"The number of distinct real roots of $$\left| {\matrix{ {\sin x} & {\cos x} & {\cos x} \cr {\cos x} & {\sin x} & {\cos x} \cr {\cos x} & {\cos x} & {\sin x} \cr } } \right| = 0$$ in the interval $- {\pi \over 4} \le x \le {\pi \over 4}$ is :"

Question

Accepted Answer

"$$\left| {\matrix{ {\sin x} & {\cos x} & {\cos x} \cr {\cos x} & {\sin x} & {\cos x} \cr {\cos x} & {\cos x} & {\sin x} \cr } } \right| = 0, - {\pi \over 4} \le x \le {\pi \over 4}$$

Apply : ${R_1} \to {R_1} - {R_2}$ & ${R_2} \to {R_2} - {R_3}$

$\Rightarrow$ $$\left| {\matrix{ {\sin x - \cos x} & {\cos x - \sin x} & 0 \cr 0 & {\sin x - \cos x} & {\cos x - \sin x} \cr {\cos x} & {\cos x} & {\sin x} \cr } } \right| = 0$$

$\Rightarrow$ $${(\sin x - \cos x)^2}\left| {\matrix{ 1 & { - 1} & 0 \cr 0 & 1 & { - 1} \cr {\cos x} & {\cos x} & {\sin x} \cr } } \right| = 0$$

$\Rightarrow$ ${(\sin x - \cos x)^2}(\sin x + 2\cos x) = 0$

$\therefore$ $x = {\pi \over 4}$"

The number of distinct real roots

of $$\left| {\matrix{ {\sin x} & {\cos x} & {\cos x} \cr {\cos x} & {\sin x} & {\cos x} \cr {\cos x} & {\cos x} & {\sin x} \cr } } \right| = 0$$ in the interval $- {\pi \over 4} \le x \le {\pi \over 4}$ is :

Solution

About this question

Drill 25 more like these. Every day. Free.

The number of distinct real roots of $$\left| {\matrix{ {\sin x} & {\cos x} & {\cos x} \cr {\cos x} & {\sin x} & {\cos x} \cr {\cos x} & {\cos x} & {\sin x} \cr } } \right| = 0$$ in the interval $- {\pi \over 4} \le x \le {\pi \over 4}$ is :

Solution

About this question

More Matrices and Determinants questions

Drill 25 more like these. Every day. Free.

The number of distinct real roots

of $$\left| {\matrix{ {\sin x} & {\cos x} & {\cos x} \cr {\cos x} & {\sin x} & {\cos x} \cr {\cos x} & {\cos x} & {\sin x} \cr } } \right| = 0$$ in the interval $- {\pi \over 4} \le x \le {\pi \over 4}$ is :