# 2007 H2 Math

## Paper 1

1. $x<-3$ or $-2< x <-1$ or $x>7$.
2. (i) $fg$ does not exist since $R_g = [0,\infty) \not \subseteq (-\infty,3)\cup (3,\infty) = D_f$.
$gf: x \mapsto \frac{1}{(x-3)^2}, \quad \textrm{for } x \in \mathbb{R},x\neq 3$.
(ii) $f^{-1}(x)=\frac{1}{x}+3$, $D_{f^{-1}} = R_f = (-\infty,0) \cup (0,\infty)$.
3. (b) $w=-1+2i$.
4. $I = \frac{2}{3} (1+2\mathrm{e}^{-\frac{3}{4}t})$.
For large values of $t$, the current approaches $\frac{2}{3}$.
5. $A=2, B=3$.
Transformation 1: Translate the graph of $y=\frac{1}{x}$ along the negative $x$-axis direction by 2 units.
Transformation 2: Scale the resulting graph by a scale factor of 3 parallel to the $y$-axis.
6. (i) Since $\overrightarrow{OA}\cdot\overrightarrow{OB}=0, OA$ is perpendicular to $OB$.
(ii) $\overrightarrow{OM} = \frac{1}{3} \begin{pmatrix} 4 \\ 2 \\ 5 \end{pmatrix}$.
(iii) $\sqrt{35} \textrm{ units}^2$.
7. (i) $r\mathrm{e}^{-i\theta}$.
8. (i) $\left ( \frac{5}{2}, \frac{3}{2}, \frac{11}{2} \right )$.
(ii) $78.8^\circ$.
(iii) $\frac{4\sqrt{14}}{7}$.
9. (i) $\alpha = 0.619, \beta = 1.512$.
10. (ii) Since $r=\frac{2}{3}, |r|<1$. Hence the geometric series is convergent and $S_\infty = 3a$.
(iii) $\{ n: 6 \leq n \leq 13, n \in \mathbb{Z} \}$.
11. (i)
(iii) $\frac{2}{5} \textrm{ units}^2$.

## Paper 2

1. $7.65. 2. (ii)$1-\frac{1}{(N+1)^2}$. (iii) As$N\to \infty$,$\frac{1}{(N+1)^2}\to 0$so$1-\frac{1}{(N+1)^2}\to 1$. Hence the series is convergent and the sum to infinity is$1$. (iv)$1-\frac{1}{N^2}$. 3. (i)$(1+x)^n = 1 + nx + \frac{n(n-1)}{2}x^2 + \frac{n(n-1)(n-2)}{6}x^3+\ldots$(ii)$8 – 3x + \frac{387}{16}x^2 – \frac{1151}{128}x^3+\ldots$(iii) Expansion is valid for$-\frac{1}{\sqrt{2}} < x < \frac{1}{\sqrt{2}}$. 4. (i)$\int^{\frac{5}{3}}_0 \sin^2 x \mathrm{ d}x = \frac{5\pi}{6}+\frac{\sqrt{3}}{8}$.$\int^{\frac{5}{3}}_0 \cos^2 x \mathrm{ d}x = \frac{5\pi}{6}-\frac{\sqrt{3}}{8}$. (ii) (a)$(\pi – 2) \textrm{ units}^2$. (b)$5.391 \textrm{ units}^3$. 5. (Out of syllabus) 6. 0.933. (i) *0.717. (ii) *0.616 7. Unbiased estimate of population mean = 30.84. Unbiased estimate of population variance = 33.7.$p$-value = 0.0382. Reject$H_0$. We have sufficient evidence at 5% level of significance to conclude that the mean time for a student to complete the project exceeds 30 hours. No assumptions are needed since the samplae size is large. By Central Limit Theorem, the distribution of$\overline{X}$is normally distributed approximately. 8. (i) 0.395. (ii) 0.160. (iii) 0.392. (iv) It is because the event in (ii) is a subset of the event in (iii). 9. (i) (a) 479 001 600. (b) 46 080. (ii)(a) 39 916 800. (b) 86 400. (c) 240. 10. (i)$\frac{1}{64}$.$\frac{21}{256}.\frac{13}{17}$. 11.$x=-0.260t+66.2$.$x=-11.8.$The linear model is not suitable as the value of$x$, the concentration, should not be negative. (i)$r=-0.994$. This indicates a strong negative linear correlation between$\ln x$and$t$. (ii)$\ln x = -0.0123t+4.62.t=155\$ minutes.