JEE MAIN - Mathematics (2024 - 30th January Evening Shift - No. 7)
Let $$R=\left(\begin{array}{ccc}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{array}\right)$$ be a non-zero $$3 \times 3$$ matrix, where $$x \sin \theta=y \sin \left(\theta+\frac{2 \pi}{3}\right)=z \sin \left(\theta+\frac{4 \pi}{3}\right) \neq 0, \theta \in(0,2 \pi)$$. For a square matrix $$M$$, let trace $$(M)$$ denote the sum of all the diagonal entries of $$M$$. Then, among the statements:
(I) Trace $$(R)=0$$
(II) If trace $$(\operatorname{adj}(\operatorname{adj}(R))=0$$, then $$R$$ has exactly one non-zero entry.
Explanation
$$\begin{aligned} & x \sin \theta=y \sin \left(\theta+\frac{2 \pi}{3}\right)=z \sin \left(\theta+\frac{4 \pi}{3}\right) \neq 0 \\ & \Rightarrow x, y, z \neq 0 \end{aligned}$$
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
Comments (0)
