Question

The modulus of the conjugate of \( z = \frac{-2 + i}{(1 - 2i)^2} \) is:

• A. \( \frac{1}{5} \)
• B. \( \frac{1}{\sqrt{5}} \)
• C. \( \frac{1}{25} \)
• D. \( \sqrt{5} \)

Hint:

To find the modulus of the conjugate of a complex number, first find the conjugate and then use the formula \( \left| \frac{a}{b} \right| = \frac{|a|}{|b|} \) to compute the modulus.

Answer 1

The Correct Option is B

We are given \( z = \frac{-2 + i}{(1 - 2i)^2} \), and we need to find the modulus of the conjugate of \( z \). Step 1: First, compute the denominator: \[ (1 - 2i)^2 = (1^2 - 2 \times 1 \times 2i + (2i)^2) = 1 - 4i + (-4) = -3 - 4i \] Step 2: Now, the conjugate of \( z \) is: \[ \overline{z} = \frac{-2 - i}{(-3 + 4i)} \] Step 3: To find the modulus of the conjugate of \( z \), we use the formula for the modulus of a complex number \( \frac{a}{b} \), which is \( \frac{|a|}{|b|} \). The modulus of the numerator is: \[ | -2 - i | = \sqrt{(-2)^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5} \] The modulus of the denominator is: \[ | -3 + 4i | = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \] Step 4: Therefore, the modulus of the conjugate of \( z \) is: \[ \left| \overline{z} \right| = \frac{\sqrt{5}}{5} = \frac{1}{\sqrt{5}} \] Thus, the correct answer is \( \frac{1}{\sqrt{5}} \).