For the reaction, \(\mathbf{X}(s) \rightleftharpoons \mathbf{Y}(s)+\mathbf{Z}(g)\), the plot of \(\ln \frac{p_{\mathbf{Z}}}{p^{\theta}}\) versus \(\frac{10^{4}}{T}\) is given below (in solid line), where \(p_{\mathbf{Z}}\) is the pressure (in bar) of the gas \(\mathbf{Z}\) at temperature \(T\) and \(p^{\theta}=1\) bar.

(Given, \(\frac{\mathrm{d}(\ln K)}{\mathrm{d}\left(\frac{1}{T}\right)}=-\frac{\Delta H^{\theta}}{R}\), where the equilibrium constant, \(K=\frac{p_{z}}{p^{\theta}}\) and the gas constant, \(\mathrm{R}=8.314 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\) )
The value of standard enthalpy, $∆{\mathrm{H}}^{\mathrm{o}}$ (in $\mathrm{kJ} {\mathrm{mol}}^{-1}$) for the given reaction is ___.

Paragraph:
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

The correct statement(s) about of the following reaction sequence is(are)

The value of \(\log _{10} \mathrm{~K}\) for a reaction \(A \rightleftharpoons B\) is (Given: \(\Delta_{\mathrm{r}} H_{298 \mathrm{~K}}^{\circ}=-54.07 \mathrm{~kJ} \mathrm{~mol}^{-1}, \Delta_{\mathrm{r}} S_{298 \mathrm{~K}}^{\circ}\)\(=10 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}\) and \(R=8.314 \mathrm{JK}^{-1} \mathrm{~mol}^{-1} ; 2.303 \times 8.314 \times 298\)\(=5705\) )

Paragraph:
Properties such as boiling point, freezing point and vapour pressure of a pure solvent change when solute molecules are added to get homogeneous solution. These are called colligative properties. Applications of colligative properties are very useful in day-to-day life. One of its examples is the use of ethylene glycol and water mixture as anti-freezing liquid in the radiator automobiles.
A solution \(M\) is prepared by mixing ethanol and water. The mole fraction of ethanol in the mixtrue is \(0.9\)
Given Freezing point depression constant of water \(\left(k^{\text {water }}\right)=1.86 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}\)
Freezing point depression constant of ethanol \(\left(k_f^{\text {ethanol }}\right)=2.0 \mathrm{Kkg} \mathrm{mol}^{-1}\)
Boiling point elevation constant of water \(\left(k_b^{\text {water }}\right)=0.52 \mathrm{Kkg} \mathrm{mol}^{-1}\)
Boiling point elevation constant of ethanol \(\left(k_b^{\text {ethanol }}\right)=1.2 \mathrm{Kkg} \mathrm{mol}^{-1}\)
Standard freezing point of water \(=273 \mathrm{~K}\)
Standard freezing point of ethanol \(=155.7 \mathrm{~K}\)
Standard boiling point of water \(=373 \mathrm{~K}\)
Standard boiling point of ethanol \(=351.5 \mathrm{~K}\)
Vapour pressure of pure water \(=328 \mathrm{~mm}\) of \(\mathrm{Hg}\)
Vapour pressure of pure ethanol \(=40 \mathrm{~mm}\) of \(\mathrm{Hg}\)
Molecular weight of water \(=18 \mathrm{~g} \mathrm{~mol}^{-1}\)
Molecular weight of ethanol \(=46 \mathrm{~g} \mathrm{~mol}^{-1}\)
In answering the following questions, consider the solutions to be ideal dilute solutions and solutes to be non-volatile and non-dissociative.

Question:
Water is added to the solution \(M\) such that the mole fraction of water in the solution becomes \(0.9\) The boiling point of this solutions is

The coordination number of \(\mathrm{Al}\) in the crystalline state of \(\mathrm{AlCl}_3\) is

Paragraph:
The hydrogen-like species \(\mathrm{Li}^{2+}\) is in a spherically symmetric state \(S_1\) with one radial node. Upon absorbing light the ion undergoes transition to a state \(S_2\). The state \(S_2\) has one radial node and its energy is equal to the ground state energy of the hydrogen atom.
Question:
The orbtial angular momentum quantum number of the state \(S_2\) is

Geometrical shapes of the complexes formed by the reaction of \(\mathrm{Ni}^{2+}\) with \(\mathrm{Cl}^{-}\), \(\mathrm{CN}^{-}\)and \(\mathrm{H}_2 \mathrm{O}\), respectively, are

If \(\frac{\sin ^4 x}{2}+\frac{\cos ^4 x}{3}=\frac{1}{5}\), then

Aluminium reacts with sulfuric acid to form aluminium sulfate and hydrogen. What is the volume of hydrogen gas in liters $\left(\mathrm{L}\right)$ produced at $300 \mathrm{K}$ and $1.0$ atm pressure, when $5.4 \mathrm{g}$ of aluminium and $50.0 \mathrm{mL}$ of $5.0 \mathrm{M}$ sulfuric acid are combined for the reaction? (Use molar mass of aluminium as $27.0 \mathrm{g} {\mathrm{mol}}^{-1}$, $\mathrm{R}=0.082 \mathrm{atm} \mathrm{L} {\mathrm{mol}}^{-1}{\mathrm{K}}^{-1}$)

Let $T_1$ and $T_2$ be two distinct common tangents to the ellipse $E:\frac{x^2}{6}+\frac{y^2}{3}=1$ and the parabola $P:y^2=12x$. Suppose that the tangent $T_1$ touches $P$ and $E$ at the points $A_1$ and $A_2$, respectively and the tangent $T_2$ touches $P$ and $E$ at the points $A_4$ and $A_3$, respectively. Then which of the following statements is(are) true?

Match the following.
 

	List – I		List – II
(A)	Let $y\left(x\right)=\cos \left(3{\cos }^{-1}x\right), x ϵ \left[-1, 1\right], x\ne \pm \frac{\sqrt{3}}{2}.$ Then $\frac{1}{y\left(x\right)} \left\{\left(x^2-1\right)\frac{d^{2}y\left(x\right)}{d{x}^2}+x\frac{dy\left(x\right)}{dx}\right\}$ equals	(P)	1
(B)	Let $A_1, A_2, \dots , A_n\left(n>2\right)$ be the vertices of a regular polygon of n sides with its centre at the origin. Let $\overrightarrow{a_k}$ be the position vector of the point $A_k, k=1, 2, \dots n.$ If $\left|{\sum }_{k=1}^{n-1}\left(\mathrm{ }\overrightarrow{a_k}\times \mathrm{ }\overrightarrow{a_{k+1}}\right)\right|=\left|{\sum }_{k=1}^{n-1}\left(\mathrm{ }\overrightarrow{a_k} \cdot \mathrm{ }\overrightarrow{a_{k+1}}\right)\right|$ , then the minimum value of n is	(Q)	2
(C)	If the normal from the point $P\left(h, 1\right)$ on the ellipse $\frac{x^2}{6}+\frac{y^2}{3}=1$ is perpendicular to the line $x+y=8,$ then the value of h is	(R)	8
(D)	Number of positive solutions satisfying the equation ${\tan }^{-1}\left(\frac{1}{2x+1}\right)+{\tan }^{-1}\left(\frac{1}{4x+1}\right)={\tan }^{-1}\left(\frac{2}{x^2}\right)$ is	(S)	9

In a Young's double slit experiment, the slit separation $\mathrm{d}$ is $0.3\mathrm{}\mathrm{m}\mathrm{m}$ and the screen distance $\mathrm{D}$ is $1\mathrm{m}.$ A parallel beam of light of wavelength $600\mathrm{}\mathrm{n}\mathrm{m}$ is incident on the slits at angle $\alpha$ as shown in figure. On the screen, the point $\mathrm{O}$ is equidistant from the slits and distance $\mathrm{P}\mathrm{O}$ is $11.0\mathrm{}\mathrm{m}\mathrm{m}.$ Which of the following statement(s) is/are correct?