Paper 2

1. $-2 < x < 1$
2. (i) $f(x) = 0.215x^2 – 0.490x + 3.281$
   (ii) $(1.14, \infty)$
3. (i) $y=\displaystyle – \frac{1}{p^3} x + \frac{3}{p}$.
   (ii) $Q(3p^2, 0)$, $R \displaystyle \left( 0, \frac{3}{p} \right)$.
   (iii) $\displaystyle y^2 = \frac{27}{8x}$.
4. (i) $1-3x^2 + 4x^4 + \ldots$
   (iia) $a-a^3+\displaystyle \frac{4}{5}a^5$, 0.540.
   (iib) 0.475.
5. (ii) $[0, 2]$.
   (iii) $a = 2 + \sqrt{5}$.
6. $\displaystyle \sum_{r=1}^n \cos r\theta = \frac{1}{2\sin {\textstyle\frac{1}{2} \theta} } \left( \sin ({\textstyle n + \frac{1}{2}}) \theta – \sin {\textstyle \frac{1}{2}\theta }\right)$.

Paper 2

4. (ai) $\displaystyle \frac{1}{4} – \left( \frac{1}{2}n^2 + \frac{1}{2} n + \frac{1}{4} \right) \mathrm{e}^{-2n}$
   (aii) $\displaystyle \frac{1}{4}$
   (b) $(2 \pi^2 – 4 \pi)$