QJEEM24Mac841: JEE Main 2024 Mathematics - Class 12 JEEM24Mac841 Solution: Let be a non-zero matrix, where . For a square matrix , let…

$\begin{aligned}& x \sin \theta=y \sin \left(\theta+\frac{2 \pi}{3}\right)=z \sin \left(\theta+\frac{4 \pi}{3}\right) \neq 0 \\& \Rightarrow x, y, z \neq 0\end{aligned}$

Also,

$\begin{aligned}& \sin \theta+\sin \left(\theta+\frac{2 \pi}{3}\right)+\sin \left(\theta+\frac{4 \pi}{3}\right)=0 \forall \theta \in \mathrm{R} \\& \Rightarrow \frac{1}{\mathrm{x}}+\frac{1}{\mathrm{y}}+\frac{1}{\mathrm{z}}=0 \\& \Rightarrow \mathrm{xy}+\mathrm{yz}+\mathrm{zx}=0\end{aligned}$

(i) $\quad \operatorname{Trace}(\mathrm{R})=\mathrm{x}+\mathrm{y}+\mathrm{z}$

If $x+y+z=0$ and $x y+y z+z x=0$

$\Rightarrow \mathrm{x}=\mathrm{y}=\mathrm{z}=0$

Statement (i) is False

(ii) $\quad \operatorname{Adj}(\operatorname{Adj}(\mathrm{R}))=|\mathrm{R}| \mathrm{R}$

Trace $(\operatorname{Adj}(\operatorname{Adj}(\mathrm{R})))$

$=x y z(x+y+z) \neq 0$

Statement (ii) is also False