(a) From \(\Delta\) AMC, \(\sin 60^\circ = \frac{\sqrt{3}}{2}\) \(\cos 30^\circ = \frac{\sqrt{3}}{2}\) \(\tan 60^\circ = \frac{\sqrt{3}}{1} = \sqrt{3}\) Hence, \(\frac{2 \tan 60^\circ + \cos 30^\circ}{\sin 60} = \frac{2 \times \sqrt{3} + \frac{\sqrt{3}}{2}}{\frac{\sqrt{3}}{2}}\) = \(\frac{4\sqrt{3} + \sqrt{3}}{2} \div \frac{\sqrt{3}}{2}\) = \(\frac{5\sqrt{3}}{2} \times \frac{2}{\sqrt{3}} = 5\) (b) In \(\Delta\) POT, \(\tan 36^\circ = \frac{|TO|}{1050}\) \(|OT| = 1050 \tan 36^\circ = 1050 \times 0.7265\) = 762.87m In \(\Delta\) POB, \(\tan 41^\circ = \frac{|OB|}{1050}\) \(|OB| = 1050 \tan 41^\circ = 1050 \times 0.8693\) = 912.751m \(|TB| = 912.751m - 762.87m = 149.88m\) Hence, the height of the control tower = 150 m (to the nearest meter). (ii) In \(\Delta\) POB, \(\cos 41^\circ = \frac{1050}{|PB|}\) \(|PB| = \frac{1050}{\cos 41^\circ} = \frac{1050}{0.7547}\) = 1,391.28m The shortest distance between the aeroplane and the base of the control tower = 1,391m (to the nearest metre).