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

Solving the Complex Number Equation 

We are given the equation:

\[ |z|^3 + 2z^2 + 4\bar{z} - 8 = 0 \] and \[ |z|^3 + 2\bar{z}^2 + 4z - 8 = 0 \]

We subtract the two equations:

\[ 2(z^2 - \bar{z}^2) + 4(\bar{z} - z) = 0 \]

Now factor the expression:

\[ (z - \bar{z}) \left[ 2(z + \bar{z}) - 4 \right] = 0 \]

Let \( z = 1 + \lambda i \), where \( \lambda \) is a real number. We can now calculate \( |z| \) and \( \bar{z} \):

\[ |z| = \sqrt{1 + \lambda^2}, \quad \bar{z} = 1 - \lambda i \]

Substitute these into the equation:

\[ (1 + \lambda^2)^{\frac{3}{2}} + 2(1 - \lambda^2) = 4 \]

Now, factor and simplify:

\[ (1 + \lambda^2) \left[ \sqrt{1 + \lambda^2} - 2 \right] = 0 \]

Thus, we solve for \( \lambda^2 \):

\[ \lambda^2 = 3 \]

Answer 2

Complex Number Equation Solution 

We are given the following two equations:

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

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

By subtracting the two equations, we get:

\[ 2(z^2 - \bar{z}^2) + 4(\bar{z} - z) = 0 \]

Factoring the expression:

\[ (z - \bar{z})[2(z + \bar{z}) - 4] = 0 \]

We have two possibilities for \( z \):

1. \( z = \bar{z} \) (Not possible)

2. \( 4x = 4 \quad \Rightarrow x = 1 \)

Let \( z = 1 + \lambda i \), then:

\[ |z| = \sqrt{1 + \lambda^2}, \quad \bar{z} = 1 - \lambda i \]

Substituting into the equation:

\[ (1 + \lambda^2)^{\frac{3}{2}} + 2(1 - \lambda^2 + 2\lambda i + 4(1 - \lambda i) - 8) = 0 \]

We simplify this to:

\[ (1 + \lambda^2)^{\frac{3}{2}} + 2(1 - \lambda^2) = 4 \]

Next, we get:

\[ (1 + \lambda^2)^{\frac{3}{2}} = 2(1 + \lambda^2) \]

Which simplifies to:

\[ (1 + \lambda^2)[\sqrt{1 + \lambda^2} - 2] = 0 \]

Solving for \( \lambda^2 \), we get:

\[ \lambda^2 = 3 \]

Now, let's evaluate the following options:

(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) \( |z + 1|^2 = |1 + \lambda i + 1|^2 = 4 + \lambda^2 = 4 + 3 = 7 \)

Correct Answer:

The correct answer is option 2:

\[ (P) \rightarrow (2), \quad (Q) \rightarrow (1), \quad (R) \rightarrow (3), \quad (S) \rightarrow (5) \]