Question

The modulus of the complex number \(\frac{(1+i)^2(1+3i)}{(2-6i)(2-2i)}\) is

• A. \(\frac{1}{\sqrt2}\)
• B. \(\frac{4}{\sqrt2}\)
• C. \(\frac{\sqrt2}{4}\)
• D. \(\frac{2}{\sqrt2}\)

Hint:

When calculating the modulus of a complex fraction, first find the modulus of the numerator and denominator separately, then divide them. This can simplify the process of solving modulus problems with complex fractions.

Answer 1

The Correct Option is C

The correct answer is: (C): \( \frac{\sqrt{2}}{4} \)

We are tasked with finding the modulus of the complex number:

\[ \frac{(1+i)^2(1+3i)}{(2-6i)(2-2i)} \]

Step 1: Simplify the numerator

First, expand \( (1 + i)^2 \):

\[ (1 + i)^2 = 1^2 + 2i + i^2 = 1 + 2i - 1 = 2i \]

Now, multiply this result by \( (1 + 3i) \):

\[ (2i)(1 + 3i) = 2i + 6i^2 = 2i - 6 = -6 + 2i \]

Step 2: Simplify the denominator

Now, expand \( (2 - 6i)(2 - 2i) \):

\[ (2 - 6i)(2 - 2i) = 2(2) - 2(2i) - 6i(2) + 6i(2i) = 4 - 4i - 12i + 12i^2 = 4 - 16i - 12 = -8 - 16i \]

Step 3: Find the modulus of the fraction

The complex number is now:

\[ \frac{-6 + 2i}{-8 - 16i} \]

We want to find the modulus of this complex number. The modulus of a complex number \[ \frac{a + bi}{c + di} \] is given by:

\[ \frac{|a + bi|}{|c + di|} \]

First, calculate the modulus of the numerator \( |-6 + 2i| \):

\[ |-6 + 2i| = \sqrt{(-6)^2 + (2)^2} = \sqrt{36 + 4} = \sqrt{40} = 2\sqrt{10} \]

Next, calculate the modulus of the denominator \( |-8 - 16i| \):

\[ |-8 - 16i| = \sqrt{(-8)^2 + (-16)^2} = \sqrt{64 + 256} = \sqrt{320} = 8\sqrt{5} \]

Step 4: Final simplification

The modulus of the complex number is:

\[ \frac{2\sqrt{10}}{8\sqrt{5}} = \frac{\sqrt{10}}{4\sqrt{5}} = \frac{\sqrt{2}}{4} \]

Conclusion:
The modulus of the complex number is \( \frac{\sqrt{2}}{4} \), so the correct answer is (C): \( \frac{\sqrt{2}}{4} \).