# 2008 H2 Math Solutions

## Math Repository

We have digitized the answers and solutions from 2007 to 2022 at
Math Repo.

In particular, you can start navigating the 2008 answers and full worked solutions from
2008P1Q1.

They are designed to be digital and mobile friendly: we hope you will have a good experience there.

## Math Pro

Done with your TYS but Prelims still too difficult? Try out the questions
at Math Pro where we tweak the past TYS questions.

We are still under construction, but 6 pure math topics have been completed so far, with more to come!

## Paper 1

1. $a=2.73$
2. (Out of syllabus)

3. (i) $\overrightarrow{OP} = \begin{pmatrix}6 \\ 3 \\ -3 \end{pmatrix}$
(ii) $\theta = 87.8^{\circ}$
(iii) Area $=15\sqrt{3} \textrm{ units}^2$

4. (i) $y=\frac{3}{2}\ln(x^2+1)+c$
(ii) $y=\frac{3}{2}\ln(x^2+1)+2$
(iii) Gradient of every solution curve approaches 0

5. (i) $\frac{\pi}{9}$
(ii) $\frac{x^{n+1}}{n+1}\left( \ln x – \frac{1}{n+1} \right) +c$
$\frac{e^{n+1}}{n+1} + \frac{1-e^{n+1}}{(n+1)^2}$

6. (i) $a=2, b=\frac{3}{4}$
(ii) $1+4x+8x^2+\ldots$
7. $x=6.09, y=12.6$ gives a maximum area

8. (i) $-8$
(ii) $a=-3, b=6$
(iii) $z=1\pm\sqrt{3}i$ or $z=-\frac{1}{2}$

9. (ii) $y=f(x)$ is a horizontal line parallel to the $x$-axis

10. (ai) $n=34$. 1 October 2011
(aii) $\$6.08$(b)$\$310$
(c) 81 months

11. $\left(-\frac{4}{11}, -\frac{4}{11}, \frac{7}{11} \right)$
$\mathbf{r} = \begin{pmatrix}-1\\-1\\0\end{pmatrix}+\lambda\begin{pmatrix}1\\1\\1\end{pmatrix}, \lambda \in \mathbb{R}$

## Paper 2

1. (ii) $x+x^2+\frac{x^3}{3}+\ldots$
(iv) $-1.96 < x < 1.56$

2. (i) 0.999
(ii) $\frac{4}{15} \pi \textrm{ units}^2$
(iii) $x=\frac{2}{3}$

3. (a)$|p|=1, \arg p = 2\theta$
$\theta=\frac{\pi}{5}, \frac{2\pi}{5}$

4. (ii)$f^{-1}(x)=4+\sqrt{x-1}$, $D_{f^{-1}}=(1, \infty)$
(iv)$y=x$
$x=\frac{9+\sqrt{13}}{2}$
5. (Out of syllabus)
6. (Out of syllabus)

7. (i) 0.5
(ii) 0.512
(iii) $\frac{7}{64}$

8. (i) 0.970
(iii) $P(4.8,7.6)$ The remaining points are such that $t$ is increasing at a decreasing rate with respect to $x$, which is consistent with a logarithmic model.
(iv) $t=1.42472 + 4.39656 \ln x$
(v) 8.32
(vi) The estimate is not reliable as $x=8.0$ lies outside the given date range from 1.2 to 6.9.
9. (Out of syllabus)

10. (i) 120
(ii) 9
(iii) 210
(iv) 485

11. (i) 0.0385
(ii) $E(Y)=110, Var(Y)=576$
$a=3, b=-40$