Let z be complex number satisfying $$|z|^3 + 2z^2 + 4\bar{z} - 8 = 0$$, 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:

Let $$z = x + i\,y$$ with $$y \neq 0$$. Then $$\bar z = x - i\,y$$ and $$|z| = \sqrt{x^{2}+y^{2}}$$.

Rewrite the given equation separating real and imaginary parts:
$$|z|^{3} + 2z^{2} + 4\bar z - 8 = 0$$

Compute each term:
$$z^{2} = (x + i\,y)^{2} = x^{2} - y^{2} + 2 i x y$$
$$2z^{2} = 2(x^{2} - y^{2}) + 4 i x y$$
$$4\bar z = 4x - 4 i y$$

Substituting in the equation:
$$(x^{2}+y^{2})^{3/2} + \bigl[\,2(x^{2}-y^{2}) + 4x - 8\,\bigr] + i\bigl[\,4xy - 4y\,\bigr] = 0$$

Since the imaginary part must vanish,
$$4xy - 4y = 0 \;\;\Rightarrow\;\; y(x-1)=0.$$
Given $$y \neq 0$$, we get $$x = 1$$.

Put $$x = 1$$ in the real part. Let $$r^{2} = x^{2} + y^{2} = 1 + y^{2}$$.
Then
$$(1+y^{2})^{3/2} + \bigl[\,2(1 - y^{2}) + 4 - 8\,\bigr] = 0$$
$$\Rightarrow (1+y^{2})^{3/2} - 2y^{2} - 2 = 0$$
$$\Rightarrow (1+y^{2})^{3/2} = 2(1+y^{2}).$$

Set $$t = 1 + y^{2} \; (>0).$$ The equation becomes $$t^{3/2} = 2t$$.
Divide by $$t$$: $$t^{1/2} = 2 \;\;\Rightarrow\;\; t = 4.$$
Hence $$1 + y^{2} = 4 \;\;\Rightarrow\;\; y^{2} = 3 \;\;\Rightarrow\;\; y = \pm\sqrt{3}.$$

Therefore the admissible complex numbers are $$z = 1 \pm i\sqrt{3}.$$ All required moduli are identical for both choices, so we can use either one. Take $$z = 1 + i\sqrt{3}$$ for calculation.

Case P: $$|z|^{2} = (1)^{2} + (\sqrt{3})^{2} = 4.$$

Case Q: $$z - \bar z = (1+i\sqrt{3}) - (1-i\sqrt{3}) = 2i\sqrt{3},$$
$$|z - \bar z|^{2} = |2i\sqrt{3}|^{2} = (2\sqrt{3})^{2} = 12.$$

Case R: $$z + \bar z = (1+i\sqrt{3}) + (1-i\sqrt{3}) = 2,$$
$$|z + \bar z|^{2} = |2|^{2} = 4,$$
$$|z|^{2} + |z + \bar z|^{2} = 4 + 4 = 8.$$

Case S: $$z + 1 = 1 + i\sqrt{3} + 1 = 2 + i\sqrt{3},$$
$$|z + 1|^{2} = 2^{2} + (\sqrt{3})^{2} = 4 + 3 = 7.$$

Thus we match the values with List-II:
$$|z|^{2} \; (P) \longrightarrow 4 \; (2)$$
$$|z - \bar z|^{2} \; (Q) \longrightarrow 12 \; (1)$$
$$|z|^{2} + |z + \bar z|^{2} \; (R) \longrightarrow 8 \; (3)$$
$$|z + 1|^{2} \; (S) \longrightarrow 7 \; (5)$$

The correct option is:
Option B: (P) → (2), (Q) → (1), (R) → (3), (S) → (5).