Consider a $70\%$ efficient hydrogen-oxygen fuel cell working under standard conditions at $1$ bar and $298 \mathrm{K}$. Its cell reaction is ${\mathrm{H}}_2\left(\mathrm{g}\right)+\frac{1}{2}{\mathrm{O}}_2\left(\mathrm{g}\right)\to {\mathrm{H}}_2\mathrm{O}\left(\mathrm{l}\right)$ The work derived from the cell on the consumption of $1.0\times {10}^{-3 }\mathrm{mol}$ of ${\mathrm{H}}_2\left(\mathrm{g}\right)$ is used to compress $1.00 \mathrm{mol}$ of a monoatomic ideal gas in a thermally insulated container. What is the change in the temperature (in $\mathrm{K}$ ) of the ideal gas? The standard reduction potentials for the two half-cells are given below. \(O _2(g)+4 H ^{+}( aq )+4 e ^{-} \rightarrow 2 H _2 O ( l ),\) \(E ^0=1.23 V\) $2{\mathrm{H}}^{+}\left(\mathrm{aq}\right)+2{\mathrm{e}}^{-}\to {\mathrm{H}}_2\left(\mathrm{g}\right), {\mathrm{E}}^0=0.00 \mathrm{V}$ Use $\mathrm{F}=96500 \mathrm{C} {\mathrm{mol}}^{-1},\mathrm{R}=8.314 \mathrm{J} {\mathrm{mol}}^{-1}{\mathrm{K}}^{-1}$.

Reduction of the metal centre in aqueous permanganate ion involves

Amongst the following, the total number of compounds soluble in aqueous \(\mathrm{NaOH}\) is

A sample \((5.6 \mathrm{~g})\) containing iron is completely dissolved in cold dilute \(\mathrm{HCl}\) to prepare a \(250 \mathrm{~mL}\) of solution. Titration of \(25.0 \mathrm{~mL}\) of this solution requires \(12.5 \mathrm{~mL}\) of \(0.03 \mathrm{M} \mathrm{KMnO}_{4}\) solution to reach the end point. Number of moles of \(\mathrm{Fe}^{2+}\) present in \(250 \mathrm{~mL}\) solution is \(\mathbf{x} \times 10^{-2}\) (consider complete dissolution of \(\mathrm{FeCl}_{2}\) ). The amount of iron present in the sample is \(\mathbf{y} \%\) by weight.
(Assume: \(\mathrm{KMnO}_{4}\) reacts only with \(\mathrm{Fe}^{2+}\) in the solution Use: Molar mass of iron as \(56 \mathrm{~g} \mathrm{~mol}^{-1}\) )
The value of $\mathrm{y}$ is________ .

Addition of excess aqueous ammonia to a pink coloured aqueous solution of $MC{l}_2.6H_{2}O \left(X\right)$ and $N{H}_{4}Cl$ gives an octahedral complex Y in the presence of air. In aqueous solution, complex Y behaves as 1 : 3 electrolyte. The reaction of X with excess HCl at room temperature results in the formation of a blue coloured complex Z. The calculated spin only magnetic moment of X and Z is 3.87 B.M., whereas it is zero for complex Y.
Among the following options, which statement(s) is(are) correct ?

LIST-I contains reactions and LIST-II contains major products

List - I	List - II
(A)	(P)
(B)	(Q)
(C)	(R)
(D)	(S)
	(T)

Match each reaction in LIST-I with one or more products in LIST-II and choose the correct option

Isomers of hexane, based on their branching, can be divided into three distinct classes as shown in the figure.

The correct order of their boiling point is

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

Let \(M\) and \(N\) be two \(3 \times 3\) non-singular skew-symmetric matrices such that \(M N=N M\). If \(P^T\) denotes the transpose of \(P\), then \(M^2 N^2\left(M^T N\right)^{-1}\left(M N^{-1}\right)^T\) is equal to

Let \(k\) be a positive real number and let \(\begin{aligned} A & =\left[\begin{array}{ccc}2 k-1 & 2 \sqrt{k} & 2 \sqrt{k} \\ 2 \sqrt{k} & 1 & -2 k \\ -2 \sqrt{k} & 2 k & -1\end{array}\right] \text { and } \\ B & =\left[\begin{array}{ccc}0 & 2 k-1 & \sqrt{k} \\ 1-2 k & 0 & 2 \sqrt{k} \\ -\sqrt{k} & -2 \sqrt{k} & 0\end{array}\right]\end{aligned}\)
If \(\operatorname{det}(\operatorname{adj} A)+\operatorname{det}(\operatorname{adj} B)=10^6\), then \([k]\) is equal to
[Note : adj \(M\) denotes the adjoint of a square matrix \(M\) and \([k]\) denotes the largest integer less than or equal to \(k\) ].

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Tollen's reagent is used for the detection of aldehyde when a solution of \(\mathrm{AgNO}_3\) is added to glucose with \(\mathrm{NH}_4 \mathrm{OH}\) then gluconic acid is formed
\(\mathrm{Ag}^{+}+e^{-} \rightarrow \mathrm{Ag} E_{\text {red }}^{\circ}=0.8 \mathrm{~V} \)
\( \mathrm{C}_6 \mathrm{H}_{12} \mathrm{O}_6+\mathrm{H}_2 \mathrm{O} \rightarrow \text { Gluconic acid }\)\(\left(\mathrm{C}_6 \mathrm{H}_{12} \mathrm{O}_7\right)+2 \mathrm{H}^{+}+2 e^{-} ; -E_{\mathrm{red}}^{\circ}=\) \(-0.05 \mathrm{~V} \)
\( \mathrm{Ag}\left(\mathrm{NH}_3\right)_2^{+}+e^{-} \rightarrow \mathrm{Ag}(s)+2 \mathrm{NH}_3 ;\) \(E_{\mathrm{red}}^{\circ}=0.337 \mathrm{~V}\)
[Use \(2.303 \times \frac{R T}{F}=0.0592\) and \(\frac{F}{R T}=38.92\) at \(298 \mathrm{~K}\) ]
Question:
Ammonia is always added in this reaction. Which of the following must be incorrect?

Let ${\mathrm{L}}_1$ and ${\mathrm{L}}_2$ denotes the lines
$\overrightarrow{\mathrm{r}}=\widehat{\mathrm{i}}+\mathrm{\lambda }\left(-\widehat{\mathrm{i}}+2\widehat{\mathrm{j}}+2\widehat{\mathrm{k}}\right),\mathrm{\lambda }\in \mathrm{R}$ and $\overrightarrow{\mathrm{r}}=\mathrm{\mu }\left(2\widehat{\mathrm{i}}-\widehat{\mathrm{j}}+2\widehat{\mathrm{k}}\right),\mathrm{\mu }\in \mathrm{R}$
respectively. If ${\mathrm{L}}_3$ is a line which is perpendicular to both ${\mathrm{L}}_1$ and ${\mathrm{L}}_2$ and cuts both of them, then which of the following options describe(s) ${\mathrm{L}}_3$ ?

Let \(f(x)=x^2\) and \(g(x)=\sin x\) for all \(x \in R\). Then, the set of all \(x\) satisfying \((\) fogogof \()(x)=(\operatorname{gogof})(x)\), where \((f \circ g)(x)=f(g(x))\) is