(i) The perimeter of the shaded region is given by the arc length \(r \cdot x\) plus the length of the arc \(2r \cdot \theta\), where \(\theta = \frac{\pi}{2} - x\). The total perimeter is \(r \cdot x + 2r \cdot \left( \frac{\pi}{2} - x \right) = \frac{1}{2} \cdot 2\pi r\). Simplifying gives \(x = \cos^{-1} \left( \frac{\pi}{4 + 4x} \right)\).

(ii) Calculate \(\cos^{-1} \left( \frac{\pi}{4 + 4 \cdot 1} \right) \approx 1.318\) and \(\cos^{-1} \left( \frac{\pi}{4 + 4 \cdot 1.5} \right) \approx 1.047\). Since \(1.047 < x < 1.318\), x lies between 1 and 1.5.

(iii) Using the iterative formula:

\(x_1 = \cos^{-1} \left( \frac{\pi}{4 + 4 \cdot 1} \right) = 1.3183\)

\(x_2 = \cos^{-1} \left( \frac{\pi}{4 + 4 \cdot 1.3183} \right) = 1.2146\)

\(x_3 = \cos^{-1} \left( \frac{\pi}{4 + 4 \cdot 1.2146} \right) = 1.2099\)

\(x_4 = \cos^{-1} \left( \frac{\pi}{4 + 4 \cdot 1.2099} \right) = 1.2109\)

\(x_5 = \cos^{-1} \left( \frac{\pi}{4 + 4 \cdot 1.2109} \right) = 1.2106\)

The value of x correct to 2 decimal places is 1.21.