Question

$\left| \begin{matrix} 1 & 1 & 1 \\ \cos(nx) & \cos(n + 1)x & \cos(n + 2)x \\ \sin(nx) & \sin(n + 1)x & \sin(n + 2)x \end{matrix} \right|$ is not depend.

• A. On x
• B. On n
• C. Both on x and n
• D. None of these

Answer 1

Answer: (B)

On n

Answer 2

$\mathbf{\Delta}\mathbf{=}\left| \begin{matrix} \mathbf{1} & \mathbf{1} & \mathbf{1} \\ \mathbf{\cos}\mathbf{n}\mathbf{x} & \mathbf{\cos}\mathbf{(}\mathbf{n + 1)x} & \mathbf{\cos}\mathbf{(}\mathbf{n + 2)x} \\ \mathbf{\sin}\mathbf{n}\mathbf{x} & \mathbf{\sin}\mathbf{(}\mathbf{n + 1)x} & \mathbf{\sin}\mathbf{(}\mathbf{n + 2)x} \end{matrix} \right|$

Applying $C_{1} \rightarrow C_{1} + C_{3} - (2\cos x)C_{2}$

2(1 - \cos x) & 1 & 1 \\ 0 & \cos(n + 1)x & \cos(n + 2)x \\ 0 & \sin(n + 1)x & \sin(n + 2)x \end{matrix} \right|$$ $$\Delta = 2(1 - \cos x)\lbrack\cos(n + 1)x\sin(n + 2)x - \cos(n + 2)x\sin(n + 1)x\rbrack$$ $$\Delta = 2(1 - \cos x)\lbrack\sin(n + 2 - n - 1)x\rbrack = 2\sin x(1 - \cos x)$$ i.e., $\Delta$ is independent of n.