Iron in the asteroid 16 Psyche is said to be valued at $8973$ quadrillion euros $\left(\text{EUR}\right)$, where one quadrillion $={10}^{15}$.

James believes the asteroid is approximately spherical with radius $113 \text{km}$. He uses this information to estimate its volume.

Three towns, $\text{A}$, $\text{B}$ and $\text{C}$ are represented as coordinates on a map, where the $x$ and $y$ axes represent the distances east and north of an origin, respectively, measured in kilometres.

Town $\text{A}$ is located at $(-6, -1)$ and town $\text{B}$ is located at $(8, 6)$. A road runs along the perpendicular bisector of $\text{[AB]}$. This information is shown in the following diagram.

The diagram below shows a helicopter hovering at point $\text{H}$, $380 \text{m}$ vertically above a lake. Point $\text{A}$ is the point on the surface of the lake, directly below the helicopter.

Minta is swimming at a constant speed in the direction of point $\text{A}$. Minta observes the helicopter from point $\text{C}$ as she looks upward at an angle of $25°$. After $15$ minutes, Minta is at point $\mathrm{B}$ and she observes the same helicopter at an angle of $40°$.

The Voronoi diagram below shows three identical cellular phone towers, $\text{T}1, \text{T}2$ and $\text{T}3$. A fourth identical cellular phone tower, $\text{T}4$ is located in the shaded region. The dashed lines in the diagram below represent the edges in the Voronoi diagram.

Horizontal scale: $1$ unit represents $1 \text{km}$.
Vertical scale: $1$ unit represents $1 \text{km}$.

Tim stands inside the shaded region.

Tower $\text{T2}$ has coordinates $(-9, 5)$ and the edge connecting vertices $\text{A}$ and $\text{B}$ has equation $y=3$.

Points $\text{A}$ and $\text{B}$ have coordinates $\left(1, 1, 2\right)$ and $\left(9, m, -6\right)$ respectively.

The line $L$, which passes through $\text{B}$, has equation $\mathit{r}=\left(\begin{array}{c}-3\\ -19\\ 24\end{array}\right)+s\left(\begin{array}{c}2\\ 4\\ -5\end{array}\right)$.

A piece of candy is made in the shape of a solid hemisphere. The radius of the hemisphere is $6 \text{mm}$.

In this question, all lengths are in metres and time is in seconds.

Consider two particles, $P_1$ and $P_2$, which start to move at the same time.

Particle $P_1$ moves in a straight line such that its displacement from a fixed-point is given by $s\left(t\right)=10-\frac{7}{4}{t}^2$, for $t\ge 0$.

A farmer owns a triangular field $\text{ABC}$. The length of side $\left[\text{AB}\right]$ is $85 \text{m}$ and side $\left[\text{AC}\right]$ is $110 \text{m}$. The angle between these two sides is $55°$.

The following diagram shows a triangle $\text{ABC}$.

$\text{AC}=15 \text{cm}, \text{BC}=10 \text{cm}$, and $\text{A}\hat{\text{B}}\text{C}=\theta$.

Let $\sin \text{C}\hat{\text{A}}\text{B}=\frac{\sqrt{3}}{3}$.

Joey is making a party hat in the form of a cone. The hat is made from a sector, $\text{AOB}$, of a circular piece of paper with a radius of $18 \text{cm}$ and $\text{AÔB}=\theta$ as shown in the diagram.

To make the hat, sides $\text{[OA]}$ and $\text{[OB]}$ are joined together. The hat has a base radius of $6.5 \text{cm}$.

There are four stations used by the fire wardens in a national forest.

On the following Voronoi diagram, the coordinates of the stations are $\text{A}(6, 2), \text{B}(14, 2), \text{C}(18, 6)$ and $\text{D}(10.8, 11.6)$ where distances are measured in kilometres.

The dotted lines represent the boundaries of the regions patrolled by the fire warden at each station. The boundaries meet at $\text{P}(10, 6)$ and $\text{Q}(13, 7)$.

To reduce the areas of the regions that the fire wardens patrol, a new station is to be built within the quadrilateral $\text{ABCD}$. The new station will be located so that it is as far as possible from the nearest existing station.

The Voronoi diagram is to be updated to include the region around the new station at $\text{P}$. The edges defined by the perpendicular bisectors of $\text{[AP]}$ and $\text{[BP]}$ have been added to the following diagram.

The owner of a convenience store installs two security cameras, represented by points $\text{C1}$ and $\text{C2}$. Both cameras point towards the centre of the store’s cash register, represented by the point $\text{R}$.

The following diagram shows this information on a cross-section of the store.

The cameras are positioned at a height of $3.1 \text{m}$, and the horizontal distance between the cameras is $6.4 \text{m}$. The cash register is sitting on a counter so that its centre, $\text{R}$, is $1.0 \text{m}$ above the floor.

The distance from Camera $1$ to the centre of the cash register is $2.8 \text{m}$.

An inclined railway travels along a straight track on a steep hill, as shown in the diagram.

The locations of the stations on the railway can be described by coordinates in reference to $x, y$, and $z$-axes, where the $x$ and $y$ axes are in the horizontal plane and the $z$-axis is vertical.

The ground level station $\text{A}$ has coordinates $(140, 15, 0)$ and station $\text{B}$, located near the top of the hill, has coordinates $(20, 5, 250)$. All coordinates are given in metres.

Station $\text{M}$ is to be built halfway between stations $\text{A}$ and $\text{B}$.

Points A(3, 1), B(3, 5), C(11, 7), D(9, 1) and E(7, 3) represent snow shelters in the Blackburn National Forest. These snow shelters are illustrated in the following coordinate axes.

Horizontal scale: 1 unit represents 1 km.

Vertical scale: 1 unit represents 1 km.

The Park Ranger draws three straight lines to form an incomplete Voronoi diagram.

A garden includes a small lawn. The lawn is enclosed by an arc $\text{AB}$ of a circle with centre $\text{O}$ and radius $6 \text{m}$, such that $\text{AÔB}=135°$. The straight border of the lawn is defined by chord $\text{[AB]}$.

The lawn is shown as the shaded region in the following diagram.

Ollie has installed security lights on the side of his house that are activated by a sensor. The sensor is located at point C directly above point D. The area covered by the sensor is shown by the shaded region enclosed by triangle ABC. The distance from A to B is 4.5 m and the distance from B to C is 6 m. Angle AĈB is 15°.

The Bermuda Triangle is a region of the Atlantic Ocean with Miami $\left(\text{M}\right)$, Bermuda $\left(\text{B}\right)$, and San Juan $\left(\text{S}\right)$ as vertices, as shown on the diagram.

The distances between $\text{M}$, $\text{B}$ and $\text{S}$ are given in the following table, correct to three significant figures.

Let $\stackrel{\to <!--\; \to \; -->}{\text{OA}}=\left(\begin{array}{c}2\\ 1\\ 3\end{array}\right)$ and $\stackrel{\to <!--\; \to \; -->}{\text{AB}}=\left(\begin{array}{c}1\\ 3\\ 1\end{array}\right)$, where O is the origin. L₁ is the line that passes through A and B.

The position vectors of points P and Q are i $+$ 2 j $-<!--\; -\; -->$ k and 7i $+$ 3j $-<!--\; -\; -->$ 4k respectively.

Money boxes are coin containers used by children and come in a variety of shapes. The money box shown is in the shape of a cylinder. It has a radius of 4.43 cm and a height of 12.2 cm.

A solid right circular cone has a base radius of 21 cm and a slant height of 35 cm.
A smaller right circular cone has a height of 12 cm and a slant height of 15 cm, and is removed from the top of the larger cone, as shown in the diagram.

Emily’s kite ABCD is hanging in a tree. The plane ABCDE is vertical.

Emily stands at point E at some distance from the tree, such that EAD is a straight line and angle BED = 7°. Emily knows BD = 1.2 metres and angle BDA = 53°, as shown in the diagram

T is a point at the base of the tree. ET is a horizontal line. The angle of elevation of A from E is 41°.

A vertical pole stands on horizontal ground. The bottom of the pole is taken as the origin, $\text{O}$, of a coordinate system in which the top, $\text{F}$, of the pole has coordinates $(0, 0, 5.8)$. All units are in metres.

The pole is held in place by ropes attached at $\text{F}$.

One of the ropes is attached to the ground at a point $\text{A}$ with coordinates $(3.2, 4.5, 0)$. The rope forms a straight line from $\text{A}$ to $\text{F}$.

The straight metal arm of a windscreen wiper on a car rotates in a circular motion from a pivot point, $\text{O}$, through an angle of $140°$. The windscreen is cleared by a rubber blade of length $46 \text{cm}$ that is attached to the metal arm between points $\text{A}$ and $\text{B}$. The total length of the metal arm, $\text{OB}$, is $56 \text{cm}$.

The part of the windscreen cleared by the rubber blade is shown unshaded in the following diagram.

A cylinder with radius $r$ and height $h$ is shown in the following diagram.

The sum of $r$ and $h$ for this cylinder is 12 cm.

A balloon in the shape of a sphere is filled with helium until the radius is 6 cm.

The volume of the balloon is increased by 40%.

Let $\mathrm{\theta <!--\; \theta \; -->}$ be an obtuse angle such that $\text{sin}\mathrm{\theta <!--\; \theta \; -->}=\frac{3}{5}$.

Let $f\left(x\right)={\text{e}}^x\text{sin}x-<!--\; -\; -->\frac{3x}{4}$.

Two schools are represented by points $\text{A}(2, 20)$ and $\text{B}(14, 24)$ on the graph below. A road, represented by the line $R$ with equation $-x+y=4$, passes near the schools. An architect is asked to determine the location of a new bus stop on the road such that it is the same distance from the two schools.

A line, $L_1$, has equation $r=\left(\begin{array}{c}-<!--\; -\; -->3\\ 9\\ 10\end{array}\right)+s\left(\begin{array}{c}6\\ 0\\ 2\end{array}\right)$. Point $\text{P}\left(15\text{,}9\text{,}c\right)$ lies on $L_1$.

Point A has coordinates (−4, −12, 1) and point B has coordinates (2, −4, −4).

The line L passes through A and B.

A solid glass paperweight consists of a hemisphere of diameter 6 cm on top of a cuboid with a square base of length 6 cm, as shown in the diagram.

The height of the cuboid, x cm, is equal to the height of the hemisphere.

A line $L$ passes through points $\text{A}(-<!--\; -\; -->3,4,2)$ and $\text{B}(-<!--\; -\; -->1,3,3)$.

The line $L$ also passes through the point $\text{C}(3,1,p)$.

Consider the vectors a = $\left(\begin{array}{c}0\\ 3\\ p\end{array}\right)$ and b = $\left(\begin{array}{c}0\\ 6\\ 18\end{array}\right)$.

Find the value of $p$ for which a and b are

Yao drains the oil from his motorbike into two identical cuboids with rectangular bases of width $10$ cm and length $40$ cm. The height of each cuboid is $5$ cm.

The oil from the motorbike fills the first cuboid completely and the second cuboid to a height of $2$ cm. The information is shown in the following diagram.

The following diagram shows triangle ABC, with $\text{AB}=3\text{ cm}$, $\text{BC}=8\text{ cm}$, and $\mathrm{A}\stackrel{^<!--\; ^\; -->}{\mathrm{B}}\mathrm{C}=\frac{\mathrm{\pi <!--\; \pi \; -->}}{3}$.

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°.

Two fixed points, A and B, are 40 m apart on horizontal ground. Two straight ropes, AP and BP, are attached to the same point, P, on the base of a hot air balloon which is vertically above the line AB. The length of BP is 30 m and angle BAP is 48°.