(3)

Let the displacement of particle be $x$.

Given, Kinetic energy = Potential energy

$\frac{1}{2}K(A^2-x^2)=\frac{1}{2}K{x}^2$

or      $A^2-x^2=x^2$

or     $A^2=2x^2$

$\Rightarrow x=\frac{A}{\sqrt{2}}$

(3)

The equation for the S.H.M. is

$y=A\sin \omega t=A\sin \left(2\pi nt\right)$                                            ...(i)

where, $A$ is amplitude, $\omega$ is angular frequency and $y$ is displacement at time $t$.
The formula of potential energy is

$U=\frac{1}{2}k{A}^2{\sin }^2\omega t$

$U=\frac{1}{2}k{A}^2\left(\frac{{\displaystyle 1-\cos 2\omega t}}{{\displaystyle 2}}\right)=\frac{1}{2}k{A}^2\left[\frac{{\displaystyle 1-\cos 2\pi \left(2n\right)t}}{{\displaystyle 2}}\right]$                         ...(ii)

                                                                                   $(\because {\sin }^2\theta =\frac{1-\cos 2\theta }{2})$  

From eqn. (i) and (ii), we get

Frequency of potential energy = $2n$