Pls solve these three questions

Dear Student
50

$$\begin{gathered} Let particle moving in x direction with velocity V = Initial Velocity vector \\ Now its direction changes by \theta but magnitude will be same as V only = Final velocity vector \\ Also direction between initial and final velocity will be 180-\theta \\ So \\ \left(a\right) \\ Magnitude of change in velocity =\left|\overrightarrow{v_f}-\overrightarrow{v_i}\right| \\ \left|\overrightarrow{v_f}-\overrightarrow{v_i}\right| =\sqrt{V^2+V^2+2\times V\times V\times \cos \left(180-\theta \right)} \\ \left|\overrightarrow{v_f}-\overrightarrow{v_i}\right| =\sqrt{2V^2-2V^2 \cos \left(\theta \right)} \\ \left|\overrightarrow{v_f}-\overrightarrow{v_i}\right| =\sqrt{2V^2(1-\cos \theta )} \\ Now, (1-\cos \theta ) = 2{\sin }^2\frac{\theta }{2} \\ \left|\overrightarrow{v_f}-\overrightarrow{v_i}\right| =\sqrt{2V^2\times 2{\sin }^2\frac{\theta }{2} } =2V\sin \frac{\theta }{2} \\ b) \\ Change in magnitude of velocity \\ \left|\overrightarrow{v_f}\right|-\left|\overrightarrow{v_i}\right| = V-V=0 \end{gathered}$$
Please post other questions in different thread.

Regards