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

$$\begin{gathered} f\left(\frac{5x-3y}{2}\right)=\frac{5f\left(x\right)-3f\left(y\right)}{2} \\ \Rightarrow f\left(\frac{5x-3y}{5-3}\right)=\frac{5f\left(x\right)-3f\left(y\right)}{5-3} \end{gathered}$$

so, f(x) must be a linear function
$f\left(x\right)=ax+b\Rightarrow f\left(x\right)=2x+3$
period of  $\sin (2x+3)is\pi$