Question

The modulus of \( (1 + i\sqrt{3})(2 + 2i) \) / \( (\sqrt{3} - i) \) is:

• A. 2
• B. 4
• C. \( 3\sqrt{2} \)
• D. \( 2\sqrt{2} \)

Hint:

The modulus of a product of complex numbers is the product of their moduli. Similarly, the modulus of a quotient is the quotient of their moduli.

Answer 1

The Correct Option is D

Step 1: First, calculate the modulus of the product and the denominator. 
The modulus of a complex number \( a + bi \) is \( \sqrt{a^2 + b^2} \). 
Step 2: For \( (1 + i\sqrt{3}) \), the modulus is: \[ |1 + i\sqrt{3}| = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2. \] 
Step 3: For \( (2 + 2i) \), the modulus is: \[ |2 + 2i| = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}. \] 
Step 4: For \( (\sqrt{3} - i) \), the modulus is: \[ |\sqrt{3} - i| = \sqrt{(\sqrt{3})^2 + (-1)^2} = \sqrt{3 + 1} = \sqrt{4} = 2. \] 
Step 5: The modulus of the entire expression is the product of the moduli of the numerator divided by the modulus of the denominator: \[ \left| \frac{(1 + i\sqrt{3})(2 + 2i)}{\sqrt{3} - i} \right| = \frac{|1 + i\sqrt{3}| \cdot |2 + 2i|}{|\sqrt{3} - i|} = \frac{2 \cdot 2\sqrt{2}}{2} = 2\sqrt{2}. \]