(i) To find angle BOC, use the formula for the angle in radians: \(\theta = 2 \tan^{-1} \left( \frac{1}{2} \right)\). Calculating gives \(\theta = 0.9273\) radians.

(ii) First, find the radius OB using the Pythagorean theorem: \(OB = \sqrt{10^2 + 5^2} = \sqrt{125} = 11.2\) cm.

Next, calculate the arc length BXC using \(s = r\theta\), where \(r = \sqrt{125}\) and \(\theta = 0.9273\). Thus, \(s = 11.2 \times 0.9273 = 10.4\) cm.

The perimeter of the shaded region is the arc length plus the side BC of the square: \(10.4 + 10 = 20.4\) cm.

(iii) The area of the sector BXC is given by \(\frac{1}{2} r^2 \theta\). Substituting the values, \(\frac{1}{2} \times 11.2^2 \times 0.9273 = 7.96\) cm².