Columns 1, 2 and 3 contain starting materials, reaction conditions and type of reactions respectively.

Column 1	Column 2	Column 3
(I) Toulene	(i) \(NaOH\) \( /Br _2\)	(P) Condensation
(II) Acetophenone	(ii) \(B r_2 / h v\)	(Q) Carboxylation
(III) Benzaldehyde	(iii) \(\left( CH _3 CO \right)_2\) \(O / CH _3\) \(COOK\)	(R) Substitution
(IV) Phenol	(iv) \(NaOH\) \(/CO _2\)	(S) Haloform

The only CORRECT combination that gives two different carboxylic acids is

The compound (s) that exhibit (s) geometrical isomerism is (are)

The number of $-{\mathrm{CH}}_{2-}$ (methylene) groups in the product formed from the following reaction sequence is _____ .

The correct statements (s) about intermolecular forces is(are)

The ${\mathrm{K}}_{\mathrm{sp}}$ of ${\text{Ag}}_2{\text{CrO}}_4$ is $1.1\times 10^{-12}$ at $298 \mathrm{K}$. The solubility (in $\mathrm{mol} {\mathrm{L}}^{-1}$) of ${\text{Ag}}_2\mathrm{CrO}{}_4$ in a $0.1 \mathrm{M} {\text{AgNO}}_3$ solution is

When ${\mathrm{O}}_2$ is adsorbed on a metallic surface, electron transfer occurs from the metal to ${\mathrm{O}}_2$ . The TRUE statement(s) regarding this adsorption is(are)

Electrolysis of dilute aq. \(\mathrm{NaCl}\) solution was carried out by passing \(10 \mathrm{~mA}\) current. The time required to liberate \(0.01\) mole of \(\mathrm{H}_2\) gas at the cathode is (1 Faraday \(=96500 \mathrm{C} \mathrm{mol}^{-1}\) )

The concentration of \(R\) in the reaction \(R \longrightarrow P\) was measured as a function of time and the following data is obtained :
\(\begin{array}{lllll}{[R] \text { (molar) }} & 1.0 & 0.75 & 0.40 & 0.10 \\ t \text { (min) } & 0.0 & 0.05 & 0.12 & 0.18\end{array}\)
The order of the reaction is

Let ${\text{S}}_{\text{n}}=\underset{\text{k}=1}{\overset{4\text{n}}{\Sigma }}{\left(-1\right)}^{\frac{\text{k}\left(\text{k}+1\right)}{2}}{\text{k}}^2$. Then S_n can take value(s)

Two uniform strings of mass per unit length \(\mu\) and \(4 \mu\), and length \(L\) and \(2 L\), respectively, are joined at point \(\mathrm{O}\), and tied at two fixed ends \(\mathrm{P}\) and \(\mathrm{Q}\), as shown in the figure. The strings are under a uniform tension \(T\). If we define the frequency \(v_0=\frac{1}{2 L} \sqrt{\frac{T}{\mu}}\), which of the following statement(s) is(are) correct?

In a circuit, a metal filament lamp is connected in series with a capacitor of capacitance \(\mathrm{C} \mu F\) across a \(200 \mathrm{~V}, 50 \mathrm{~Hz}\) supply. The power consumed by the lamp is \(500 \mathrm{~W}\) while the voltage drop across it is \(100 \mathrm{~V}\). Assume that there is no inductive load in the circuit. Take \(r m s\) values of the voltages. The magnitude of the phaseangle (in degrees) between the current and the supply voltage is \(\varphi\). Assume, \(\pi \sqrt{3} \approx 5\).
The value of $\varphi$ is _____.

The standard state Gibb's free energies of formation of  $\mathrm{C}$ (graphite) and $\mathrm{C}$ (diamond) at $\mathrm{T}=298\mathrm{ }\mathrm{K}$ are
${\mathrm{\Delta }}_{\mathrm{f}}{\mathrm{G}}^{\mathrm{o}}\left[\mathrm{C} (\mathrm{graphite}\right]=0\mathrm{ }\mathrm{kJ}\mathrm{ }{\mathrm{mol}}^{-1}$
${\mathrm{\Delta }}_{\mathrm{f}}{\mathrm{G}}^{\mathrm{o}}\left[\mathrm{C} \left(\mathrm{diamond}\right)\right]=2.9\mathrm{ }\mathrm{kJ}\mathrm{ }{\mathrm{mol}}^{-1}$
The standard state means that the pressure should be 1 bar, and substance should be pure at a given temperature. The conversion of graphite [$\mathrm{C}$ (graphite)] to diamond [$\mathrm{C}$ (diamond)] reduces its volume by $2\times {10}^{-6}\mathrm{ }{\mathrm{m}}^3 {\mathrm{mol}}^{-1}$ . If $\mathrm{C}$ (graphite) is converted to $\mathrm{C}$ (diamond) isothermally at $\mathrm{T}=298 \mathrm{K}$, the pressure at which $\mathrm{C}$ (graphite) is in equilibrium with $\mathrm{C}$ (diamond), is
[Useful information:
\(1 J=1 kg m ^2 s^{-2}, 1 Pa=1 kg m ^{-1} s^{-2} ;\) \(1 bar =10^5 Pa]\)

Paragraph:
In hexagonal system of crystals, a frequently encountered arrangement of atoms is described as a hexagonal prism. Here, the top and bottom of the cell are regular hexagons and three atoms are sandwiched in between them. A space-filling model of this structure, called hexagonal close-packed (HCP), is constituted of a sphere on a flat surface surrounded in the same plane by six identical spheres as closely as possible. Three spheres are then placed over the first layer so that they touch each other and represent the second layer. Each one of the three spheres touches three spheres of the bottom layer. Finally, the second layer is covered with a third layer that is identical to the bottom layer in relative position. Assume radius of every sphere to be ' \(r\) '.
Question:
The number of atoms in this HCP unit cell is