# SPM Additional Mathematics (Model Test Paper)

SPM Additional Mathematics (Model Test Paper)

Section B
[40 marks]
Answer any four questions from this section.

Question 7
(a)$\text{Prove that}{\left(\frac{\mathrm{cos}ec\text{}x-\mathrm{sec}x}{\mathrm{sec}x\text{}\mathrm{cos}ec\text{}x}\right)}^{2}=1-\mathrm{sin}2x$
[3 marks]
(b)(i) Sketch the graph of y = 1 – sin2x for 0 ≤ x ≤ 2π.

(b)(ii) Hence, using the same axes, sketch a suitable straight line to find the number of solutions to the equation $2-{\left(\frac{\mathrm{cos}ec\text{}x-\mathrm{sec}x}{\mathrm{sec}x\text{}\mathrm{cos}ec\text{}x}\right)}^{2}=\frac{x}{\pi }$for 0 ≤ x ≤ 2π.
State the number of soultions.
[7 marks]

$\begin{array}{l}LHS\\ ={\left(\frac{\mathrm{cos}ec\text{}x-\mathrm{sec}x}{\mathrm{sec}x\text{}\mathrm{cos}ec\text{}x}\right)}^{2}\\ ={\left(\frac{\overline{)\mathrm{cos}ec\text{}x}}{\mathrm{sec}x\text{}\overline{)\mathrm{cos}ec\text{}x}}-\frac{\overline{)\mathrm{sec}x}}{\overline{)\mathrm{sec}x}\text{}\mathrm{cos}ec\text{}x}\right)}^{2}\\ ={\left(\frac{1}{\mathrm{sec}x}-\frac{1}{\mathrm{cos}ec\text{}x}\right)}^{2}\\ ={\left(\mathrm{cos}x-\mathrm{sin}x\right)}^{2}\\ ={\mathrm{cos}}^{2}x-2\mathrm{cos}x\mathrm{sin}x+{\mathrm{sin}}^{2}x\\ =1-\mathrm{sin}2x\text{(RHS)}\end{array}$
$\begin{array}{l}2-{\left(\frac{\mathrm{cos}ec\text{}x-\mathrm{sec}x}{\mathrm{sec}x\text{}\mathrm{cos}ec\text{}x}\right)}^{2}=\frac{x}{\pi }\\ 2-\left(1-\mathrm{sin}2x\right)=\frac{x}{\pi }←\overline{)\text{From 7(a)}}\\ 2-y=\frac{x}{\pi }←\overline{)\text{From 7(b)(i)}}\\ y=2-\frac{x}{\pi }\text{(suitable straight line)}\end{array}$