The value of the determinant
$\left|\begin{array}{ccc}a^{2} & a & 1 \\ \cos n x & \cos (n+1) x & \cos (n+2) x \\ \sin n x & \sin (n+1) x & \sin (n+2) x\end{array}\right|$ is independent of
(a) $n$
(b) $\mathrm{a}$
(c) $x$
(d) none of these
(a) $n$
Let $A=\mathrm{nx}, B=(\mathrm{n}+1) \mathrm{x}, C=(\mathrm{n}+2) \mathrm{x}$
$\Rightarrow C-B=\mathrm{x}, B-A=\mathrm{x}, C-A=2 \mathrm{x}$
Thus, the given determinant is
\begin{tabular}{|lll}
$\mathrm{a}^{2}$ & $\mathrm{a}$ & 1
$\begin{array}{lll}\mathrm{a}^{2} & \mathrm{a} & 1\end{array}$
$\begin{array}{lll}\cos A & \cos B & \cos C\end{array}$
$\sin A \quad \sin B \quad \sin C \mid$
$=\mathrm{a}^{2}(\cos B \sin C-\cos C \sin B)-\mathrm{a} \times(\cos A \sin C-\cos C \sin A)+1 \times(\cos A \sin B-\sin A \cos B)$
$=\mathrm{a}^{2} \sin (C-B)-\mathrm{a} \sin (C-A)+\sin (B-A)$
$=\mathrm{a}^{2} \sin \mathrm{x}-\mathrm{a} \sin 2 \mathrm{x}+\sin \mathrm{x} \quad[$ Independent of $\mathrm{n}]$
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