Paper 1

1. (i) $a=-3.594, b = -5.187, c=7.303$.
   (ii) $x=-0.589$.
   (iii) $y=-5.19x+7.30$.
2. (ii) $-1.73 < x < 0.414$ or $x > 1.73$.
3. (i) $\frac{1}{n} \left \{ f\left( \frac{1}{n} \right) + f\left( \frac{2}{n} \right) + \cdots + f\left( \frac{n}{n} \right) \right \}$ is the total area of $n$ rectangles.
   These rectangles approximate the area under the curve, which is given by $\displaystyle \int_0^1 f(x) \; \mathrm{d}x$.
   As $n \to \infty$, the approximation becomes exact and $\lim_{n \to \infty} \frac{1}{n} \left \{ f\left( \frac{1}{n} \right) + f\left( \frac{2}{n} \right) + \cdots + f\left( \frac{n}{n} \right) \right \} = \displaystyle \int_0^1 f(x) \; \mathrm{d}x$. (ii) $\frac{3}{4}$.
4. $k=\frac{1}{32}$.
5. (i) A scaling of scale factor $\frac{1}{4}$ parallel to the $y$-axis.
   A translation of 3 units along the positive $x$-axis direction.
6. (i) $y= 2x-2x^2 + \frac{8}{3}x^3 + \ldots$
   (ii) $a=2, b=\frac{5}{3}, c=-\frac{3}{5}$.
   Coefficient of $x^4 = -\frac{104}{27}$.
7. (i) $\overrightarrow{OC} = \frac{3}{5}\mathbf{a}$, $\overrightarrow{OD} = \frac{5}{11}\mathbf{b}$.
   (ii) $l_{AD}: \mathbf{r} = (1-\mu) \mathbf{a} + \frac{5}{11} \mu \mathbf{b}, \mu \in \mathbb{R}$.
   (iii) $\overrightarrow{OE} = \frac{9}{20}\mathbf{a} + \frac{1}{4} \mathbf{b}$.
   $AE:ED = 11:9$.
8. (i) $\{ T \in \mathbb{R}: 59 \leq T \leq 77 \}$.
   (ii) $\{ t \in \mathbb{R}: 63.9 \leq T \leq 74.4 \}$.
   (iii) 11 s.
9. (a) $w=a\pm i \frac{1}{\sqrt{3}}a$.
10. (iii) $b=\frac{\pi}{4}, a=0$, $\frac{\pi^3+8\pi^2-32\pi}{16\sqrt{2}}$.
11. (ii) $\tan \theta = \sqrt{2}$, $\left ( \left( \frac{2}{3} \right)^{\frac{3}{2}}, \frac{2}{\sqrt{3}} \right )$.
    (iii) 0.884.
    (iv) $a=\frac{3}{\sqrt{2}}$.

Paper 2

1. (i) Maximum $h=32$ cm.
   (ii) $t=160-40\sqrt{16-\frac{h}{2}}$, 46.9 years.
2. (i) $73.4^\circ$.
   (ii) $(3,1,-10),(\frac{13}{7},-\frac{5}{7},-\frac{46}{7})$.
   $(\frac{17}{7},\frac{1}{7},-\frac{58}{7})$.
   (iii) $36x-2y+11z=-4$.
3. (a)(i) Since every horizontal line $y=k, k\in \mathbb{R}$ intersects the graph of $y=f(x)$ at most once, $f$ is one-one and it has an inverse.
   (a)(ii) $f^{-1}(x) = \sqrt{1-\frac{1}{x}}$, $D_{f^{-1}} = (-\infty,0)$.
   (b) $\left (-\infty, 1-\frac{\sqrt{3}}{2} \right ] \; \cup \; \left [ 1 + \frac{\sqrt{3}}{2}, \infty \right )$.
4. (i) $A=1,B=-1$.
   (ii) $S_n = \frac{1}{3} - \frac{1}{2n+3}$.
   (iii) $499$.
5. (i) 0.224.
   (ii) 0.150.
   (iii) 0.824.
6. (i) 0.4.
   (ii) 0.185.
   (iii) $0.165 \leq P(A' \cap B' \cap C') \leq 0.365$.
7. (ii)(a) -0.9807
   (ii)(b) -0.9748
   (ii)(c) -0.9986
   (iii) $P=-0.147\sqrt{h}+34.8$.
   (iv) $P=-2.66\sqrt{m}+34.8$.
8. (i) 10080.
   (ii) 10079.
   (iii) 720.
   (iv) 5760.
   
9. (i) 0.0127.
   (ii) 0.0524.
   (iii) 0.742.