(i) The area of sector OAB is \(\frac{1}{2} \times 11^2 \times \alpha\).

The area of sector OCD is \(\frac{1}{2} \times 5^2 \times \alpha\).

The area of the shaded region ABDC is \(\frac{1}{2} \times 11^2 \times \alpha - \frac{1}{2} \times 5^2 \times \alpha\).

Thus, \(k = \frac{\frac{1}{2} \times 11^2 \times \alpha - \frac{1}{2} \times 5^2 \times \alpha}{\frac{1}{2} \times 5^2 \times \alpha} = \frac{11^2 - 5^2}{5^2} = \frac{96}{25}\) or 3.84.

(ii) The perimeter of the shaded region ABDC is \(11\alpha + 5\alpha + 6 + 6 = 16\alpha + 12\).

The perimeter of the unshaded region OCD is \(5\alpha + 5 + 5 = 5\alpha + 10\).

Setting the perimeter of the shaded region equal to twice the perimeter of the unshaded region gives:

\(16\alpha + 12 = 2(5\alpha + 10)\)

\(16\alpha + 12 = 10\alpha + 20\)

\(6\alpha = 8\)

\(\alpha = \frac{4}{3}\) or 1.33.