SPM Additional Mathematics 2017, Paper 1 (Question 14 – 16)

Question 15 (4 marks):
It is given that cos α = t where t is a constant and 0^o ≤ α ≤ 90^o.
Express in terms of t
(a) sin (180^o + α),
(b) sec 2α.

Solution:
(a)




$\begin{array}{l}\text{sin}\left({180}^o+\alpha \right)\\ =\sin 180\cos \alpha +\cos 180\sin \alpha \\ =0-\sin \alpha \\ =-\sin \alpha \\ =-\sqrt{1-t^2}\end{array}$

(b)
$\begin{array}{l}\sec 2\alpha =\frac{1}{\cos 2\alpha }\\ =\frac{1}{2{\cos }^2\alpha -1}\\ =\frac{1}{2t^2-1}\end{array}$

Question 16 (3 marks):
Diagram 7 shows two sectors AOD and BOC of two concentric circles with centre O.

Diagram 7

The angle subtended at the centre O by the major arc AD is 7α radians and the perimeter of the whole diagram is 50 cm.
Given OB = r cm, OA = 2OB and ∠BOC = 2α, express r in terms of α.


Solution:

$\begin{array}{l}\text{Length of major arc }AOD\\ =2r\times 7\alpha \\ =14r\alpha \\ \\ \text{Length of minor arc }BOC\\ =r\times 2\alpha \\ =2r\alpha \\ \\ \text{Perimeter of the whole diagram}\\ =50\text{ cm}\\ 14r\alpha +2r\alpha +r+r=50\\ 16r\alpha +2r=50\\ 8r\alpha +r=25\\ r\left(8\alpha +1\right)=25\\ r=\frac{25}{8\alpha +1}\end{array}$