Q1
Answer
Find, to 1 decimal place, the values of $\theta$ in the range $0 \leq \theta \leq 180^{\circ}$ for which $4\sqrt{3}\sin(3\theta+20^{\circ})=4\cos(3\theta+20^{\circ})$
$3.3, 63.3, 123.3$
Q2
Answer
A student attempts to solve the equation $2\cos2x+\sqrt{3}=0$ for $0 \leq x \leq 180^{\circ}$ as follows: $$\begin{align*} 2\cos2x &= -\sqrt{3}\\ \cos 2x &= -\dfrac{\sqrt{3}}{2}\\ 2x &= \cos^{-1}\left(-\dfrac{\sqrt{3}}{2}\right)\\ 2x &= 150^{\circ}\\ x &= 75^{\circ}\\ x &= 360^{\circ}-75^{\circ} = 295^{\circ}\ & \mathrm{out\ of\ range} \end{align*}$$ Identify the mistake made by the student and hence write down the correct solution(s) to the equation
Final line is the mistake, $2x = 360 - 150$ and $x = 105$ is the second solution
Q3
Answer
Prove from first principles that the derivative of $5x^3$ is $15x^2$.
$\displaystyle\lim_{h\rightarrow 0} \dfrac{5(x+h)^3 - 5x^3}{h}$ expand and simplify
Q4
Answer
The diagram shows the plan of a school running track. It consists of two straight sections, which are the opposite sides of a rectangle, and two semicircular sections, each of radius $r$ m. The length of the track is 300 m and it can be assumed to be very narrow.
- Show that the internal area, $A$ m$^2$, is given by $A = 300r - \pi r^2$.
- Hence find in terms of $\pi$ the maximum value of the internal area. You do not have to justify that the value is a maximum.
- Perimeter: $300 - 2\pi r + 2x \Rightarrow 2x = 300 - 2\pi r$ and area: $\pi r^2 + 2rx = 300r - \pi r^2$
- $\dfrac{22500}{\pi}$
Q5
Answer
A boat $Q$ is 7 km from another boat $P$ on a bearing of $327^{\circ}$. Another boat $R$ is 15 km from boat $P$ on a bearing of $041^{\circ}$.
- Find the distance between boats $Q$ and $R$ to 1 decimal place.
- Find the bearing of boat $R$ from boat $Q$.
- $14.7$ km
- $068^{\circ}$