Question

The value of the determinant

\[\begin{vmatrix}a^2 & a & 1 \\ \cos nx & \cos \left( n + 1 \right) x & \cos \left( n + 2 \right) x \\ \sin nx & \sin \left( n + 1 \right) x & \sin \left( n + 2 \right) x\end{vmatrix}\text{ is independent of}\]

 

Options

• n

• a

• x

• none of these

Answer 1

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