Let $z$  be complex number satisfying  $\left|z^3\right|+2z^2+4\overline{z}-8=0,$ where $\overline{z}$  denotes the complex conjugate of $z.$  If the imaginary part of $z$  be nonzero.
Match each entry in Column – I to the correct entries in Column – II.
 

Column - I		Column - II
A)	${\left|z\right|}^2$ is equal to	P)	12
B)	${|z-\overline{z}|}^2$ is equal to	Q)	4
C)	${\left|z\right|}^2+{|z+\overline{z}|}^2$ is equal to	R)	8
D)	${|z+1|}^2$ is equal to	S)	10
		T)	7

 $\because {\left|z\right|}^3+z^2+4\overline{z}-8=0\mathrm{............}\left(1\right)$
Take conjugate both sides
 $\Rightarrow {\left|z\right|}^3+2{\overline{z}}^2+4z-8=0\mathrm{............}\left(2\right)$
By (1) – (2)
 $\Rightarrow 2(z^2-{\overline{z}}^2)+4(\overline{z}-z)=0$
 $\Rightarrow \boxed{z+\overline{z}=2}\mathrm{............}\left(3\right)$
 $\Rightarrow |z+\overline{z}|=2\mathrm{............}\left(4\right)$
Let  $z=x+iy$
 $\therefore x=1$
 $\therefore z=1+iy$
Put in (1)
 $\Rightarrow {(1+y^2)}^{3/2}+2(1-y^2+2iy)+4(1-iy)-8$
 $=0$
 $\Rightarrow {(1+y^2)}^{3/2}=2(1+y^2)$
 $\Rightarrow \sqrt{1+y^2}=2=\left|z\right|$
Also  $y=\pm \sqrt{3}$
 $\therefore z=1\pm i\sqrt{3}$
 $\Rightarrow z-\overline{z}=\pm 2i\sqrt{3}$
 $\Rightarrow |z-\overline{z}|=2\sqrt{3}$
 $\Rightarrow {|z-\overline{z}|}^2=12$
Now  $z+1=2+i\sqrt{3}$
${|z+1|}^2=4+3=7$ 
 $\therefore \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}A}\to Q;B\to P;C\to R;D\to T$