Write down the radius of the planet.

The volume of the planet can be expressed in the form ${\pi \cdot a \cdot 10^k \, \text{km}^3}$ where ${1 \leq a < 10}$ and ${k \in \mathbb{Z}}$.

Find the value of ${a}$ and the value of ${k}$.

Calculate the percentage of the population of Ottawa that speak English but not French.

Calculate the number of people in Ottawa that speak both English and French.

Write down your answer to part (b) in the form $a \times 10^k$ where $1 \leqslant a < 10$ and $k  \in \mathbb{Z}$.

Show that ${b}=21$.

The third term in the expansion is the mean of the second term and the fourth term in the expansion.

Find the possible values of $x$.

Find the value of ${A}$ and the value of ${B}$.

Hence, expand $\frac{2+7x}{(1+2x)(1-x)}$ in ascending powers of $x$, up to and including the term in $x^2$.

Give a reason why the series expansion found in part (b) is not valid for $x=\frac{3}{4}$.

Find the solution of the system of equations when ${a = 2}$.

Expand and simplify $(1-a)^3$ in ascending powers of $a$.

By using a suitable substitution for $a$, show that

$1-3\cos^2 x+3\cos^2 2x - \cos^3 2x = 8\sin^6 x$.

Show that $\int_0^m f(x) dx = \frac{32}{7} \sin^7m$ where $m$ is a positive real constant.

By considering the area of two triangles and the area of the sector show that

${\cos\alpha \sin\alpha < \alpha < \frac{\sin\alpha}{\cos\alpha}}$.

Hence show that ${\lim_{\alpha \to 0} \frac{\alpha}{{\sin \alpha}} = 1}$.

Let ${z^n} = 1$, $z \in \mathbb{C}$, $n \in \mathbb{N}$, $n \geqslant 5$. Working in modulus/argument form find the $n$ solutions to this equation.

Represent these $n$ solutions on an Argand diagram. Let their positions be denoted by $P_0, P_1, P_2, \ldots P_{n - 1}$ placed in order in an anticlockwise direction round the circle, starting on the positive $x$-axis. Show the positions of $P_0, P_1, P_2$, and $P_{n - 1}$.

Show that the length of the line segment$P_0P_1$ is $2\sin\frac{\pi }{n}$.

Hence, write down the total length of the perimeter of the regular $n$-sided polygon $P_0P_1P_2 \ldots P_{n - 1}P_0$.

Using part (b) find the limit of this perimeter as $n \to \infty$.

Find the total area of this ${n}$ sided polygon.

Show that $p = \pm \frac{1}{\sqrt{3}}$.

Hence or otherwise, show that the series is convergent.

Given that $p > 0$ and $S_\infty = 3 + \sqrt{3}$, find the value of $x$.

Now consider the case where the series is arithmetic with common difference $d$.

Show that $p=\frac{2}{3}$.

Write down $d$ in the form $k \ln x$, where $k \in \mathbb{Q}$.

The sum of the first $n$ terms of the series is $\ln\left(\frac{1}{{x^3}}\right)$.

Find the value of $n$.