National 5 Homework with Worked Solutions:Pupil Version · " National 5 Mathematics Revision Homework with Worked Solutions" to make photocopies of the pages for their own or their

National 5 Mathematics Revision Homework

with Worked Solutions

Alexander Forrest

Doggy

Typewritten Text

Doggy

Typewritten Text

Pupil Version : Questions only

Published in the United Kingdom by:

Beagle Bytes PO Box 6766

Fochabers IV30 9AY

Copyright © Alexander Forrest, 2013

All rights reserved. The copyright holder authorises ONLY purchasers of " National 5 Mathematics Revision Homework with Worked Solutions" to make photocopies of the pages for their own or their student's immediate use within the teaching context. No part of this book may be reproduced in any other form by photocopying or any electronic or mechanical means, including information storage or retrieval systems, without prior permission in writing from the publisher.

The right of Alexander Forrest to be identified as the author of this work has been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

First comb edition printed 2013 in the United Kingdom. A catalogue record for this book is available from the British Library. ISBN 978-0-9576916-0-5 Although every precaution has been taken in the preparation of this book, the publisher and author assume no responsibility for errors or omissions. Neither is any liability assumed for damages resulting from the use of this information contained herein.

Contents Mathematics (National 5) Expressions and Formulae........................................................2 Mathematics (National 5) Relationships .............................................................................3 Mathematics (National 5) Applications ...............................................................................4 Arcs & Sectors ...................................................................................................................5 Brackets .............................................................................................................................6 Completing the Square.......................................................................................................7 Equations and Inequalities .................................................................................................8 Factorisation 1....................................................................................................................9 Factorisation 2..................................................................................................................10 Formulae ..........................................................................................................................11 Fractions 1 .......................................................................................................................12 Fractions 2 .......................................................................................................................13 Line of Best Fit .................................................................................................................14 Money & Finance – 1 .......................................................................................................15 Money & Finance – 2 .......................................................................................................16 Properties of Shapes........................................................................................................17 Pythagoras’ Theorem.......................................................................................................18 Quadratic Check-up .........................................................................................................19 Scale Factor and Area......................................................................................................20 Scale Factor and Volume.................................................................................................21 Similar Triangles ..............................................................................................................22 Simultaneous Equations..................................................................................................23 Standard Deviation & Boxplots.........................................................................................24 The Straight Line 1...........................................................................................................25 The Straight Line 2...........................................................................................................26 Surds & Indices ................................................................................................................27 Trigonometry ...................................................................................................................28 Trig Equations ..................................................................................................................29 Trig – Area of triangle and exact values ...........................................................................30 Trig graphs .......................................................................................................................31 Vectors .............................................................................................................................34 Volume.............................................................................................................................36 Solutions ..........................................................................................................................37 An attempt has been made to match these homework sheets to the relevant outcomes of the new Scottish National 4 and National 5 units, but it is stressed that this is the author’s interpretation only.

Pupil Version : Questions Only

Page 2 National 5 Mathematics Revision Homework © A Forrest 2013

Mathematics (National 5) Expressions and Formulae Nat 5 Topic Homework Sheet

E&F 1.1 Working with surds Surds & Indices

Pythagoras' Theorem

E&F 1.1 Simplifying expressions using the

laws of indices Surds & Indices

E&F 1.2 Working with algebraic expressions

involving expansion of brackets Brackets

E&F 1.2 Factorising an algebraic expression Factorisation 1 Factorisation 2

Quadratic Check-up

E&F 1.2 Completing the square in a quadratic expression with unitary x2 coefficient

Completing the square Quadratic Check-up

E&F 1.3 Reducing an algebraic fraction to its

simplest form Fractions 1 Fractions 2

E&F 1.3 Applying one of the four operations to

algebraic fractions

Fractions 1 Fractions 2 Formulae

E&F 1.4 Determining the gradient of a straight

line, given two points The Straight Line 2

E&F 1.4 Calculating the length of arc or the area

of a sector of a circle Arcs & Sectors

Properties of shapes

E&F 1.4 Calculating the volume of a standard

solid Volume

E&F 1.4 (Rounding to a given number of

significant figures) Formulae Volume

E&F 2.1 Interpreting a situation where

mathematics can be used and identifying a strategy

Fractions 2

E&F 2.2 Explaining a solution and/or relating it to

context Fractions 2

Mathematics (National 4) Expressions and Formulae

E&F 1.1 Using the distributive law in an

expression with a numerical common factor to produce a sum of terms

Brackets

E&F 1.1 Simplifying an expression which has

more than one variable Brackets Formulae

E&F 1.1 Evaluating an expression or a formulae

which has more than one variable Brackets

E&F 1.1 Calculating the gradient of a straight line from horizontal and vertical distances

The Straight Line 1


National 5 Mathematics Revision Homework © A Forrest 2013 Page 3

Mathematics (National 5) Relationships Nat 5 Topic Homework Sheet

Rel 1.1 Determining the equation of a straight

line, given the gradient The Straight Line 1 The Straight Line 2

Rel 1.1 Working with linear equations and

inequations Equations and Inequalities

Rel 1.1 Working with simultaneous equations Simultaneous Equations

Rel 1.1 Changing the subject of a formula Formulae

Rel 1.2 Recognise and determine the equation of

a quadratic function from its graph Completing the square

Rel 1.2 Sketching a quadratic function Completing the square

Rel 1.2 Identifying features of a quadratic

function Quadratic Check-up

Rel 1.3 Working with quadratic equations Quadratic Check-up

Rel 1.4 Applying the Pythagoras’ theorem Pythagoras' Theorem

Rel 1.4 Applying the properties of shapes to

determine an angle involving at least two steps


Rel 1.4 Using similarity Scale Factor and Area

Scale Factor and Volume Similar Triangles

Rel 1.5 Working with the graphs of trigonometric

functions Trig Equations

Rel 1.5 Working with trigonometric relationships

in degrees

Trig Equations Trig – Area of triangle and exact values

Trig graphs

Rel 2.1 Interpreting a situation where



Rel 2.2 Explaining a solution and relating it to

context Properties of shapes

Mathematics (National 4) Relationships

Rel 1.1 Drawing and recognising a graph of a

linear equation. The Straight Line 1

Rel 1.1 Solving linear equations. Equations and Inequalities

Rel 1.1 Changing the subject of a formula. Formulae

Rel 1.2 Using Pythagoras’ theorem Pythagoras' Theorem

Rel 1.4 Drawing and applying a best-fitting

straight line Line of Best Fit



Mathematics (National 5) Applications Nat 5 Topic Homework Sheet

Apps 1.1 Calculating the area of a triangle using

trigonometry Trigonometry

Trig – Area of triangle and exact values

Apps 1.1 Using the sine and cosine rules to find a

side or angle Trigonometry

Apps 1.1 Using bearings with trigonometry Trigonometry

Apps 1.2 Working with 2D vectors Vectors

Apps 1.2 Working with 3D coordinates Vectors

Apps 1.2 Using vector components Vectors

Apps 1.3 Working with percentages Money & Finance - Worksheet 1

Apps 1.3 Working with fractions Fractions 1 Formulae

Apps 1.4 Comparing data sets using statistics Standard Deviation & Boxplots

Apps 1.4 Forming a linear model from a given set

of data Line of Best Fit

Apps 2.1 Interpreting a situation where


Money & Finance - Worksheet 2

Apps 2.2 Explaining a solution and/or relating it to

context Money & Finance - Worksheet 2

Vectors



60°

6cm

O

A

B

Arcs & Sectors 1) Calculate the area of sector OAB. 2) Calculate the length of arc AB. (Give your answers to 2 dp.) 3) Find angles BCA and BOA . (Give your answers to 1 dp.)

4) What is the length of chord AB? (Give your answer to 2 dp.) 5) What is the area of ΔABC ? (Give your answer to 2 dp.)

OA

B

C7cm

6 cm

Nat 5 E&F 1.4



Brackets

1) Simplify the following :

a) 5x2 – 3x2 b) 2y2 –6y2 c) - z2 - z2 d) k2 - k2

e) 5(m – n) – 3( m + n) f) 3( a + b) = 2( a –5b)

2) Find the sum of :

a) 2x + 3y and -6x + 12y b) p – 3q – r and 5p + 3q + r

c) 4c – 6d + 5e and –2c –3d –7e d) x2 - x - 4 and 3x2 – x + 5

3) Simplify the following:

a) 4x2 – 4x –2x2 – 2x b) k( 2k – 1) – 2k( k + 2)

c) x(x2 – 3) –2x(x +2) + (x2 + x) d) 2(x2-x) +3(x2 + x) –(x2 + 2x)

4) Simplify the following:

a) (a + 2)(b + 5) b) (x –1)( y – 4) c) (x + 3)(2x2 – x – 3)

d) (a + b)2 – ( a – b)2 e) (2x + 3)2 – (3x + 2)2 f)

5) Writing 2.01 as 2 + 0.01, show that 2.012 = 4.04 to two decimal places.

6) Prove that ( a + b)(a2 – ab + b2) = a3 + b3 .

2 21 1

x xx x

Nat 4 E&F 1.1



Completing the Square 1 ) Write each expression in the form (x + p )2 + q

2 2 2

2 2 2

) 6 10 b) 2 3 c) 8 10

) 10 5 e) 18 81 f) 40 1

a x x y y z z

d a a b b c c

2) Write each expression in the form p - (x + q )2 3) Write each expression in completed square form

2 2 2

2 2 2

) 2 4 1 b) 3 12 5 c) 5 30 18

) 2 2 1 e) 4 12 5 f) 2 3 6

a x x y y z z

d a a b b c c

4)

For each function : (i) Write down the equation of the axis of symmetry; (ii) Write down the coordinates of the turning point; and (iii) Sketch the graph.

2 2

22

) 3 2 b) 2 3 4

1c) 1 ) 10 2 2 3

2

a y x y x

y x d y x

2 2 2

2 2 2

) 4 2 - b) 5 4 - c) 2 -

) 6 3 - e) 2 - - f) 4 - 6 -

a x x x x x x

d x x x x x x x

Nat 5 E&F 1.2



Equations and Inequalities

1) 6 8 26

2) 5 -15 30

3) 7 - 4 8 7 - 29

4) 3(2 6) 4( 8)

5) 3(2 - 3 2 ) 5(12 - 8) - 6

Solve

x

x

x x

x x

y y y y

6) 2 8 26

7) 3 -15 30

8) 7 4 8 7 27

9) 3(2 - 6) 4( - 8)

10) 3(2 2 ) 5(12 - 8)

x

x

x x

x x

x x

11) The graph below shows the equation 3x +5 < 2x + 1. Which of the two points A( -6,2) or B( -3,1) lie within the solution set ?

Nat 5 E&F 1.2 Nat 5

Rel 1.2 Nat 5

Rel 1.1

Nat 4 Rel 1.1



2

2

2

2

2

2

2

2

2

2

) 3 2

) 14 45

) 10 25

) 5 36

) 9 8

) 6 7

) 4 45

) 71 72

) 24 11

) 25 10

a x x

b x x

c b b

d x x

e c c

f r r

g x x

h z z

i x x

j a a

2

2

2

2

2

2

2

2

2

2

) 2 5 3

) 6 7 2

) 9 6 1

) 2 19 9

) 5 23 10

) 10 19 6

) 2 1

) 8 10 3

) 1 3 18

) 15 7 2

a x x

b a a

c c c

d x x

e c c

f r r

g x x

h z z

i x x

j p p

Factorisation 1 1) Remove the brackets 2) Copy and complete 3) Factorise

4) Factorise 5) Factorise (Difference of two squares)

2

2

2

2

) 7 12 ( 4)( )

) 6 5 ( )( )

) 8 15 ( 3)( )

) 12 20 ( )( )

a x x x x

b x x x x

c x x x x

d x x x

2 2

2

2

2 2

2

)

) 36

) 4

) 16 9

) 4 100

a x t

b x

c y

d x y

e c

) ( 4)( 5)

) (2 5)( 1)

) ( 5)( 5)

) ( 4)(3 15)

a x x

b x x

c a a

d x x



Factorisation 2 Factorise Fully

2 2

2 2

2 2

2 2

2 2

2

2

2

2

4

2

3

2 2

2 2

) 8 18

) 1

) 3 2

) 4

) 5 9 2

) 6 25 9

) 6 5 1

) 2 3 1

) 6 5 6

) 16

) 3 6 3

) 4

) 5 20

)

a x y

b a b

c a ab b

d a b c

e x xy y

f x x

g x x

h x x

i x x

j x

k x x

l x x

m x y

n ab ac

Expand

2

2

) 2 3 3 4

) 3 2

) 2 5 3 2

) 2 3

) 5 3

a x y x y

b x y x y

c x y x y

d x y

e x y

Simplify

2 2

2 2 2 2

2 2

2

2

) 3 2 5 6 9

) 4 3 5 3 2 5 4

) 3 2 3

) 75 25

2 3 1)

6 3

a x x x x

b a c d a c d

c x y x y

d

x xe

x x

Nat 5 E&F 1.2



Formulae 1) The cost of tuition, £c, is £20 per hour, plus a booking fee of £5.

a) Write a formula to express the cost of tuition for x hours. b) Use the formula to calculate the cost of 12 hours tuition.

2) The formula Ek = ½mv2 is used to calculate the kinetic energy (Ek) of an object

with mass (m) kilograms travelling at velocity (v) metres per second.

Use the formula to calculate the energy of an object of mass 10 kg travelling at velocity 15m/s .

3) Use Newton’s Law of Gravitation,

2

MmF G

R

to calculate the gravitational force between Earth and the Moon.

Give your answer correct to 3 significant figures.

M = 5.9722x 1024 Kg (Mass of the Earth.) m = 7.3477x 1022 Kg ( Mass of the Moon.) G = 6.672 x 10-11 N(m/kg)2 (Gravitational Constant .) R = 384403 km (Distance between Earth and the Moon.)

4) Rearrange the formula v = u + at to make t the subject.

5) Rearrange the formula 2

MmF G

R to make R the subject.

6) A square and a triangle both have the same base and an area of 81 cm2 . Calculate the dimensions of the triangle.

Nat 5 E&F 1.2

81cm2

81cm2



Fractions 1

2

Simplify the following :

12 20 32a 12x 72y1) 2) 3) 4) 5)

16 25 16ab 48 81y

Change to mixed numbers:

47 22 32 63 396) 7) 8) 9) 10)

9 7 16 21 15

Change to improper (top heavy) fractions:

12 3 6 4 9

1) 1 12) 2 13) 5 14) 3 15) 17 16 7 15 18


1 1 1 3 2 3 116) 17) 18) 3 2

2 3 4 5 15 5 4

2 1 1 219) 4 3 20)

9 3 x y

1 2 1 5 2 5 221. 22. 3 23. 3

3 3 4 12 3 12 3

1 2 4 1 2 4 124. 25. 4 2 -

3 3 30 3 3 30 15

Nat 5 Rel 1.1

Nat 5 Apps 1.3

Nat 5 E&F 1.3 E&F 1.4

Nat 4 Rel 1.1

Nat 4 E&F 1.1

Nat 5 E&F 1.3 E&F 1.1

Nat 5 Apps 1.3



Fractions 2


2 3

2 2 2

2

4 6 12 8 14 101) 2) 3)

2 4 2

8 16 18 12 34) 5) 6)

2 6 2

3 3 47) 8) 9)

2 1 2

x a a b

x y a a x

a xy xz

p a x y

p p a x y

Solve the equations:

2 6 4 3 2 3 3 10

10) =10 11) = 12) = 3 4 4 4 7

x a a a

13) A fox broke into a henhouse and killed half of the chickens. It then took another one and left. A mink then slunk in and killed half of the remaining flock, leaving 6 terrified chickens. How many chickens were there before the break in?

Nat 5

E&F 1.3 2.1 2.2



Line of Best Fit Test scores for an S4 maths and physics test are shown below: Pupil 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Maths 70 44 15 78 48 90 49 32 81 55 68 86 66 71 77Physics 62 50 16 83 66 42 88 25 59 50 87 95 77 78 82 They are plotted on a scatter graph.

Maths and Physics scores

0

10

20

30

40

50

60

70

80

90

100

0 10 20 30 40 50 60 70 80 90 100

Maths Score

Ph

ysic

s S

core

a) Is there a correlation between scoring well in maths and physics? Explain your answer. b) Draw a line of best fit on the scatter graph and use it to find an equation linking the

physics and maths test results for this data set. c) Use your equation to predict the physics test score for a pupil who scored 55 in the

maths test.

Nat 5 Apps 1.3

Nat 5 Apps 1.4



Money & Finance – 1 1) Glenfox Lodge was valued at £365,000 on 30th April 2001. If appreciation is 4% per annum, what is the value of Glenfox Lodge on 30th April 2011? 2) A car was bought for £ 8000 in 1998. Each year, it depreciated in value by 20%. What was the car worth 4 years later? 3) Calculate the compound interest earned on: a) £500 for 3 years at 6% per annum. b) £875 for 2 years at 4.5% per annum c) £550 for 7 months at 6.2% 4) Greg Gregson earns £ 296 per week. He pays 6% superannuation. a) How much superannuation does he pay per week? b) How much superannuation does he pay per year? National Insurance is payable at 2% on the first £56 earned and 9% on the rest. c) How much National Insurance does he pay per week? d) How much National Insurance does he pay per year?

He also pays £39.31 tax and £2.80 union dues each week. e) What is his weekly net pay? 5) HMRC Income tax rates

2010-11

2011-12

Basic rate £0 - £37,400

at 20 per cent. £0 - £35,000

at 20 per cent.

Higher rate £37,401 - £150,000

at 40 per cent. £35,001 - £150,000

at 40 per cent.

Additional rate Over £150,000 at 50 per cent.

Over £150,000 at 50 per cent.

Charlie Charleson has a personal allowance of £7,475.Her salary is £44,350 . a) Calculate her annual income tax for the tax years 2010/11 and 2011/12. b) How much more tax does she have to pay in 2011/2012? c) Express this increase as a percentage of her annual income tax for 2010/11. d) Calculate her monthly income tax for 2011/12.

Nat 4 Rel 1.4

Nat 5 Apps 1.3



Money & Finance – 2 1) You wish to buy a new car which has a cash price of £ 13,335. The following options are available:

Which is the best option? Give reasons for your answer.

2) A high street electrical retailer offers a television set for £700.

Hire purchase is available on the product for a 10% deposit and 2 years of monthly payments. The total amount payable is £826.72.

(i) Calculate the monthly repayment due.

(ii) Express the cost of the hp as a percentage of the cost of the television set.

Finance Package Deposit £ 2738.63 Plus 35 monthly payments of £199 and walk away. Final GFV payment £5234.40 to keep the vehicle / future trade in.

Bank of Bod Loan £13 500 at fixed rate of 3.2% pa for 36 months.



Properties of Shapes 1) Find the area of quadrilateral ACOB in terms of r and x by considering it as two right-

angled triangles. 2) Hence show that an expression in terms of r and x for the shaded area is:

Shaded area = 2 tan180

xr x

x

r

x

AO

C

B

Remember to justify all the steps of your working and write your solution so that it is clearly understood by anyone who reads it.

Does the solution read well? Is there sufficient explanation? The reader should be led, line by line, through your argument.

Nat 5 Rel 1.4 2.1 2.2

Nat 5 E&F 1.4

Nat 5 Apps 2.1 2.2



Pythagoras’ Theorem 1) Use Pythagoras’ Theorem to find x for each triangle. Round any decimals answers to 1 decimal place. Give the answers to h, i and j as exact values in their simplest form. a) b) c) d) e) f)

x

93

x11

8

g) h) i)

4

x

21

j)

1515

x

2) Is it possible to have a right-angled triangle with sides 5, 7 and 11 cm ?

Give reasons for your answer.

x6

3

x

12

5

x

12

15

7

x

13

18

x57

x

15



Quadratic Check-up

1) Solve each of the following quadratic equations by factorisation:

2) Solve the following by using the quadratic formula a

acbbx

2

42

Give answers, where they exist, to 2 dp. 3) Sketch each of the following quadratics on separate diagrams. Clearly mark the roots and y-intercept on each diagram.

2

2

2

2

2

2

2

2

) 3 2 0

) 2 0

) 4 3 0

) 2 1 0

) 3 15 18 0

) 3 2 1 0

) 4 2 6 0

) 2 14 36 0

a x x

b x x

c x x

d x x

e x x

f x x

g x x

h x x

2

2

2

2

2

2

2

2

) 4 3 5 0

) 2 5 2 0

) 5 3 0

) 2 2 1 0

) 2 2 0

) 7 8 0

) 3 3 3 0

) 2 0

a x x

b x x

c x x

d x x

e x x

f x x

g x x

h x x

2

2

2

2

2

) 12

) 2 2 12

) 3 2 8

) 2 8

) 2

a y x x

b y x x

c y x x

d y x x

e y x

Nat 4 Rel 1.2

Nat 5 Rel 1.4

Nat 5 E&F 1.1



Scale Factor and Area 1) A photograph is enlarged in the ratio 4:1.

What is the area of the enlargement?

2) What is the area of the larger circle?

2 cm 4 cm 3) What is the area of the smaller similar triangle? 2 cm 4 cm 4) A photograph is reduced in the ratio 2:3.

What is the area of the smaller photograph?

8.5 cm2 x cm2

8.5 cm2

x cm2

x cm2 5 cm2

x cm 2

Nat 5 Rel1.2 Rel 1.3

Nat 5 E&F 1.2

24 cm 2



Scale Factor and Volume 1) What is the volume of the enlargement?

3 cm 9 cm 2) What is the volume of the larger can?

2 cm 4 cm 3) Orangi is a new drink sold in similar triangular cartons. What is the volume of the smaller carton? 5cm 20 cm

15 cm3

x cm3

150 cm3

x cm3

640 cm3

x cm3

Nat 5 Rel 1.4



Nat 5 Rel 1.4

Similar Triangles For each of the following isosceles triangles:

Sketch the two similar triangles; and Find the value of the missing length x.

1)

20 cm

x cm

5 cm

4 cm

2)

12 cm

x cm

6 cm2 cm



Simultaneous Equations

1) Use the graph to write down the solution to the simultaneous equations y= 3x +5 and y = -2x.

2) Solve the equations: a) b) c)

3 13

5 3 7

x y

x y

2 5 39

3 23

x y

x y

3 4 7

4 3 24

x y

x y

3) The total cost of an order for three hamburger meals and two chicken meals is £19.45

The total cost of an order for two hamburger meals and three chicken meals is £19.30

Find the individual costs for each meal.

Nat 5 Rel 1.4

Nat 5 Rel 1.1



Standard Deviation & Boxplots 1) The number of punishment exercises handed out to pupils in various schools on

a certain day were counted in a survey:

8 12 18 23 26 34 65

a) Calculate the standard deviation. b) Calculate a 5 figure summary of the data. c) Calculate the range. d) Calculate the interquartile range. e) Calculate the semi – interquartile range. f) Draw a boxplot of the data.

2) A second survey was carried out one week later.

7 14 17 24 28 40 70

a) Calculate the standard deviation. b) Calculate a 5 figure summary of the data. c) Calculate the range. d) Calculate the interquartile range. e) Calculate the semi – interquartile range. f) Draw a boxplot of the data on the same axis as the one drawn in question

(1)(f) above.

3) What do you notice when you compare the two sets of data?

22 /

-1x x n

sn



The Straight Line 1 1) Find the gradient, y-intercept and equation of each of the following lines : a) b) 2) Write down the equation of the line which passes through the point (0, 5) with gradient 3. 3) For each of the following, copy and complete the table and sketch each line. a) y = 4x – 3

b) 2y - 6x – 4 = 0

x -5 -4 -3 -2 -1 0 1 2 3 4 5 y

x -5 -4 -3 -2 -1 0 1 2 3 4 5 y

y

x1 2 3 4 5 – 1 – 2 – 3 – 4 – 5

1

2

3

4

5

– 1

– 2

– 3

– 4

– 5

y

x1 2 3 4 5 – 1 – 2 – 3 – 4 – 5

1

2

3

4

5

– 1

– 2

– 3

– 4

– 5

Nat 5 Apps 1.4



The Straight Line 2 1) Find the gradient of the lines joining: a) P(-1, 7) to Q( 5,10) b) A(-6,-2) to B( 2, -4) c) V( 1,-2) to W( 5, 0) What can you say about the lines PQ and VW? 2) Write down the equation of the line which passes through (0, 5) with gradient 3. 3) For each of the following, plot 3 or more points and sketch each line: a) y = 4x – 3 b) 6y + 3x – 18 = 0 4) Write down the equation of the graph below:

Nat 5 Rel 1.1

Nat 4 Rel 1.1

Nat 4 E&F 1.1



Surds & Indices 1) Simplify the following : 2) Rationalise the denominator, expressing your answer in its simplest form: 3) Calculate each of the following, expressing your answer with a rationalised

denominator in its simplest form: 4) Calculate: a) 34 b) 53 c) 75 x 35 d) 2-3 e) 4-5 f) 34 x 34 g) (53 )4 h) 75 ÷ 73 5) Write out the following as roots:

a) 31/2 b) 5-1/3 c) 72/3 d) 2-3/4 e) 45/8

) 0.0196 ) 40 ) 200 ) 189 ) 8181 a b c d e

7 56 3 5 3) ) ) ) )

15 12 6 8 6 8 2 8a b c d e

22 1 56 3 2 3) ) ) ) 3 2 3

3 15 12 6 8

a b c d

Nat 5 Rel 1.1

Nat 5 E&F 1.4



Trigonometry 1) Calculate the missing values a) b) c) 2) Calculate the bearing of the robot from the jogger.

50° 110°

12 cm

x cm

70° 7 cm

8 cm

x cm

50° 45°

7 cm x cm

168 m150 m

252 m

Nat 5 E&F 1.1

Nat 5 Apps 1.1



Trig Equations

11) Write down the equation of the following graph and state the values of x that give the solution y =1 for 0 450 x .

Nat 5 Rel 1.5

Solve

1) sin 0.5 0 360

2) cos 1 0 720

3) 2cos 1 0 360

4) 3cos 1 0 720

5) 8cos 5 1 0 360

6) 12sin 8 3 90 360

7) 2sin 2 1 0 0 360

8) 2cos

x x

x x

x x

x x

x x

x x

x x

2

2 1 0 0 720

9) 2cos 1 0 360

10) 2tan2 1 0 0 720

x x

x x

x x



Trig – Area of triangle and exact values 1) Calculate the area of the triangle.

7 cm 50° 8 cm

2) Given that the area of the triangle below is 74 cm2, calculate the angle θ°. Give your answer to 1 dp.

15 cm

θ°

10 cm

3) Find the exact values of a) sin 135° b) cos 225° c) tan240° d) sin210° e) cos 300° f) tan315°

Nat 5 Rel 1.5



Trig graphs For each of the following six graphs, write down its equation, amplitude and period.

1) 2)

Nat 5 Rel 1.5



3)

4)



5)

6.



Vectors 1) Write out each of the following vectors in component form.

2) In the following picture, a Helicopter (H) must take on some fuel at (F) before

rescuing the canoeist (C). North is in the direction of the y axis. Distance is in units.

Given the points C( -3, 1) F(0,0) and H( 3 ,3) :-

a) Work out the individual components of the helicopter flight and sum them to find the resultant vectorHC

.

b) Use the answer to 2a to state the direction and how many units the helicopter has

travelled from its original starting place.

c) Sketch the vector diagram of the helicopter flight, clearly showing the resultant vector.



3) Write down the co-ordinates of corners B, C, D, E, F and G.

. 4)

2

2 (3, -1, 5)

1

The vector is applied to the point Z to make

b ZP .

What are the co-ordinates of P?

5)

2 5 2 8

2 24 2 0

1 13 11 1

a b c d

Calculate a + b + c - d.

Nat 5 Apps 1.2 2.2



Volume Calculate the volume of the following, giving your answers correct to 2 significant figures : 1) 2)

3)

4)

7cm

16cm

6 cm

7cm

22 cm

The volume of this globe is 430 cm3. What is its radius?

2

2

4 33

1

3

sphere

cone

cylinder

V r

V r h

V r h


Documents

National 5 Homework with Worked Solutions:Pupil Version · " National 5 Mathematics Revision Homework with Worked Solutions" to make photocopies of the pages for their own or their