$\mathrm{If} \mathrm{f}\left(\mathrm{x}\right) \mathrm{satisfies} \mathrm{the} \mathrm{relation} \mathrm{f}\left(\frac{5\mathrm{x}-3\mathrm{y}}{2}\right)=\frac{5\mathrm{f}\left(\mathrm{x}\right)-3\mathrm{f}\left(\mathrm{y}\right)}{2}\forall \mathrm{x},\mathrm{y}\in \mathrm{R}, \mathrm{and} \mathrm{f}\left(0\right)=3 \mathrm{and} \mathrm{f}\text{'}\left(0\right)=2,$ then the period of sin(f(x)) is

$\begin{array}{l}\mathrm{Given} \mathrm{f}\left(\frac{5\mathrm{x}-3\mathrm{y}}{2}\right)=\frac{5\mathrm{f}\left(\mathrm{x}\right)-3\mathrm{f}\left(\mathrm{y}\right)}{2}\\ \mathrm{or} \mathrm{f}\left(\frac{5\mathrm{x}-3\mathrm{y}}{5-3}\right)=\frac{5\mathrm{f}\left(\mathrm{x}\right)-3\mathrm{f}\left(\mathrm{y}\right)}{5-3}\end{array}$

which satisfies section formula for abscissa on L.H.S. and for ordinate on R.H.S. Hence, f(x) must be the linear function 

(as only straight line satisfies such section formula).

$\begin{array}{l}\mathrm{Hence}, \mathrm{f}\left(\mathrm{x}\right)=\mathrm{ax}+\mathrm{b}\\ \mathrm{But} \mathrm{f}\left(0\right)=3 \mathrm{or} \mathrm{b}=3, \mathrm{f}\text{'}\left(0\right)=2 \mathrm{or} \mathrm{a}=2.\\ \mathrm{Thus}, \mathrm{f}\left(\mathrm{x}\right)=2\mathrm{x}+3 \mathrm{or} \mathrm{Period} \mathrm{of} \sin \left(\mathrm{f}\left(\mathrm{x}\right)\right)=\sin \left(2\mathrm{x}+3\right) \mathrm{is} \mathrm{\pi }.\end{array}$