Explanation

(d)
$\because a,b,c,d,e\$and$\ f\$are in GP.
$\therefore a = a,b = ar,c = ar^{2},d = ar^{3},e = ar^{4}$and$\ f = ar^{5}$
$\therefore\left| \begin{matrix}
a^{2} & d^{2} & x \\
b^{2} & e^{2} & y \\
c^{2} & f^{2} & z
\end{matrix} \right| = \left| \begin{matrix}
a^{2} & a^{2}r^{6} & x \\
a^{2}r^{2} & a^{2}r^{8} & y \\
a^{2}r^{4} & a^{2}r^{10} & z
\end{matrix} \right|$
$= a^{4}r^{6}\left| \begin{matrix}
1 & 1 & x \\
r^{2} & r^{2} & y \\
r^{4} & r^{4} & z
\end{matrix} \right| = 0$
Thus, the given determinant is independent of $x,y$ and $z$.