Given below are two statements : Statement I : The number of...

Statement I: Number of compounds in which sulphur does not follow the octet rule (from $SO_2$, $SO_3$, $SF_4$, $SF_6$ and $H_2S$).

$SO_2$: The formal-charge-minimised structure $O=S=O$ places $10$ electrons around sulphur (expanded octet).
$SO_3$: With three $S=O$ double bonds, sulphur has $12$ electrons around it (expanded octet).
$SF_4$: $4$ bond pairs $+ 1$ lone pair $= 10$ electrons around $S$ (expanded octet).
$SF_6$: $6$ bond pairs $= 12$ electrons around $S$ (expanded octet).
$H_2S$: $2$ bond pairs $+ 2$ lone pairs $= 8$ electrons around $S$ (obeys the octet rule).

Depending on whether formal-charge-minimised structures are used for $SO_2$ and $SO_3$, either $2$ ($SF_4, SF_6$) or $4$ ($SO_2, SO_3, SF_4, SF_6$) species violate the octet rule, but the number is never $3$.
Hence, Statement I is false.

Statement II: Among the given sets, exactly one contains only molecules having one lone pair on the central atom.

Number of lone pairs on the central atom, $\text{LP} = \dfrac{\text{valence electrons of central atom} - \text{number of bonded monovalent atoms}}{2}$:

$H_2O$: $\dfrac{6 - 2}{2} = 2$ lone pairs
$ClF_3$: $\dfrac{7 - 3}{2} = 2$ lone pairs
$SF_4$: $\dfrac{6 - 4}{2} = 1$ lone pair
$NH_3$: $\dfrac{5 - 3}{2} = 1$ lone pair
$BrF_5$: $\dfrac{7 - 5}{2} = 1$ lone pair
$XeF_4$: $\dfrac{8 - 4}{2} = 2$ lone pairs

Checking the sets:
$\{H_2O\ (2),\ ClF_3\ (2),\ SF_4\ (1)\}$: not all have $1$ lone pair.
$\{NH_3\ (1),\ BrF_5\ (1),\ SF_4\ (1)\}$: all have exactly $1$ lone pair.
$\{BrF_5\ (1),\ ClF_3\ (2),\ XeF_4\ (2)\}$: not all have $1$ lone pair.
$\{XeF_4\ (2),\ ClF_3\ (2),\ H_2O\ (2)\}$: not all have $1$ lone pair.

Exactly one set satisfies the condition, so Statement II is true.

Hence, Statement I is false but Statement II is true, which corresponds to option $(4)$.