Question

The modulus and amplitude of $\frac{\left(1+2i\right)}{1-{\left(1-i\right)}^2}$ are

• A. $\sqrt{2}$ and $\frac{\mathrm{\pi }}{6}$
• B. $1$ and $0$
• C. $1$ and $\frac{\mathrm{\pi }}{3}$
• D. $1$ and $\frac{\mathrm{\pi }}{4}$

Answer 1

The correct option is B

$1$ and $0$

Explanation for the correct option:

Find the modulus and amplitude of the given complex number

In the question, a complex number $\frac{\left(1+2i\right)}{1-{\left(1-i\right)}^2}$ is given.

Rewrite the given complex number using the binomial theorem as follows,

$$\begin{gathered} \Rightarrow \frac{\left(1+2i\right)}{1-{\left(1-i\right)}^2}=\frac{\left(1+2i\right)}{1-\left(1-2i-1\right)} \\ =\frac{\left(1+2i\right)}{\left(1+2i\right)} \\ =1 \end{gathered}$$

According to the Euler's representation,

$\Rightarrow \frac{\left(1+2i\right)}{1-{\left(1-i\right)}^2}=e^{i·0}$

So, the modulus of the given complex number is $1$.

The amplitude of the given complex is $0$.

Therefore, the modulus and amplitude of the given complex number are $1$ and $0$ respectively.

Hence, correct option is $B$