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 _________.

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When a metal rod \(M\) is dipped into an aqueous colourless concentrated solution of compound \(\mathrm{N}\), the solution turns light blue. Addition of aqueous \(\mathrm{NaCl}\) to the blue solution gives a white precipitate \(\mathrm{O}\). Addition of aqueous \(\mathrm{NH}_3\) dissolves \(\mathrm{O}\) and gives an intense blue solution.
Question:
The final solution contains

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 \()\).

The correct statement(s) regarding the binary transition metal carbonyl compounds is (are) (Atomic numbers: $Fe = 26, Ni = 28$ )

Match each of the diatomic molecules in Column I with its property/properties in Column II.

Column I	Column II
A. \(B _2\)	p. Paramagnetic
B. \(N_2\)	q. Undergoes oxidation
C. \(O _2^{-}\)	r. Undergoes reduction
D. \(O _2\)	s. Bond order
	t. Mixing of 's' and 'p' orbitals

The volume (in \(\mathrm{mL}\) ) of \(0.1 \mathrm{M} \mathrm{AgNO}_3\) required for complete precipitation of chloride ions present in \(30 \mathrm{~mL}\) of \(0.01 \mathrm{M}\) solution of \(\left[\mathrm{Cr}\left(\mathrm{H}_2 \mathrm{O}_{)_5} \mathrm{Cl}\right] \mathrm{Cl}_2\right.\), as silver chloride is close to

For the process \(\mathrm{H}_2 \mathrm{O}(\mathrm{l})\) (1 bar, \(\left.273 \mathrm{~K}\right) \longrightarrow \mathrm{H}_2 \mathrm{O}(\mathrm{g})\) (1 bar, \(\left.373 \mathrm{~K}\right)\), the correct set of thermodynamic parameters is

Let $S$ be the set of all complex numbers $z$ satisfying $\left|z^2+z+1\right|=1$. Then which of the following statements is/are TRUE?

Let \(a_0, a_1, \ldots, a_{23}\) be real numbers such that
\(\left(1+\frac{2}{5} x\right)^{23}=\sum_{i=0}^{23} a_i x^i\)
for every real number \(x\). let \(a_r\) be the largest among the numbers \(a_j\) for \(0 \leq j \leq 23\).
The the value of \(r\) is ______

Consider a disc rotating in the horizontal plane with a constant angular speed \(\omega\) about its centre \(\mathrm{O}\). The disc has a shaded region on one side of the diameter and an unshaded region on the other side as shown in the figure. When the disc is in the orientation as shown, two pebbles \(P\) and \(Q\) are simultaneously projected at an angle towards \(R\). The velocity of projection is in the \(y\) - \(z\) plane and is same for both pebbles with respect to the disc. Assume that (i) they land back on the disc before the disc has completed \(1 / 8\) rotation, (ii) their range is less than half the disc radius, and (iii) \(\omega\) remains constant throughout. Then

Let $f :R\to R$ and $g :R\to R$ be respectively given by $f\left(x\right)=\left|x\right|+1$ and $g\left(x\right)=x^2+1.$ Define $h :R\to R$ by $h\left(x\right)= \left\{\begin{array}{ccc}\max \left\{f\left(x\right), g\left(x\right)\right\}& if& x \le 0\\ \min \left\{f\left(x\right), g\left(x\right)\right\}& if& x>0\end{array}\right.$
Then number of points at which $h\left(x\right)$ is not differentiable is_________

A $10\mathrm{}\mathrm{c}\mathrm{m}$ long perfectly conducting wire $\mathrm{P}\mathrm{Q}$ is moving, with a velocity $1\mathrm{}\mathrm{c}\mathrm{m}/\mathrm{s}$ on a pair of horizontal rails of zero resistance. One side of the rails is connected to an inductor $\mathrm{L}=1\mathrm{}\mathrm{m}\mathrm{H}$ and a resistance $\mathrm{R}=1\mathrm{\Omega }$ as shown in figure. The horizontal rails, $\mathrm{L}$ and $\mathrm{R}$ lie in the same plane with a uniform magnetic field $\mathrm{B}=1\mathrm{}\mathrm{T}$ perpendicular to the plane. If the key $\mathrm{S}$ is closed at certain instant, the current in the circuit after $1$ milli second is $\mathrm{x}\times {10}^{-3}\mathrm{A},$ where the value of $\mathrm{x}$ is_______.
[Assume the velocity of wire $\mathrm{P}\mathrm{Q}$ remains constant $\left(1\mathrm{c}\mathrm{m}/\mathrm{s}\right)$ after key $\mathrm{S}$ is closed. Given: ${\mathrm{e}}^{-1}=0.37,$ where $\mathrm{e}$ is base of the natural logarithm]

Complete reaction of acetaldehyde with excess formaldehyde, upon heating with conc. \(\mathrm{NaOH}\) solution, gives \(\mathbf{P}\) and \(\mathbf{Q}\). Compound \(\mathbf{P}\) does not give Tollens' test, whereas \(\mathbf{Q}\) on acidification gives positive Tollens' test. Treatment of \(\mathbf{P}\) with excess cyclohexanone in the presence of catalytic amount of \(p\)-toluenesulfonic acid (PTSA) gives product \(\mathbf{R}\).
Sum of the number of methylene groups \(\left(-\mathrm{CH}_2-\right)\) and oxygen atoms in \(\mathbf{R}\) is ________.