Let be a complex number satisfying where denotes the complex conjugate of . Let the imaginary part of be non-zero. Match each entry in List-I to the correct entries in List-II. [2023]

Question

Let $z$ be a complex number satisfying $| z |^{3} + 2 z^{2} + 4 \bar{z} - 8 = 0,$ where $\bar{z}$ denotes the complex conjugate of $z$ . Let the imaginary part of $z$ be non-zero.

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: [2023]

Smart Achievers · Accepted Answer

(2)

$Given, | z |^{3} + 2 z^{2} + 4 \bar{z} - 8 = 0 \dots (i)$

$| \bar{z} |^{3} + 2 {\bar{z}}^{2} + 4 z - 8 = 0 [Conjugate both sides]$

$\Rightarrow 2 (z^{2} - {\bar{z}}^{2}) + 4 (\bar{z} - z) = 0$

$| z |^{2} + | z + \bar{z} |^{2}$

$∵$ $z = \bar{z} (Not possible) or z + \bar{z} = 2$

$∴$ $z = 1 + b i (b \neq 0) \Rightarrow \bar{z} = 1 - b i$

${(1 + b^{2})}^{3 / 2} + 2 (1 - b^{2} + 2 b i) + 4 (1 - b i) - 8 = 0 [from (i)]$

${(1 + b^{2})}^{3 / 2} - 2 (1 + b^{2}) = 0$

$\Rightarrow (1 + b^{2}) (\sqrt{1 + b^{2}} - 2) = 0$

$∵$ $1 + b^{2} \neq 0 \Rightarrow \sqrt{1 + b^{2}} - 2 = 0 \Rightarrow b^{2} = 3$

$(P) | z |^{2} = 1 + b^{2} = 1 + 3 = 4$

$(Q) | z - \bar{z} |^{2} = | 1 + i b - 1 + i b |^{2} = 4 b^{2} = 12$

$(R) | z |^{2} + | z + \bar{z} |^{2} = 4 + | 1 + i b + 1 - i b |^{2} = 4 + 4 = 8$

$(S) | z + 1 |^{2} = | 1 + 1 + i b |^{2} = 4 + b^{2} = 4 + 3 = 7$

Solution

1 (P) → (1), (Q) → (3), (R) → (5), (S) → (4)

2 (P) → (2), (Q) → (1), (R) → (3), (S) → (5)

3 (P) → (2), (Q) → (4), (R) → (5), (S) → (1)

4 (P) → (2), (Q) → (3), (R) → (5), (S) → (4)

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