DP Mathematics: Applications and Interpretation Questionbank

Find the velocity vector at time $t$.

Show that the magnitude of the velocity of the ice-skater at time $t$ is given by

$a {\text{e}}^{bt}\sqrt{\left(1+b^2\right)}$.

Find the magnitude of the velocity of the ice-skater when $t=2$.

Find the value of $a$ and the value of $b$.

At a point $\text{P}$, the ice-skater is skating parallel to the $y$-axis for the first time.

Find $\text{OP}$.

Find the size of $\text{AĈB}$.

Find $\text{DE}$.

Find the area of triangle $\text{DCE}$.

Estimate $\text{DF}$. You may assume the highway has a width of zero.

Given that $\sin 60°=\frac{\sqrt{3}}{2}$, show that the area of the base of the box is equal to $\frac{3\sqrt{3}{x}^2}{2}$.

Given that the total external surface area of the box is $1200 {\text{cm}}^2$, show that the volume of the box may be expressed as $V=300\sqrt{3}x-\frac{9}{4}{x}^3$.

Verify that $G$ satisfies the handshaking lemma.

Show that $G$ cannot be redrawn as a planar graph.

State, giving a reason, whether $G$ contains an Eulerian circuit.

Describe, in words, the effect of the transformation $T$.

Show that the points $\text{A}(1, 4), \text{B}(4, 8), \text{C}(8, 5), \text{D}(5, 1)$ form a square.

Determine the area of this square.

Find the coordinates of $\text{A', B', C', D'}$, the points to which $\text{A, B, C, D}$ are transformed under $T$.

Show that $\text{A' B' C' D'}$ is a parallelogram.

Determine the area of this parallelogram.

Use Dijkstra’s algorithm to find the shortest time to travel from $\text{C}$ to $\text{F}$, clearly showing how the algorithm has been applied.

Hence write down the shortest path from $\text{C}$ to $\text{F}$.

A new road is constructed that allows Christine to travel from $\text{H}$ to $\text{A}$ in $t$ minutes. If Christine starts from home and uses this new road her minimum travel time to $\text{A}$ is reduced, but her minimum travel time to $\text{F}$ remains the same.

Find the possible values of $t$.

Find the maximum possible value of $e$.

Find an expression for $e\text{'}$ in terms of $e$.

Given that the complement of $G$ is also planar and connected, find the possible values of $e$.

$H$ is a simple graph with $v$ vertices and $e$ edges.

Given that both $H$ and its complement are planar and connected, find the maximum possible value of $v$.

Given that $\text{A}\hat{\text{B}}\text{C}$ is acute, find $\sin \theta$.

Find $\cos \left(2\times \text{C}\hat{\text{A}}\text{B}\right)$.

Find an expression for the velocity of $P_1$ at time $t$.

Particle $P_2$ also moves in a straight line. The position of $P_2$ is given by $\mathit{r}=\left(\begin{array}{c}-1\\ 6\end{array}\right)+t\left(\begin{array}{c}4\\ -3\end{array}\right)$.

The speed of $P_1$ is greater than the speed of $P_2$ when $t>q$.

Find the value of $q$.

Express $\overrightarrow{\text{AB}}$ in terms of $m$.

Find the value of $m$.

Consider a unit vector $\mathit{u}$, such that $\mathit{u}=p\mathit{i}-\frac{2}{3}\mathit{j}+\frac{1}{3}\mathit{k}$, where $p>0$.

Point $\text{C}$ is such that $\overrightarrow{\text{BC}}=9\mathit{u}$.

Find the coordinates of $\text{C}$.

Show that $\text{AC}=\tan \theta$.

Find the value of $\theta$ when the area of the shaded region is equal to the area of sector $\text{OADB}$.

Write down the value of the iron in the form $a\times {10}^k$ where $1\le a<10 , k\in \mathrm{\mathbb{Z}}$.

Calculate James’s estimate of its volume, in ${\text{km}}^3$.

The actual volume of the asteroid is found to be $6.074\times {10}^6 {\text{km}}^3$.

Find the percentage error in James’s estimate of the volume.

Calculate the length of $\text{BD}$.

Show that angle $\text{EDC}=48.0°$, correct to three significant figures.

Calculate the area of triangle $\text{BDC}$.

Pedro draws a circle, with centre at point $\text{E}$, passing through point $\text{C}$. Part of the circle is shown in the diagram.

Show that point $\text{A}$ lies outside this circle. Justify your reasoning.

Find the area of the field.

The farmer would like to divide the field into two equal parts by constructing a straight fence from $\text{A}$ to a point $\text{D}$ on $\text{[BC]}$.

Find $\text{BD}$. Fully justify any assumptions you make.

Find the speed of the helicopter.

Find the distance of the helicopter from the communications tower at $t=0$.

Find the bearing on which the helicopter is travelling.

Nymphenburg Palace in Munich has extensive grounds with $9$ points of interest (stations) within them.

These nine points, along with the palace, are shown as the vertices in the graph below. The weights on the edges are the walking times in minutes between each of the stations and the total of all the weights is $105$ minutes.

Anders decides he would like to walk along all the paths shown beginning and ending at the Palace (vertex A).

Use the Chinese Postman algorithm, clearly showing all the stages, to find the shortest time to walk along all the paths.

Find the area of campus managed by office $\text{C}$.

State a further assumption that must be made in order to use area covered as a measure of whether or not the manager of office $\text{C}$ is responsible for more students than the managers of offices $\text{A}$ and $\text{B}$.

A new office is to be built within the triangle formed by $\text{A}$, $\text{B}$ and $\text{C}$, at a point as far as possible from the other three offices.

Find the distance of this office from each of the other offices.

Hence or otherwise find the areas managed by offices $\text{A}$ and $\text{B}$.

Find an expression for the acceleration of the ball at time $t$.

Show that the ball is moving in a circle with its centre at $\text{O}$ and state the radius of the circle.

Hence show that the velocity of the ball is always perpendicular to the position vector of the ball.

Find an expression for the velocity of the ball at time $t$.

Find the value of $t$ at the instant the string breaks.

How many complete revolutions has the ball completed from $t=0$ to the instant at which the string breaks?

Write down ${\mathit{A}}_n$, in terms of $n$.

Write down ${\mathit{A}}_1$.

The total surface of the candy is coated in chocolate. It is known that $1$ gram of the chocolate covers an area of $240 {\text{mm}}^2$.

Calculate the weight of chocolate required to coat one piece of candy.

Calculate the total surface area of one piece of candy.

Write down the coordinates of tower $\text{T4}$.

Tower $\text{T1}$ has coordinates $(-13, 3)$.

Find the gradient of the edge of the Voronoi diagram between towers $\text{T1}$ and $\text{T2}$.

A triangular field $\text{ABC}$ is such that $\text{AB}=56 \text{m}$ and $\text{BC}=82 \text{m}$, each measured correct to the nearest metre, and the angle at $\text{B}$ is equal to $105°$, measured correct to the nearest $5°$.

Calculate the maximum possible area of the field.

Explain why Tim will receive the strongest signal from tower $\text{T4}$.

Hence find the area of the triangle $\text{ABC}$.

Find $\overrightarrow{\text{CB}}$.

Find $\overrightarrow{\text{CA}}\times \overrightarrow{\text{CB}}$.

Find $\overrightarrow{\text{CA}}$.

Use Prim’s algorithm to find the weight of the minimum spanning tree of the subgraph of $G$ obtained by deleting $\text{A}$ and starting at $\text{B}$. List the order in which the edges are selected.

Hence find a lower bound for the travelling time needed to visit all the oil rigs.

Describe how an improved lower bound might be found.

Write down the adjacency matrix, $\mathit{M}$, for this graph.

Find the number of ways that the ant can start at the vertex $\text{A}$, and walk along exactly $6$ edges to return to $\text{A}$.

Assuming the submarine travels in a straight line, write down an equation for the line along which it travels.

Find $\text{OP}$.

Find the coordinates of $\text{P}$.

A storage container consists of a box of length $90 \text{cm}$, width $42 \text{cm}$ and height $34 \text{cm}$, and a lid in the shape of a half-cylinder, as shown in the diagram. The lid fits the top of the box exactly. The total exterior surface of the storage container is to be painted.

Find the area to be painted.

Find the coordinates of station $\text{M}$.

A footpath is to be laid around the curved side of the lawn. Find the length of the footpath.

Find the area of the lawn.

Find the distance between stations $\text{A}$ and $\text{B}$.

Find the distance from $\text{A}$ to $\text{C}$.

Find Minta’s speed, in metres per hour.

Find the equation of the perpendicular bisector of $\text{[AB]}$ . Give your equation in the form $y=mx+c$.

Write down the size of the angle of depression from $\text{H}$ to $\text{C}$.

Write down the height of station $\text{M}$, in metres, above the ground.

Find the distance from $\text{B}$ to $\text{C}$.

In the case that $p<0$, determine whether the lines intersect.

Find the possible value(s) for $p$.

Instead of having to catch the bus to the village, Musab’s sister offers to drop him off at any junction and pick him up at any other junction of his choice.

Explain which junctions Musab should choose as his starting and finishing points.

Musab starts and finishes from the village bus-stop at $\text{A}$. Determine the total distance Musab will need to walk.

Find the coordinates of the image of the point $(6, -2)$.

A triangle $L$ with vertices lying on the $xy$ plane is transformed by $T$.

Explain why both $L$ and its image will have exactly the same area.

Given that $T:\left(\begin{array}{c}p\\ q\end{array}\right)\mapsto 2\left(\begin{array}{c}p\\ q\end{array}\right)$, find the value of $p$ and the value of $q$.

Find the size of angle $\text{A}\hat{\text{O}}\text{B}$ in degrees.

Find the distance between points $\text{A}$ and $\text{B}$.

Calculate the length of arc $\text{AC}$.

Write down the height of the endpoint of the minute hand above the bookshelf at $6:00$ am.

Calculate the area of the shaded sector, $\text{AOC}$.

Find the size of angle $\theta$ in degrees.

Show that the new station should be built at $\text{P}$.

Find the area of the Bermuda Triangle.

Write down the perimeter of the base of the hat in terms of $\mathrm{\pi }$.

Calculate the value of $\theta$, the measure of angle $\text{MŜB}$.

Hence draw the missing boundaries of the region around $\text{P}$ on the following diagram.

Write down the equation of the perpendicular bisector of $\text{[PC]}$.

Find the value of $\theta$.

Find the surface area of the outside of the hat.

The following diagram shows a corner of a field bounded by two walls defined by lines $L_1$ and $L_2$. The walls meet at a point $\text{A}$, making an angle of $40°$.

Farmer Nate has $7 \text{m}$ of fencing to make a triangular enclosure for his sheep. One end of the fence is positioned at a point $\text{B}$ on $L_2$, $10 \text{m}$ from $\text{A}$. The other end of the fence will be positioned at some point $\text{C}$ on $L_1$, as shown on the diagram.

He wants the enclosure to take up as little of the current field as possible.

Find the minimum possible area of the triangular enclosure $\text{ABC}$.

Find the expression $\frac{dV}{d\theta }$.

Find the value of $t$ when the ship will be closest to the lighthouse.

Solve algebraically $\frac{dV}{d\theta }=0$ to find the value of $\theta$ that will maximize the volume, $V$.

Find the position vector $\overrightarrow{\text{OS}}$ of the ship at time $t$ hours.

Show that $r=\frac{6}{2+\theta }$.

An alarm will sound if the ship travels within $20$ kilometres of the lighthouse.

State whether the alarm will sound. Give a reason for your answer.

Find an expression for $V$ in terms of $\theta$.

Find the angle of depression from $\text{M}$ to $\text{N}$.

Find $\text{CV}$.

Hence or otherwise, show that the volume of the reservoir is $29 600 {\text{m}}^3$.

To avoid water leaking into the ground, the five interior sides of the reservoir have been painted with a watertight material.

Find the area that was painted.

By finding an appropriate value, determine whether Joshua is correct.

Find a single matrix $\mathit{P}$ that defines a transformation that represents the overall change in position.

Write down each of the transformations in matrix form, clearly stating which matrix represents each transformation.

Three drones are initially positioned at the points $\text{A}$, $\text{B}$ and $\text{C}$. After performing the movements listed above, the drones are positioned at points $\text{A}\prime$, $\text{B}\prime$ and $\text{C}\prime$ respectively.

Show that the area of triangle $\text{ABC}$ is equal to the area of triangle $\text{A}\prime \text{B}\prime \text{C}\prime$ .

Find a single transformation that is equivalent to the three transformations represented by matrix $\mathit{P}$.

Find ${\mathit{P}}^2$.

Hence state what the value of ${\mathit{P}}^2$ indicates for the possible movement of the drone.

Use Prim’s algorithm, starting at vertex $\text{B}$, to find the weight of the minimum spanning tree of the remaining graph. You should indicate clearly the order in which the algorithm selects each edge.

Hence, for the case where $x<9$, find a lower bound for Daniel’s travel time, in terms of $x$.

Write down a Hamiltonian cycle in $W$.

$p$.

The tourist office in the city has received complaints about the lack of cleanliness of some routes between the attractions. Corinne, the office manager, decides to inspect all the routes between all the attractions, starting and finishing at $\text{H}$. The sum of the weights of all the edges in graph $W$ is $(92+x)$.

Corinne inspects all the routes as quickly as possible and takes $2$ hours.

Find the value of $x$ during Corinne’s inspection.

$q$.

Find the least value of $x$ for which the edge $\text{AF}$ will definitely not be used by Daniel.

$r$.

Use the nearest neighbour algorithm to find two possible cycles.

Find the best upper bound for Daniel’s travel time.

Hence state the value of the upper bound for Daniel’s travel time for the value of $x$ found in part (e)(i).

The front view of a doghouse is made up of a square with an isosceles triangle on top.

The doghouse is $1.35 \text{m}$ high and $0.9 \text{m}$ wide, and sits on a square base.

The top of the rectangular surfaces of the roof of the doghouse are to be painted.

Find the area to be painted.

Town $\text{C}$ is due north of town $\text{A}$ and the road passes through town $\text{C}$.

Find the $y$-coordinate of town $\text{C}$.

Find $\text{FÂO}$, the angle the rope makes with the ground.

Find the length of the rope connecting $\text{A}$ to $\text{F}$.

Find the equation of the line that the road follows.

Find $\overrightarrow{\text{AF}}$.

Find the length of the rope.

Find $\text{FÂO}$, the angle the rope makes with the platform.

Write down an expression for the position vector $\mathit{r}$ of the ship, $t$ hours after $1:00 \text{pm}$.

Find the time at which the bearing of the ship from the harbour is $045˚$.

Calculate the length of the arc made by $\text{B}$, the end of the rubber blade.

Determine the area of the windscreen that is cleared by the rubber blade.

Determine the angle of depression from Camera $1$ to the centre of the cash register. Give your answer in degrees.

Calculate the distance from Camera $2$ to the centre of the cash register.

Without further calculation, determine which camera has the largest angle of depression to the centre of the cash register. Justify your response.

Write down the adjacency matrix for this network.

Determine the number of different walks of length $5$ that start and end at the same vertex.

Use Sam’s measurements to calculate the area of the sector. Give your answer to four significant figures.

Show that the equation of the resulting line does not depend on $m$ or $c$.

Find the vectors $\mathit{a}$ and $\mathit{b}$ in terms of $m$ and/or $c$.

Find the velocity vector of $P$.

Show that the acceleration vector of $P$ is never parallel to the position vector of $P$.

Find the area of the shaded segment.

Find the angle $\text{AÔB}$.

Find the area of the field that can be reached by the goat.

Find the angle $\text{AÔB}$.

Find the area of the shaded segment.

Find the area of the field that can be reached by the goat.

Find an expression for$h\left(x\right)$.

Find the value of $a$.

Briefly explain the two differences in the application of Prim’s and Kruskal’s algorithms for finding a minimum spanning tree in a weighted connected graph.

State the total minimum length of the tracks that have to be tarmacked.

Using Prim’s algorithm starting at vertex F.

Hence, state, with a reason, what can be deduced about the values of $p$, $q$, $r$.

Find the equation of the line which would complete the Voronoi cell containing site E.

Give your answer in the form $ax+by+d=0$ where $a$, $b$, $d\in \mathbb{Z}$.

Find with a reason, the value of $x$.

If the total length of the minimal spanning tree is 14, find the value of $s$.

Find the value of $b$.

By considering the image of $(0, 0)$, find $\mathit{q}$.

By considering the image of $(1, 0)$ and $(0, 1)$, show that

$\mathit{P}=\left(\begin{array}{cc}\frac{\sqrt{3}}{4} & \frac{1}{4}\\ -\frac{1}{4} & \frac{\sqrt{3}}{4}\end{array}\right)$.

Write down the matrix $\mathit{S}$.

Use $\mathit{P}=\mathit{R}\mathit{S}$ to find the matrix $\mathit{R}$.

Write down an equation satisfied by $\left(\begin{array}{c}a\\ b\end{array}\right)$.

In the context of the question, explain the significance of the Voronoi cell containing site E.

Find CÂB.

Hence find the angle and direction of the rotation represented by $\mathit{R}$.

Find the midpoint of $\text{[BD]}$.

Find the equation of $\text{(XZ)}$.

Using Prim’s algorithm starting at vertex A.

Sketch the tracks that are to be tarmacked.

Point B on the ground is 5 m from point E at the entrance to Ollie’s house. He is 1.8 m tall and is standing at point D, below the sensor. He walks towards point B.

Find the distance Ollie is from the entrance to his house when he first activates the sensor.

Using Kruskal’s algorithm.

Find the value of $a$ and the value of $b$.

Determine the exact length of $\text{[YZ]}$.

Given that the exact length of $\text{[XY]}$ is $\sqrt{32}$, find the size of $\text{XŶZ}$ in degrees.

Calculate the gradient of the line segment AE.

Hence find the area of triangle $\text{XYZ}$.

Given that v is perpendicular to B, find the value of $d$.

The force, F, produced by P moving in the magnetic field is given by the vector equation F = $a$v × B, $a\in {\mathbb{R}}^{+}$.

Given that | F | = 14, find the value of $a$.

Sketch a new Voronoi diagram to represent this new situation.

The diagram below is part of a Voronoi diagram.

Diagram not to scale

A and B are sites with B having the co-ordinates of (4, 6). L is an edge; the equation of this perpendicular bisector of the line segment from A to B is $y=-2x+9$

Find the co-ordinates of the point A.

Sketch a Voronoi diagram to represent this information.

State, with a reason, if all five children now own an equal amount of land.

Find how much land an older child has lost.

Find the area of the youngest child’s land.

Find the area of the image of P.

Find the area of triangle X.

State what the shape of the land, owned by the youngest child, is.

Describe fully the geometrical transformation represented by X.

Describe fully the geometrical transformation represented by B.

Hence find the area of triangle Y.

Find the matrix X.

Describe fully the geometrical transformation represented by A.

Find the coordinates of triangle X.

Matrix A represents a combination of transformations:            

A stretch, with scale factor 3 and y-axis invariant;
Followed by a stretch, with scale factor 4 and x-axis invariant;
Followed by a transformation represented by matrix C.

Find matrix C.

Find the initial speed of the ball.

Find the angle of elevation of the ball as it is launched.

For the path of the ball, find an expression for $y$ in terms of $x$.

Find the equation of the perpendicular bisector of $\left[\text{IF}\right]$.

Sketch this new Voronoi diagram showing the regions within the metropolitan area which are closest to each town.

Write down the number of days it takes for the pollution to reach the last town.

A sewer inspector needs to plan the shortest possible route through each of the connections between different locations. Determine an appropriate start point and an appropriate end point of the inspection route.

Note that the fact that each location is connected to itself does not correspond to a sewer that needs to be inspected.

Find the location of the toxic waste dump, given that this location is not on the edge of the metropolitan area.

Make one possible criticism of the council’s choice of location.

By deleting $\text{D}$, use the deleted vertex algorithm to find the lower bound for the installation cost of the cycle.

Given that the coordinates of one vertex of the new Voronoi diagram are $(20, 37.5)$, find the coordinates of the other two vertices within the metropolitan area.

A car departs from a point due north of Hamilton. It travels due east at constant speed to a destination point due North of Gaussville. It passes through the Edison, Isaacopolis and Fermitown districts. The car spends $30\%$ of the travel time in the Isaacopolis district.

Find the distance between Gaussville and the car’s destination point.

Find which town is last to be polluted. Justify your answer.

Find the cost per metre of installing this cable.

State why the cost for installing the cable between $\text{A}$ and $\text{F}$ would be higher than between the other buildings.

By using Kruskal’s algorithm, find the minimum spanning tree for $S$, showing clearly the order in which edges are added.

Hence find the minimum installation cost for the cables that would allow all the buildings to be part of the computer network.

State why a path that forms a Hamiltonian cycle does not always form an Eulerian circuit.

Starting at $\text{D}$, use the nearest neighbour algorithm to find the upper bound for the installation cost of a computer network in the form of a Hamiltonian cycle.

Note: Although the graph is not complete, in this instance it is not necessary to form a table of least distances.

The model assumes that each town is positioned at a single point. Describe possible circumstances in which this modelling assumption is reasonable.

Sketch a Voronoi diagram showing the regions within the metropolitan area that are closest to each town.

Find $\text{AF}$.

Show that the lower bound you have obtained is not the best possible for the solution to the travelling salesman problem for G.

Give an upper bound for the travelling salesman problem for the graph above.

Hence, find a lower bound for the travelling salesman problem for G.

Find the total number of Hamiltonian cycles in G, starting at vertex A. Explain your answer.

Find a minimum spanning tree for the subgraph obtained by deleting A from G.

Find the number of distinct walks of length 4 beginning and ending at A.

Name an algorithm that will allow Sameer to find the lowest cost road system.

Find the lowest cost road system and state the cost of building it. Show clearly the steps of the algorithm.

Write down the adjacency matrix for G.

What is the total weight of the tree?

The diagram below shows a weighted graph.

Use Prim’s algorithms to find a minimal spanning tree, starting at J. Draw the tree, and find its total weight.

Apply Prim’s algorithm to the weighted graph given below to obtain the minimal spanning tree starting with the vertex A.

Find the weight of the minimal spanning tree.

In this part, marks will only be awarded if you show the correct application of the required algorithms, and show all your working.

In an offshore drilling site for a large oil company, the distances between the planned wells are given below in metres.

It is intended to construct a network of paths to connect the different wells in a way that minimises the sum of the distances between them.

Use Prim’s algorithm, starting at vertex 3, to find a network of paths of minimum total length that can span the whole site.

The weights of the edges in a simple graph G are given in the following table.

Use Prim’s Algorithm, starting with vertex F, to find and draw the minimum spanning tree for G. Your solution should indicate the order in which the edges are introduced.

The weights of the edges of a complete graph G are shown in the following table.

Starting at B, use Prim’s algorithm to find and draw a minimum spanning tree for G. Your solution should indicate the order in which the vertices are added. State the total weight of your tree.

Use Prim’s algorithm to draw a minimum spanning tree starting at M.

Use Giovanni's diagram to calculate the length of AX.

Use Giovanni's diagram to find the length of BX, the horizontal displacement of the Tower.

Find the percentage error on Giovanni’s diagram.

Giovanni adds a point D to his diagram, such that BD = 45 m, and another triangle is formed.

Find the angle of elevation of A from D.

Find, in terms of a and b $\overrightarrow{\text{OF}}$.

Find, in terms of a and b $\overrightarrow{\text{AF}}$.

Find an expression for $\overrightarrow{\text{OD}}$ in terms of a, b and $\lambda$;

Find an expression for $\overrightarrow{\text{OD}}$ in terms of a, b and $\mu$.

Show that $\mu =\frac{1}{13}$, and find the value of $\lambda$.

Deduce an expression for $\overrightarrow{\text{CD}}$ in terms of a and b only.

Given that area $\mathrm{\Delta }\text{OAB}=k(\text{area }\mathrm{\Delta }\text{CAD})$, find the value of $k$.

Find a vector equation for L₁.

The vector $\left(\begin{array}{c}2\\ p\\ 0\end{array}\right)$ is perpendicular to $\overrightarrow{\text{AB}}$. Find the value of p.

Show that $\text{sin}\theta =\frac{\sqrt{15}}{4}$.

Find the two possible values for the length of the third side.

The vector equation of line $L$ is given by r $=\left(\begin{array}{c}-1\\ 3\\ 8\end{array}\right)+t\left(\begin{array}{c}4\\ 5\\ -1\end{array}\right)$.

Point P is the point on $L$ that is closest to the origin. Find the coordinates of P.

Find a vector equation of the line that passes through P and Q.

The line through P and Q is perpendicular to the vector 2i $+$ nk. Find the value of $n$.

find the value of $a$;

determine the coordinates of the point of intersection P.

Solve the equation ${\sec }^{2}x+2\tan x=0,0⩽x⩽2\pi$.

Find the volume of the money box.

A second money box is in the shape of a sphere and has the same volume as the cylindrical money box.

Find the diameter of the second money box.

Find the slant height of the cone-shaped container.

Show that the total surface area of the cone-shaped container is 314 cm², correct to three significant figures.

Find the height, $h$, of this cylinder-shaped container.

The factory director wants to increase the volume of coconut water sold per container.

State whether or not they should replace the cone-shaped containers with cylinder‑shaped containers. Justify your conclusion.

Find the size of angle $\text{C}\stackrel{\wedge }{\text{A}}\text{D}$.

Find the size of angle $\text{A}\stackrel{\wedge }{\text{C}}\text{D}$.

Calculate the radius of the base of the cone which has been removed.

Write down the angle of elevation of B from E.

Find the vertical height of B above the ground.

Find the value of $\sin \frac{\pi }{4}+\sin \frac{3\pi }{4}+\sin \frac{5\pi }{4}+\sin \frac{7\pi }{4}+\sin \frac{9\pi }{4}$.

Show that $\frac{1-\cos 2x}{2\sin x}\equiv \sin x,x\ne k\pi$ where $k\in \mathbb{Z}$.

Use the principle of mathematical induction to prove that

$\sin x+\sin 3x+\dots +\sin (2n-1)x=\frac{1-\cos 2nx}{2\sin x},n\in {\mathbb{Z}}^{+},x\ne k\pi$ where $k\in \mathbb{Z}$.

Hence or otherwise solve the equation $\sin x+\sin 3x=\cos x$ in the interval .

Calculate the area of triangle EAD.

Calculate the total volume of the barn.

Calculate the length of MN.

Calculate the length of AE.

Show that Farmer Brown is incorrect.

Calculate the total length of metal required for one support.

Write down the height of the cylinder.

Find the total volume of the trash can.

Find the height of the cylinder, h , of the new trash can, in terms of r.

Show that the volume, V cm³ , of the new trash can is given by

$V=110\pi r^3$.

Using your graphic display calculator, find the value of r which maximizes the value of V.

The designer claims that the new trash can has a capacity that is at least 40% greater than the capacity of the original trash can.

State whether the designer’s claim is correct. Justify your answer.

Use the cosine rule to show that $r^2-12\sqrt{3}r+144-p^2=0$.

Calculate the two corresponding values of PQ.

Hence, find the area of the smaller triangle.

Determine the range of values of $p$ for which it is possible to form two triangles.

Find the set of values of $k$ that satisfy the inequality .

The triangle ABC is shown in the following diagram. Given that , find the range of possible values for AB.

Calculate the volume of this pan.

Find the radius of the sphere in cm, correct to one decimal place.

Find the value of $a$.

Find the temperature that the pizza will be 5 minutes after it is taken out of the oven.

Calculate, to the nearest second, the time since the pizza was taken out of the oven until it can be eaten.

In the context of this model, state what the value of 19 represents.

Write down an equation for the area, $A$, of the curved surface in terms of $r$.

Find $\frac{\text{d}A}{\text{d}r}$.

Find the value of $r$ when the area of the curved surface is maximized.

Show that the volume of a cone shaped glass is $160\text{ c}{\text{m}}^3$, correct to 3 significant figures.

Calculate the radius, $r$, of a hemisphere shaped glass.

Find the cost of $100\text{ c}{\text{m}}^3$ of chocolate mousse.

Show that there is $20\text{ c}{\text{m}}^3$ of orange paste in each special dessert.

Find the total cost of the ingredients of one special dessert.

Find the value of $x$.

Show that $\text{BD}=93\text{ m}$ correct to the nearest metre.

Calculate angle $\mathrm{B}\hat{\mathrm{C}}\mathrm{D}$.

Find the area of ABCD.

Calculate Abdallah’s estimate for the area.

Find the percentage error in Abdallah’s estimate.

Write down the value of x.

Calculate the volume of the paperweight.

1 cm³ of glass has a mass of 2.56 grams.

Calculate the mass, in grams, of the paperweight.

Write down a formula for $A$, the surface area to be coated.

Express this volume in $\text{c}{\text{m}}^3$.

Write down, in terms of $r$ and $h$, an equation for the volume of this water container.

Show that $A=\pi r^2+\frac{1000000}{r}$.

Find $\frac{\text{d}A}{\text{d}r}$.

Using your answer to part (e), find the value of $r$ which minimizes $A$.

Find the value of this minimum area.

Find the least number of cans of water-resistant material that will coat the area in part (g).

the length of BD;

the size of angle DAB;

the area of triangle ABD;

the area of quadrilateral ABCD;

the length of AP;

the length of the fence, BP.

Let a = $\left(\begin{array}{c}2\\ k\\ -1\end{array}\right)$ and b = $\left(\begin{array}{c}-3\\ k+2\\ k\end{array}\right)$, $k\in \mathbb{R}$.

Given that a and b are perpendicular, find the possible values of $k$.

Calculate the volume of the balloon.

Calculate the radius of the balloon following this increase.

Write down an equation for the area of ABCDE using the above information.

Show that the equation in part (a)(i) simplifies to $3x^2+19x-414=0$.

Find the length of CD.

Show that angle $\mathrm{B}\hat{\mathrm{A}}\mathrm{E}={67.4}^{\circ }$, correct to one decimal place.

Find the length of the perimeter of ABCDE.

Calculate the length of CF.

Starting at A , use the nearest neighbour algorithm to find an upper bound for the travelling salesman problem for $G$.

By first deleting vertex A , use the deleted vertex algorithm together with Kruskal’s algorithm to find a lower bound for the travelling salesman problem for $G$.

Find an expression for the volume of water $V({\text{m}}^3)$ in the trough in terms of $\theta$.

Find and simplify an expression for a • b in terms of $t$.

Hence or otherwise, find the values of $t$ for which the angle between a and b is obtuse .

Find the value of $\text{tan}\theta$.

Line $L$ passes through the origin and has a gradient of $\text{tan}\theta$. Find the equation of $L$.

Find the derivative of $f$.

The following diagram shows the graph of $f$  for 0 ≤ $x$ ≤ 3. Line $M$ is a tangent to the graph of $f$ at point P.

Given that $M$ is parallel to $L$, find the $x$-coordinate of P.

State, with a reason, whether or not G has an Eulerian circuit.

Use Kruskal’s algorithm to find a minimum spanning tree for G, stating its total weight. Indicate clearly the order in which the edges are added.

Use a suitable algorithm to show that the minimum time in which Mr José can get from A to E is 13 minutes.

Find the minimum time it takes Mr José to patrol the resort if he has to walk along every road at least once, starting and ending at A. State clearly which roads need to be repeated.

State what feature of G ensures that G has an Eulerian trail.

State what feature of G ensures that G does not have an Eulerian circuit.

Write down an Eulerian trail in G.

Starting and finishing at B, find a solution to the Chinese postman problem for G.

Calculate the total weight of the solution.

Calculate $\frac{\text{d}\theta }{\text{d}t}$ when $\theta =\frac{\pi }{3}$.

In the context of graph theory, explain briefly what is meant by a circuit;

In the context of graph theory, explain briefly what is meant by an Eulerian circuit.

The graph $G$ has six vertices and an Eulerian circuit. Determine whether or not its complement $G^{\prime }$ can have an Eulerian circuit.

Find an example of a graph $H$, with five vertices, such that $H$ and its complement $H^{\prime }$ both have an Eulerian trail but neither has an Eulerian circuit. Draw $H$ and $H^{\prime }$ as your solution.

Draw the weighted graph $H$.

Starting at library D use the nearest-neighbour algorithm, to find an upper bound for Mathilde’s minimum travelling distance. Indicate clearly the order in which the edges are selected.

By first removing library C, use the deleted vertex algorithm, to find a lower bound for Mathilde’s minimum travelling distance.

(i)     Draw the graph ${\kappa }_4$ including the weights of all the edges.

(ii)     Use the nearest-neighbour algorithm, starting at vertex ${\text{A}}_1$, to find a Hamiltonian cycle.

(iii)     Hence, find an upper bound to the travelling salesman problem for this weighted graph.

Consider the graph ${\kappa }_5$. Use the deleted vertex algorithm, with ${\text{A}}_5$ as the deleted vertex, to find a lower bound to the travelling salesman problem for this weighted graph.

(i)     Use the nearest-neighbour algorithm, starting at vertex ${\text{A}}_1$, to find a Hamiltonian cycle.

(ii)    Hence find and simplify an expression in $n$, for an upper bound to the travelling salesman problem for this weighted graph.

By splitting the weight of the edge ${\text{A}}_i{\text{A}}_j$ into two parts or otherwise, show that all Hamiltonian cycles of ${\kappa }_n$ have the same total weight, equal to the answer found in (c)(ii).

Find the slant height of the cone-shaped container.

By deleting $W$, use the deleted vertex algorithm to find a lower bound for the length of a route that visits every shop, starting and finishing at $W$.

Starting from $W$, use the nearest-neighbour algorithm to find a route which gives an upper bound for this problem and calculate its length.

Write down the minimum number of edges in $G$.

Find the maximum number of edges in $G$.

Find the maximum number of edges in $G$, given that $G$ contains an Eulerian circuit.

Explain why $2e=kf$.

Find the value of $f$ when $v=9$ and $k=3$.

Find the possible values of $f$ when $v=13$.

Calculate the volume of oil drained from Yao’s motorbike.

Yao then pours all the oil from the cuboids into an empty cylindrical container. The height of the oil in the container is $30$ cm.

Find the internal radius, $r$, of the container.

Find the gradient of $L_1$.

Find the coordinates of point $\text{C}$.

Find the equation of $L_2$. Give your answer in the form $ax+by+d=0$, where $a, b, d\in \mathrm{\mathbb{Z}}$.

Find the value of $k$.

Find the distance between points $\text{M}$ and $\text{N}$.

Given that the length of $\text{AM}$ is $\sqrt{45}$, find the area of triangle $\text{ANC}$.

Find the length of $\text{AD}$.

Find the size of $\text{A}\hat{\text{B}}\text{C}$.

Calculate the area of the triangular plot of land $\text{ABC}$.

Determine whether the rope can extend into the triangular plot of land, $\text{ACD}$. Justify your answer.

Write down an expression for $V$, the volume (cm³) of the speaker, in terms of $r$, $l$ and $\mathrm{\pi }$.

Write down an equation for the surface area of the speaker in terms of $r$, $l$ and $\mathrm{\pi }$.

Given the design constraint that $l=\frac{150-2\mathrm{\pi }{r}^2}{\mathrm{\pi }r}$, show that $V=150r-\frac{2\mathrm{\pi }{r}^3}{3}$.

Find $\frac{\text{d}V}{\text{d}r}$.

Using your answer to part (d), show that $V$ is a maximum when $r$ is equal to $\sqrt{\frac{75}{\mathrm{\pi }}} \text{cm}$.

Find the length of the cylinder for which $V$ is a maximum.

Calculate the maximum value of $V$.

Use your answer to part (f) to identify the shape of the speaker with the best quality of sound.

state the height of the aircraft at this point.

Find DB.

Find $c$.

Find $\left(f\circ f\right)\left(x\right)$.

Point D is also on L and has coordinates (8, 4, −9).

Find the area of triangle OCD.

Hence or otherwise, find the area of the triangle ABC.

Show that the two submarines would collide at a point P and write down the coordinates of P.

Starting at A, use Prim’s algorithm to find and draw the minimum spanning tree for G.

Your solution should indicate clearly the way in which the tree is constructed.

Find the time at which the aircraft is 4 km above the ground.

Show that $\overrightarrow{\text{AB}}=\left(\begin{array}{c}6\\ 8\\ -5\end{array}\right)$

Show that $\overrightarrow{\text{AB}}=\left(\begin{array}{c}2\\ -1\\ 1\end{array}\right)$.

Let $\overrightarrow{\text{AC}}=\left(\begin{array}{c}3\\ 0\\ 0\end{array}\right)$. Find $\mathrm{B}\hat{\mathrm{A}}\mathrm{C}$.

an Eulerian trail.

Given that the velocity of the aircraft, after the adjustment of the angle of descent, is $\left(\begin{array}{c}-150\\ -50\\ a\end{array}\right)\text{km}{\text{h}}^{-1}$, find the value of $a$.

Find the area of triangle ABC.

Write down the value of angle OBA.

Find $\overrightarrow{\text{PQ}}$.

Hence or otherwise find the shortest distance from R to the line through P and Q.

Six equilateral triangles, each with side length 3 cm, are arranged to form a hexagon.
This is shown in the following diagram.

The vectors p , q and r are shown on the diagram.

Find p•(p + q + r).

Write down the adjacency matrix of the graph.

an Eulerian circuit.

Find the number of walks of length 5 from P to Q.

Write down an Eulerian circuit in G starting at P.

Find the angle between $\overrightarrow{\text{AB}}$ and $\overrightarrow{\text{AC}}$.

Find the value of $p$.

Let SR = $x$. Use the cosine rule to show that $x^2-\left(76\text{cos}{43}^{\circ }\right)x+420=0$.

verify that it would pass directly over the airport.

Hence or otherwise, find the obtuse angle formed by the tangent line to $f$ at $x=8$ and the tangent line to $f$ at $x=2$.

Find the distance between the two submarines at this time.

The point D has coordinates $(q^2,0,q)$. Given that $\overrightarrow{\text{DC}}$ is perpendicular to $L$, find the possible values of $q$.

Draw a graph to represent this map.

parallel.

Find the gradient of $L$.

Find the value of $y$.

Find the number of walks of length 4 from E to F.

Find u.

Find the vector $\overrightarrow{\text{AC}}$.

Find a vector equation for L.

Find the area of triangle PQR.

Find a vector equation for $L$.

Show the direct flights between the cities as a graph.

Using your answer to part (b), find the number of different ways to travel from and return to city A in exactly 6 flights.

By deleting vertex A, use the deleted vertex algorithm to find a lower bound for the cost of the trip.

Define an Eulerian circuit.

Find DC.

The following diagram shows the chord [AB] in a circle of radius 8 cm, where $\text{AB}=12\text{ cm}$.

Find the area of the shaded segment.

In triangle ABC, AB = 5, BC = 14 and AC = 11.

Find all the interior angles of the triangle. Give your answers in degrees to one decimal place.

find the time at which it would fly directly over the airport.

A second line, $L_2$, is parallel to $L_1$ and passes through (1, 2, 3).

Write down a vector equation for $L_2$.

Find the value of t when the two submarines are closest together.

Find the vector $\overrightarrow{\text{AB}}$.

Find the value of t when submarine B passes through P.

When the ship reaches D, it changes direction and travels directly to the island at 50 km per hour. At the same time as the ship changes direction, a boat starts travelling to the island from a point B. This point B lies on (AC), between A and C, and is the closest point to the island. The ship and the boat arrive at the island at the same time. Find the speed of the boat.

Find the direct distance of the aircraft from the airport at this point.

Write down a vector equation for the displacement, r of the aircraft in terms of $t$.

The magnitudes of two vectors, u and v, are 4 and $\sqrt{3}$ respectively. The angle between u and v is $\frac{\pi }{6}$.

Let w = u − v. Find the magnitude of w.

perpendicular.

Find the acute angle between $y=x$ and $L$.

Hence, write down $f^{-1}\left(x\right)$.

Find $\left|\overrightarrow{\text{AB}}\right|$.

Show that $\overrightarrow{\text{AC}}=\left(\begin{array}{c}8\\ -10\\ -1\end{array}\right)$.

Point C (k , 12 , −k) is on L. Show that k = 14.

Find $\overrightarrow{\text{OB}}\bullet \overrightarrow{\text{AB}}$.

Find $\left|\overrightarrow{\text{PQ}}\right|$.

Find the angle between PQ and PR.

Find $\left|\overrightarrow{\text{AB}}\right|$.

List the degrees of each of the vertices.

Write down the adjacency matrix for this graph.

Use the nearest neighbour algorithm to find an upper bound for the cost of the trip.

State whether or not it is possible to travel from and return to city A in exactly 6 flights, having visited each of the other 5 cities exactly once. Justify your answer.

Draw the graph of G.

Explain why it is not possible to have a Hamiltonian cycle in G.

Which pairs of distinct vertices have more than 15 walks of length 3 between them?

Define a Hamiltonian cycle.

Find the values of $a$ and $b$.

Find the volume of this log.

Draw and label the angle of depression on the diagram.

Boat A is situated 10km away from boat B, and each boat has a marine radio transmitter on board. The range of the transmitter on boat A is 7km, and the range of the transmitter on boat B is 5km. The region in which both transmitters can be detected is represented by the shaded region in the following diagram. Find the area of this region.

The shape in the following diagram is formed by adding a semicircle with diameter [AC] to the triangle.

Find the exact perimeter of this shape.

Show that $\text{AC}=7\text{ cm}$.

Find the area of the shaded region.

Find AB.

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 $A$ cm² and the perimeter be $P$ cm. Given that $A=P$, find the value of $r$.

Helen is building a cabin using cylindrical logs of length 2.4 m and radius 8.4 cm. A wedge is cut from one log and the cross-section of this log is illustrated in the following diagram.

Find the volume of this log.

Use the cosine rule to find the two possible values for AC.

Find the length of EB.

Find the value of $r$.

Show that $\alpha =4\arcsin \frac{1}{4}$.

Use Giovanni’s diagram to show that angle ABC, the angle at which the Tower is leaning relative to the
horizontal, is 85° to the nearest degree.

Find the perimeter of sector OABC.

Hence find the value of $r$ given that the shaded area is equal to 4.

Find 50° in radians.

Write down the exact value of $\theta$ in radians.

On the diagram, draw and label with an x the angle of depression of B from P.

Find the area of triangle OBC in terms of $r$ and θ.

Find AM.

Find the length of arc ABC.

Find the area of sector OABC.

Find an expression for the shaded area in terms of $\alpha$, $\theta$ and $r$.

Find the difference between the areas of the two possible triangles ABC.