The diameter of a spherical planet is $6 \times 10^4$ km.

By using the substitution $u=\sec x$ or otherwise, find an expression for $\int_0^{\frac{\pi}{3}} \sec^n x \tan x, dx$ in terms of $n$, where $n$ is a non-zero real number.

The following diagram shows quadrilateral ABCD. ${AB} = 11\,cm,\, {BC} = 6\,cm,\, B\widehat{A}D = 100^\circ, \text{ and } CB\widehat{D} = 82^\circ$

A farmer is placing posts at points $A$, $B$, and $C$ in the ground to mark the boundaries of a triangular piece of land on his property.

From point $A$, he walks due west 230 metres to point $B$. From point $B$, he walks 175 metres on a bearing of $063°$ to reach point $C$.

This is shown in the following diagram.

The farmer wants to divide the piece of land into two sections. He will put a post at point $D$, which is between $A$ and $C$. He wants the boundary $BD$ to divide the piece of land such that the sections have equal area.

This is shown in the following diagram.

The voltage $v$ in a circuit is given by the equation

$v(t) = 3\sin(100\pi t)$

where $t \geqslant 0$ is measured in seconds.

The current $i$ in this circuit is given by the equation

$i(t) = 2\sin(100\pi(t + 0.003))$

The power $p$ in this circuit is given by $p(t) = v(t) \times i(t)$.

The average power $p_{av}$ in this circuit from $t = 0$ to $t = T$ is given by the equation

$p_{av}(T) = \frac{1}{T}\int_0^T p(t)dt$

where $T > 0$.

The following diagram shows a right triangle ABC. Point D lies on AB such that CDbisects AĈB.

AĈD = $θ$ and AC = 14 cm

A buoy is floating in the sea and can be seen from the top of a vertical cliff. A boat is travelling from the base of the cliff directly towards the buoy.

The top of the cliff is 142 m above sea level. Currently the boat is 100 metres from the buoy and the angle of depression from the top of the cliff to the boat is 64°.

The following diagram shows triangle $ABC$, with $AB=10$, $BC=x$ and $AC=2x$. Given that $\cos^{\hat{}}C = \frac{3}{4}$, find the area of the triangle. Give your answer in the form $\frac{p\sqrt{q}}{2}$ where $p,q \in \mathbb{Z}^+$.

[ \begin{{align*}} \text{{A sector of a circle with radius }} r \text{{ cm , where }} r > 0 \text{{, is shown on the following diagram.}} \ \text{{The sector has an angle of 1 radian at the centre.}} \end{{align*}} ]

[ \text{{}} ]

[ \text{{Let the area of the sector be }} A \text{{ cm}}^2 \text{{ and the perimeter be }} P \text{{ cm. Given that }} A = P \text{{, find the value of }} r\text{{.}} ]

A sector of a circle with radius ({r}) cm, where (r > 0), is shown on the following diagram. The sector has an angle of 1 radian at the centre.

Let the area of the sector be (A) cm^2 and the perimeter be (P) cm. Given that (A = P), find the value of (r).

The following diagram shows the quadrilateral ABCD. AB = 6.73 cm, BC = 4.83 cm, BĈD = 78.2° and CD = 3.80 cm.