# 2015 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 2007 answers and full worked solutions from 2015P1Q1. 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. (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.