First slide

Introduction to Determinants

Question

If $- π / 4 \leq x \leq π / 4$ ,then the number of distinct real roots of $|\begin{matrix} \sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x \end{matrix}| = 0$ is

Moderate

A

0
B

2
C

1
D

3

Solution

Using $C_{1} \to C_{1} + C_{2} + C_{3}$ ,
$\begin{matrix} Δ = |\begin{matrix} \sin x + 2 \cos x & \cos x & \cos x \\ \sin x + 2 \cos x & \sin x & \cos x \\ \sin x + 2 \cos x & \cos x & \sin x \end{matrix}| \\ = (\sin x + 2 \cos x) |\begin{array}{ccc} 1 & \cos x & \cos x \\ 1 & \sin x & \cos x \\ 1 & \cos x & \sin x \end{array}| \end{matrix}$
Applying $R_{2} \to R_{2} - R_{1} and R_{3} \to R_{3} - R_{1}$ , we get
$\begin{matrix} Δ = (\sin x + 2 \cos x) |\begin{array}{ccc} 1 & \cos x & \cos x \\ 0 & \sin x - \cos x & 0 \\ 0 & 0 & \sin x - \cos x \end{array}| \\ = (\sin x + 2 \cos x) (\sin x - \cos x)^{2} \end{matrix}$
Thus, $Δ = 0 \Rightarrow \tan x = - 2 or \tan x = 1$
As $- π / 4 \leq x \leq π / 4$ , we get $- 1 \leq \tan x \leq 1$
$∴ \tan x = 1 \Rightarrow x = π / 4$

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