Question

Let \( z = 1 + i \) and \( z_1 = \frac{1 + \bar{z}}{z(1 - z) + 1} \), where \( \bar{z} \) denotes the conjugate of \( z \). Then\(\frac{12}{\pi} \arg(z_1)\) is equal to:

Answer 1

Correct Answer: 3

We are given the complex number \( z = 1 + i \) and the equation:

\[ z_1 = \frac{i + \overline{z}(1 - i)}{\overline{z}(1 - z)}, \]

where \(\overline{z}\) denotes the complex conjugate of \(z\). We are tasked to find \(12 \pi \cdot \arg(z_1)\).

Step 1: Compute the complex conjugate of \(z\)

The complex conjugate of \(z = 1 + i\) is:

\[ \overline{z} = 1 - i. \]

Step 2: Simplify the numerator

The numerator is \(i + \overline{z}(1 - i)\). Substitute \(\overline{z} = 1 - i\):

\[ \overline{z}(1 - i) = (1 - i)(1 - i). \]

Expand the product:

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

Thus, the numerator becomes:

\[ i + (-2i) = -i. \]

Step 3: Simplify the denominator

The denominator is \(\overline{z}(1 - z)\). Substitute \(\overline{z} = 1 - i\) and \(z = 1 + i\):

\[ \overline{z}(1 - z) = (1 - i)(1 - (1 + i)) = (1 - i)(-i). \]

Expand the product:

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

Step 4: Simplify \(z_1\)

Substitute the simplified numerator and denominator into the expression for \(z_1\):

\[ z_1 = \frac{-i}{-(1 + i)} = \frac{i}{1 + i}. \]

To simplify further, multiply the numerator and denominator by the complex conjugate of the denominator, \(1 - i\):

\[ z_1 = \frac{i(1 - i)}{(1 + i)(1 - i)}. \]

Expand the denominator:

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

Expand the numerator:

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

Thus:

\[ z_1 = \frac{1 + i}{2} = \frac{1}{2} + \frac{i}{2}. \]

Step 5: Find the argument of \(z_1\)

The complex number \(z_1 = \frac{1}{2} + \frac{i}{2}\) has real part \(\frac{1}{2}\) and imaginary part \(\frac{1}{2}\). The argument is given by:

\[ \arg(z_1) = \tan^{-1}\left(\frac{\text{Imaginary part}}{\text{Real part}}\right) = \tan^{-1}\left(\frac{\frac{1}{2}}{\frac{1}{2}}\right) = \tan^{-1}(1). \]

Since \(\tan^{-1}(1) = \frac{\pi}{4}\), we have:

\[ \arg(z_1) = \frac{\pi}{4}. \]

Step 6: Compute \(12 \pi \cdot \arg(z_1)\)

Finally, multiply the argument by \(12 \pi\):

\[ 12 \pi \cdot \arg(z_1) = 12 \pi \cdot \frac{\pi}{4} = 3. \]

Conclusion

The value of \(12 \pi \cdot \arg(z_1)\) is:

\[ \boxed{3}. \]