Question

If complex numbers \( z_1, z_2, \ldots , z_n \) satisfy the equation \( 4z^2 + \bar{z} = 0 \), then \( \sum_{i=1}^{n} |z_i|^2 \) is equal to:

• A. \( \dfrac{3}{16} \)
• B. \( \dfrac{3}{64} \)
• C. \( \dfrac{9}{64} \)
• D. \( \dfrac{1}{16} \)

Hint:

For equations involving \(z\) and \(\bar z\):
Always convert to \(x+iy\) form
Equate real and imaginary parts separately

Answer 1

The Correct Option is A

Concept: To solve equations involving a complex number and its conjugate, write \[ z = x + iy \quad \text{and} \quad \bar{z} = x - iy \] and equate real and imaginary parts separately. Also, \[ |z|^2 = x^2 + y^2 \]
Step 1: Substitute \( z = x + iy \) in the given equation. \[ 4z^2 + \bar{z} = 0 \] \[ 4(x+iy)^2 + (x-iy) = 0 \]
Step 2: Simplify and separate real and imaginary parts. \[ 4(x^2 - y^2 + 2ixy) + x - iy = 0 \] Real part: \[ 4(x^2 - y^2) + x = 0 \quad \cdots (1) \] Imaginary part: \[ 8xy - y = 0 \] \[ y(8x - 1) = 0 \quad \cdots (2) \]
Step 3: Solve the cases from equation (2). Case 1: \( y = 0 \) From (1): \[ 4x^2 + x = 0 \] \[ x(4x + 1) = 0 \] \[ x = 0 \quad \text{or} \quad x = -\frac{1}{4} \] Thus, \[ z = 0,\; -\frac{1}{4} \] Case 2: \( 8x - 1 = 0 \Rightarrow x = \frac{1}{8} \) Substitute in (1): \[ 4\left(\frac{1}{64} - y^2\right) + \frac{1}{8} = 0 \] \[ \frac{1}{16} + \frac{1}{8} - 4y^2 = 0 \] \[ \frac{3}{16} = 4y^2 \] \[ y^2 = \frac{3}{64} \] \[ y = \pm \frac{\sqrt{3}}{8} \] Thus, \[ z = \frac{1}{8} \pm i\frac{\sqrt{3}}{8} \]
Step 4: Compute \( \sum |z_i|^2 \). \[ |0|^2 = 0 \] \[ \left|-\frac{1}{4}\right|^2 = \frac{1}{16} \] \[ \left|\frac{1}{8} \pm i\frac{\sqrt{3}}{8}\right|^2 = \frac{1}{64} + \frac{3}{64} = \frac{1}{16} \] There are two such roots. \[ \sum |z_i|^2 = 0 + \frac{1}{16} + \frac{1}{16} + \frac{1}{16} = \frac{3}{16} \] \[ \boxed{\sum_{i=1}^{n} |z_i|^2 = \frac{3}{16}} \]