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Let $z$ be complex number satisfying $|z|^3+2 z^2+4 z-8=0$, where $\bar{z}$ denotes the complex conjugate of $z$. Let the imaginary part of $z$ be nonzero.
Match each entry in List-$I$ to the correct entries in List-$II$.
| List-$I$ | List-$II$ |
| ($P$) $|z|^2$ is equal to | ($1$) $12$ |
| ($Q$) $|z-\bar{z}|^2$ is equal to | ($2$) $4$ |
| ($R$) $|z|^2+|z+\bar{z}|^2$ is equal to | ($3$) $8$ |
| ($S$) $|z+1|^2$ is equal to | ($4$) $10$ |
| ($5$) $7$ |
The correct option is:
$(\mathrm{A})(\mathrm{P}) \rightarrow(1)(\mathrm{Q}) \rightarrow(3)(\mathrm{R}) \rightarrow(5)(\mathrm{S}) \rightarrow(4)$
$(\mathrm{P}) \rightarrow(2)(\mathrm{Q}) \rightarrow(1)(\mathrm{R}) \rightarrow(3) (S) \rightarrow (5)$
$(P) \rightarrow (2) (Q) \rightarrow (4) (R) \rightarrow (5) (S) \rightarrow (1)$
$(\mathrm{P}) \rightarrow(2)(\mathrm{Q}) \rightarrow(3)(\mathrm{R}) \rightarrow(5)(\mathrm{S}) \rightarrow(4)$
Solution
$\because|z|^3+2 z^2+4 \bar{z}-8=0$ $. . . . . . (1)$
Take conjugate both sides
$\Rightarrow|z|^3+2 z^2+4 z-8=0$ $. . . . . . .(2)$
$\operatorname{By}(1)-(2)$
$\Rightarrow 2\left(z^2-\bar{z}^2\right)+4(\bar{z}-z)=0$
$\Rightarrow z+\bar{z}=2$ $. . . . . .(3)$
$\Rightarrow|z+\bar{z}|=2$ $. . . . . (4)$
Let $z=x+i y$
$\therefore x=1 \quad \therefore z=1+\mathrm{i}$
Put in $(1)$
$\Rightarrow\left(1+y^2\right)^{3 / 2}+2\left(1-y^2+2 i y\right)+4(1-i y)-8=0$
$\Rightarrow\left(1+y^2\right)^{3 / 2}=2\left(1+y^2\right)$
$\Rightarrow \sqrt{1+y^2}=2=|z|$
Also $\mathrm{y}= \pm \sqrt{3}$
$\therefore z=1 \pm i \sqrt{3}$
$\Rightarrow z-\bar{z}= \pm 2 i \sqrt{3}$
$\Rightarrow|z-\bar{z}|=2 \sqrt{3}$
$\Rightarrow|z-\bar{z}|^2=12$
Now $z+1=2+i \sqrt{3}$
$|z+1|^2=4+3=7$
$\therefore P \rightarrow 2 ; Q \rightarrow 1 ; R \rightarrow 3 ; S \rightarrow 5$
$\therefore$ Option $[B]$ is correct.
