The reagent(s) for the following conversion,

is/are

The total number of compounds having at least one bridging oxo group among the molecules given below is
\(\text N _2\text O _5,\)\(\text P _4\text O _6,\)\(\text P _4\text O _8,\)\(\text H _4\text P _2\text O _5,\)\(\text H _5\text P _3\text O _{10},\) \(\text H _2\text S_2\text O _3,\text H _2\text S_2\text O _5\)

Using the data provided, calculate the multiple bond energy \(\left(\mathrm{kJ} \mathrm{mol}^{-1}\right)\) of a \(\mathrm{C} \equiv \mathrm{C}\) bond in \(\mathrm{C}_{2} \mathrm{H}_{2}\). That energy is (take the bond energy of a \(\mathrm{C}-\mathrm{H}\) bond as \(350 \mathrm{~kJ} \mathrm{~mol}^{-1}\) ) \(2 \mathrm{C}(\mathrm{s})+\mathrm{H}_{2}(\mathrm{~g}) \longrightarrow \mathrm{HC}=\mathrm{CH}(\mathrm{g}) ; \Delta \mathrm{H}=\) \(225 \mathrm{~kJ} \mathrm{~mol}^{-1}\) \(2 \mathrm{C}(\mathrm{s}) \longrightarrow 2 \mathrm{C}(\mathrm{g}) ; \Delta \mathrm{H}=1410 \mathrm{~kJ} \mathrm{~mol}^{-1}\) \(\mathrm{H}_{2}(\mathrm{~g}) \longrightarrow 2 \mathrm{H}(\mathrm{g}) ; \Delta \mathrm{H}=330 \mathrm{~kJ} \mathrm{~mol}^{-1}\)

If the distribution of molecular speeds of a gas is as per the figure shown below, then the ratio of the most probable, the average, and the root mean square speeds, respectively, is

Assuming $2\mathrm{s}\mathrm{ }–\mathrm{ }2\mathrm{p}$ mixing is NOT operative, the paramagnetic species among the following is

For the given close packed structure of a salt made of cation $\mathrm{X}$ and anion $\mathrm{Y}$ shown below (ions of only one face are shown for clarity), the packing fraction is approximately (packing fraction $=\frac{\mathrm{packing} \mathrm{efficiency}}{100}$)

The product formed in the reaction of $\mathrm{S}\mathrm{O}\mathrm{C}{\mathrm{l}}_2$ with white phosphorous is

The correct statement(s) about $\mathrm{C}{\mathrm{r}}^{2+}$ and $\mathrm{M}{\mathrm{n}}^{3+}$ is (are)
[Atomic number of Cr = 24 and Mn = 25]

Paragraph:

Football teams \(T_{1}\) and \(T_{2}\) have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probabilities of \(T_{1}\) winning, drawing and losing a game against \(T_{2}\) are \(\frac{1}{2}, \frac{1}{6}\) and \(\frac{1}{3}\), respectively. Each team gets \(3\) points for a win, \(1\) point for a draw and \(0\) point for a loss in a game. Let \(X\) and \(Y\) denote the total points scored by teams \(T_{1}\) and \(T_{2}\), respectively, after two games.


Question:

\(P(X>Y)\) is

One end of a spring of negligible unstretched length and spring constant $k$ is fixed at the origin $(0,0)$. A point particle of mass $m$ carrying a positive charge $q$ is attached at its other end. The entire system is kept on a smooth horizontal surface. When a point dipole $\overrightarrow{p}$ pointing towards the charge $q$ is fixed at the origin, the spring gets stretched to a length $l$ and attains a new equilibrium position (see figure below). If the point mass is now displaced slightly by $\mathrm{\Delta }l\ll l$ from its equilibrium position and released, it is found to oscillate at frequency $\frac{1}{\delta }\sqrt{\frac{k}{m}},$ The value of $\delta$ is _______.

Paragraph :
When liquid medicine of density \(\rho\) is to be put in the eye, it is done with the help of a dropper. As the bulb on the top of the dropper is pressed, a drop forms at the opening of the dropper. We wish to estimate the size of the drop.
We first assume that the drop formed at the opening is spherical because that requires a minimum increase in its surface energy. To determine the size, we calculate the net vertical force due to the surface tension \(T\) when the radius of the drop is \(R\). When this force becomes smaller than the weight of the drop, the drop gets detached from the dropper.
Question :
If the radius of the opening of the dropper is \(r\), the vertical force due to the surface tension on the drop of radius \(R\) (assuming \(r< < R\) ) is

For a reaction, $A\rightleftharpoons P$ , the plots of $\left[A\right]\mathrm{a}\mathrm{n}\mathrm{d}\left[P\right]$ with time at temperatures $T_1\mathrm{a}\mathrm{n}\mathrm{d}{T}_2$ are given below.

If $T_2>T_1$ , the correct statement(s) is (are)
(Assume $∆H^{⊝}\mathrm{a}\mathrm{n}\mathrm{d}∆S^{⊝}$ are independent of temperature and ratio of $\ln K\mathrm{a}\mathrm{t}{T}_1\mathrm{t}\mathrm{o}\ln K\mathrm{a}\mathrm{t}{T}_2$ is greater than $T_2/T_1$ . Here H, S , G and K are enthalpy, entropy, Gibbs energy and equilibrium constant, respectively.)

A curve passes through the point $\left(1,\frac{\pi }{6}\right)$. Let the slope of the curve at each point $\left(\text{x},\text{y}\right)$ be $\frac{\text{y}}{\text{x}}+\text{sec}\left(\frac{\text{y}}{\text{x}}\right),\; x > 0$. Then the equation of the curve is