Question

Let z be a complex number satisfying |z|³ + 2z² +4z̄ - 8= 0 , where z̄ 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 - z̄|2 is equal to	(2)	4
(R)	|z|2 + |z - z̄|2 is equal to	(3)	8
(S)	|z + 1|2 is equal to	(4)	10
		(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

The Correct Option is B

\(|z|^3+2z^2+4\bar{z}-8=0\)
\(|z|^3+2\bar{z}^2+4z-8=0\)
_______________________________
\(2(z^2-\bar{z}^2)+4(\bar{z}-z)=0\)
\((z-\bar{z})[2(z+\bar{z})-4]=0\)
\(z=1+\lambda i\Rightarrow |z|=\sqrt{1+\lambda^2}\Rightarrow\bar{z}=1-\lambda i\)
\(\Rightarrow (1+\lambda^2)^{\frac{3}{2}}+2(1-\lambda^2)=4\)
\((1+\lambda^2)[\sqrt{1+\lambda^2}-2]=0\)
\(\Rightarrow \lambda^2=3 \)

Answer 2

\(|z|^3+2z^2+4\bar{z}-8=0\)
\(|z|^3+2\bar{z}^2+4z-8=0\)
_______________________________
\(2(z^2-\bar{z}^2)+4(\bar{z}-z)=0\)
\((z-\bar{z})[2(z+\bar{z})-4]=0\)
\(z=\bar{z}\)(Not possible) or \(4x=4,x=1\)
\(z=1+\lambda i\Rightarrow |z|=\sqrt{1+\lambda^2}\Rightarrow\bar{z}=1-\lambda i\)
\(\Rightarrow (1+\lambda^2)^{\frac{3}{2}}+2(1-\lambda^2+2\lambda i+4(1-\lambda i)-8=0\)
\(\Rightarrow (1+\lambda^2)^{\frac{3}{2}}+2(1-\lambda^2)=4\)
\(\Rightarrow (1+\lambda^2)^{\frac{3}{2}}=2(1+\lambda^2)\)
\((1+\lambda^2)[\sqrt{1+\lambda^2}-2]=0\)
\(\Rightarrow \lambda^2=3\)
Now
\((P) \ |Z|^2=1+\lambda^2=1+3=4\)
\((Q)\ |z-\bar{z}|=|1+\lambda i-(1-\lambda i)|^2=|2\lambda i|^2=4\lambda^2=12\)
\((R)\ |z|^2+|z+\bar{z}|^2=4+|(1+\lambda i)+(1-\lambda i)|^2=4+4=8\)
\((S)|z+1|^2=|1+\lambda i+1|^2=4+\lambda^2=4+3=7\)
So correct Answer is option 2 
\((P) →(2) (Q)→ (1) (R) →(3) (S) →(5) \)