(a) The modulus of a product is the product of the moduli: \( |zw| = |z| \cdot |w| \).

For \( z = 2 \left( \cos \frac{\pi}{5} + i \sin \frac{\pi}{5} \right) \), \( |z| = 2 \).

For \( w = 8 \left( \cos \frac{2k\pi}{5} – i \sin \frac{2k\pi}{5} \right) \), \( |w| = 8 \).

\[ |zw| = 2 \cdot 8 = 16 \] A1

[1 mark]

(b) The argument of a product is the sum of the arguments: \( \arg(zw) = \arg(z) + \arg(w) \). M1

For \( z \), \( \arg(z) = \frac{\pi}{5} \).

For \( w = 8 \left( \cos \frac{2k\pi}{5} – i \sin \frac{2k\pi}{5} \right) = 8 \left( \cos \frac{2k\pi}{5} + i \sin \left(-\frac{2k\pi}{5}\right) \right) \), \( \arg(w) = -\frac{2k\pi}{5} \).

\[ \arg(zw) = \frac{\pi}{5} – \frac{2k\pi}{5} = \frac{(1 – 2k)\pi}{5} \] A1

[2 marks]

(c) Given \( zw \in \mathbb{Z} \), so \( zw \) is a real integer.

(i) For \( zw \) to be real, \( \arg(zw) \) must be a multiple of \( \pi \):

\[ \frac{(1 – 2k)\pi}{5} = n\pi \Rightarrow 1 – 2k = 5n, \quad n \in \mathbb{Z} \] M1

\[ k = \frac{1 – 5n}{2} \]

Since \( k \in \mathbb{Z}^+ \), test values for \( n \):

For \( n = 0 \): \( k = \frac{1}{2} \), not an integer.

For \( n = -1 \): \( k = \frac{1 – 5(-1)}{2} = \frac{6}{2} = 3 \).

For \( n = -2 \): \( k = \frac{1 – 5(-2)}{2} = \frac{11}{2} \), not an integer.

For \( n = -3 \): \( k = \frac{1 – 5(-3)}{2} = 8 \).

The smallest positive integer \( k \) is 3. A1

[2 marks]

(ii) For \( k = 3 \):

\[ \arg(zw) = \frac{(1 – 2 \cdot 3)\pi}{5} = \frac{-5\pi}{5} = -\pi \]

\[ zw = |zw| \left( \cos(-\pi) + i \sin(-\pi) \right) = 16 \left( \cos \pi + i \sin(-\pi) \right) = 16 (-1 + i \cdot 0) = -16 \] M1 A1

[2 marks]

Total [7 marks]