JEE Main 2024 — 4 April, Shift 1 — Question 7

$\text{ }\!\!~\!\!\text{ }\left| \begin{matrix}1 & \sqrt{2}\text{sin}\alpha & \sqrt{2}\text{cos}\alpha \\1 & \text{sin}\alpha & -\text{cos}\alpha \\1 & \text{cos}\alpha & \text{sin}\alpha \\\end{matrix} \right|=0$

\begin{array}{*{35}{r}}{} & ~\Rightarrow 1-\sqrt{2}\text{sin}\alpha \left( \text{sin}\alpha +\text{cos}\alpha \right)+\sqrt{2}\text{cos}\alpha \left( \text{cos}\alpha -\text{sin}\alpha \right)=0 \\{} & ~\Rightarrow 1+\sqrt{2}\text{cos}2\alpha -\sqrt{2}\text{sin}2\alpha =0 \\{} & \text{cos}2\alpha -\text{sin}2\alpha =-\frac{1}{\sqrt{2}} \\{} & \text{cos}\left( 2\alpha +\frac{\pi }{4} \right)=-\frac{1}{2} \\{} & 2\alpha +\frac{\pi }{4}=2n\pi \pm \frac{2\pi }{3} \\{} & \alpha +\frac{\pi }{8}=\text{n}\pi \pm \frac{\pi }{3} \\{} & \text{n}=0 \\{} & x=\frac{\pi }{3}-\frac{\pi }{8}=\frac{5\pi }{24} \\\end{array}