First slide

Introduction to Determinants

Question

If $Δ = |\begin{array}{ccc} \sin θcos ϕ & \sin θsin ϕ & \cos θ \\ \cos θcos ϕ & \cos θsin ϕ & - \sin θ \\ - \sin θsin ϕ & \sin θcos ϕ & 0 \end{array}|$ then

Moderate

A

$∆$ is independent of $θ$
B

$∆$ is independent of $ϕ$
C

$∆$ is a constant
D

${\frac{dΔ}{dθ}]}_{θ = π / 2} = 0$

Solution

Applying $C_{1} \to C_{1} - (\cot ϕ) C_{2}$ , we get
$Δ = |\begin{array}{ccc} 0 & \sin θsin ϕ & \cos θ \\ 0 & \cos θsin ϕ & - \sin θ \\ - \sin θ / \sin ϕ & \sin θcos ϕ & 0 \end{array}|$
$= - \frac{\sin θ}{\sin ϕ} [- \sin {ϕsin}^{2} θ - \cos^{2} θsin ϕ] [expanding alone C_{1}]$
$= \sin θ, which is independent of ϕ$ .
Also, ${\frac{dΔ}{dθ} = \cos θ \Rightarrow \frac{dΔ}{dθ}]}_{θ = π / 2} = \cos (π / 2) = 0$

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