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 $=\frac{\mathrm{packing} \mathrm{efficiency}}{100}$)

A closed tank has two compartments A and B, both filled with oxygen (assumed to be ideal gas). The partition separating the two compartments is fixed and is a perfect heat insulator (Figure$1$). If the old partition is replaced by a new partition which can slide and conduct heat but does not allow the gas to leak across (Figure $2$), the volume (in ${\mathrm{m}}^3$) of the compartment A after the system attains equilibrium is _________.

Among the following, the correct statement(s) is(are)

Paragraph I: An organic compound \(\mathrm{P}\) with molecular formula \(\mathrm{C}_5 \mathrm{H}_{18} \mathrm{O}_2\) decolorizes bromine water and also shows positive iodoform test. \(\mathrm{P}\) on ozonolysis followed by treatment with \(\mathrm{H}_2 \mathrm{O}_2\) gives \(\mathrm{Q}\) and \(\mathrm{R}\). While compound \(\mathrm{Q}\) shows positive iodoform test, compound \(\mathrm{R}\) does not give positive iodoform test. \(\mathrm{Q}\) and \(\mathrm{R}\) on oxidation with pyridinium chlorochromate (PCC) followed by heating give \(\mathrm{S}\) and \(\mathrm{T}\), respectively. Both \(\mathrm{S}\) and \(\mathrm{T}\) show positive iodoform test.
Complete copolymerization of 500 moles of \(\mathrm{Q}\) and 500 moles of \(\mathrm{R}\) gives one mole of a single acyclic copolymer \(\mathrm{U}\).
[Given, atomic mass: \(\mathrm{H}=1, \mathrm{C}=12, \mathrm{O}=16\)]
Question: The molecular weight of \(\mathbf{U}\) is

The unbalanced chemical reactions given in List I show missing reagent or condition (?) which are provided in list II. match List I with List II and select the correct answer using the code given below the lists :

List - I	List - II
A. \(PbO _2+ H _2 SO _4 \xrightarrow{?}\) \(PbSO _4 + O _2+\) other product	P. \(\text{NO}\)
B. \(Na _2 S_2 O _3+ H _2 O \xrightarrow{?}\) \(NaHSO _4~+\) other product	Q. \(\text{I}_2\)
C. \(N _2 H _4 \xrightarrow{?} N_2 +\) other product	R. Warm
D. \(XeF _2 \xrightarrow{?} Xe ~+\) other product	S. \(\text{Cl}_2\)

For the reaction sequence given below, the correct statement(s) is(are)

(In the options, \(X\) is any atom other than carbon and hydrogen, and it is different in \(\mathbf{P}, \mathbf{Q}\) and \(\mathbf{R}\) )

Statement I Alkali metals dissolve in liquid ammonia to give blue solutions. Statement II Alkali metals in liquid ammonia give solvated species of the type \(\left(M\left(\mathrm{NH}_3\right)_n\right]^{+}(M=\) alkali metals \()\).

In a metal deficient oxide sample, \(\mathbf{M}_{\mathbf{x}} \mathbf{Y}_2 \mathbf{O}_4\) ( \(\mathbf{M}\) and \(\mathbf{Y}\) are metals), \(\mathbf{M}\) is present in both +2 and +3 oxidation states and \(\mathbf{Y}\) is in +3 oxidation state. If the fraction of \(\mathbf{M}^{2+}\) ions present in \(\mathbf{M}\) is \(\frac{1}{3}\), the value of \(\mathbf{X}\) is ________.

Let $-\frac{\pi }{6}<\theta <-\frac{\pi }{12}$ . Suppose ${\alpha }_1$ and ${\beta }_1$ are the roots of the equation $x^2-2x\sec \theta +1=0$ and ${\alpha }_2$ and ${\beta }_2$ are the roots of the equation $x^2+2x\tan \theta -1=0.$ If ${\alpha }_1>{\beta }_1$ and ${\alpha }_2>{\beta }_2$ , then ${\alpha }_1+{\beta }_2$ equals

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) ${\mathrm{L}}_3$ ?

Amongst the following, the total number of compounds whose aqueous solution turns red litmus paper blue is
\(\begin{array}{llll}
\mathrm{KCN} & \mathrm{K}_2 \mathrm{SO}_4 & \left(\mathrm{NH}_4\right)_2 \mathrm{C}_2 \mathrm{O}_4 & \mathrm{NaCl} \\
\mathrm{Zn}\left(\mathrm{NO}_3\right)_2 & \mathrm{FeCl}_3 & \mathrm{~K}_2 \mathrm{CO}_3 & \mathrm{NH}_4 \mathrm{NO}_3 \\
\mathrm{LiCN} & & &
\end{array}\)

Paragraph:

A box \(B_{1}\) contains \(1\) white ball, \(3\) red balls and \(2\) black balls. Another box \(B_{2}\) contains \(2\) white balls, \(3\) red balls and \(4\) black balls. A third box \(B_{3}\) contains \(3\) white balls, \(4\) red balls and \(5\) black balls.


Question:

If \(1\) ball is drawn from each of the boxes \(B_{1}, B_{2}\) and \(B_{3}\), the probability that all \(3\) drawn balls are of the same colour is

The number of seven digit integers, with sum of the digits equal to 10 and formed by using the digits 1,2 and 3 only, is