# Add Maths 2010 May-June Paper 11

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 $$0^{\circ} \leqslant x \leqslant 180^{\circ}$$.