Bridging GCSE and A-level Maths Student Book

Contents Revision section

Practice section

Revision section

Practice section

4

77

43

101

1.1 Surds

4

77

6.1 Trigonometry and triangles

43

101

1.2 Indices

6

78

6.2 The area of any triangle

45

103

2 Algebra 1

9

79

6.3 Solving a trigonometric equation

47

105

2.1 Basic algebra

9

79

2.2 Solving linear equations

11

80

50

107

2.3 Forming expressions

13

81

7.1 The magnitude and direction of a vector

50

107

3 Coordinate geometry 1

15

84

7.2 Position vectors

54

109

3.1 Straight-line graphs

15

84

8 Differentiation

59

110

3.2 Finding the equation of a line

18

85

3.3 Mid-points and distances

21

86

8.1 Estimating the gradient of a curve

59

110

3.4 Intersections of lines

24

87

8.2 The rules of differentiation

63

111

8.3 Applying differentiation

66

113

26

89

4.1 Solving a quadratic equation by factorising

70

115

26

89

9.1 The rules of integration

70

115

4.2 Using the quadratic formula

28

90

9.2 Applying integration

73

116

4.3 Further equation solving

30

92

5 Coordinate geometry 2

33

95

5.1 Transformations of graphs

33

95

5.2 Sketching curves

36

96

5.3 Intersection points of graphs

40

98

1 Surds and indices

4 Algebra 2

William Collins’ dream of knowledge for all began with the publication of his fi rst book in 1819.

A self-educated mill worker, he not only enriched millions of lives, but also founded a flourishing publishing house. Today, staying true to this spirit, Collins books are packed with inspiration, innovation and practical expertise. They place you at the centre of a world of possibility and give you exactly what you need to explore it.

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An imprint of HarperCollinsPublishers The News Building 1 London Bridge Street London SE1 9GF Browse the complete Collins catalogue at www.collins.co.uk © HarperCollinsPublishers Limited 2017 10 9 8 7 6 5 4 3 2 1

6 Trigonometry

7 Vectors

9 Integration

Practice section

77

Answers

119

Exam paper

137

Mark Rowland asserts his moral right to be identified as the author of this work.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the Publisher or a licence permitting restricted copying in the United Kingdom issues by the Copyright Licensing Agency Ltd., 90 Tottenham Court Road, London W1T 4LP. British Library Cataloguing in Publication Data A catalogue record for this publication is available from the British Library.

Commissioned by Jennifer Hall Project editor Amanda Redstone Project managed by Emily Hooton Edited by Caroline Petherick Proofread by Karen Williams Answers checked by Steven Matchett Cover design by We are Laura Internal design by Graham Brasnett Typesetting by Jouve India Private Limited Illustrations by Kathy Baxendale and Ann Paganuzzi Production by Rachel Weaver Printed and bound by

978-0-00-820501-0

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Introduction Welcome to Collins Bridging GCSE and A Level Maths Student Book. This book helps you to progress smoothly onwards from GCSE Maths with detailed examples and plenty of practice in the key areas needed for success at A Level.

&ACTORISE EACH GROUP OF TERMS x(x x

4

x x 3O EITHER x OR x 3OLVE FOR x x 2 OR x 3

Algebra 2

Checkpoint Check that x 3 is a solution to the equation 3x 2 7x 6 0.

3(3)2 7(3) 6 27 21 6 0 (checked)

Moving on to A Level )N ! ,EVEL QUESTIONS YOU OFTEN NEED TO FORM AND SOLVE A QUADRATIC EQUATION AS YOU WORK TOWARDS THE SOLUTION OF A MORE COMPLICATED PROBLEM

A Level Example 2 4HE DIAGRAM SHOWS A SQUARE LAWN ABCD WITH SIDE LENGTH x METRES AND A RECTANGULAR PATH EFGH WHERE EF x METRES AND FG x METRES C

A Level Alert!

H

Consolidate your knowledge of difficult GCSE topics with graded worked examples.

4.1 Solving a quadratic equation by factorising

A

What you should already know:

4HE LAWN AND THE PATH HAVE EQUAL AREAS

Key points Look out for the blue ‘Key points’ boxes that highlight the most important things to remember for each topic.

HOW TO FACTORISE A QUADRATIC EXPRESSION SUCH AS x2 x

s

HOW TO SOLVE A QUADRATIC EQUATION SUCH AS x2 x BY FACTORISING

Make use of any given information (in this case, the two shapes have equal area).

s

HOW TO FORM AND SOLVE A QUADRATIC EQUATION AS PART OF A MORE COMPLEX QUESTION

4O lND THE PERIMETER OF THE PATH YOU lRST NEED TO lND THE VALUE OF x

Handy hint

Revisiting GCSE

7ORKING IN METRES THE AREA OF THE SQUARE LAWN IS x2 AND THE AREA OF THE RECTANGULAR PATH IS x x

You have to make one side equal to 0 in order to solve the equation by factorisation.

4HE TWO AREAS ARE EQUAL SO

!T '#3% YOU SAW THAT FACTORISING HELPS YOU TO SOLVE A QUADRATIC EQUATION

x x x2

x2 x x x2

%XPAND THE BRACKETS

GCSE Example 1

3IMPLIFY

x2 x x2

3OLVE THE EQUATION x2 x

3UBTRACT x2 FROM EACH SIDE

x2 x

-ULTIPLY THE lRST AND LAST NUMBERS

TO SIMPLIFY THE FACTORISATION

8m

x x

&ACTORISE AND SOLVE

Handy hint

H

THIS QUADRATIC

Use the factorisation method you learnt at GCSE.

18

2m E

x IS NOT POSSIBLE BECAUSE THEN EACH SIDE OF THE RECTANGLE WOULD HAVE A NEGATIVE LENGTH FOR EXAMPLE EF

Write 7x as 9x 2x rather than 2x 9x to avoid issues with negative signs.

AND 3O 3x 2 7x 6 3 9

x 2 x 2 x 6

F 8m

3O x EF AND FG

Handy hint

5SE AND BECAUSE 18

G

2m

3O x OR x

&IND TWO NUMBERS WITH PRODUCT AND WHICH COMBINE TO MAKE THE MIDDLE NUMBER IN THE EXPRESSION

26

A Level Alert!

You may need give a reason why a particular solution should be ignored.

x2 x

$IVIDE EACH TERM BY

(ERE IS ONE WAY TO FACTORISE THE EXPRESSION x2 x

Handy hint

Make sure you answer the question being asked (in this case, to find the perimeter of the path).

EFGH IS AN M BY M RECTANGLE AND SO HAS PERIMETER METRES

ALGEBRA 2

8

Practice: Differentiation

y

PS

PS

A (1, 1)

0

The gradient of the tangent to the curve at P is m, where m is an integer.

2 The diagram shows part of the curve y 4x2 3 which passes through the point P(1, 7).

PS

a Write down the gradient of the curve at point R.

ii R(1.5, 12)

iii The point S where x 冑2, giving your answer to 3 significant figures.

b Use differentiation from first principles to dy show that if y x4 then 4x 3 . dx

For questions 3 to 10, use the (n 1) rule to differentiate each equation.

y

x

Not drawn accurately

d Write down the gradient of this curve at the point where x 1.

4 By calculating GradPQ, where Q is a point on the given curve, obtain an estimate for the gradient of each curve at point P. a Curve: y 2x2 x 4, point P(2, 10), point Q(2.5, 14).

a y x2 4x 1

b y (2x 1)(2x 3)

c y x3 6x2 5

d y x3(x 4)

e y x(x 3)(x 5)

f y x(x2 1)2

4 Find the gradient of the curves with these equations at the points indicated. a y x 2x 1 at the point P where x 1 1 b y x 4 3x 2 4x at the point Q where 2 x 2

Q

P

Handy hint Use the symmetry of the curve.

dy for the curves with dx these equations. Factorise each answer as far as possible.

3 Find an expression for

3

c By finding the gradients of the chords PQ and RP, show that m 2.

x

a Find a similar expression for (x h)4.

P

7 The diagram shows part of the curve y 2x2.

The line y 5x 3 intersects this curve at points P and Q. The dotted line is the tangent to the curve at P. Point 0 is the origin.

The gradient of the curve at point P is m, where m is an integer.

P (1, 7)

a Use each of these points to find an 0 estimate for the gradient of the curve at P. In each case, state whether the answer gives an underor over-estimate for this gradient.

1 Use differentiation from first principles to show that: dy a if y 3x2 then 6x dx dy b if y x2 5x then 2x 5. dx 2 It is given that (x h)3 x3 3x2h 3xh2 h3.

x

b By expressing x 2 4x 7 in the form (x a)2 b, for constants a and b, find the coordinates of point R.

y

x

y

By choosing a suitable 0 2 pair of points A and B on this curve either side of P, determine the gradient of the curve at point P correct to 2 decimal places.

P (3, 4) R

x

6 The diagram shows part of the curve 3 y 2 x which

d Write down the value of m and hence find the equation of the tangent to the curve at P.

8.2 The rules of differentiation P

passes through the point P with x-coordinate 2.

a Find the gradient of PB.

c Use your answers to a and b to write down the value of m.

y

Point R is the minimum point on this curve.

Q (4, 7)

0

5 The diagram shows part of the curve 1 y 4 x2 5 which

By choosing a suitable 0 3 pair of points A and B on this curve either side of P, show that the gradient of the curve at P is 1.5, correct to 1 decimal place.

B (3, 9)

b Find the gradient of AP.

The tangent at P intersects the line OQ at a point which has a positive x-coordinate. c Explain why m 3.

passes through the point P with x-coordinate 3.

3 The diagram shows the curve y x 2 4x 7 which passes through the points P(3, 4), Q(4, 7) and R. y

Not drawn accurately

b Curve: y 3x 4x2, point P(2, 10), point Q(1.75, 7). c Curve: y 2x2 1, point P( 2, 9), point Q with x-coordinate 1.5. d Curve: y 冑2x2 7 , x 0 point P has x-coordinate 1, point Q has y-coordinate 5.

b State, with a reason, which of the answers found in part a gives the best estimate for the gradient of the curve at P.

1 The diagram shows part of the curve y x2 which passes through the points A(1, 1), P(2, 4), and B(3, 9). The tangent to the curve at P has been drawn.

Q(0.5, 4)

27

ALGEBRA 2

8.1 Estimating the gradient of a curve

i

Avoid the common misconceptions that students regularly make with ‘Common errors’ boxes and discover useful ways to check your workings with ‘Checkpoint’ boxes.

F 3x – 4 m

Working

P (2, 4)

Common errors and Checkpoints

E

&IND THE PERIMETER OF THE PATH

In this section you will learn:

Handy hints and A level Alerts Find valuable ‘Handy hint’ boxes throughout the book and understand methods that are specific to A Level with ‘A Level Alert!’ boxes.

B xm

Handy hint

s

4HERE ARE SEVERAL METHODS YOU COULD USE TO SOLVE THIS EQUATION BUT FACTORISATION IS GENERALLY THE FASTEST

Find out how the strategies you learnt at GCSE are extended and explored with A Level worked examples.

You need to think your way through a problem. Ask yourself questions such as ‘what do I need to find first in order to find the perimeter?’

G x–2m

Working

Moving on to A Level

(3x 2)(x 3) 0 does not mean x 2 is a solution to this equation.

4HE EQUATION x2 x CAN BE WRITTEN AS

D

Revisiting GCSE

Common error

&ULLY FACTORISE THE EXPRESSION x x

0

x

The equation of the tangent to the curve at P is y mx c, where m is an integer and c 0. a By comparing the tangent at P with the line PQ, explain why m 5.

2

c y (x 3)(x2 9) at the point R where y 0 1 d y x 3 x 1 at the point S where this 8 curve intersects the line y x 5 A curve has equation y 2x2 5x 3.

Find the gradient of this curve at each of its roots.

b Use algebra to find the coordinates of P and Q. 110

PRACTICE: DIFFERENTIATION

PRACTICE: DIFFERENTIATION

111

Practice section Increase your confidence and improve your skills with comprehensive practice sections, packed with questions, dedicated to each topic. Look out for questions that have the problem solving icon PS . These questions test your ability to reason and think your way through a problem.

Exam practice Ensure you are ready to start your A Level course by taking the tear-out exam paper at the back of the book.

Answers Find all the answers to the practice exercises at the back of the book. The exam paper answers are available on request from education@harpercollins.co.uk

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7

Vectors

7.1 The magnitude and direction of a vector What you should already know:

 3   4   2 how to add and subtract column vectors, for example:       .  2  1  1   2 how to multiply a column vector by a number, for example: 3   .  5

• •

In this section you will learn: •

how to find the magnitude and direction of a vector.

•

how to write a vector in i – j form.

Revisiting GCSE  2 You know that the column vector   means ‘move 2 steps right and 3 steps up’ and that you can use  3 bold lower-case letters to represent vectors.

GCSE Example 1  4  3 The vectors a   and b    3  4 Work out: a

2a b

b

3a 2b.

Working a

You work out 2a by multiplying each number in vector a by 2.  4 a    3

50

so

 2 4  8 2a 

 2 3   6

VECTORS

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So

 8  3  2a + b   +    6  –4

Handy hint Add the top numbers: 8 3 11.

 8 3   6 4

Add the bottom numbers: 6 ( 4) 6 4 2.

 11    2 b

 3  4 3a – 2b 3  – 2   -4  3  12  6       9   8 

Common error

 12 6    9 ( 8 )  6    17

 2 You can draw a directed line to represent the vector a   .  3  2 Since   means ‘move 2 steps right and 3 steps up’, this vector  3 has horizontal component 2 and vertical component 3.

 12  6  is not equal to  6 .  9   8  1 

a

3 magnitude

a

2 θ

A vector is completely described by its magnitude and direction.

The diagram shows the vector a.

positive horizontal

Key point •

The magnitude of a is the length of the line representing a.

•

You write a for the magnitude of a.

•

The direction of a is the angle θ between this line and the positive horizontal.

•

You can assume all directions are measured against the positive horizontal and lie in the range 0° θ , 360°.

You use the components of a vector to calculate its magnitude and direction.

A Level Example 2  3 The vector AB   .  2

a Find the magnitude of AB , giving the answer to 1 decimal place.

b Find the direction of AB , giving the answer to the nearest degree.

Handy hint

The vector AB is the directed line from point A to point B. B 2

Working

 3 Sketch the vector AB   , showing its components and  2 direction θ.

θ A

3

The triangle shown is right-angled.

VECTORS

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a

Use Pythagoras’ theorem to calculate the magnitude of AB.

AB 32 22 13 3.605…

AB 3.6 (1 decimal place).

b

Use trigonometry to calculate angle θ. opposite tan θ adjacent

2 3

So θ tan −1

Common error 32 + 22 is not equal to 3 2.

Add the squares then square root the answer.

Handy hint Use tan because you are given the opposite and adjacent sides to θ in the triangle.

( 32 )

33.69…°

AB has direction 34° (nearest degree).

Key point  x The vector   has:  y • •

 x magnitude   x 2 y 2  y y n θ , x 0. direction θ, where tan x

 0 1  At A Level, vectors are also written using a combination of the vectors i   and j =   . 1  0 1   0  3 For example, you can write   as 3   4   , or 3i 4j.  0 1   4  3 3i 4j is the vector   written in i – j form.  4

Key point  x  y  xi y j

↑ column vector

↑ i j form

The vector xi yj has: •

magnitude xi yyjj x 2 y 2

•

direction θ, where tan nθ

52

y , x 0. x

VECTORS

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A Level Example 3 For the vectors a 2i 3j, b 4i 8j and c 4i pj, it is given that 3a 2b qc 0, where p and q are constants.

Handy hint

Find the magnitude of c, giving the answer in simplified surd form.

0 0i 0j is the zero vector.

Working Find the components of 3a 2b qc.

Handy hint

3a 2b qc 3(2i 3j) 2(4i 8j) q(4i pj)

You may find it helpful to write a, b and c as column vectors.

(6i 9j) (8i 16j) (4qi qpj) Combine like components: (6 8 4q)i ( 9 16 qp)j Simplify: (14 4q)i (7 qp)j So (14 4q)i (7 qp)j 0i 0j Equate i components: 14 4q 0 Solve for q:

Handy hint

q 3.5

Use the value of q to find the value of p.

Equate j components: 7 3.5p 0 Solve for p:

p 2

So c 4i 2j and hence c 4 22 2

20 2 5 At A Level, particularly when studying Mechanics, it is important to be able to find the components of a vector from its magnitude and direction.

A Level Example 4 The vector q 4i j and the vector r has magnitude 4 and direction 65°.

Find the magnitude and direction of q r. Give answers to 1 decimal place.

Working You first need to find the components of r.

Let r xi yj where components x and y are to be found. r = 4

The sketch shows the magnitude and direction of r.

Use trigonometry on this right-angled triangle to find x and y. x cos 65° so x 4 cos 65° 4 1.6904… y sin 65° so y 4 sin 65° 4 3.6252…

y

65° x

So, to 2 decimal places, r 1.69i 3.63j.

Handy hint

Working to 2 decimal places, calculate the components of q r.

Round components to 2 decimal places so that the final answers are correct to 1 decimal place.

q r ( 4i j) (1.69i 3.63j) 2.31i 4.63j So q r

( 2.31)2 4.632

26.773 5.1742… q r 5.2 (1 decimal place) Sketch the vector q r to find its direction θ.

4.63

q+r

Common error

α

Do not enter –2.312 + 4.632 on your calculator. Use brackets with negative numbers.

θ

–2.31

Common error q + r is not equal to q + r because q and r have different directions.

VECTORS

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From the sketch, θ 180° α where the acute angle α is such that 4.63 tan α 2.31

( )

α tan 1 4.63 2.31 63.4844…° θ 180° α Calculate θ : 180° 63.4844…° 116.5155…° q r has direction 116.5° (1 decimal place).

Calculate α :

Common error 4.663 –2.331 To find α, ignore signs on the components. tan α is not equal to

Key point •

The vector a with direction θ has: horizontal component a cos θ vertical component a sin θ.

•

Always sketch a vector when finding its direction.

The diagrams explain how to find the direction θ of a vector a when θ is not acute. In each diagram, α is an acute angle.

180° θ 270°

90° θ 180°

270° θ 360°

θ

θ

α

α

a

a

a

α

θ

Direction: θ = 180° – α

Direction: θ = 180° + α

Direction: θ = 360° – α

Taking it further You will use vectors to solve problems in Mechanics. Forces are vector quantities and can be added or subtracted by writing them in component form.

7.2 Position vectors What you should already know: •

how to use the triangle law of vector addition: AB BC AC .

•

how to solve geometrical problems using vectors.

In this section you will learn: •

how to find the position vector of a point.

•

how to use position vectors to find the distance between two points and solve geometrical problems.

At GCSE, you solved geometrical problems using vectors.

54

VECTORS

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GCSE Example 5 B

In triangle ABC, M is the midpoint of AB and N is the midpoint of AC. a

Show that MN is parallel to BC.

b

Find the ratio of the area of triangle AMN to the area of the trapezium MNCB.

M

Working a

b

A You need to express MN in terms of the vector BC. N 1 M is the midpoint of AB so MA BA .

2 Handy hint 1 Similarly, AN AC .

Use vectors to express the information 2

you are given about M and N. By the triangle law of addition: MN MA AN 1 1 BA AC 2 2 Handy hint 1 ( BA AC ) Think of + as meaning ‘followed by’ 2

followed

by AN is equivalent MA 1

Now use the triangle law again: BC to MN .

2 1 So MN BC . This means MN has the same direction as BC but is half its length.

2 This proves MN is parallel to BC. From the result in part a, ABC is a scale factor 2 enlargement of AMN. So area ABC 4 area AMN.

C

Handy hint Area scale factor (length scale factor)2.

Hence, area of MNCB 3 area AMN. Triangle AMN and trapezium MNCB have areas in the ratio 1 : 3.

At A Level, you need to work with position vectors.

Key point • •

The position vector of the point P is the vector OP , where O is the origin.  a  OP   is equivalent to saying P has coordinates (a, b).  b

For example, the point with coordinates P(2, 3) has position  2 vector OP   , as shown in the diagram.  3 Using position vectors, you can find the vector joining two points.

If A and B are any two points then, by the triangle law of addition, OA AB OB. y

P

3

2

23 1

0

1

2

3

x

VECTORS

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You can rewrite this equation as:

AB OB OA.

y

A

B

x

0

Key point

For any two points A and B, AB OB OA.

You can use this result to solve geometrical problems.

A Level Example 6 Relative to the origin O, points A and B have coordinates (1, 2) and (4, 2), respectively.

a Find AB .

A Level Alert!

The word ‘respectively’ is used frequently at A Level. It means you pair items off in the obvious order, so here, A(1, 2) and B(4, 2).

Point C has coordinates (13, k), where k is a positive constant.

Given that AC 13: b

find the exact perimeter of triangle ABC. Working a

First find the components of AB .  4   1  OA   and OB   are the position vectors of A and B, respectively.  2  2

AB OB OA  4  1      2  2

Checkpoint

 3    4 

Verify this answer by finding the length AB using the distance formula – see Section 3.3.

2 AB 32 ( 4 )

So:

y

25 5 b

Sketch triangle ABC, showing all known lengths.

Use the information AC 13 to find k:

AC OC OA  13  1       k   2  12    k 2

56

A 13 C 0

x

5

B

VECTORS

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Express AC 13 as an equation involving k:

2 AC 122 ( k 2)

Handy hint You can still sketch the triangle even though you do not know where to place C.

122 ( k 2) 13 2

So:

Square each side: 122 (k 2)2 132 So:

Handy hint

(k 2)2 169 144

You could solve the equation (k 2)2 25 by expanding the brackets to form a quadratic, but taking square roots is quicker.

25 Square root each side:

k 2 5

Solve for k:

k 7 or k 3

So k 7, because k is positive.  12  Hence, AC   7 2

Handy hint Underline and make use of all the important information in the question (in this case, that k is positive).

 12    5

Now find BC so that you can calculate BC .

BC BA AC

AB AC

where

 3  AB    4

 3  12      4   5

Handy hint

BA = –AB since BA and AB have equal lengths but opposite directions.

y

A 13

 9    9

BC 92 92

So:

C x

5

0

9 2

9 2 The perimeter of triangle ABC 5 9 2 13

B

18 9 2 units.

You can use position vectors to prove geometric formulae.

A Level Example 7 The diagram shows points A and B with coordinates (x1, y1) and (x2, y2), respectively.

y

B

M is the mid-point of AB.

 x x 2 y1 y 2  .

Use vectors to prove that the coordinates of M are  1 , 2   2

M A

Working  x 2   x1  OA   and OB   .  y1   y2 

1 M is the mid-point of AB so AM AB .

2

You need to find an expression for the position vector OM .

By the triangle law: OM OA AM 1 OA AB 2 1

OA (OB OA ) Use AB OB OA : 2 1 1 OA OB 2 2

x

0

Handy hint Where possible, work with vectors rather than their components.

VECTORS

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1 OA OB ) 2(

Simplify:

Use components of OA and OB :

1   x1   x 2       2 y   y  1 2

Handy hint Remember that, for any point P,  a  OP =   means  b P has coordinates (a, b).

1  x1 + x 2  x x 2 y1 y 2  So OM  which proves that the coordinates of M are  1 . , 2  y + y  2   2 1 2 You might like to compare the proof about mid-points in this example with that given in Section 3.3.

Taking it further As you progress through your course you will study vectors in three dimensions. You will also learn how to use vectors to solve problems involving projectile motion.

58

VECTORS

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7

Practice: Vectors

7.1 The magnitude and direction of a vector All directions 0°    360° are measured against the positive horizontal.

i p 1 2q

b Find the vector r such that 2p 1 7q 1 4r 5 0, where 0 is the zero vector. Give your answer in i – j form.

6

Where appropriate, give magnitudes in simplified surd form unless told otherwise.

1

2

3

 3  3

 5 b    12 

 6 c    2

 6  d    2

c Find the direction of r 5 ci 1 dj, for c and d positive constants, given that r 2d.

7 PS

 5  3 Given that p   and q   find the  3  5 magnitude and direction of these vectors. a p 1 q b p  q c 3p  5q d 1 ( 5p 3q ) 15 Express these vectors in component form.

4

5

2 , direction 315°.

 2   3 The vectors p   and q  .  12   1  a By finding their directions, show that p and q are perpendicular.

Handy hint b Find PQ , where OP p

Two vectors are and OQ q.

Given vectors p 5 5i  3j, q 5 2i 1 2j: a Find the magnitude and direction of these vectors.

Give answers to 1 decimal place.

perpendicular if the angle between the lines representing them is 90°.

for example, tan 60° = 3 them is 90°.

b Find the direction of c 1 d. c Find the magnitude of c 1 d.

b The vector q, magnitude 4, direction 120°. d The vector s, magnitude

Try this question without using a calculator. 1 The vector c   and the Handy hint  3 If needed, revise vector d has magnitude 6 , your trig ratios, direction 45°. a Find the vector d. Give components in surd form.

a The vector p, magnitude 6, direction 30°. c The vector r, magnitude 5, direction 270°.

a Given that p 13, where p 5 5i 1 aj, find the value of the positive constant a. b Given that the direction of q bi 12 j is 60°, find the value of b.

Find the magnitude and direction of these vectors. When not exact, give angles to 1 decimal place. a

ii 3p  2q iii 4q  3p

8 PS

The diagram shows triangle UVW. The line WU passes through the origin O and makes an angle of 45° against the positive x-axis.

OU 5 8 cm and OW 5 3 cm.

Handy hint For part c use 3 3 3

( 3 1) .

y

U V

45° 0 W

PRACTICE: VECTORS

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x

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The lines OV and WU are perpendicular and the area of triangle UVW is 22 cm2.

5

The diagram shows points A, B and C. y

C

a Find these vectors. Give answers in i – j form, leaving components as surds.

ii OW iii OV i OU

B

b Find the exact perimeter of triangle UVW.

A

7.2 Position vectors 1

Relative to the origin O, point A has coordinates (1, 1)and point B has coordinates (2, 3).

a Write down the position vectors of the points with these coordinates. i A(3, 2) ii B(4, 3) iii C

( 3 3, 3 3)

a Find the magnitude and direction of OA .

Point C is such that AC 2 AB .

b Find the magnitude and direction of OC.

b Which of these three position vectors has the greatest magnitude? c On a single diagram, sketch the position vectors of points A, B and C. 2

x

0

For each of these pairs of points find:

ii AB . i AB Give answers in simplified surd form where appropriate.

c Show that triangle AOC has area 5 square units.

6

The diagram shows triangle ABC, where, relative to the origin O, the position vectors PS of points A and B are 3i 1 j and i 1 7j, respectively. y

a A(7, 2), B(3, 4)

B

b A(3, 6), B(5, 2) c A 3

( 8, 1) , B ( 18, 3)

C

Points A, B and C have coordinates A(2, 3), B(1, 4) and C(7, 6). a Show that A, B and C are collinear:

i by finding the vectors AB Handy hint and BC Collinear means ii by finding the equation of the line which passes through points A and B.

4

A, B and C lie on the same straight line.

b Comment on which of the methods used in part a you found easier to use.

a Given that AB 5 where A(2, 7) and B(k, 4), find the possible values of k. b Show, on a single diagram, point A and the two possible positions of point B

A 0

x

Given that AC 8i kj and BC 4i 2kj. a Show that k 5 2. b Find the exact lengths of the three sides of the triangle ABC. Give each answer in the form is an integer.

k , where k

26 , where  5 angle c Show that cos θ 26 ABC. d Using calculator accuracy, find the area of triangle ABC.

c Which of these three points is furthest from the origin?

108

PRACTICE: VECTORS

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PS 7 Relative to the origin O, the points A, B, C

and D form a parallelogram, as shown in the diagram.

Also shown is the mid-point E of BD. y

B

c Prove that  = .

E

y

D

B

x

0

a

b Find the coordinates of point B.

9 The diagram shows a trapezium OABC, where PS O is the origin, and sides AB and OC are parallel.

C

A

a Show that AB 4i 2j.

A

Explain why AB DC .

b Starting with the equation

AE AB BE show that AE 1 ( DC BC ) .

2 1 c Hence show that AE AC and explain 2 what this result tells you about the diagonals of a parallelogram.

The diagram shows the points A, B and C. PS 8 Relative to the origin O, the position vectors of A and C are 2i 1 8j and 12i 1 3j, respectively.

A B

0

x

0

1 6 The position vectors of A and C are   and    2  2 respectively, and the y-coordinate of B is greater than 2. a Show that the position vector of B can be written in the form  6k 1  for k a positive constant.  2k 2

y

α

C

The lengths of the diagonals of this trapezium are equal.

C

b Find the coordinates of point B.

β x

c Find the interior angle ABC.

Point B on AC is such that AB 2 AC .

5 Angle AOB 5  and angle BOC = .

PRACTICE: Vectors

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8 a Hint: Apply a suitable translation to the graph of y 5 sin x.

11 a Stretch, scale factor 1 , along the x-axis from the a origin.

b x 5 78.5°, x 5 281.5°

b x 5 23.6°, x 5 156.4°

c x 5 48.6°, x 5 131.4°

c i x 5 11.8°, x 5 78.2°

9 a 230°

iii x 5 15.7°, x 5 104.3°

b 210°

c x 5 233.1°, x 5 306.9°

12 a x 5 112.0°, x 5 248.0°

10 a Hint: Sketch the graph of y 5 sin x and the 5 line y 5 4 on the same diagram. b i Not possible as cos x  1 for all x.

b i x 5 37.3°, x 5 82.7°

b x 5 2319.5°, x 5 2220.5°, x 540.5°, x 5 139.5° c x 5 2135.6°, x 5 244.4°, x 5 224.4°, x 5 315.6°

iii Not possible as sin x  21 for all x.

d x 5 2246.4°, x 5 2113.6°, x 5 113.6°, x 5 246.4°

iv x 5 41.4°, x 5 318.6°

7

Vectors

7.1 The magnitude and direction of a vector a 3 2 , θ 45°

6 7

b 13,  5 112.6°

3

4

2 2 , θ 135°

a 8 2 , θ 45°

b

c 16,  5 180°

d 16 , θ 90°

 1  0 c r   d s    1  5 a Hint: work out direction of q minus direction of p.

a i 9.1,  5 6.3°

ii 17.0,  5 310.2°

3i 3 j

)

a i 4 2i 4 2 j

ii

3 2 i 3 2 j 2 2

b Hint: Use Pythagoras to find the lengths UV and VW. Answer: Perimeter 16 4 5

7.2 Position vectors 1

Answers to part a rounded 1 decimal place.

c 30°

iii 2 2i 2 2 j

 2  b q    2 3

b 2 5 5

8

b b52

 6 cos 45  3  a d     or  6 sin 45   3  b 60°

(

15

 6 cos 30°  3 3   a p    6 sin 30°   3 

a a 5 12

c 2 1 3

c 2 10 , θ 341.6° d 2 2 , θ 210° 2

ii x 5 56.0°, x 5 124.0°

13 a x 5 2286.3°, x 5 273.7°, x 5 73.7°, x 5 286.3°

ii 270°

1

ii x 5 5.9°, x 5 39.1°

 3  4 ai   ii    2  3

b OB (magnitude 5 5)

 3 3 iii    3 3

iii 18.4,  5 112.4° b r 5 26i 2 2j

ANSWERS

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c

y

5

C

a

2 , θ 135°

b 5 2 , θ 45°

A

6

b

x

0

c Hint: first show that OA and OC are perpendicular.

a Hint: Use OC OA AC and OC OB BC .

AB 52 , BC 32 , AC 68

c Hint: Use the cosine rule b 2 a 2 c 2 2ac cos Bˆ .

1 d Hint: Use 2 ac sin Bˆ . Area 5 20 square units

B

2

3

7

 10 a i   2 

ii 2 26

 8 b i    4

ii 4 5

Answer: E is the midpoint of AC so the diagonals of a parallelogram bisect each other.

8 a Hint: Use AC OC OA to find AC .

 2 c i   ii 3 2  4

a i Hint: Show that BC 2 AB .

ii Hint: show that an equation for the line through A and B is 3y 5 x 1 11. b You probably found the vector approach more straightforward. It has the advantage in telling you that the ratio AB : BC 5 1 : 2 4

a k 5 22 or k 5 6 b

y

A

b (6, 6) c Hint: Use the cosine rule to find the exact values of cos  and cos .

9 a Hint: Use the fact that AB is parallel to  6 .

OC    2

b Hint: Show AC 5 and then solve the 6k 1  . equation   2k 2 5

Answer: B(4, 3)

c Hint: Find AB , BC and then use the

B2

B1

a Hint: Use the definition of a parallelogram.

1

b Hint: Use the results AB DC and BE 2 BD .

c Hint: Use the result AB DC .

cosine rule on triangle ABC. Answer: Angle ABC 5 135° 0

x

c Point A

132

ANSWERS

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