If the freezing point of a 0.01 molal aqueous solution of a cobalt (III) chloride-ammonia complex (which behaves as a strong electrolyte) is $-{0.0558}^{\mathrm{o}}\mathrm{C}$ , the number of chloride(s) in the coordination sphere of the complex is
$[{\mathrm{K}}_{\mathrm{f}}\mathrm{ }\mathrm{o}\mathrm{f}\mathrm{ }\mathrm{w}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}=1.86\mathrm{ }\mathrm{K}\mathrm{ }\mathrm{k}\mathrm{g}\mathrm{ }\mathrm{m}\mathrm{o}{\mathrm{l}}^{-1}]$

Match the following

An aqueous solution of X is added slowly to an aqueous solution of Y as shown in List I. The variation in conductivity of these reactions is given in List II. Match List I with List II and select the correct using answer the code given below the lists :

List - I	List - II
a. \(\underset{\quad\quad~~ X }{( C _2 H _5)_3 N}+\underset{ Y }{CH _3 COOH}\)	p. Conductivity decreases and then increases
b. \(\underset{\quad\quad~~ X }{KI (0.1 M )}+\underset{ Y }{AgNO _3(0.01 M )}\)	q. Conductivity decreases and then does not change much
c. \(\underset{\text X }{CH _3 COOH}+\underset{\text Y }{KOH}\)	r. Conductivity increases and then does not change much
d. \(\underset{\text X }{NaOH}+\underset{\text Y }{HI}\)	s. Conductivity does not change much and then increases

For the following compounds the correct statement(s) with respect to nucleophilic substitution reactions is(are)

The compound(s) which react(s) with ${\mathrm{NH}}_3$ to give boron nitride $\left(\mathrm{BN}\right)$ is(are)

Consider the following cell reaction,
\(2 \mathrm{Fe}(\mathrm{s})+\mathrm{O}_2(g)+4 \mathrm{H}^{+}(a q) \longrightarrow \)
\( 2 \mathrm{Fe}^{2+}(a q)+2 \mathrm{H}_2 \mathrm{O}(l), E^{\circ}=1.67 \mathrm{~V} \)
\( \text {At } {\left[\mathrm{Fe}^{2+}\right]=10^{-3} \mathrm{M}, \mathrm{P}\left(\mathrm{O}_2\right)=0.1 \mathrm{~atm}}\)
and \(\mathrm{pH}=3\), the cell potential at \(25^{\circ} \mathrm{C}\) is

Match the compounds in LIST-$\mathrm{I}$ with the observations in LIST-$\mathrm{II}$, and choose the correct option.

	LIST-$\mathrm{I}$		LIST-$\mathrm{II}$
$\left(\mathrm{I}\right)$	Aniline	$\left(\mathrm{P}\right)$	Sodium fusion extract of the compound on boiling with ${\mathrm{FeSO}}_4$, followed by acidification with conc. ${\mathrm{H}}_2{\mathrm{SO}}_4$, gives Prussian blue color.
$\left(\mathrm{II}\right)$	$\mathrm{o}-$Cresol	$\left(\mathrm{Q}\right)$	Sodium fusion extract of the compound on treatment with sodium nitroprusside gives blood red color.
$\left(\mathrm{III}\right)$	Cysteine	$\left(\mathrm{R}\right)$	Addition of the compound to a saturated solution of ${\mathrm{NaHCO}}_3$ results in effervescence.
$\left(\mathrm{IV}\right)$	Caprolactam	$\left(\mathrm{S}\right)$	The compound reacts with bromine water to give a white precipitate.
		$\left(\mathrm{T}\right)$	Treating the compound with neutral ${\mathrm{FeCl}}_3$ solution produces violet color.

Paragraph:
Let \(V_r\) denotes the sum of the first \(r\) terms of an arithmetic progression \((A P)\) whose first term is \(r\) and the common difference is \((2 r-1)\). Let \(T_r=V_{r+1}-V_r-2\) and \(Q_r=T_{r+1}-T_r\) for \(r=1,2, \ldots\)
Question:
The sum \(V_1+V_2+\ldots+V_n\) is

Electrons with de-Broglie wavelength \(\lambda\) fall on the target in an X-ray tube. The cut-off wavelength of the emitted X-rays is

Consider $4$ boxes, where each box contains $3$ red balls and $2$ blue balls. Assume that all $20$ balls are distinct. In how many different ways can $10$ balls be chosen from these $4$ boxes so that from each box at least one red ball and one blue ball are chosen?

Paragraph:
There are some deposits of nitrates and phosphates in earth's crust. Nitrates are more soluble in water. Nitrates are difficult to reduce under the laboratory conditions but microbes do it easily. Ammonia forms large number of complexes with transition metal ions. Hybridisation easily explains the ease of sigma donation capability of \(\mathrm{NH}_3\) and \(\mathrm{PH}_3\). Phosphine is a flammable gas and is prepared from white phosphorus.
Question:
Among the following, the correct statement is

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 \(350 \mathrm{~kJ} \mathrm{~mol}^{-1}\) ) \(2 \mathrm{C}(\mathrm{s})+\mathrm{H}_{2}(\mathrm{~g}) \longrightarrow \mathrm{HC}=\mathrm{CH}(\mathrm{g}) ; \Delta \mathrm{H}=\) \(225 \mathrm{~kJ} \mathrm{~mol}^{-1}\) \(2 \mathrm{C}(\mathrm{s}) \longrightarrow 2 \mathrm{C}(\mathrm{g}) ; \Delta \mathrm{H}=1410 \mathrm{~kJ} \mathrm{~mol}^{-1}\) \(\mathrm{H}_{2}(\mathrm{~g}) \longrightarrow 2 \mathrm{H}(\mathrm{g}) ; \Delta \mathrm{H}=330 \mathrm{~kJ} \mathrm{~mol}^{-1}\)

One Indian and four American men and their wives are to be seated randomly around a circular table. Then, the conditional probability that the Indian man is seated adjacent to his wife given that each American man is seated adjacent to his wife, is