What is the modulus of `| (1+2i)/(1-(1-i)^(2))|` ?

Options

A. A. 1
B. B. `sqrt(5)`
C. C. `sqrt(3)`
D. D. 5

Answer

To find the modulus of the expression \(\left| \frac{1 + 2i}{1 - (1 - i)^2} \right|\), we will solve it step by step.

### Step 1: Simplify the denominator
First, we need to simplify the denominator \(1 - (1 - i)^2\).

Calculating \((1 - i)^2\):
\[ (1 - i)^2 = 1^2 - 2 \cdot 1 \cdot i + i^2 = 1 - 2i + (-1) = 0 - 2i = -2i \]

Now substitute this back into the denominator:
\[ 1 - (1 - i)^2 = 1 - (-2i) = 1 + 2i \]

### Step 2: Rewrite the expression
Now we can rewrite the original expression:
\[ \frac{1 + 2i}{1 + 2i} \]

### Step 3: Simplify the fraction
Since the numerator and the denominator are the same, we can simplify:
\[ \frac{1 + 2i}{1 + 2i} = 1 \]

### Step 4: Find the modulus
The modulus of a complex number \(z\) is given by \(|z|\). Since we have simplified our expression to \(1\):
\[ |1| = 1 \]

### Conclusion
Thus, the modulus of the given expression is:
\[ \boxed{1} \]