Question

Let $z$ be a complex number satisfying $|z|^3 + 2z^2 + 4\overline{z} - 8 = 0$, where $\overline{z}$ 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
(Q) $	z-\overline{z}
(R) $	z
(S) $	z+1
(5) 7

The correct option is:

• A. (P)→(1) (Q)→(3) (R)→(5) (S)→(4)
• B. (P)→(2) (Q)→(1) (R)→(3) (S)→(5)
• C. (P)→(2) (Q)→(4) (R)→(5) (S)→(1)
• D. (P)→(2) (Q)→(3) (R)→(5) (S)→(4)

Answer 1

Answer: (B)

(P)→(2) (Q)→(1) (R)→(3) (S)→(5)

Answer 2

1. Write $z = x + iy$ ($y\ne0$).

2. The imaginary part of

   $$|z|^3 + 2z^2 + 4\overline{z} - 8 = 0$$

   gives: $4y(x-1)=0$ ⟹ $x=1$.

3. Substitute $x=1$ into the real part:

   $$(1+y^2)^{3/2} + 2(1-y^2) + 4 - 8 = 0 \quad \Longrightarrow \quad (1+y^2)^{3/2} = 2(1+y^2).$$

   Dividing both sides by $(1+y^2)$ (since $y^2\ge0$) yields:

   $$\sqrt{1+y^2}=2 \quad \Longrightarrow \quad 1+y^2=4 \quad \Longrightarrow \quad y^2 = 3.$$

   Thus, $z=1\pm i\sqrt{3}$.

4. Now determine each expression:

   • (P) $|z|^2$:

     $$|z|^2 = 1^2 + (\sqrt{3})^2 = 1+3 = 4 \quad \Rightarrow \; \text{matches (2) [4]}.$$

   • (Q) $|z-\overline{z}|^2$:

     $$z-\overline{z} = 2iy \quad \Longrightarrow \quad |2iy|^2 = 4y^2 = 4\cdot 3 = 12 \quad \Rightarrow \; \text{matches (1) [12]}.$$

   • (R) $|z|^2+|z+\overline{z}|^2$:

     $$z+\overline{z} = 2x = 2 \quad \Longrightarrow \quad |z+\overline{z}|^2 =4.$$

     So,

     $$4+4=8 \quad \Rightarrow \; \text{matches (3) [8]}.$$

   • (S) $|z+1|^2$:

     $$z+1 = (1+1) + iy = 2 + iy \quad \Longrightarrow \quad |2+iy|^2 = 2^2+y^2 = 4+3=7 \quad \Rightarrow \; \text{matches (5) [7]}.$$

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