The other end of the spring is connected to another solid sphere of radius R and density 3 ρ The complete arrangement is placed in a liquid of density 2ρ and is allowed to reach equilibrium. The correct statement(s) is (are)

Two moles of ideal helium gas are in a rubber balloon at \(30^{\circ} \mathrm{C}\). The balloon is fully expandable and can be assumed to require no energy in its expansion. The temperature of the gas in the balloon is slowly changed to \(35^{\circ} \mathrm{C}\). The amount of heat required in raising the temperature is nearly (take \(R=8.31 \mathrm{~J} / \mathrm{mol} . \mathrm{K}\) )

The value of the integral $\underset{0}{\overset{\frac{1}{2}}{\int }}\frac{1+\sqrt{3}}{{\left({\left(x+1\right)}^2{\left(1-x\right)}^6\right)}^{\frac{1}{4}}} dx$ is