Question

The determinant $\left| \begin{matrix} \cos(\theta + \varphi) & –\sin(\theta + \varphi) & \cos 2\varphi \\ \sin\theta & \cos\theta & \sin\varphi \\ –\cos\theta & \sin\theta & \cos\varphi \end{matrix} \right|$ is –

• A. 0
• B. Independent of θ
• C. Independent of φ
• D. Independent of both θ and φ

Answer 1

Answer: (B)

Independent of θ

Answer 2

∆ = cos (θ + φ). [ cos θ cos φ – sin θ sin φ]

+ sin(θ + φ).[sin θ cos φ + cos θ sin φ] + cos2φ.[sin²θ + cos²θ]

∆ = [cos² (θ + φ) + sin² (θ + φ)] + cos(2φ)

∆ = 1 + cos (2φ)

∴ ∆ is independent of θ