Explanation

(b)
We have, $\mathrm{\Delta}(x) = \left| \begin{matrix}
x^{n} & \sin x & \cos x \\
n! & \sin\frac{n\pi}{2} & \cos\frac{n\pi}{2} \\
a & a^{2} & a^{3}
\end{matrix} \right|$
$\frac{d^{n}}{dx^{n}}\left\lbrack \mathrm{\Delta}(x) \right\rbrack = \left| \begin{matrix}
\frac{d^{n}}{dx^{n}}x^{n} & \frac{d^{n}}{dx^{n}}\sin x & \frac{d^{n}}{dx^{n}}\cos x \\
n! & \sin\left( \frac{n\pi}{2} \right) & \cos\left( \frac{n\pi}{2} \right) \\
a & a^{2} & a^{3}
\end{matrix} \right|$
$(\because Differentiation\ of\ R_{2\ }and\ R_{3}\ are\ zero)$
$\left| \begin{matrix}
n! & \sin\left( x + \frac{n\pi}{2} \right) & \cos\left( x + \frac{n\pi}{2} \right) \\
n! & \sin\left( \frac{n\pi}{2} \right) & \cos\left( \frac{n\pi}{2} \right) \\
a & a^{2} & a^{3}
\end{matrix} \right|$
$\Rightarrow \left\lbrack \mathrm{\Delta}^{n}(x) \right\rbrack_{x = 0} = \left| \begin{matrix}
n! & \sin\left( 0 + \frac{n\pi}{2} \right) & \cos\left( 0 + \frac{n\pi}{2} \right) \\
n! & \sin\left( \frac{n\pi}{2} \right) & \cos\left( \frac{n\pi}{2} \right) \\
a & a^{2} & a^{3}
\end{matrix} \right|$
$= \left| \begin{matrix}
n! & \sin\left( \frac{n\pi}{2} \right) & \cos\left( \frac{n\pi}{2} \right) \\
n! & \sin\left( \frac{n\pi}{2} \right) & \cos\left( \frac{n\pi}{2} \right) \\
a & a^{2} & a^{3}
\end{matrix} \right|$
$= 0\ \ (\because R_{1}\ and\ R_{2}\ are\ identical)$