Question

Modulus of the complex number \[ \frac{(1+i)^{10}}{(2i-4)^4} \] is equal to

• A. \(\frac{2}{25}\)
• B. \(-\frac{2}{25}\)
• C. \(\frac{1}{25}\)
• D. \(-\frac{1}{25}\)

Hint:

For modulus, use \(|z^n|=|z|^n\) and \(\left|\frac{z_1}{z_2}\right|=\frac{|z_1|}{|z_2|}\).

Answer 1

The Correct Option is A

Step 1: Use property: \[ \left|\frac{z_1}{z_2}\right|=\frac{|z_1|}{|z_2|} \] and \[ |z^n|=|z|^n \]

Step 2: \[ |1+i|=\sqrt{1^2+1^2}=\sqrt{2} \] So, \[ |(1+i)^{10}|=(\sqrt{2})^{10}=2^5=32 \]

Step 3: \[ |2i-4|=|-4+2i|=\sqrt{(-4)^2+2^2} \] \[ =\sqrt{16+4}=\sqrt{20}=2\sqrt{5} \] \[ |(2i-4)^4|=(2\sqrt{5})^4=16\times 25=400 \]

Step 4: \[ \left|\frac{(1+i)^{10}}{(2i-4)^4}\right| = \frac{32}{400} \] \[ =\frac{2}{25} \] \[ \boxed{\frac{2}{25}} \]