First slide

Introduction to Determinants

Question

Let
$Δ (θ) = |\begin{array}{ccc} 1 & \sin θ & 1 \\ - \sin θ & 1 & \sin θ \\ - 1 & - \sin θ & 1 \end{array}|, 0 \leq θ \leq 2 π$
Solution of $∆ (θ) = 3$ is

Moderate

A

$\{\frac{π}{2}, \frac{3 π}{2}\}$
B

$\{\frac{π}{4}, \frac{3 π}{4}, \frac{5 π}{4}, \frac{7 π}{4}\}$
C

$\{\frac{π}{4}, \frac{3 π}{4}\}$
D

$\{\frac{π}{4}, \frac{π}{2}, \frac{3 π}{4}, π\}$

Solution

Using $C_{1} \to C_{1} + C_{3}$ we get
$\begin{aligned} Δ (θ) = |\begin{array}{ccc} 2 & \sin θ & 1 \\ 0 & 1 & \sin θ \\ 0 & - \sin θ & 1 \end{array}| \\ = 2 (1 + \sin^{2} θ) \end{aligned}$
$\begin{array}{l} ∴ Δ (θ) = 3 \Rightarrow 2 \sin^{2} θ = 1 \\ \Rightarrow \sin θ = \pm \frac{1}{\sqrt{2}} \Rightarrow θ = \frac{π}{4}, \frac{3 π}{4}, \frac{5 π}{4}, \frac{7 π}{4} \end{array}$

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