Explanation

(b)
$\because\ {\cos{(315\ \pi + x) = ( - 1)}}^{315}\cos{x = - \cos x}$
$\therefore\ 4\cos^{3}{x - 4\cos^{2}{x - \cos{(315\ \pi + x) = 1}}}$
$\Rightarrow 4\cos^{3}{x - 4\cos^{2}{x + \cos{x - 1 = 0}}}$
$\Rightarrow (4\cos^{2}{x + 1)(\cos{x - 1) = 0}}$
$\Rightarrow \cos{x = 1,\ 4\cos^{2}{x + 1 \neq 0}}$
$\Rightarrow \cos{x = \cos 0}$
$\Rightarrow x = 2n\pi,\ n \in I$
$\therefore x = 2\pi,\ 4\pi,\ 6\pi,\ 8\pi,\ldots,100\pi\ \ (\because 0 < x < 315)$
$(ie,\ 100\pi < 315 < 101\pi)$
Required arithmetic mean
$= \frac{2\pi + \ 4\pi + \ 6\pi + \ 8\pi + ... + 100\pi}{50}$
$= \frac{2\pi(1 + 2 + 3 + 4 + ... + 50)}{50}$
$= \frac{2\pi \bullet \frac{50}{2} \bullet 51}{50} = 51\pi$