d
\(\begin{array}{l}
\,Here,\,\bar R = 4\sin \left( {2\pi t} \right)\hat i + 4\cos \left( {2\pi t} \right)\hat j\\
The\,velocity\,of\,the\,particle\,is\\
\bar v = \frac{{d\bar r}}{{dt}} = \frac{d}{{dt}}\left[ {4\sin \left( {2\pi t} \right)\hat i + 4\cos \left( {2\pi t} \right)\hat j} \right]\\
 = \,8\pi \cos \left( {2\pi t} \right)\hat i - 8\pi \sin \left( {2\pi t} \right)\hat j\\
Its\,magnitude\,is
\end{array}\)

\(\begin{array}{l}
\left| {\bar v} \right| = \sqrt {{{\left( {8\pi \cos \left( {2\pi t} \right)} \right)}^2} + {{\left( { - 8\pi \sin \left( {2\pi t} \right)} \right)}^2}} \\
\,\,\,\,\,\, = \sqrt {64{\pi ^2}{{\cos }^2}\left( {2\pi t} \right)64{\pi ^2}{{\sin }^2}\left( {2\pi t} \right)} \\
\,\,\,\,\,\, = \sqrt {64{\pi ^2}\left[ {{{\cos }^2}\left( {2\pi t} \right) + {{\sin }^2}\left( {2\pi t} \right)} \right]} \\
\,\,\,\,\,\, = \sqrt {64{\pi ^2}} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {as\,{{\sin }^2}\theta  + {{\cos }^2}\theta  = 1} \right)\\
\,\,\,\,\,\, = 8\pi \,m/s
\end{array}\)