Statement I Band gap in germanium is small.
Statement II The energy spread of each germanium atomic energy level is infinitesimally small.

An electrochemical cell is fueled by the combustion of butane at 1 bar and 298 K . Its cell potential is \(\frac{\boldsymbol{X}}{\boldsymbol{F}} \times 10^3\) volts, where \(\boldsymbol{F}\) is the Faraday constant. The value of \(\boldsymbol{X}\) is \(\qquad\) .
Use : Standard Gibbs energies of formation at 298 K are : \(\Delta_f G_{\mathrm{CO}_2}^{\circ}=-394 \mathrm{~kJ} \mathrm{~mol}^{-1}\);
\(\Delta_f G_{\text {water }}^{\circ}=-237 \mathrm{~kJ} \mathrm{~mol}^{-1} ; \Delta_f G_{\text {butane }}^{\circ}\) \(=-18 \mathrm{~kJ} \mathrm{~mol}^{-1}\)

Calamine, malachite, magnetite and cryolite, respectively are:

In the following reactions, $\mathbf{P},\mathbf{Q},\mathbf{R}$, and $\mathbf{S}$ are the major products.




The correct statement(s) about $\mathbf{P},\mathbf{Q},\mathbf{R}$, and $\mathbf{S}$ is(are)

The standard reduction potential data at $25°C$ is given below.
\(E ^o\left( Fe ^{3+}, Fe ^{2+}\right)=+0.77 V ; \)
\( E ^o\left( Fe ^{2+}, Fe \right)=-0 . 44 V \)
\( E ^o\left( Cu ^{2+}, Cu \right)=+0.34 V ; \)
\( E ^o\left( Cu ^{+}, Cu \right)=+0.52 V \)
\( E ^o\left[ O _2(g)+4 H ^{+}+4 e ^{-} \rightarrow 2 H _2 O \right]=\) \(+1.23~V \)
\( E ^o\left[ O _2(g)+2 H _2 O +4 e^{-} \longrightarrow \text { 4OH}^{-}\right]=\) \(+0.40~V \)
\( E ^o\left( Cr ^{3+}, Cr \right)=-0.74 V ; \)
\( E ^o\left( Cr ^{2+}, Cr \right)=-0.91 V\)
Match E^o of the redox pair in List I with the values given in List II and select the correct answer using the code given below the lists :
 

	List I		List II
A.	$E^o\left({\mathit{\text{Fe}}}^{3+},\mathit{\text{Fe}}\right)$	P.	$-0.18V$
B.	$E^o\left(4{\text{H}}_2\text{O}\rightleftharpoons 4H^{+}+4{\mathit{\text{OH}}}^{-}\right)$	Q.	$-0.4V$
C	$E^o\left({\mathit{\text{Cu}}}^{2+}+\mathit{\text{Cu}}\to 2{\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{Cu}}^{+}\right)$	R.	$-0.04V$
D.	$E^o\left({\mathit{\text{Cr}}}^{3+},{\mathit{\text{Cr}}}^{2+}\right)$	S.	$-0.83V$

The major product of the following reaction is

Consider the strong electrolytes ${\mathrm{Z}}_{\mathrm{m}}{\mathrm{X}}_{\mathrm{n}}, {\mathrm{U}}_{\mathrm{m}}{\mathrm{Y}}_{\mathrm{p}}$ and ${\mathrm{V}}_{\mathrm{m}}{\mathrm{X}}_{\mathrm{n}}$. Limiting molar conductivity of ${\mathrm{U}}_{\mathrm{m}}{\mathrm{Y}}_{\mathrm{p}}$ and ${\mathrm{V}}_{\mathrm{m}}{\mathrm{X}}_{\mathrm{n}}$ are $250$ and $440 \mathrm{S} {\mathrm{cm}}^2 {\mathrm{mol}}^{-1}$, respectively. The value of $\left(\mathrm{m}+\mathrm{n}+\mathrm{p}\right)$ is Given:

Ion	\(Z ^{ n +}\)	\(U ^{ p +}\)	\(V ^{ n +}\)	\(X ^{ m }\)	\(Y ^{ m -}\)
\(\lambda^0(\text{S cm} ^2\) \(\text{mol}^{-1})\)	50.0	25.0	100.0	80.0	100.0

${\lambda }^0$ is the limiting molar conductivity of ions The plot of molar conductivity $\left(\Lambda \right)$ of ${\mathrm{Z}}_{\mathrm{m}}{\mathrm{X}}_{\mathrm{n}}$ vs ${\mathrm{c}}^{\frac{1}{2}}$ is given below. If the numerical value has more than two decimal places, truncate/round-off the value to TWO decimal places.

\[
\text { Match the statements/expressions in Column I with the open intervals in Column II. }
\]

A wire, which passes through the hole in a small bead, is bent in the form of quarter of a circle. The wire is fixed vertically on ground as shown in the figure. The bead is released from near the top of the wire and it slides along the wire without friction. As the bead moves from A to B, the force it applies on the wire is

Let \(f(x)=2+\cos x\) for all real \(x\).
Statement I For each real \(t\), there exists a point \(c\) in \([t, t+\pi]\) such that \(f^{\prime}(c)=0\).
Statement II \(f(t)=f(t+2 \pi)\) for each real \(t\).

 The diameter of a cylinder is measured using a vernier calipers with no zero error. It is found that the zero of the vernier scale lies between $5.10 \mathrm{cm}$ and $5.15 \mathrm{cm}$ of the main scale. The vernier scale has $50$  divisions equivalent to $2.45 \mathrm{cm}$. The ${24}^{\mathrm{th}}$ division of the vernier scale exactly coincides with one of the main scale divisions. The diameter of the cylinder is, 

Let $b_i>1$ for $i=1, 2,\dots .,101.$ Suppose ${\log }_e{b}_1,{\log }_e{b}_2,\dots ..,{\log }_e{b}_{101}$ are in Arithmetic Progression (A.P.) with the common difference ${\log }_{e}2.$ Suppose $a_1, a_2,\dots .,a_{101}$ are in A.P. such that $a_1=b_1$ and $a_{51}=b_{51}.$ If $t=b_1+b_2+\dots +b_{51}$ and $s=a_1+a_2+\dots +a_{51}$ then

An ideal monoatomic gas is confined in a horizontal cylinder by a spring loaded piston (as shown in the figure).Initially the gas is at temperature ${\mathrm{T}}_1$ , pressure ${\mathrm{P}}_1$ and volume ${\mathrm{V}}_1$ and the spring is in its relaxed state. The gas is then heated very slowly to temperature ${\mathrm{T}}_2$ , pressure ${\mathrm{P}}_2$ and volume ${\mathrm{V}}_2$ . During this process the piston moves out by a distance $\mathrm{x}$ . Ignoring the friction between the piston the cylinder, the correct statement(s) is (are)