1. Show that \(\frac{1}{1-\cos \theta}+\frac{1}{1+\cos \theta}=2 \operatorname{cosec}^{2} \theta\). [3]

2. Express \(\lg a+3 \lg b-3\) as a single logarithm. [3]

3. (a) Shade the region corresponding to the set given below each Venn diagram.

(b) Given that \(P=\left\{p: \tan p=1\right.\) for \(\left.0^{\circ} \leqslant p \leqslant 540^{\circ}\right\}\), find \(n (P)\).

4. (a) Solve the equation \(16^{3 x-2}=8^{2 x}\)

(b) Given that \(\frac{\sqrt{a^{\frac{4}{3}} b^{-\frac{2}{5}}}}{a^{-\frac{1}{3}} b^{\frac{3}{5}}}=a^{p} b^{q},\) find the value of \(p\) and of \(q\)

5. (i)

On the diagram above, sketch the curve \(y=1+3 \sin 2 x\) for \(0^{\circ} \leqslant x \leqslant 180^{\circ}\)

(ii)

On the diagram above, sketch the curve \(y=|1+3 \sin 2 x|\) for \(0^{\circ} \leqslant x \leqslant 180^{\circ}\).

(iii) Write down the number of solutions of the equation \(|1+3 \sin 2 x|=1\) for \(0^{\circ} \leqslant x \leqslant 180^{\circ}\).