In a one-litre flask, $6$ moles of $\mathrm{A}$ undergoes the reaction $\mathrm{A}( \mathrm{g})\rightleftharpoons \mathrm{P}( \mathrm{g})$. The progress of product formation at two temperatures (in Kelvin), ${\mathrm{T}}_1$ and ${\mathrm{T}}_2$, is shown in the figure: If ${\mathrm{T}}_1=2{\mathrm{T}}_2$ and$\left({\mathrm{\Delta G}}_2^{\mathrm{o}}-{\mathrm{\Delta G}}_1^{\mathrm{o}}\right)={\mathrm{RT}}_2\mathrm{lnx}$ , then the value of $\mathrm{x}$ is
[ ${\mathrm{\Delta G}}_1^{\mathrm{o}}$ and ${\mathrm{\Delta G}}_2^{\mathrm{o}}$ are standard Gibb's free energy change for the reaction at temperatures ${\mathrm{T}}_1$ and ${\mathrm{T}}_2$, respectively.]

The ammonia is prepared by treating ammonium sulphate with calcium hydroxide is completely used by ${\mathrm{NiCl}}_2\mathbf{·}6{\mathrm{H}}_2\mathrm{O}$ to form a stable coordination compound. Assume that, both the reactions are $100\%$ completed. If $1584 \mathrm{g}$ of ammonium sulphate and $952 \mathrm{g}$ of ${\mathrm{NiCl}}_2\mathbf{·}6{\mathrm{H}}_2\mathrm{O}$ are used in the preparation, the combined weight (in grams) of gypsum and the nickel ammonia coordination compound produced is ____. Atomic weights in $\mathrm{g}\mathrm{ }{\mathrm{mol}}^{-1}$: $$\begin{gathered} \mathrm{H}=1;\mathrm{ }\mathrm{N}=14\mathrm{ } \\ \mathrm{O}=16; \mathrm{S}=32\mathrm{ } \\ \mathrm{Cl}=35.5;\mathrm{ }\mathrm{Ca}=40\mathrm{ } \\ \mathrm{Ni}=59 \end{gathered}$$

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Upon heating \(\mathrm{KClO}_{3}\) in the presence of catalytic amount of \(\mathrm{MnO}_{2}\), a gas \(\mathbf{W}\) is formed. Excess amount of \(\mathbf{W}\) reacts with white phosphorus to give \(\mathbf{X}\). The reaction of \(\mathbf{X}\) with pure \(\mathrm{HNO}_{3}\) gives \(\mathbf{Y}\) and \(\mathbf{Z}\).
Question:
\(\mathbf{W}\) and \(\mathbf{X}\) are, respectively

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

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When a metal rod \(M\) is dipped into an aqueous colourless concentrated solution of compound \(\mathrm{N}\), the solution turns light blue. Addition of aqueous \(\mathrm{NaCl}\) to the blue solution gives a white precipitate \(\mathrm{O}\). Addition of aqueous \(\mathrm{NH}_3\) dissolves \(\mathrm{O}\) and gives an intense blue solution.
Question:
The metal \(\operatorname{rod} M\) is

Fusion of $\mathrm{M}\mathrm{n}{\mathrm{O}}_2$ with $\mathrm{K}\mathrm{O}\mathrm{H}$ in presence of ${\mathrm{O}}_2$ produces a salt $\mathrm{W}.$ Alkaline solution of $\mathrm{W}$ upon electrolytic oxidation yields another salt $\mathrm{X}.$ The manganese containing ions present in $\mathrm{W}$ and $\mathrm{X},$ respectively, are $\mathrm{Y}$ and $\mathrm{Z}.$ Correct statement(s) is (are)

Columns 1, 2 and 3 contain starting materials, reaction conditions and type of reactions respectively.

Column 1	Column 2	Column 3
(I) Toulene	(i) \(NaOH\) \( /Br _2\)	(P) Condensation
(II) Acetophenone	(ii) \(B r_2 / h v\)	(Q) Carboxylation
(III) Benzaldehyde	(iii) \(\left( CH _3 CO \right)_2\) \(O / CH _3\) \(COOK\)	(R) Substitution
(IV) Phenol	(iv) \(NaOH\) \(/CO _2\)	(S) Haloform

The only CORRECT combination that gives two different carboxylic acids is

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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:
The vapour pressure of the solution \(M\) is

An organic compound undergoes first-order decomposition. The time taken for its decomposition to \(1 /\) 8 and \(1 / 10\) of its initial concentration are \(t_{1 / 8}\) and \(t_{1 / 10}\) respectively. What is the value of \(\left[\frac{t_{1 / 8}}{t_{1 / 10}}\right] \times 10\) ? \(\left(\log _{10} 2=0.3\right)\)

Match List I with List II and select the correct answer using the code given below the lists :
 

	List - I		List - II
A.	${\left(\frac{1}{{\text{y}}^2}\left(\frac{\text{cos}\left({\text{tan}}^{-1}\text{y}\right)+\text{y sin}\left({\text{tan}}^{-1}\text{y}\right)}{\text{cot}\left({\text{sin}}^{-1}\text{y}\right)+\text{tan}\left({\text{sin}}^{-1}\text{y}\right)}\right)+{\text{y}}^4\right)}^{1/2}takes\; value$	P.	$\frac{1}{2}\sqrt{\frac{5}{3}}$
B.	​If cosx + cosy + cosz = 0 = sinx + siny + sinz then possible value of cos $\frac{\text{x}-\text{y}}{2}is$	Q.	$\sqrt{2}$
C.	If cos $\left(\frac{\pi }{4}-\text{x}\right)$ cos2x + sinx sin 2x secx = cosx sin2x secx + cos $\left(\frac{\pi }{4}+\text{x}\right)$ cos2x then possible value of secx is	R.	$\frac{1}{2}$
D.	If cot $\left({\text{sin}}^{-1}\sqrt{1-{\text{x}}^2}\right)=\text{sin}\left({\text{tan}}^{-1}\left(\text{x}\sqrt{6}\right)\right)\text{,}x\ne 0\text{,}$  then possible value of x is	S.	1

Let $E$ be the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$. For any three distinct points $P,Q$ and $Q^{\text{'}}$ on $E$, let $M\left(P,Q\right)$ be the mid-point of the line segment joining $P$ and $Q$, and $M\left(P,Q^{\text{'}}\right)$ be the mid-point of the line segment joining $P$ and $Q^{\text{'}}$. Then the maximum possible value of the distance between $M\left(P,Q\right)$ and $M\left(P,Q^{\text{'}}\right)$, as $P,Q$ and $Q^{\text{'}}$ vary on $E$, is _____.

An infinitely long wire, located on the \(z\)-axis, carries a current \(I\) along the \(+z\)-direction and produces the magnetic field \(\vec{B}\). The magnitude of the line integral \(\int \vec{B} \cdot \overrightarrow{d l}\) along a straight line from the point \((-\sqrt{3} a, a, 0)\) to \((a, a, 0)\) is given by
[ \(\mu_0\) is the magnetic permeability of free space.]

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Let \(U_1\) and \(U_2\) be two urns such that \(U_1\) contains 3 white and 2 red balls and \(U_2\) contains only 1 white ball. A fair coin is tossed. If head appears then 1 ball is drawn at random from \(U_1\) and put into \(U_2\). However, if tail appears then 2 balls are drawn at random from \(U_1\) and put into \(U_2\). Now, 1 ball is drawn at random from \(U_2\).Question:
The probability of the drawn ball from \(U_2\) being white is