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HL Paper 1

A garden has a triangular sunshade suspended from three points $A (2, 0, 2), B (8, 0, 2)$ and $C (5, 4, 3)$ , relative to an origin in the corner of the garden. All distances are measured in metres.

Find $\vec{CA}$ .

[1]

a.i.

Find $\vec{CB}$ .

[1]

a.ii.

Find $\vec{CA} \times \vec{CB}$ .

[2]

Hence find the area of the triangle $ABC$ .

[2]

The position of a helicopter relative to a communications tower at the top of a mountain at time $t$ (hours) can be described by the vector equation below.

$r = (\begin{matrix} 20 \\ - 25 \\ 0 \end{matrix}) + t (\begin{matrix} 4.2 \\ 5.8 \\ - 0.5 \end{matrix})$

The entries in the column vector give the displacements east and north from the communications tower and above/below the top of the mountain respectively, all measured in kilometres.

Find the speed of the helicopter.

[2]

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

[2]

Find the bearing on which the helicopter is travelling.

[2]

An ant is walking along the edges of a wire frame in the shape of a triangular prism.

The vertices and edges of this frame can be represented by the graph below.

Write down the adjacency matrix, $M$ , for this graph.

[3]

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

[2]

The following diagram shows the graph $G$ .

Verify that $G$ satisfies the handshaking lemma.

[3]

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

[3]

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

[2]

An engineer plans to visit six oil rigs $(A - F)$ in the Gulf of Mexico, starting and finishing at $A$ . The travelling time, in minutes, between each of the rigs is shown in the table.

The data above can be represented by a graph $G$ .

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

[4]

a.i.

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

[2]

a.ii.

Describe how an improved lower bound might be found.

[1]

Points in the plane are subjected to a transformation $T$ in which the point $(x, y)$ is transformed to the point $(x', y')$ where

$[\begin{matrix} x' \\ y' \end{matrix}] = [\begin{matrix} 3 & 0 \\ 0 & 2 \end{matrix}] [\begin{matrix} x \\ y \end{matrix}]$ .

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

[1]

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

[3]

b.i.

Determine the area of this square.

[1]

b.ii.

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

[2]

b.iii.

Show that $A' B' C' D'$ is a parallelogram.

[3]

b.iv.

Determine the area of this parallelogram.

[2]

b.v.

A geometric transformation $T : (\begin{matrix} x \\ y \end{matrix}) \mapsto (\begin{matrix} x' \\ y' \end{matrix})$ is defined by

$T : (\begin{matrix} x' \\ y' \end{matrix}) = (\begin{matrix} 7 & - 10 \\ 2 & - 3 \end{matrix}) (\begin{matrix} x \\ y \end{matrix}) + (\begin{matrix} - 5 \\ 4 \end{matrix})$ .

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

[2]

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

[3]

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

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

[2]

A ship $S$ is travelling with a constant velocity, $v$ , measured in kilometres per hour, where

$v = (\begin{matrix} - 12 \\ 15 \end{matrix})$ .

At time $t = 0$ the ship is at a point $A (300, 100)$ relative to an origin $O$ , where distances are measured in kilometres.

A lighthouse is located at a point $(129, 283)$ .

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

[1]

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

[6]

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.

[2]

The diagram below shows a network of roads in a small village with the weights indicating the distance of each road, in metres, and junctions indicated with letters.

Musab is required to deliver leaflets to every house on each road. He wishes to minimize his total distance.

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

[5]

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.

[2]

Two lines $L_{1}$ and $L_{2}$ are given by the following equations, where $p \in ℝ$ .

$L_{1} : r = (\begin{matrix} 2 \\ p + 9 \\ - 3 \end{matrix}) + λ (\begin{matrix} p \\ 2 p \\ 4 \end{matrix})$

$L_{2} : r = (\begin{matrix} 14 \\ 7 \\ p + 12 \end{matrix}) + μ (\begin{matrix} p + 4 \\ 4 \\ - 7 \end{matrix})$

It is known that $L_{1}$ and $L_{2}$ are perpendicular.

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

[3]

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

[4]

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 $A$ , making an angle of $40 °$ .

Farmer Nate has $7 m$ of fencing to make a triangular enclosure for his sheep. One end of the fence is positioned at a point $B$ on $L_{2}$ , $10 m$ from $A$ . The other end of the fence will be positioned at some point $C$ 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 $ABC$ .

The following diagram shows a frame that is made from wire. The total length of wire is equal to $15 cm$ . The frame is made up of two identical sectors of a circle that are parallel to each other. The sectors have angle $θ$ radians and radius $r cm$ . They are connected by $1 cm$ lengths of wire perpendicular to the sectors. This is shown in the diagram below.

The faces of the frame are covered by paper to enclose a volume, $V$ .

Show that $r = \frac{6}{2 + θ}$ .

[2]

Find an expression for $V$ in terms of $θ$ .

[2]

b.i.

Find the expression $\frac{d V}{d θ}$ .

[3]

b.ii.

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

[2]

b.iii.

At $1 : 00 pm$ a ship is $1 km$ east and $4 km$ north of a harbour. A coordinate system is defined with the harbour at the origin. The position vector of the ship at $1 : 00 pm$ is given by $(\begin{matrix} 1 \\ 4 \end{matrix})$ .

The ship has a constant velocity of $(\begin{matrix} 1.2 \\ - 0.6 \end{matrix})$ kilometres per hour ( ${km h}^{- 1}$ ).

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

[1]

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

[4]

Consider the following directed network.

Write down the adjacency matrix for this network.

[2]

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

[3]

A vertical pole stands on a sloped platform. The bottom of the pole is used as the origin, $O$ , of a coordinate system in which the top, $F$ , of the pole has coordinates $(0, 0, 5.8)$ . All units are in metres.

The pole is held in place by ropes attached at $F$ .

One of these ropes is attached to the platform at point $A (3.2, 4.5, - 0.3)$ . The rope forms a straight line from $A$ to $F$ .

Find $\vec{AF}$ .

[1]

Find the length of the rope.

[2]

Find $FÂO$ , the angle the rope makes with the platform.

[5]

The position vector of a particle, $P$ , relative to a fixed origin $O$ at time $t$ is given by

$\vec{OP} = (\begin{matrix} \sin (t^{2}) \\ \cos (t^{2}) \end{matrix})$ .

Find the velocity vector of $P$ .

[2]

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

[5]

A submarine is located in a sea at coordinates $(0.8, 1.3, - 0.3)$ relative to a ship positioned at the origin $O$ . The $x$ direction is due east, the $y$ direction is due north and the $z$ direction is vertically upwards.

All distances are measured in kilometres.

The submarine travels with direction vector $(\begin{matrix} - 2 \\ - 3 \\ 1 \end{matrix})$ .

The submarine reaches the surface of the sea at the point $P$ .

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

[2]

Find the coordinates of $P$ .

[3]

b.i.

Find $OP$ .

[2]

b.ii.

The cost adjacency matrix below represents the distance in kilometres, along routes between bus stations.

All the values in the matrix are positive, distinct integers.

It is decided to electrify some of the routes, so that it will be possible to travel from any station to any other station solely on electrified routes. In order to achieve this with a minimal total length of electrified routes, Prim’s algorithm for a minimal spanning tree is used, starting at vertex A.

The algorithm adds the edges in the following order:

AB AC CD DE.

There is only one minimal spanning tree.

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

[2]

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

[2]

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

[2]

A particle P moves with velocity v = $\left( {\begin{array}{*{20}{c}} { - 15} \\ 2 \\ 4 \end{array}} \right)$ in a magnetic field, B = $\left( {\begin{array}{*{20}{c}} 0 \\ d \\ 1 \end{array}} \right)$ , $d \in \mathbb{R}$ .

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

[2]

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 $a$ .

[4]

The equation of the line $y = m x + c$ can be expressed in vector form $r = a + λ b$ .

The matrix $M$ is defined by $(\begin{matrix} 6 & 3 \\ 4 & 2 \end{matrix})$ .

The line $y = m x + c$ (where $m \neq - 2$ ) is transformed into a new line using the transformation described by matrix $M$ .

Find the vectors $a$ and $b$ in terms of $m$ and/or $c$ .

[2]

Find the value of $det M$ .

[1]

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

[4]

The diagram shows a sector, $OAB$ , of a circle with centre $O$ and radius $r$ , such that $AÔB = θ$ .

Sam measured the value of $r$ to be $2 cm$ and the value of $θ$ to be $30 °$ .

It is found that Sam’s measurements are accurate to only one significant figure.

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

[2]

Find the upper bound and lower bound of the area of the sector.

[3]

Find, with justification, the largest possible percentage error if the answer to part (a) is recorded as the area of the sector.

[3]

The above diagram shows the weighted graph G.

Write down the adjacency matrix for G.

[1]

a.i.

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

[3]

a.ii.

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.

[5]

Sameer is trying to design a road system to connect six towns, A, B, C, D, E and F.

The possible roads and the costs of building them are shown in the graph below. Each vertex represents a town, each edge represents a road and the weight of each edge is the cost of building that road. He needs to design the lowest cost road system that will connect the six towns.

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

[1]

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

[7]

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.

Let G be a weighted graph with 6 vertices L, M, N, P, Q, and R. The weight of the edges joining the vertices is given in the table below:

For example the weight of the edge joining the vertices L and N is 3.

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

[5]

What is the total weight of the tree?

[1]

In the following diagram, $\overrightarrow {{\text{OA}}}$ = a, $\overrightarrow {{\text{OB}}}$ = b. C is the midpoint of [OA] and $\overrightarrow {{\text{OF}}} = \frac{1}{6}\overrightarrow {{\text{FB}}}$ .

N17/5/MATHL/HP1/ENG/TZ0/09

It is given also that $\overrightarrow {{\text{AD}}} = \lambda \overrightarrow {{\text{AF}}}$ and $\overrightarrow {{\text{CD}}} = \mu \overrightarrow {{\text{CB}}}$ , where $\lambda ,{\text{ }}\mu \in \mathbb{R}$ .

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

[1]

a.i.

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

[2]

a.ii.

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

[2]

b.i.

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

[2]

b.ii.

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

[4]

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

[2]

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

[5]

Consider the lines ${l_1}$ and ${l_2}$ defined by

${l_1}:$ r $= \left( {\begin{array}{*{20}{c}} { - 3} \\ { - 2} \\ a \end{array}} \right) + \beta \left( {\begin{array}{*{20}{c}} 1 \\ 4 \\ 2 \end{array}} \right)$ and ${l_2}:\frac{{6 - x}}{3} = \frac{{y - 2}}{4} = 1 - z$ where $a$ is a constant.

Given that the lines ${l_1}$ and ${l_2}$ intersect at a point P,

find the value of $a$ ;

[4]

determine the coordinates of the point of intersection P.

[2]

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

[2]

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

[2]

Use the principle of mathematical induction to prove that

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

[9]

Hence or otherwise solve the equation $\sin x + \sin 3x = \cos x$ in the interval $0 < x < \pi$ .

[6]

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.

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.

Let a = $\left( {\begin{array}{*{20}{c}} 2 \\ k \\ { - 1} \end{array}} \right)$ and b = $\left( {\begin{array}{*{20}{c}} { - 3} \\ {k + 2} \\ k \end{array}} \right)$ , $k \in \mathbb{R}$ .

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

The lengths of two of the sides in a triangle are 4 cm and 5 cm. Let θ be the angle between the two given sides. The triangle has an area of $\frac{{5\sqrt {15} }}{2}$ cm².

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

[1]

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

[6]

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

Three points in three-dimensional space have coordinates A(0, 0, 2), B(0, 2, 0) and C(3, 1, 0).

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

[1]

a.i.

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

[1]

a.ii.

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

[4]

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.

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 $r$ .

The vectors a and b are defined by a = $\left( {\begin{array}{*{20}{c}} 1 \\ 1 \\ t \end{array}} \right)$ , b = $\left( {\begin{array}{*{20}{c}} 0 \\ { - t} \\ {4t} \end{array}} \right)$ , where $t \in \mathbb{R}$ .

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

[2]

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

[4]

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 50° in radians.

[1]

Find the volume of this log.

[4]