The reversible expansion of an ideal gas under adiabatic and isothermal conditions is shown in the figure. Which of the following statement(s) is (are) correct?

Consider a reaction \(a G+b H \rightarrow\) products. When concentration of both the reactants \(G\) and \(H\) is doubled, the rate increases by eight times. However, when concentration of \(G\) is doubled keeping the concentration of \(H\) fixed, the rate is doubled. The overall order of the reaction is

Match the compounds in Column I with their characteristic test(s)/reactions(s) given in Column II. Indicate your answer by darkening the appropriate bubbles of the \(4 \times 4\) matrix given in the ORS.

Atom $\mathrm{X}$ occupies the fcc lattice sites as well as alternate tetrahedral voids of the same lattice. The packing efficiency (in $\%$) of the resultant solid is closest to

Statement I Alkali metals dissolve in liquid ammonia to give blue solutions. Statement II Alkali metals in liquid ammonia give solvated species of the type \(\left(M\left(\mathrm{NH}_3\right)_n\right]^{+}(M=\) alkali metals \()\).

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\(\mathbf{X}\) and \(\mathbf{Y}\) are two volatile liquids with molar weights of \(10 \mathrm{~g} \mathrm{~mol}^{-1}\) and \(40 \mathrm{~g} \mathrm{~mol}^{-1}\) respectively. Two cotton plugs, one soaked in \(\mathbf{X}\) and the other soaked in \(\mathbf{Y}\), are simultaneously placed at the ends of a tube of length \(L=24 \mathrm{~cm}\), as shown in the figure. The tube is filled with an inert gas at \(1\) atmosphere pressure and a temperature of \(300 \mathrm{~K}\). Vapours of \(\mathbf{X}\) and \(\mathbf{Y}\) react to form a product which is first observed at a distance \(\mathbf{d} \mathrm{~cm}\) from the plug soaked in \(\mathbf{X}\). Take \(\mathbf{X}\) and \(\mathbf{Y}\) to have equal molecular diameters and assume ideal behaviour for the inert gas and the two vapours.



Question:

The experimental value of \(\mathbf{d}\) is found to be smaller than the estimate obtained using Graham's law. This is due to

On complete hydrogenation, natural rubber produces

If the value of Avogadro number is $6.023\times {10}^{23}\mathrm{ }\mathrm{m}\mathrm{o}{\mathrm{l}}^{–1}$ and the value of Boltzmann constant is $1.380\times {10}^{-23}\mathrm{ }\mathrm{J}{\mathrm{K}}^{-1}$ , then the number of significant digits in the calculated value of the universal gas constant is

Consider the equation ${\int }_1^e\frac{{\left({\log }_{e}x\right)}^{{\displaystyle \frac{1}{2}}}}{x{\left(a-{\left({\log }_{e}x\right)}^{{\displaystyle \frac{3}{2}}}\right)}^2}dx=1,a\in \left(-\infty ,0\right)\cup \left(1,\infty \right)$. Which of the following statements is/are TRUE?

For $x\in \mathrm{ℝ}$, let ${\tan }^{-1}\left(x\right)\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$. Then the minimum value of the function $f:\mathrm{ℝ}\to \mathrm{ℝ}$ defined by $f\left(x\right)={\int }_0^{x{\tan }^{-1}x}\frac{e^{\left(t-\cos t\right)}}{1+t^{2023}}dt$ is

An electric dipole with dipole moment $\frac{{\mathrm{p}}_0}{\sqrt{2}}\left(\widehat{\mathrm{i}}+\widehat{\mathrm{j}}\right)$ is held fixed at the origin $\mathrm{O}$ in the presence of an uniform electric field of magnitude ${\mathrm{E}}_0.$ If the potential is constant on a circle of radius $\mathrm{R}$ centered at the origin as shown in figure, then the correct statement(s) is/are:
( ${\mathrm{\epsilon }}_0$ is permittivity of free space, $\mathrm{R}>>$ dipole size)

The correct option(s) related to the extraction of iron from its ore in the blast furnace operating in the temperature range $900-1500 \mathrm{K}$ is(are)

Let \(f\) be a real-valued function defined on the interval \((0, \infty)\), by \(f(x)=\ln x+\int_0^x \sqrt{1+\sin t} d t\). Then which of the following statement(s) is (are) true ?