Question

The value of the determinant

$\left|\begin{array}{ccc}{a}^2& a& 1\\ \cos nx& \cos \left(n+1\right)x& \cos \left(n+2\right)x\\ \sin nx& \sin \left(n+1\right)x& \sin \left(n+2\right)x\end{array}\right|\mathrm{is}\mathrm{independent}\mathrm{of}$

(a) n
(b) a
(c) x
(d) none of these

Answer 1

(a) n

$$\begin{gathered} \mathrm{Let}A=\mathrm{nx},B=\left(\mathrm{n}+1\right)\mathrm{x},C=\left(\mathrm{n}+2\right)\mathrm{x} \\ \Rightarrow C-B=\mathrm{x},B-A=\mathrm{x},C-A=2\mathrm{x} \\ \mathrm{Thus},\mathrm{the}\mathrm{given}\mathrm{determinant}\mathrm{is} \\ \left|{\mathrm{a}}^2\mathrm{a}1 \\ \cos A\cos B\cos C \\ \sin A\sin B\mathit{}\sin C\right| \\ ={\mathrm{a}}^2\left(\cos B\sin C-\cos C\sin B\right)-\mathrm{a}\times \left(\cos A\sin C-\cos C\sin A\right)+1\times \left(\cos A\sin B-\sin A\cos B\right) \\ ={\mathrm{a}}^2\sin \left(C-B\right)-\mathrm{a}\sin \left(C-A\right)+\sin \left(B-A\right) \\ ={\mathrm{a}}^2\sin \mathrm{x}-\mathrm{a}\sin 2\mathrm{x}+\sin \mathrm{x}\left[\mathrm{Independent}\mathrm{of}\mathrm{n}\right] \end{gathered}$$