First slide

Introduction to Determinants

Question

If $f (θ) = |\begin{array}{ccc} \sin θ & \cos θ & \sin θ \\ \cos θ & \sin θ & \cos θ \\ \cos θ & \sin θ & \sin θ \end{array}|$ , then

Moderate

A

$f (θ) = 0$ has exactly 2 real solutions in $[0, π]$
B

$f (θ) = 0$ has exactly 3 real solutions in $[0, π]$
C

range of function $\frac{f (θ)}{1 - \sin 2 θ} is [- \sqrt{2}, \sqrt{2}]$
D

range of function $\frac{f (θ)}{\sin 2 θ - 1} is [- 3, 3] is [- 3, 3]$

Solution

$\begin{matrix} f (θ) = \sin^{3} θ + \cos^{3} θ - \cos θsin θ (\sin θ + \cos θ) \\ = (\sin θ + \cos θ)^{3} - 4 \sin θcos θ (\sin θ + \cos θ) \\ = (\sin θ + \cos θ) [1 - \sin 2 θ] \end{matrix}$
Now, $f (θ) = 0$
$\Rightarrow \tan θ = - 1 or \sin 2 θ = 1$
$\Rightarrow f (θ) = 0$ has 2real solutions in $[0, π]$
Also, $\frac{f (θ)}{1 - \sin 2 θ} = \sin θ + \cos θ \in [- \sqrt{2}, \sqrt{2}]$

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