The Voronoi diagram below shows four supermarkets represented by points with coordinates$\text{A}(0,0)$, $\text{B}(6,0)$, $\text{C}(0,6)$ and $\text{D}(2,2)$. The vertices $\text{X}$, $\text{Y}$, $\text{Z}$ are also shown. All distancesare measured in kilometres.

The equation of $\text{(XY)}$ is $y=2-x$ and the equation of $\text{(YZ)}$ is $y=0.5x+3.5$.

The coordinates of $\text{Y}$ are $(-1,3)$ and the coordinates of $\text{Z}$ are $(7,7)$.

A town planner believes that the larger the area of the Voronoi cell $\text{XYZ}$, the more people willshop at supermarket $\text{D}$.

Consider the function$f\left(x\right)=\frac{48}{x}+k{x}^2-<!--\; -\; -->58$, where x > 0 and k is a constant.

The graph of the function passes through the point with coordinates (4 , 2).

P is the minimum point of the graph of f (x).

Let $f\left(x\right)=12\text{cos}x-<!--\; -\; -->5\text{sin}x,-<!--\; -\; -->\mathrm{\pi <!--\; \pi \; -->}⩽<!--\; ⩽\; -->x⩽<!--\; ⩽\; -->2\mathrm{\pi <!--\; \pi \; -->}$,be a periodic function with$f\left(x\right)=f\left(x+2\mathrm{\pi <!--\; \pi \; -->}\right)$

The following diagram shows the graph of$f$.

There is a maximum point at A. The minimum value of $f$ is −13 .

A ball on a spring is attached to a fixed point O. The ball is then pulled down and released, so that it moves back and forth vertically.

The distance, d centimetres, of the centre of the ball from O at time t seconds, is given by

$d\left(t\right)=f\left(t\right)+17,0⩽<!--\; ⩽\; -->t⩽<!--\; ⩽\; -->5.$

Let$f\left(x\right)=2\text{sin}\left(3x\right)+4$ for$x\in <!--\; \in \; -->\mathbb{R}$.

Let$g\left(x\right)=5f\left(2x\right)$.

The function $g$ can be written in the form $g\left(x\right)=10\text{sin}\left(bx\right)+c$.

A wind turbine is designed so that the rotation of the blades generates electricity. The turbine is built on horizontal ground and is made up of a vertical tower and three blades.

The point $\text{A}$ is on the base of the tower directly below point $\text{B}$ at the top of the tower. The height of the tower, $\text{AB}$, is $90 \text{m}$. The blades of the turbine are centred at $\text{B}$ and are each of length $40 \text{m}$. This is shown in the following diagram.

The end of one of the blades of the turbine is represented by point $\text{C}$ on the diagram. Let $h$ be the height of $\text{C}$ above the ground, measured in metres, where $h$ varies as the blade rotates.

Find the

The blades of the turbine complete $12$ rotations per minute under normal conditions, moving at a constant rate.

The height, $h$, of point $\text{C}$ can be modelled by the following function. Time, $t$, is measuredfrom the instant when the blade $\text{[BC]}$ first passes $\text{[AB]}$ and is measured in seconds.

$h\left(t\right)=90-40 \cos \left(72t°\right),t\ge 0$

Looking through his window, Tim has a partial view of the rotating wind turbine. The positionof his window means that he cannot see any part of the wind turbine that is more than $\mathbf{100}\mathbf{}\mathbf{\text{m}}$above the ground. This is illustrated in the following diagram.

A factory produces shirts. The cost, C, in Fijian dollars (FJD), of producing x shirts can be modelled by

C(x) = (x− 75)² + 100.

The cost of production should not exceed 500 FJD. To do this the factory needs to produce at least 55 shirts and at most s shirts.

Let f(x) = ax² − 4x − c. A horizontal line, L , intersects the graph of f at x = −1 and x = 3.

The graph of the quadratic function$f\left(x\right)=\frac{1}{2}\left(x-2\right)\left(x+8\right)$intersects the $y$-axis at $\left(0,c\right)$.

The vertex of the function is $\left(-3,-12.5\right)$.

The equation $f\left(x\right)=12$ has two solutions. The first solution is $x=-10$.

Let $T$ be the tangent at $x=-3$.