The monomer (X) involved in the synthesis of Nylon 6,6 gives positive carbylamine test. If 10 moles of \(\mathbf{X}\) are analyzed using Dumas method, the amount (in grams) of nitrogen gas evolved is ______ .
Use: Atomic mass of N (in amu) = 14

Molar volume \(\left(V_m\right)\) of a van der Waals gas can be calculated by expressing the van der Waals equation as a cubic equation with \(V_m\) as the variable. The ratio (in \(\mathrm{mol} \mathrm{dm}^{-3}\) ) of the coefficient of \(V_m^2\) to the coefficient of \(V_m\) for a gas having van der Waals constants \(a=6.0 \mathrm{dm}^6 \mathrm{~atm} \mathrm{~mol}^{-2}\) and \(b=0.060 \mathrm{dm}^3 \mathrm{~mol}^{-1}\) at 300 K and 300 atm is _____ .
Use: Universal gas constant \((R)=0.082 \mathrm{dm}^3 \mathrm{~atm} \mathrm{~mol}{ }^{-1} \mathrm{~K}^{-1}\)

Passing \(\mathrm{H}_2 \mathrm{~S}\) gas into a mixture of \(\mathrm{Mn}^{2+}, \mathrm{Ni}^{2+}, \mathrm{Cu}^{2+}\) and \(\mathrm{Hg}^{2+}\) ions in an acidified aqueous solution precipitates

The compound(s) formed upon combustion of sodium metal in excess air is (are)

The major product(s) of the following reaction is/are:

The correct statement(s) about the following sugars \(X\) and \(Y\) is (are)

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The reaction of compound \(\mathbf{P}\) with \(\mathrm{CH}_{3} \mathrm{MgBr}\) (excess) in \(\left(\mathrm{C}_{2} \mathrm{H}_{5}\right)_{2} \mathrm{O}\) followed by addition of \(\mathrm{H}_{2} \mathrm{O}\) gives \(\mathbf{Q}\). The compound \(\mathbf{Q}\) on treatment with \(\mathrm{H}_{2} \mathrm{SO}_{4}\) at \(0^{\circ} \mathrm{C}\) gives \(\mathbf{R}\). The reaction of \(\mathbf{R}\) with \(\mathrm{CH}_{3} \mathrm{COCl}\) in the presence of anhydrous \(\mathrm{AlCl}_{3}\) in \(\mathrm{CH}_{2} \mathrm{Cl}_{2}\) followed by treatment with \(\mathrm{H}_{2} \mathrm{O}\) produces compound \(\mathbf{S}\). [Et in compound \(\mathbf{P}\) is ethyl group]


Question :
The reactions, \(\mathbf{Q}\) to \(\mathbf{R}\) and \(\mathbf{R}\) to \(\mathbf{S}\), are

${}_{92}{}^{238}\mathrm{U}$ is known to undergo radioactive decay to form ${}_{82}{}^{206}\mathrm{Pb}$ by emitting alpha and bet a particles. A rock initially contained $68\times {10}^{-6}\mathrm{g}$ of ${}_{92}{}^{238}\mathrm{U}$. If the number of alpha particles that it would emit during its radioactive decay of ${}_{92}{}^{238}\mathrm{U}$ to ${}_{82}{}^{206}\mathrm{Pb}$ in three half-lives is $\mathrm{Z}\times {10}^{18}$, then what is the value of $\mathrm{Z}$?

Let \(\mathbb{R}^3\) denote the three-dimensional space. Take two points \(P=(1,2,3)\) and \(Q=(4,2,7)\). Let \(\operatorname{dist}(X, Y)\) denote the distance between two points \(X\) and \(Y\) in \(\mathbb{R}^3\). Let
\(\begin{gathered}S=\left\{X \in \mathbb{R}^3:(\operatorname{dist}(X, P))^2-(\operatorname{dist}(X, Q))^2=50\right\} \text { and } \\T=\left\{Y \in \mathbb{R}^3:(\operatorname{dist}(Y, Q))^2-(\operatorname{dist}(Y, P))^2=50\right\} .\end{gathered}\)
Then which of the following statements is (are) TRUE?

Let $f:\left[0,2\right]\to R$ be a function which is continuous on $\left[0,2\right]$ and is differentiable on $\left(0,2\right) f\left(0\right)=1$. Let $F\left(x\right)=\underset{0}{\overset{x^2}{\int }}f\mathrm{ }\left(\sqrt{t}\right)dt$ for $x\in \left[0,2\right].$ If $F^{\text{'}}\left(x\right)=f^{\text{'}}\left(x\right)$ for all $x\in \left(0,2\right),$ then $F\left(2\right)$ equals

In the figure, the inner (shaded) region $A$ represents a sphere of radius $r_A=1$, within which the electrostatic charge density varies with the radial distance $r$ from the center as ${\rho }_A=kr$, where $k$ is positive. In the spherical shell $B$ of outer radius $r_B$, the electrostatic charge density varies as ${\rho }_B=\frac{2k}{r}$. Assume that dimensions are taken care of. All physical quantities are in their SI units.

Which of the following statement(s) is/(are) correct?

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 \(R=8.31 \mathrm{~J} / \mathrm{mol} . \mathrm{K}\) )

For $x\in \mathrm{ℝ}$, let $y\left(x\right)$ be a solution of the differential equation $\left(x^2-5\right)\frac{dy}{dx}-2xy=-2x{\left(x^2-5\right)}^2$ such that $y\left(2\right)=7$. Then the maximum value of the function $y\left(x\right)$ is