The oxidation number of \(\mathrm{Mn}\) in the product of alkaline oxidative fusion of \(\mathrm{MnO}_2\) is

The correct statement(s) about the following sugars \(X\) and \(Y\) is (are)

Match each of the compounds is Column I with its characteristic reaction(s) is Column II.

Extraction of copper from copper pyrite $\left(\mathrm{C}\mathrm{u}\mathrm{F}\mathrm{e}{\mathrm{S}}_2\right)$ involves

Among the following complexes, the total number of diamagnetic species is_____
\(\left[\mathrm{Mn}\left(\mathrm{NH}_3\right)_6\right]^{3+},\left[\mathrm{MnCl}_6\right]^{3-},\left[\mathrm{FeF}_6\right]^{3-},\left[\mathrm{CoF}_6\right]^{3-},\)\(\left[\mathrm{Fe}\left(\mathrm{NH}_3\right)_6\right]^{3+}\), and \(\left[\mathrm{Co}(\mathrm{en})_3\right]^{3+}\)
[Given, atomic number: \(\mathrm{Mn}=25, \mathrm{Fe}=26, \mathrm{Co}=27\);
\(\text { en } \left.=\mathrm{H}_2 \mathrm{NCH}_2 \mathrm{CH}_2 \mathrm{NH}_2\right]\)

The total number of diprotic acids among the following is
\(\begin{array}{lll}\mathrm{H}_3 \mathrm{PO}_4 & \mathrm{H}_2 \mathrm{SO}_4 & \mathrm{H}_3 \mathrm{PO}_3 \\ \mathrm{H}_2 \mathrm{CO}_3 & \mathrm{H}_2 \mathrm{~S}_2 \mathrm{O}_7 & \mathrm{H}_3 \mathrm{BO}_3 \\ \mathrm{H}_3 \mathrm{PO}_2 & \mathrm{H}_2 \mathrm{CrO}_4 & \mathrm{H}_2 \mathrm{SO}_3\end{array}\)

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)

Paragraph :
In the figure a container is shown to have a movable (without friction) piston on top. The container and the piston are all made of perfectly insulating material allowing no heat transfer between outside and inside the container. The container is divided into two compartments by a rigid partition made of a thermally conducting material that allows slow transfer of heat. The lower compartment of the container is filled with \(2\) moles of an ideal monatomic gas at \(700 K\) and the upper compartment is filled with \(2\) moles of an ideal diatomic gas at \(400 K\). The heat capacities per mole of an ideal monatomic gas are \(C_{V}=\frac{3}{2} R, C_{P}=\frac{5}{2} R\), and those for an ideal diatomic gas are \(C_{V}=\frac{5}{2} R, C_{P}=\frac{7}{2} R\).


Question :
Consider the partition to be rigidly fixed so that it does not move. When equilibrium is achieved, the final temperature of the gases will be

Let $f, g :\left[-1, 2\right]\to \mathbb{ }\mathbb{R}$   be continuous functions which are twice differentiable on the interval $\left(-1, 2\right)$. Let the values of $f$ and $g$ at the points $-1, 0$ and $2$ be as given in the following table:
 

	$x= -1$	$x = 0$	$x = 2$
$f\left(x\right)$	3	6	0
$g\left(x\right)$	0	1	-1

In each of the intervals $\left(-1, 0\right)$ and $\left(0, 2\right)$ the function $(f-3g)"$ never vanishes. Then the correct statement(s) is(are)

A small object of uniform density rolls up a curved surface with an initial velocity \(v\). It reaches up to a maximum height of \(\frac{3 v^2}{4 g}\) with respect to the initial\(
\text { position. The object is }
\)

Paragraph:
Scientists are working hard to develop nuclear fusion reactor. Nuclei of heavy hydrogen, \({ }_1^2 \mathrm{H}\) known as deuteron and denoted by \(D\) can be thought of as a candidate for fusion reactor. The \(D\) - \(D\) reaction is \({ }_1^2 \mathrm{H}+{ }_1^2 \mathrm{H} \rightarrow{ }_2^3 \mathrm{He}+n+\) energy. In the core of fusion reactor. A gas of heavy hydrogen is fully ionized into deuteron nuclei and electrons. This collection of \({ }_1^4 \mathrm{H}\) nuclei and electrons is known as plasma. The nuclei move randomly in the reactor core and occasionally come close enough for nuclear fusion to take place. Usually, the temperatures in the reactor core are too high and no material wall can be used to confine the plasma. Special techniques are used which confine the plasma for a time \(t_0\) before the particles fly away from the core. If n is the density (number/volume) of deutrons, the product t \(_0\) is called Lawson number. In one of the criteria, \(a\) reactor is termed successful if Lawson number is greater than \(5 \times 10^{14} \mathrm{scm}^{-3}\).
It may be helpful to use the following : Boltzmann constant \(k=8.6 \times 10^{-5} \mathrm{eV} / \mathrm{K}\); \(\frac{e^2}{4 \pi \varepsilon_0}=1.44 \times 10^{-9} \mathrm{eVm}\)
Question:
Assume that two deuteron nuclei in the core of fusion reactor at temperature \(T\) are moving towards each other, each with kinetic energy \(1.5 \mathrm{kT}\), when the separation between them is large enough to neglect Coulomb potential energy. Also neglect any interaction from other particles in the core. The minimum temperature \(T\) required for them to reach a separation of \(4 \times 10^{-15} \mathrm{~m}\) is in the range

\(\mathrm{NiCl}_{2}\left\{\mathrm{P}\left(\mathrm{C}_{2} \mathrm{H}_{5}\right)_{2}\left(\mathrm{C}_{6} \mathrm{H}_{5}\right)\right\}_{2}\) exhibits temperature depend-ent magnetic behaviour (paramagnetic/diamagnetic). The coordination geometries of \(\mathrm{Ni}^{2+}\) in the paramagnetic and diamagnetic states are respectively

In a conductometric titration, small volume of titrant of higher concentration is added stepwise to a larger volume of titrate of much lower concentration, and the conductance is measured after each addition.
The limiting ionic conductivity \(\left(\Lambda_0\right)\) values (in \(\mathrm{mS} \mathrm{m}^2 \mathrm{~mol}^{-1}\)) for different ions in aqueous solutions are given below:

For different combinations of titrates and titrants given in List-I, the graphs of 'conductance' versus 'volume of titrant' are given in List-II.
Match each entry in List-I with the appropriate entry in List-II and choose the correct option.