The volume of a hemisphere, V, is given by the formula

V = $\sqrt{\frac{4S^3}{243\mathrm{\pi <!--\; \pi \; -->}}}$,

where S is the total surface area.

The total surface area of a given hemisphere is 350 cm².

The following table shows the probability distribution of a discrete random variable $A$, in terms of an angle $\mathrm{\theta <!--\; \theta \; -->}$.

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

A line $L_1$ passes through the points $\text{A}(0,1,8)$ and $\text{B}(3,5,2)$.

Given that $L_1$ and $L_2$ are perpendicular, show that $p=2$.

The points $\text{A}$ and $\text{B}$ have position vectors $\left(\begin{array}{c}-<!--\; -\; -->2\\ 4\\ -<!--\; -\; -->4\end{array}\right)$ and $\left(\begin{array}{c}6\\ 8\\ 0\end{array}\right)$ respectively.

Point $\text{C}$ has position vector $\left(\begin{array}{c}-<!--\; -\; -->1\\ k\\ 0\end{array}\right)$. Let $\text{O}$ be the origin.

Find, in terms of $k$,

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%.

A cylindrical container with a radius of 8 cm is placed on a flat surface. The container is filled with water to a height of 12 cm, as shown in the following diagram.

A heavy ball with a radius of 2.9 cm is dropped into the container. As a result, the height of the water increases to $h$ cm, as shown in the following 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°.

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

AĈD = θ and AC = 14 cm

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 type of candy is packaged in a right circular cone that has volume $\text{100 c}{\text{m}}^{\text{3}}$ and vertical height 8 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.

The following diagram shows triangle ABC, with AB = 6 and AC = 8.

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 $\text{C}(3,1,p)$.

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

A lampshade, in the shape of a cone, has a wireframe consisting of a circular ring and four straight pieces of equal length, attached to the ring at points A, B, C and D.

The ring has its centre at point O and its radius is 20 centimetres. The straight pieces meet at point V, which is vertically above O, and the angle they make with the base of the lampshade is 60°.

This information is shown in the following diagram.

Three airport runways intersect to form a triangle, ABC. The length of AB is 3.1 km, AC is 2.6 km, and BC is 2.4 km.

A company is hired to cut the grass that grows in triangle ABC, but they need to know the area.

The following diagram shows a ball attached to the end of a spring, which is suspended from a ceiling.

The height, $h$ metres, of the ball above the ground at time $t$ seconds after being released can be modelled by the function $h\left(t\right)=0.4 \cos \left(\mathrm{\pi }t\right)+1.8$ where $t\ge 0$.

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.

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

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.

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

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

Consider $f\left(x\right)=4 \sin x+2.5$ and $g\left(x\right)=4 \sin \left(x-\frac{3\pi }{2}\right)+2.5+q$, where $x\in \mathrm{ℝ}$ and $q>0$.

The graph of $g$ is obtained by two transformations of the graph of $f$.

Consider $f\left(x\right)=4 \cos x\left(1-3 \cos 2x+3 {\cos }^2 2x-{\cos }^3 2x\right)$.

AC is a vertical communications tower with its base at C.

The tower has an observation deck, D, three quarters of the way to the top of the tower, A.

From a point B, on horizontal ground 250 m from C, the angle of elevation of D is 48°.

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 $\mathit{r}=\left(\begin{array}{c}-3\\ -19\\ 24\end{array}\right)+s\left(\begin{array}{c}2\\ 4\\ -5\end{array}\right)$.

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 $t\ge 0$.

Let $\sin <!--\; \; -->\mathrm{\theta <!--\; \theta \; -->}=\frac{\sqrt{5}}{3}$, where $\mathrm{\theta <!--\; \theta \; -->}$ is acute.

A triangular postage stamp, ABC, is shown in the diagram below, such that $\text{AB}=5\text{ cm},\mathrm{B}\stackrel{^<!--\; ^\; -->}{\mathrm{A}}\mathrm{C}={34}^{\circ <!--\; \circ \; -->},\mathrm{A}\stackrel{^<!--\; ^\; -->}{\mathrm{B}}\mathrm{C}={26}^{\circ <!--\; \circ \; -->}$ and $\mathrm{A}\stackrel{^<!--\; ^\; -->}{\mathrm{C}}\mathrm{B}={120}^{\circ <!--\; \circ \; -->}$.

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

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

The line L passes through A and B.

Consider the function $f$ defined by $f\left(x\right)=6+6 \cos x$, for $0\le x\le 4\pi$.

The following diagram shows the graph of $y=f\left(x\right)$.

The graph of $f$ touches the $x$-axis at points $\text{A}$ and $\text{B}$, as shown. The shaded region is enclosed by the graph of $y=f\left(x\right)$ and the $x$-axis, between the points $\text{A}$ and $\text{B}$.

The right cone in the following diagram has a total surface area of $12\mathrm{\pi }$, equal to the shaded area in the previous diagram.

The cone has a base radius of $2$, height $h$, and slant height $l$.

A calculator fits into a cuboid case with height 29 mm, width 98 mm and length 186 mm.

The following diagram shows a circle with centre $\text{O}$ and radius $r$.

Points $\text{A}$ and $\text{B}$ lie on the circumference of the circle, and $\text{A}\hat{\text{O}}\text{B}=1 \text{radian}$.

The perimeter of the shaded region is $12$.

Note: In this question, distance is in metres and time is in seconds.

Two particles $P_1$ and $P_2$ start moving from a point A at the same time, along different straight lines.

After $t$ seconds, the position of $P_1$ is given by r = $\left(\begin{array}{c}4\\ -<!--\; -\; -->1\\ 3\end{array}\right)+t\left(\begin{array}{c}1\\ 2\\ -<!--\; -\; -->1\end{array}\right)$.

Two seconds after leaving A, $P_1$ is at point B.

Two seconds after leaving A, $P_2$ is at point C, where $\stackrel{\to <!--\; \to \; -->}{\text{AC}}=\left(\begin{array}{c}3\\ 0\\ 4\end{array}\right)$.

A ladder on a fire truck has its base at point B which is 4 metres above the ground. The ladder is extended and its other end rests on a vertical wall at point C, 16 metres above the ground. The horizontal distance between B and C is 9 metres.

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.

A park in the form of a triangle, ABC, is shown in the following diagram. AB is 79 km and BC is 62 km. Angle A$\stackrel{\wedge <!--\; \wedge \; -->}{\text{B}}$C is 52°.

The vectors a = $\left(\begin{array}{c}4\\ 2\end{array}\right)$ and b = $\left(\begin{array}{c}k+3\\ k\end{array}\right)$ are perpendicular to each other.

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°.

Angle APB is acute.

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 $L_1$.

Consider the functions $f\left(x\right)=\sqrt{3}\sin x+\cos x$ where $0\le x\le \pi$ and $g\left(x\right)=2x$ where $x\in \mathrm{ℝ}$.

Consider the graph of the function $f\left(x\right)=2\text{sin}x$,  0 ≤ $x$ < $2\mathrm{\pi <!--\; \pi \; -->}$ . The graph of $f$ intersects the line $y=-<!--\; -\; -->1$ exactly twice, at point A and point B. This is shown in the following diagram.

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 a triangle $\text{ABC}$.

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

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

Julio is making a wooden pencil case in the shape of a large pencil. The pencil case consists of a cylinder attached to a cone, as shown.

The cylinder has a radius of r cm and a height of 12 cm.

The cone has a base radius of r cm and a height of 10 cm.