If the system of equations x + (√2sinα) y + (√2cosα)z = 0 x + (cosα)y + (sinα)z = 0

Question

If the system of equations

\(x+(\sqrt{2} \sin \alpha) y+(\sqrt{2} \cos \alpha) z=0\)

\(\mathrm{x}+(\cos \alpha) \mathrm{y}+(\sin \alpha) \mathrm{z}=0\)

\(\mathrm{x}+(\sin \alpha) \mathrm{y}-(\cos \alpha) \mathrm{z}=0\)

has a non-trivial solution, then \(\alpha \in\left(0, \frac{\pi}{2}\right)\) is equal to :

(1) \(\frac{3 \pi}{4}\)

(2) \(\frac{7 \pi}{24}\)

(3) \(\frac{5 \pi}{24}\)

(4) \(\frac{11 \pi}{24}\)

Answer 1

Correct option is (3) \(\frac{5 \pi}{24}\)

\(\left|\begin{array}{ccc} 1 & \sqrt{2} \sin \alpha & \sqrt{2} \cos \alpha \\ 1 & \sin \alpha & -\cos \alpha \\ 1 & \cos \alpha & \sin \alpha \end{array}\right|=0 \\\)

\(\Rightarrow 1-\sqrt{2} \sin \alpha(\sin \alpha+\cos \alpha)+\sqrt{2} \cos \alpha(\cos \alpha-\sin \alpha)=0\)

\( \Rightarrow 1+\sqrt{2} \cos 2 \alpha-\sqrt{2} \sin 2 \alpha=0 \)

\(\cos 2 \alpha-\sin 2 \alpha=-\frac{1}{\sqrt{2}}\)

\(\cos \left(2 \alpha+\frac{\pi}{4}\right)=-\frac{1}{2} \)

\( 2 \alpha+\frac{\pi}{4}=2 \mathrm{n} \pi \pm \frac{2 \pi}{3} \)

\( \alpha+\frac{\pi}{8}=\mathrm{n} \pi \pm \frac{\pi}{3}\)

\(\mathrm{n}=0\)

\(x=\frac{\pi}{3}-\frac{\pi}{8}=\frac{5 \pi}{24}\)