Question

If ∆ =$\left| \begin{matrix} \sin x.\cos y & \sin x.\sin y & \cos x \ \cos x.\cos y & \cos x.\sin y & - \sin x \

• \sin x.\sin y & \sin x.\cos y & 0 \end{matrix} \right|$, then ∆ is

independent of –

• A. x
• B. y
• C. Constant
• D. None of these

Answer 1

Answer: (B)

y

Answer 2

Take sin x, cos x and sin x common from R₁, R₂ and R₃ respectively

∴ ∆ = sin² x . cos x $\left| \begin{matrix} \cos y & \sin y & \cot x \ \cos y & \sin y & - \tan x \

• \sin y & \cos y & 0 \end{matrix} \right|$

Make two zeros by R₁ – R₂ = sin² x cos x

$\left| \begin{matrix} 0 & 0 & \cot x + \tan x \ \cos y & \sin y & - \tan x \

• \sin y & \cos y & 0 \end{matrix} \right|$= sin² x . cos x .

$\frac{\sin^{2}x + \cos^{2}x}{\sin x.\cos x}$ [cos² y + sin² y]

= sin x, which is independent of y.