(a) 

A segment of a circle is cut off from a rectangular board as shown in the diagram. If the radius of the circle is \(1\frac{1}{2}\) times the length of the chord; calculate, correct to 2 decimal places, the perimeter of the remaining portion. [Take \(\pi = \frac{22}{7}\)]

(b) Evaluate without using calculators or tables, \(\frac{3}{\sqrt{3}}(\frac{2}{\sqrt{3}} - \frac{\sqrt{12}}{6})\).

The frequency distribution table shows the marks obtained by 100 students in a Mathematics test.

(a) Draw the cumulative curve for the distribution.

(b) Use the graph to find the : (i) 60th percentile ; (ii) probability that a student passed the test if the pass mark was fixed at 35%.

An aeroplane flies due North from a town T on the equator at a speed of 950km per hour for 4 hours to another town P. It then flies eastwards to town Q on longitude 65°E. If the longitude of T is 15°E,

(a) represent this information in a diagram ;

(b) calculate the : (i) latitude of P, correct to the nearest degree ; (ii) distance between P and Q, correct to four significant figures. [Take \(\pi = \frac{22}{7}\); Radius of the earth = 6400km].

When one end of a ladder, LM, is placed against a vertical wall at a point 5 metres above the ground, the ladder makes an angle of 37° with the horizontal ground.

(a) Represent this information in a diagram ;

(b) Calculate, correct to 3 significant figures, the length of the ladder ;

(c) If the foot of the ladder is pushed towards the wall by 2 metres, calculate,correct to the nearest degree, the angle which the ladder nows makes with the ground.

(a) Simplify : \(\frac{1\frac{1}{4} + \frac{7}{9}}{1\frac{4}{9} - 2\frac{2}{3} \times \frac{9}{64}}\)

(b) Given that \(\sin x = \frac{2}{3}\), evaluate, leaving your answer in surd form and without using tables or calculator, \(\tan x - \cos x\).