$\left(\mathrm{a}\right) \mathrm{The} \mathrm{speed} \mathrm{of} \mathrm{the} \mathrm{sound} \mathrm{is} \mathrm{given} \mathrm{by}:$
$\mathrm{v}=\sqrt{\frac{\mathrm{\gamma P}}{\mathrm{\rho } }} ...\left(\mathrm{i}\right)$
$\mathrm{where} \mathrm{Density}, \mathrm{\rho }=\frac{\mathrm{Mass}}{\mathrm{Volume}}=\frac{\mathrm{M}}{\mathrm{V}}$
$\mathrm{M}=\mathrm{Molecular} \mathrm{weight} \mathrm{of} \mathrm{the} \mathrm{gas}$
$\mathrm{V}=\mathrm{Volume} \mathrm{of} \mathrm{the} \mathrm{gas}$
$\mathrm{Hence},$
$\mathrm{v}=\sqrt{\frac{\mathrm{\gamma PV}}{\mathrm{M}}} ...\left(\mathrm{ii}\right)$
$\mathrm{Now} \mathrm{from} \mathrm{the} \mathrm{ideal} \mathrm{gas} \mathrm{equation} \mathrm{for} \mathrm{n}=1-$
$\mathrm{PV} = \mathrm{RT}$
$\mathrm{For} \mathrm{constant} \mathrm{temperature}, \mathrm{PV} = \mathrm{Constant}$
$\mathrm{Since} \mathrm{both} \mathrm{M} \mathrm{and} \mathrm{\gamma } \mathrm{are} \mathrm{constants}, \mathrm{v} = \mathrm{Constant}$
$\mathrm{Hence}, \mathrm{at} \mathrm{a} \mathrm{constant} \mathrm{temperature}, \mathrm{the} \mathrm{speed} \mathrm{of} \mathrm{sound} \mathrm{in} \mathrm{a}$
$\mathrm{gaseous} \mathrm{medium} \mathrm{is} \mathrm{independent} \mathrm{of} \mathrm{the} \mathrm{change} \mathrm{in} \mathrm{the}$
$\mathrm{pressure} \mathrm{of} \mathrm{the} \mathrm{gas}.$
$\left(\mathrm{c}\right)$
$\mathrm{v}=\sqrt{\frac{\mathrm{\gamma P}}{\mathrm{\rho }}} ...\left(\mathrm{i}\right)$
$\mathrm{For} \mathrm{one} \mathrm{mole} \mathrm{of} \mathrm{an} \mathrm{ideal} \mathrm{gas},$
$\mathrm{PV} = \mathrm{RT}$
$\mathrm{P} =\frac{\mathrm{RT}}{\mathrm{V}} ... \left(\mathrm{ii}\right)$
$\mathrm{Substituting} \mathrm{equation} \left(\mathrm{ii}\right) \mathrm{in} \mathrm{equation} \left(\mathrm{i}\right),$
$\mathrm{v}=\sqrt{\frac{\mathrm{\gamma RT}}{\mathrm{V\rho }}}=\mathrm{v}=\sqrt{\frac{\mathrm{\gamma RT}}{\mathrm{M}}} ...\left(\mathrm{iv}\right)$
$\mathrm{Where}, \mathrm{Mass}, \mathrm{M} = \mathrm{\rho V} = \mathrm{constant}$
$\mathrm{\gamma } \mathrm{and} \mathrm{R} \mathrm{are} \mathrm{also} \mathrm{constants}.$
$\mathrm{So} \mathrm{v}\propto \sqrt{\mathrm{T}}.$

Let  ${\mathrm{v}}_{\mathrm{m}} \mathrm{and} {\mathrm{v}}_d$ be the speeds of sound in moist air and dry air respectively.

Let ${\mathrm{\rho }}_{\mathrm{m}} \mathrm{and} {\mathrm{\rho }}_{\mathrm{d}}$ be the densities of moist air and dry air respectively.

So,

$\mathrm{v}=\sqrt{\frac{\mathrm{\gamma P}}{\mathrm{\rho }}}$
$\mathrm{Hence}, \mathrm{the} \mathrm{speed} \mathrm{of} \mathrm{sound} \mathrm{in} \mathrm{moist} \mathrm{air}:$
${\mathrm{v}}_{\mathrm{m}}=\sqrt{\frac{\mathrm{\gamma P}}{{\mathrm{\rho }}_{\mathrm{m}}}} ...\left(\mathrm{i}\right)$
$\mathrm{And} \mathrm{the} \mathrm{speed} \mathrm{of} \mathrm{sound} \mathrm{in} \mathrm{dry} \mathrm{air}:$
${\mathrm{v}}_{\mathrm{d}}=\sqrt{\frac{\mathrm{\gamma P}}{{\mathrm{\rho }}_{\mathrm{d}}}} ...\left(\mathrm{ii}\right)$
$\mathrm{On} \mathrm{dividing} \mathrm{equations} \left(\mathrm{i}\right) \mathrm{and} \left(\mathrm{ii}\right),$
$\frac{{\mathrm{v}}_{\mathrm{m}}}{{\mathrm{v}}_{\mathrm{d}}}=\sqrt{\frac{\mathrm{\gamma P}}{{\mathrm{\rho }}_{\mathrm{m}}}\times \frac{{\mathrm{\rho }}_{\mathrm{d}}}{\mathrm{\gamma \rho }}}=\sqrt{\frac{{\mathrm{\rho }}_{\mathrm{d}}}{{\mathrm{\rho }}_{\mathrm{m}}}}$
$\mathrm{However}, \mathrm{the} \mathrm{presence} \mathrm{of} \mathrm{water} \mathrm{vapour} \mathrm{reduces} \mathrm{the} \mathrm{density} \mathrm{of} \mathrm{air}, \mathrm{i}.\mathrm{e}., {\mathrm{\rho }}_{\mathrm{d}}<{\mathrm{\rho }}_{\mathrm{m}.}$
$\Rightarrow {\mathrm{v}}_{\mathrm{m}}>{\mathrm{v}}_{\mathrm{d}}$