Exercises in GCSE

Mathematics
Higher Level
Robert Joinson

Sumbooks
Sum books Chester CH4 8BB

Exercises in GCSE Mathematics-Higher level

First Published 1998 Computer edition 2000

Copyright R Joinson and Sum books

This package of worksheets is sold subject to the condition

that it is used for educational purposes only on
the premises of the purchaser.

ISBN 0 9531626 2 1
 Preface
This book covers the GCSE syllabi examined for the first time in 1998.
However, not all parts of the book are used by all the examination boards.
Some areas have more questions than are needed for some pupils. Exercises
on pages 1, 8, 9, 10, 11, 14, 18, 19, and 21 contain lots of questions and are
aimed at pupils requiring a great deal of practice. These questions are graded
and it might only be necessary for some students to do the first column and
then each row when they begin to have problems. In general questions in the
same row tend to be of the same difficulty, whereas the difficulty increases
down the page.
All graphs can be accommodated on A4 size graph paper used in ‘portrait’
mode.
I would like to thank my wife Jenny and my two daughters Abigail and Hannah
for all the help and encouragement they have given me in writing this.

R Joinson August 1998

Chester
.
 Higher Level Contents

1. Estimation and Calculations

2. Standard Form
3. Rational and Irrational Numbers
4. Surds
5. Prime Factors
6. Percentages
7. Conversion Graphs
8. Fractions
9. Indices
10. Simplifying
11. Rearranging Formulae
12. Number Sequences
13. Substitution
14. Factorising
15. Trial and Improvement
16. Iteration
17. Direct and Inverse Proportion
18. Solving Equations 1
19. Solving Equations 2
20. Writing Simple Equations
21. Simultaneous Equations
22. Writing Quadratic Equations
23. Straight Line Graphs
24. Inequalities
25. Linear Inequalities
26. Linear Programming
27. Recognising Graphs
28. Graphs 1
29. Graphs 2
30. Graphs 3
31. Growth and Decay
32. Distance - Time Diagrams
33. Velocity - Time Diagrams
34. Angles and Triangles
35. Regular Polygons
36. Congruent Triangles
37. Geometry of a Circle 1
38. Geometry of a Circle 2
39. Vectors 1
40. Vectors 2
41. Similar Shapes
42. Similarity
43. Bearings
44. Constructions
45. Loci
46. Transformations 1
47. Transformations 2
48. Transformations of Graphs
49. Matrix Transformations
50. Area and Perimeter
51. Volume
52. Ratios and Scales
53. Degree of Accuracy
54. Formulae
55. Pythagoras Theorem
56. Sine, Cosine and Tangent Ratios
57. Sine and Cosine Rules
58. Areas of Triangles
59. Trigonometry - Mixed Exercise
60. Graphs of Sines, Cosines and Tangents
61. Three Dimensional Trigonometry
62. Questionnaires
63. Sampling
64. Scatter Diagrams
65. Pie Charts
66. Flowcharts 1
67. Flowcharts 2
68. Histograms 1
69. Histograms 2
70. Histograms 3
71. Mean
72. Mean, Median and Mode
73. Mean and Standard Deviation
74. Frequency Polygons
75. Cumulative Frequency 1
76. Cumulative Frequency 2
77. Probability 1
78. Probability 2
79. Relative Probability
80. Tree Diagrams
©Sum books 1998 Higher Level

1. Estimations and Calculations

In questions 1 to 36 use your calculator to work out the value of the problem. In each case
a) write down the calculator display correct to four significant figures
b) write down an estimate you can do to check your answer to part a.
c) write down your answer to part b.

1.631 × 5.482 12.62 × 3.851 9.143 × 8.639

1) --------------------------------- 2) --------------------------------- 3) ---------------------------------
4.8143 2.415 4.384
0.4831 × 6.8159 2.841 × 0.356 8.143 × 0.3142
4) --------------------------------------- 5) --------------------------------- 6) ------------------------------------
3.286 9.416 7.8154
0.7154 × 0.832 0.1843 × 0.6315 0.5714 × 0.8354
7) ------------------------------------ 8) --------------------------------------- 9) ---------------------------------------
5.416 6.4153 7.831
2.453 × 0.154 8.415 × 0.3142 15.815 × 0.2231
10) --------------------------------- 11) ------------------------------------ 12) ---------------------------------------
0.3651 0.2164 0.8145
14.264 × 27.742 27.98 × 96.41 184 × 32.415
13) --------------------------------------- 14) --------------------------------- 15) -------------------------------
0.6421 0.415 0.283
16.854 × 0.0182 27.684 × 0.8142 67.48 × 29.684
16) --------------------------------------- 17) --------------------------------------- 18) ------------------------------------
0.3145 0.0156 0.00142
2 2 2
( 27.63 ) × 14.32 4.831 × ( 0.821 ) 23.41 × ( 17.83 )
19) ---------------------------------------- 20) ---------------------------------------- 21) ----------------------------------------
27.61 0.614 126.4
2 2 2
( 15.84 ) × 1.982 105 × ( 0.325 ) ( 0.312 ) × 27.4
22) ---------------------------------------- 23) ------------------------------------ 24) -------------------------------------
0.614 × 3.28 0.761 × 26.45 0.51 × 6.72
43.1 × 6.74 14.71 × 23.42 27.41 × 0.8541
25) --------------------------------- 26) --------------------------------- 27) ------------------------------------
8.931 + 12.82 16.84 + 33.21 4.312 + 6.842
2 2
( 26.3 ) × 0.416 41.6 ( 8.42 + 3.86 ) ( 27.64 ) × 19.68
28) ------------------------------------------- 29) ------------------------------------------- 30) -------------------------------------------
2.84 ( 3.14 + 2.86 ) 0.415 2.54 ( 2.84 + 6.41 )
2 2 2 2
27.63 × ( 18.41 ) ( 28.31 ) × ( 0.843 ) 36.38 × ( 4.321 )
31) ------------------------------------------ 32) ------------------------------------------------ 33) -------------------------------------------------
0.31 ( 8.62 – 4.31 ) 0.415 0.31 ( 2.865 + 7.416 )
2 2
14.841 × 26.832 ( 41.821 ) × 0.0154 ( 0.531 ) × 0.00614
34) ------------------------------------------
- 35) ---------------------------------------------- 36) ----------------------------------------------
( 5.483 ) + 16.943
2 0.831 × 0.0132 0.8312 × 0.0154

2
( 8.64 ) + 29.83
37) If v = ------------------------------------- a) Use your calculator to find the value of v, correct to 3
0.0154
significant figures. b) What figures would you use to check the value of v? c) Write
down the answer to part b.
3
( 27.61 ) × 0.00814
38) If D = ---------------------------------------------- a) Use your calculator to find the value of D correct to 4
3.61 ( 2.48 + 5.61 )
significant figures. b) What figures would you use to check the value of D? c) Write down
the answer to part b.
©Sum books 1998 Higher Level

2. Standard Form

Exercise 1
Write down these numbers in standard form
1) 457 2) 1427 3) 9431
4) 156,321 5) 17 million 6) 0.2813
7) 0.08142 8) 0.000486 9) 0.0000097

Exercise 2
Change these numbers from standard form
5 7 4
1) 2.8 × 10 2) 6.4 × 10 3) 9.3 × 10
6 9 –2
4) 4.315 × 10 5) 8.614 × 10 6) 4.31 × 10
–6 –7 –9
7) 3.2 × 10 8) 6.84 × 10 9) 4.38 × 10

Exercise 3
Calculate each of the following leaving your answers in standard form. Round off to four
significant figures wherever necessary.
2 4 6 2
1) ( 6.4 × 10 ) × ( 3.8 × 10 ) 2) ( 5.4 × 10 ) × ( 8.3 × 10 )
–2 –3 –7 5
3) ( 4.6 × 10 ) × ( 3.4 × 10 ) 4) ( 4.3 × 10 ) × ( 8.8 × 10 )
–2 8 –7 3
5) ( 8.3 × 10 ) × ( 6.4 × 10 ) 6) ( 5.3 × 10 ) × ( 4.6 × 10 )
–2 2 –6 3
7) ( 5.7 × 10 ) ÷ ( 3.4 × 10 ) 8) ( 8.3 × 10 ) ÷ ( 5.4 × 10 )
5 –3 –7 –2
9) ( 8.4 × 10 ) ÷ ( 2.4 × 10 ) 10) ( 5.4 × 10 ) ÷ ( 4.3 × 10 )
5 6
11) 137, 000 × 10 12) 0.08123 × 10
6 –7
13) 27.31 × 4.82 × 10 14) 571.31 × 4.2 × 10
6 –4
3.841 × 10 7.41 × 10
15) ---------------------------2- 16) --------------------------
–6
-
3.182 × 10 3.54 × 10
3 7 –3 –6
( 27.41 × 10 ) × ( 2.684 × 10 ) ( 2.641 × 10 ) × ( 2.84 × 10 )
17) -----------------------------------------------------------------------
5
- 18) -------------------------------------------------------------------------
5
-
7.41 × 10 3.82 × 10
4 –6
19) 4 × 10 20) 9 × 10
Exercise 4
13
1) The distance from the earth to a star is 8 × 10 kilometres. Light travels at a speed of
5
approximately 3.0 × 10 kilometres per second.
a) How far will light travel in one year?
b) How long will light take to travel from the star to the earth? Give your answer correct to
3 significant figures.
24 22
2) The mass of the earth is 5.976 × 10 kilograms and the mass of the moon is 7.35 × 10
kilograms.
a) Write down these masses in tonnes.
b) How many times is the mass of the earth greater than the mass of the moon?
– 27 – 31
3) A neutron has a mass of 1.675 × 10 kilograms and an electron 9.109 × 10 kilograms.
a) How many electrons are needed for their mass to be equal to that of a neutron?
b) How many electrons are required to have a mass of 1 kg?
©Sum books 1998 Higher Level

3. Rational and Irrational Numbers

Exercise 1
Convert the following numbers into fractions
1) 0.25̇ 2) 0.31̇ 3) 0.54̇ 4) 0.62̇
5) 0.04̇ 6) 0.73̇ 7) 0.007̇ 8) 0.017̇
9) 0.1̇5̇ 10) 0.3̇2̇ 11) 0.5̇3̇ 12) 0.07̇2̇

Exercise 2
Which of the following numbers are rational?
1 7 1 7
1) --- 2) --- 3) ------- 4) -------
5 8 2 8
2
5) π 6) π 7) 0.23̇ 8) 3
2 2
12)  ---
4 9 5
9) ---------- 10) ------ 11) ------2-
25 49  3
5
13) 6.25 14) 2.5 15) 4 16) 4.1

Exercise 3
Which of the following are irrational?
1 1 3
1) 1 + --- 2) 1 + ------- 3) 1 + -------
3 3 2
1 2 18 12
4) 3 + ------- 5) 2 – ------- 6) ------- – -------
3 2 3 3
0 –1 –2
7) 2 + 2 +2 8) ( 1 + 2 ) × ( 1 + 2 ) 9) 2 2
10) 2.58 11) 0.37̇ 12) 1.5̇1̇
13) 8× 2 14) 12 × 2 15) 2× 3
1 1 1 1
--- 0 --- –2 --- – ---
16) 6 + 32
17) 4 × 9
2
18) 4 × 9 2
2

Exercise 4
Which of the following equations have rational answers?
2
1) 2x = 5 2) 3x = 3 3) 4x = 2
2 7 4
4) 2x = ------- 5) 5x = --- 6) 3x = -------
4 8 3
2 2 2
7) 5x = 7 8) 9x = 4 9) 4x = 17
2 1 3 2 2
10) 3x = --- 11) --- x = 12 12) 7x = 5
2 4

Exercise 5
Which two of the following are descriptions of irrational numbers?
a) A number which, in its decimal form, recurs.
b) A number written in its decimal form has a finite number of decimal places.
c) A number whose exact value cannot be found.
d) A number which can be represented by the ratio of two integers.
e) An infinite decimal which does not repeat itself.
©Sum books 1998 Higher Level

4. Surds
Exercise 1
Simplify each of the following by writing as products of whole numbers and surds
1) 8 2) 12 3) 24 4) 28
5) 108 6) 40 7) 50 8) 18
9) 48 10) 32 11) 20 12) 125
13) 200 14) 216 15) 192 16) 320

Exercise 2
Simplify
2 3 4 6
1) ------- 2) ------- 3) ------- 4) -------
2 3 4 2
14 8 9 12
5) ------- 6) ------- 7) ------- 8) -------
7 2 3 3
14 20 30 50
9) ------- 10) ------- 11) ------- 12) -------
2 2 3 5
70 39 49 63
13) ------- 14) ------- 15) ------- 16) ----------
5 3 7 21

Exercise 3
Simplify
1) 2 + 2 2 2) 3 + 3 3 3) 2 2 + 3 2
4) 8 + 2 5) 8 – 2 6) 12 – 3
7) 2 5 – 5 8) 32 – 2 2 9) 2 5 – 5
10) 3 5 – 2 5 11) 4 7 – 28 12) 500 – 3 5

Exercise 4
Simplify
2 3 3
1) 2 2 – ------- 2) 2 2 – ------- 3) 2 3 + -------
2 2 3
12 18 28
4) ------- + 2 2 5) ------- + 2 3 6) ------- – 3 7
2 3 7
30 144 49
7) ------- + 5 8) ---------- – 23 3 9) ------- – 3 7
5 2 3 7
45 60 80
10) ---------- – 5 11) ---------- + 3 5 12) ------- – 2 2
3 5 20 8
Exercise 5
Simplify
1) 6 × 3 2) 5 × 10 3) 3 × 12
4) 6 × 12 5) 7 × 2 7 6) 3 2 × 2 8
7) 2 2 × 4 12 8) 5 6 × 4 3 9) 4 2 × 3 8
10) 5 10 × 2 2 11) 7 2 × 3 12 12) 4 8 × 7 12
©Sum books 1998 Higher Level

5. Prime Factors

Exercise 1
Express the following numbers as products of their prime factors.
1) 300 2) 900 3) 630 4) 700 5) 792
6) 945 7) 1960 8) 1815 9) 1512 10) 8580
11) 2640 12) 5460 13) 3744 14) 6336 15) 9240

Exercise 2
Express each of the following numbers as products of their prime factors. In each case state the
smallest whole number it has to be multiplied by to produce a perfect square.
1) 660 2) 300 3) 450 4) 700 5) 1575
6) 2205 7) 600 8) 396 9) 1350 10) 1872
11) 4950 12) 3840 13) 8820 14) 11,760 15) 11,340

Exercise 3
Calculate the largest odd number that is a factor of each of the following.
1) 120 2) 210 3) 432 4) 416 5) 440
6) 704 7) 1144 8) 1200 9) 1840 10) 1848
11) 2464 12) 2112 13) 5880 14) 4725 15) 9240

Exercise 4
1) a) What is the highest common factor of 735 and 756?
An area of land measures 73.5 metres by 75.6 metres. It is to be divided up into square plots of
equal size.
b) What is the size of the largest squares that will fit on it?
c) How many squares will fit on it?
2) a) What is the highest common factor of 60 and 75?
b) Jane’s mum organises a party for her. She makes 60 cakes and 75 sandwiches. Everyone
at the party is allowed the same amount of food to eat. She invites as many children as
possible. How many does she invite?
3) a) What is the highest common factor of 990 and 756?
b) A shop is moving its stock. It has 9900 type A items and 7560 type B items. They have to
be packed into boxes, each box containing both item A and item B. The same number of
type A items and the same number of type B items are in each box. What is the maximum
number of boxes needed and c) how many items will be in each box?
©Sum books 1998 Higher Level

6. Percentages

1. By selling a car for £2,500, John made a profit of 25%. How much did he pay for it?
2. A company makes a profit for the year of £75,000 before tax is paid. What percentage tax
does it pay if its tax bill amounts to £11,250?
3. a) What is the total cost of a television set if it is priced at £240 plus VAT of 17 1--2- %?
b) A radio costs £21.60 in a sale. If it had a reduction of 10%, what was its original price?
c) A computer costs £998.75 including VAT at 17 1--2- %. What is its price before VAT is added?
4. Jane invests £1500 in a bank account which pays interest of 6 1--2- % per annum.
a) How much interest has she earned at the end of 1 year?
b) She has to pay tax on this interest at 22%. How much tax does she pay?
5. Jonathan earns £23,000 per year as a shop manager.
a) If he is offered a pay rise of 7 1--2- %, what will his new wages be?
b) Instead he is offered a new job by a different firm and the rate of pay is £25,400 per
annum. What percentage increase does this represent on his old wages?
6. The population of a certain town was 50,000 at the beginning of 1998. It is expected to rise
by 7% each year until the end of the year 2000. What is the expected population at the end
of this period?
7. A shopkeeper buys 35 radios for £435.75. If she sells them at £15 each, what is her
percentage profit?
8. A car was bought at the beginning of 1994. During the first year it depreciated in value by
23% and then by 9% each subsequent year. If its original price was £9,000, what was its
value at the end of 1997, to the nearest pound?
9. A can of cola has a label on it saying ‘20% extra free’.
a) If the can holds 960ml, what did the original can hold?
b) The original can cost 45p. If the company increase the price of the new can to 60p, does
this represent an increase in price? Explain your answer.
10. In the general election, Maureen Johnson got 22,016 votes, which was 43% of all the votes
cast.
a) If Anthony Jones got 19,968 votes, what percentage of the people voted for him?
b) If John Parry got 8% of the vote, how many people voted for him?
11. The cost of building a bridge in 1995 was estimated as £24 million. When it was finally
completed in 1998 its total cost amounted to £37.7 million. What was the percentage
increase?
12. It is estimated that a certain rainforest gets smaller by 8% each year. Approximately how
many years will it take to be 39% smaller?
13. A firms profits were £500,000 in 1991. In 1992 they were 15% higher. However in 1993
they were 5% lower than in the previous year. What were the profits in 1993?
14. £1000 is invested at 6% compound interest. Interest is added to the investment at the end of
each year. For how many years must the money be invested to in order to get at least £400
interest?
15. A lady wants a room to be built onto her house. Builder A quotes £11,400 which includes
VAT of 17 1--2- %. However he will reduce this by 10% if it is accepted within one week.
Builder B quotes £9000 excluding VAT. Which is the cheapest quotation and by how much?
16. VAT of 17 1--2- % is added to the cost of a computer. If the VAT is £166.25, what is the total
cost of the computer?
©Sum books 1998 Higher Level

7. Conversion Graphs

1. This graph shows the relationship between the Italian lira and the pound sterling.

60

50

Pounds (£)
40

30

20

10

0 20 40 60 80 100 120 140 160 180

Italian Lira (thousands)
From the graph determine:
a) The rate of exchange (i.e. the number of lira to the pound)
b) The number of lira that can be obtained for £22.
c) The amount of £ sterling that can be exchanged for 140,000 lira.
2. Christmas trees are priced according to their height. The table below shows some of the
prices charged last Christmas.

Height (metres) 1.0 1.5 2.0 2.5 3.0

Price (£) 8.50 11.75 15.00 18.25 21.50

Draw a conversion graph using 8cm to represent 1 metre on the vertical axis and 1cm to
represent £2 on the horizontal axis.
a) From your graph, calculate the cost of a tree measuring 2m 35cm.
b) What size tree can be purchased for £12.50?
3. Bill is a craftsman who makes wooden bowls on his lathe. He advertises that he can make
any size bowl between 20cm and 60cm diameter. In his shop he gives the price of five
different bowls as an example.

Diameter of bowl (cm) 20 30 40 50 60

Price £8.80 £20.00 £42.40 £79.00 £133.60

a) Use these figures to draw a conversion graph. Use a scale of 2cm to represent a diameter
of 10cm on the horizontal axis and 2cm to represent £10 on the vertical axis.
b) Jane has £50 to spend. From your diagram, estimate the size of bowl she can buy.
c) What is the cost of a bowl of 34cm diameter?
©Sum books 1998 Higher Level

8. Fractions

Exercise 1
Simplify into single fractions
1 2 1 2 7 7 1 1
1) --- + --- 2) --- + --- 3) ------ + ------ 4) --- + ---
5 5 3 3 10 10 5 3
1 1 1 1 3 2 4 2
5) --- + --- 6) --- – --- 7) --- + --- 8) --- – ---
8 7 3 4 5 3 5 7
7 7 3 3 7 11 3 4
9) --- + ------ 10) 1 --- – --- 11) 2 --- – ------ 12) 4 --- – ---
9 12 4 5 8 12 7 5
a 2a 4a 2a a 3a x x
13) --- + ------ 14) ------ + ------ 15) --- + ------ 16) --- + ---
7 7 5 5 4 4 5 3
x x b b 3x x 7x 2x
17) --- + --- 18) --- – --- 19) ------ – --- 20) ------ – ------
5 7 2 5 8 4 10 7
3x 4x 3a 3a 15x x 11x x
21) ------ + ------ 22) ------ + ------ 23) --------- + --- 24) --------- – ------
10 9 2 4 7 2 4 11
Exercise 2
Simplify
2 3 7 6 6 9 1 3
1) --- + --- 2) --- – --- 3) --- + --- 4) --- + ------
x x a a x x x 2x
5 3 3 1 2 3 5 4
5) ------ + ------ 6) ------ – ------ 7) ------ + ------ 8) ------ – ------
2x 2x 4x 5x 3x 2x 4a 5a
1 1 2 1 3 5 1 1
9) --- + --- 10) --- – --- 11) --- + --- 12) --- + ---
a b x y x y 5 b
2 3 4 x 4 x x 3
13) --- + --- 14) --- + --- 15) --- – --- 16) --- – ---
5 b 5 5 x 4 4 x
Exercise 3
Simplify
3 2 5 1 3 2
1) --- + ------------ 2) --- – ------------ 3) ------ – -----------
4 a+1 8 x+1 10 x – 1
x x–1 a x+1 2b 3a
4) --- + ----------- 5) --- – ------------ 6) ------ + ------
3 4 4 3 5 4
4x + 3 5x 5a 2a + 3 7x + 4 5x
7) --------------- + ------ 8) ------ + --------------- 9) --------------- – ------
9 2 4 5 3 8
x 2x – 1 4y + 3 y 7a 3a + 5
10) --- – --------------- 11) --------------- – --- 12) ------ + ---------------
2 3 6 5 4 7
6 3 7 4 5 2
13) ----------- + ----------- 14) ------------ + --------------- 15) --------------- – ---------------
x–1 x–4 x + 2 2x – 1 3x + 2 4x + 1
5 y 3x 4x 4x 5x
16) ------ + --- 17) ------ + ------ 18) ------ – ------
2x 3 a 5a a 3a
--- ( x + 1 ) – 1--6- ( x + 3 ) --- ( x + 1 ) – 2--5- ( x – 3 ) --- ( a – 2 ) + 3--7- ( a – 1 )
1 5 3
19) 5
20) 8
21) 5
3( x + 3) 5(a + 6) 11 ( x + 3 )
22) -------------------- + x 23) -------------------- – a 24) ----------------------- – 2x
4 4 4
2 4a + 1 3x + 3
25) x + --- 26) 3a + --------------- 27) 5x – ---------------
x 3 4
©Sum books 1998 Higher Level

9. Indices

Exercise 1
Simplify each of the following
2 5 4 7 3 4 9 3
1) x × x 2) y × y 3) a × a 4) b × b
2 3 4 2 3 4 7
5) 3x × 4 6) 5a × a 7) 6y × 4y 8) 7x × 5x
5 2 7 3 6 4 7 2
9) a ÷ a 10) b ÷ b 11) y ÷ y 12) y ÷ y
2 3 3 6 7 4 3 4
13) ( a ) 14) ( c ) 15) ( x ) 16) ( x )
2 2 3 4 5 4 5 3 3 3 4 4
17) x y × xy 18) a b × a b 19) x y × y 20) a b × a b
2 2 2 3 2 2 4
21) ( 4x ) 22) ( 3x ) 23) ( 4y ) 24) ( 2a )
2 5 2 5 2 5 2
25) 12a ÷ 4 26) 18x ÷ x 27) 16y ÷ 4y 28) 20a ÷ 5a
4 7 3 2 3 5
24a 18b 12a b 21x y
29) -----------
2
30) ----------- 31) ----------------- 32) ----------------
2 4
-
6a 3b 3ab 7x y
5 2 3 2 5 2 3 7 4 3
33) x × x ÷ x 34) 3a × 2a × a 35) 4y × 7y ÷ 2y 36) 4x × 3x ÷ 4x

Exercise 2
Simplify
9 9 5 7 3 4 3 5
1) x ÷ x 2) a ÷ a 3) 20y ÷ 10y 4) 4b ÷ 8b
2 –3 2 –5 3 –6 2 –2
5) y × y 6) 2x × 3x 7) 4a × 3a 8) 3a b × 2ab
3 0 –3 0 2 0 0 0
9) x × x 10) y ×y 11) 3x × 2x 12) 6a × 4a
1 1 1 1 1 1 3
--- --- – --- --- 0 --- --- – ---
13) a 2 × a 2 14) x 3 × x 3 15) b × b 2 16) y 2 × y 2
1 1 1 1 1 1 1
--- --- --- --- --- – --- --- 0
17) x 2 ÷ x 2 18) y 2 ÷ y 4 19) a 4 ÷ a 2 20) b 2 ÷ b
1
1
--- 3 1
--- 2 1
--- 3 3 ---

21) ( x 2 ) 22) ( a 4 ) 23) ( 2b 3 ) 24) ( 5y )

2

Exercise 3
Simplify
1 1 1 1
--- – --- --- – ---
1) 25 2 2) 25 2 3) 8 3 4) 27 3
2
---
3
---
1
--- 3 1 --12-
5) 8 3 6) 4 2 7) ( 4 2 ) 8) 9
3 2 3 3
--- --- --- ---
9) ( 81 ) 4 10) ( 64 ) 3 11) ( 32 ) 5 12) 25 2
3 5 1 2
– --- – --- --- ---
13) ( 25 ) 2 14) ( 64 ) 6 15) ( 6 1--4- ) 16) ( 125
--------- )
2 3
8

Exercise 4
Solve the following equations
1 1 1 1
--- --- --- ---
1) 36 x = 6 2) 81 x = 9 3) 81 x = 3 4) 64 x = 4
1 1
x x 1
--- x --- x
5) 32 = 2 6) 25 = ---
5
7) 81 2 = 3 8) 512 3 = 2
1 1 1
– ---
1
– ---
1
– ---
1 x 1
9) 32 x = ---
2
10) x 4 = ---
3
11) x 4 = ---
2
12) 16 = ---
4
©Sum books 1998 Higher Level

10. Simplifying

Exercise 1
Simplify each of the following by expanding the brackets where necessary
1) 7x – 3x 2) 5y – 7y
3) – 8y + 3y 4) – 6x – 7x
5) 4x + 3y + 5x + 6y 6) 9y + 7x – 11y – 4x
7) – 6x + 3y – 4x + 2y 8) – 7x – 6y + 3x – 4y
9) 2ab + 3b – 4a – 6ab 10) 12b – 4a + 3ab – 7a
2 2 2 2
11) 4x + 3x – 2x 12) 6y – 4y – 5y
2 2 2 2
13) 4y – 4xy + 4xy 14) – 6x + 3y – 4x
15) 3 ( x + y ) + y 16) 5 ( 2x – y ) + 2y
17) 3x + 4 ( x + y ) 18) 7x – 3 ( x + 2y )
19) 4y – 2 ( x – y ) 20) 6x – 4 ( 2x – 2y )
21) 3 ( x + y ) + 2 ( x – y ) 22) 4 ( 2x + 3y ) – ( x + 2y )
23) 5 ( 2x – 3y ) – ( 2x – y ) 24) 3 ( 2x – y ) – 2 ( 3x – y )
25) 3 ( 2x – 3 ) – 3 ( x – 4 ) 26) 5 ( 2x – 3y ) – 3 ( 5x – 2y )
27) 4x ( 2x – 3 ) – 3x ( 2x + 4 ) 28) 7x ( y – 2 ) – 6y ( 2x – 3 )
29) 4y ( 2y – 3x ) – 2x ( x – 3y ) 30) 4x ( 4y + 3x ) – 3y ( 4x – 3y )
1 1 1 1
31) --- x + --- x 32) --- x – --- x
4 3 2 3
2 1 1 2 1 2
33) --- y – --- y 34) --- y + --- y
3 2 3 6

Exercise 2
Expand and simplify
1) ( x + 1 ) ( x – 3 ) 2) ( x + 2)( x – 4)
3) ( 2x + 3 ) ( x – 7 ) 4) ( 2x + 5 ) ( 3x – 2 )
5) ( 2x – 7 ) ( 3x + 2 ) 6) ( 3x + 4 ) ( 2x – 5 )
7) ( 5x + 2 ) ( 7x – 3 ) 8) ( 4x + 8 ) ( 3x – 2 )
9) ( 4x – 4 ) ( 2x + 1 ) 10) ( 3x + 2y ) ( 2x + 3y )
11) ( 4x + y ) ( x + 3y ) 12) ( x – 4 ) ( 2x – 6 )
2 2
13) ( x + 1 ) 14) ( 3x + 2 )
2 2
15) ( 5x – 2 ) 16) ( 3x + 4 )
2 2
17) ( 5x – 6 ) 18) ( 7x + 2 )
2 2
19) ( – 3x + 2 ) 20) ( – 5x – 7 )
2 2
21) ( – 4x – 6 ) 22) ( x + y)
2 2
23) ( 2x + 3y ) 24) ( 4x – 2y )
©Sum books 1998 Higher Level

11. Re-arranging Formulae

Exercise 1
In each of the following questions, re-arrange the equation to make the letter in the bracket the
subject.
1) v = u + at (u) 2) v = u + at (a)
3) d = 3b – c (b) 4) c = pd + w (w)
5) x = 7y – z (z) 6) a = 3b + c (b)
4v + u 5y + b
7) w = --------------- (v) 8) x = --------------- (b)
3 4
9) 2x = x + b (b) 10) 6y = 3a – 2y (a)
11) p = 1--2- a + 3b (a) 12) w = 2v + 1--4- u (v)
a+b 2r – q
13) c = ------------ (d) 14) p = -------------- (s)
d 3s
15) x = 2 ( y + z ) (z) 16) a = 3 ( 3b + 4c ) (c)
17) x = 1--2- ( y + z ) (y) 18) 3a = 1--3- ( 2b + c ) (c)
19) 3x = 1--4- ( y – z ) (z) 20) 5w = 1--3- ( 3v – 2u ) (u)
21) 7x – 4y = 1--2- ( 3x + 6y ) (x) 22) 5a + 3b = 2--3- ( 3b – 2a ) (a)

Exercise 2
In each of the following questions, re-arrange the equation to make the letter in the bracket the
subject.
2
b y
1) a = ----- (b) 2) x = ----2 (z)
c z
2
4a 9
3) c = -------- (a) 4) 3v = ----2- (u)
b u
1
5) --- x
2
= 2--3- x + y (y) 6) 3
--- y
4
– 2x = y (y)

7) 7
--- x
2
= 1--2- ( x + y ) (x) 8) 4
--- b
9
= 1--4- ( b – 3c ) (b)
1 1 1 2 2 3
9) --- = --- + --- (x) 10) ------ = --- + --- (x)
x a b 3x y z
1 1 1 2 3 b
11) ------- = ------ + ------ (x) 12) ------- = ------ + --- (x)
x 2a 3b x 2y 2
2 y 1 3 6 1
13) ------ = --- – --- (z) 14) --- = --- – --- (z)
3x 2 z x y z
1 1 2
15) x = ----2- + --- (a) 16) 4y = --------2 + 3b (a)
a b 3a
3 1 1 3 6 1
17) --- = --- + --- (b) 18) --- – --- = --- (x)
b b c x y x
2y + z b – 3c
19) 3x = -------------- (y) 20) 4a = --------------- (c)
y c
1 x + 3y 2 x – 3y
21) --- = --------------- (x) 22) --- = --------------- (y)
x x y 2y
©Sum books 1998 Higher Level

12. Number Sequences

Exercise 1
Write down the next two numbers in the following sequences
1) 2, 5, 8, 11, 14, 17 . . . 2) 1, 5, 9, 13, 17, 21 . . .
3) 4, 5, 7, 10, 14, 19 . . . 4) 2, 2, 3, 5, 8, 12 . . .
5) 17, 19, 22, 26, 31, 37 . . . 6) 20, 19, 17, 14, 10, 5 . . .
7) 10, 8, 6, 4, 2, 0 . . . 8) 15, 14, 11, 6, –1, –10 . . .
9) 6, 7, 7, 6, 4, 1 . . . 10) 2, 4, 8, 16, 32, 64 . . .
11) –1, –2, –4, –7, –11, –16 . . . 12) –6, –6, –7, –9, –12, –16 . . .
13) –7, –1, 5, 11, 17, 23 . . . 14) –3, –1, 0, 0, –1, –3 . . .
15) 1, 3, 7, 15, 31, 63 . . . 16) 1, 4, 9, 16, 25, 36 . . .
17) 7, 5, 3, 1, –1, –3 . . . 18) 1, 8, 27, 64, 125, 216 . . .
19) –7, 0, 19, 56, 117, 208 . . . 20) –2, 1, 6, 13, 22, 33 . . .
21) 1, 2, 4, 8, 16, 32 . . . 22) 2, 11, 26, 47, 74, 107 . . .
23) 0, 1, 1, 2, 3, 5 . . . 24) 6, 7, 13, 20, 33, 53 . . .
25) 1, 3, 6, 10, 15, 21 . . . 26) 4, 9, 15, 22, 30, 39 . . .

Exercise 2
th
Write down the next two numbers and find the rule, in terms of the n number, for each of the
following sequences
1) 4, 7, 10, 13, 16, 19 . . . 2) –3, 2, 7, 12, 17, 22 . . .
3) –2, –6, –10, –14, –18, –22 . . . 4) 8, 3, –2, –7, –12, –17 . . .
5) 39, 31, 23, 15, 7, –1 . . . 6) 6, 11, 16, 21, 26, 31 . . .
7) –19, –12, –5, 2, 9, 16 . . . 8) 1, 4, 9, 16, 25, 36 . . .
9) 3, 6, 11, 18, 27, 38, . . . 10) –6, –3, 2, 9, 18, 29, . . .
11) 0, 1, 4, 9, 16, 25 . . . 12) 0, 2, 6, 12, 20, 30 . . .
2 4 6 8 10 12 1 2 3 4 5 6
13) --- , --- , --- , --- , ------ , ------ 14) --- , --- , --- , --- , --- , ---
3 5 7 9 11 13 2 3 4 5 6 7
1 4 9 16 25 6 9 12 15 18
15) --- , --- , --- , ------ , ------ , 4 16) 3, --- , ------ , ------ , ------ , ------
4 5 6 7 8 7 17 31 49 71

Exercise 3
th
1) a) Write down an expression for the n term of the sequence 2, 3, 4, 5, 6 . . .
b) Show algebraically that the product of any two terms in the sequence is itself a term in the
sequence.
th
2) a) Write down an expression for the n term of the sequence 3, 5, 7, 9, 11 . . .
b) Show algebraically that the product of any two terms in the sequence is itself a term in the
sequence.
th
3) a) Write down an expression for the n term of the sequence 1, 4, 9, 16, 25, 36 . . .
b) Show algebraically that the product of any two terms in the sequence is itself a term in the
sequence.
th
4) a) Write down an expression for the n term of the sequence 5, 9, 13, 17, 21, 27 . . .
b) Show algebraically that the product of any two terms in the sequence is itself a term in the
sequence.
©Sum books 1998 Higher Level

13. Substitution

Exercise 1
Evaluate each of the following, given that a = 6, b = 4 and c = –5.
1) 3a + 5b 2) 4a – 6b 3) 2a – 7b 4) 4c + 2a
2 2 2
5) 5a – 4c 6) 3a + 2b 7) 4b – 2a 8) 5a + 3c
2 2 2 2
9) ( 4b – 6c ) 10) ( 2a ) + 3b 11) 4c – 5b 12) 6a – ( 5c )

Exercise 2
1) If y = 3x + 4 a) Calculate the value of y when x = 6 b) What value of x is needed if y = 19?
2) If a = 3b – 6 a) Calculate the value of a when b = –7 b) What value of b is needed if a = 54?
3) If y = 7x + 3 a) Calculate the value of y when x = –6 b) What value of x is needed if y = –81?
4) Carol works for a garden centre and plants rose bushes in her nursery. She works out the
length of each row of bushes using the formula L = 50R+200, where L represents the length
of the row in centimetres and R is the number of bushes she plants. Use the formula to calculate
a) the length of a row containing 10 bushes
b) the number of bushes in a row 15 metres long.
1 2
5) The volume of a cone is given by the formula V = --- πr h .
3
a) Calculate the volume of a cone when π = 3.142 , r = 3cm and h = 2.5cm.
3
b) A cone has a volume of 183cm . Calculate the value of its height h, if r = 5cm and
π = 3.142 . Give your answer correct to the nearest millimetre.
PTR
6) Simple interest can be calculated from the formula I = ----------- .
100
a) If the principal (P) = £250, the time (T) = 3 years and the rate (R) = 9.5%, calculate the
interest.
b) If the interest required is £200, what principal needs to be invested for 6 years at 7%?
7) The temperature F (degrees fahrenheit) is connected to the temperature C (degrees celsius)
5
by the formula C = --- ( F – 32 ) .
9
a) Calculate C if F = – 20°F .
b) Convert – 10°C into °F
D S 1
8) A bus company uses the formula T = ------ + ------ + --- to calculate the time needed for their bus
20 60 4
journeys. D is the distance in miles and S the number of stops on the journey. If T is measured
in hours, calculate
a) the time needed for a bus journey of 10 miles with 10 stops.
b) the time needed for a bus journey of 20 miles with 4 stops.
c) During the rush hour, more people get on the bus and the extra traffic slows the bus down.
What would you do to the formula to take this into account?
©Sum books 1998 Higher Level

14. Factorising

Exercise 1
Factorise each of the following
1) 4x + 8 2) 6y – 9 3) 7b – 14a
4) xy – x 5) xy + 3x 6) 4y + 10xy
2 2 2
7) 6x + 2 8) 5x – x 9) 9x – 3x
2 2 2 2
10) a b + ab 11) 4ab – a b 12) 8ab + 6ab
2 2 2 2
13) a + ab – a 14) 3ab – ac + a 15) 5x y – 4y – 3xy
2 3 2 2
x x y xy 5x 2x
16) ----- – ----- 17) ----- – ----- 18) -------- – ------
2 4 3 6 6 3

Exercise 2
Factorise
2 2 2 2 2
1) m –n 2) a –4 3) ( xy ) – z
2 2 2 2 2
4) ( ab ) – 9 5) x y –4 6) v w – 25
2 2 2 2 2 2
7) a b – 9c 8) 25a – 9b 9) b –1
2 2 2 2
10) 2a – 50 11) 8a – 50 12) 12x – 27y
2 3 2 3 2 2 2
13) xy – 4x 14) 2xy – 8x 15) 4x y – 9x
4 4 4 4 4 2
16) x –y 17) 16x – 81y 18) 3a – 12b

Exercise 3
Factorise
2 2 2
1) x + 4x + 3 2) x + 4x + 4 3) x + 8x + 7
2 2 2
4) x + 7x + 10 5) x + 7x + 12 6) x + 11x + 30
2 2 2
7) x + 2x – 3 8) x – 2x + 3 9) x + 4x – 5
2 2 2
10) x – 2x – 8 11) x – 2x – 15 12) x – x – 12
2 2 2
13) x – 10x + 24 14) x – 8x + 15 15) x – 11x + 28

Exercise 4
Factorise
2 2 2
1) 2x + 3x + 1 2) 2x + 9x + 4 3) 2x + 7x + 3
2 2 2
4) 2x + 8x + 6 5) 2x + x – 6 6) 3x – 7x – 6
2 2 2
7) 2x – 9x + 4 8) 3x – 10x + 3 9) 3x – 14x + 8
2 2 2
10) 3x + x – 14 11) 3x + 19x + 20 12) 3x – 12x + 12
2 2 2
13) 4x + 10x + 6 14) 4x – 10x + 6 15) 4x + 13x + 3
2 2 2
16) 4x + 21x + 5 17) 5x + 13x – 6 18) 6x – 5x – 6
2 2 2
19) 6x + 5x + 1 20) 9x + 12x + 4 21) 8x + 11x + 3
2 2 2
22) 4x – 23x + 15 23) 5x – 13x – 6 24) 12x – 13x + 3
©Sum books 1998 Higher Level

15. Trial and Improvement

Exercise 1
By using the method of trial and improvement, calculate the value of x in each of the following
equations. In each case show all your workings and give each answer correct to 1 decimal place.
3 3 3
1) x = 53 2) x = 77 3) x = 101
2 2 2
4) x + x = 36 5) 2x + x = 41 6) x + 3x = 61
2 2 2
7) 5x – 3x = 120 8) 6x – 2x = 77 9) 3x – 5x = 5
3 3 3
10) x + 3x = 16 11) x – 3x = 96 12) x – 4x = 12
3 3 3
13) 2x + 4x = 96 14) 3x – 4x = 14 15) 2x – 5x = 4

Exercise 2
2
1) A square has an area of 45 cm . Using the method of trial and improvement, calculate the
length of its sides, correct to one decimal place. Show all your calculations.
2
2) A rectangle has one side 4cm longer than the other. If its area is 100 cm , use the method of
trial and improvement to calculate the length of the shorter side, correct to 1 decimal place.
Show all your calculations.
2
3) The perpendicular height of a triangle is 2cm greater than its base. If its area is 35 cm , use
the method of trial and improvement to calculate its height, correct to 1 decimal place. Show
all your calculations.
3
4) A cube has a volume of 36 cm . Use the method of trial and improvement to calculate the
length of one of its sides, correct to 1 decimal place. Show all your calculations.
3
5) A cuboid has a height and length which is 3cm longer than its width. Its volume is 150cm .
Use the method of trial and improvement to calculate its width, correct to 1 decimal place.
Show all your calculations.
2
6) The solution to the equation x + 3x = 7 lies between 1 and 2. Use the method of trial and
improvement to calculate its value, correct to 1 decimal place. Show all your calculations.
3
7) Using the method of trial and improvement, solve the equation x + 3x – 22 = 0 , correct to
one decimal place. Show all your calculations.
3
8) The solution to the equation x + 4x = 48 lies between 3 and 4. Use a method of trial and
3
improvement to find the solution to x + 4x = 48 , correct to 1 decimal place. Show all your
calculations.
©Sum books 1998 Higher Level

16. Iteration

1 1
1) Starting with x = 4 and using the iteration x n + 1 = 4 – ----- find a solution of x = 4 – ---
xn x
correct to two decimal places.
9 9
2) Starting with x = 5 and using the iteration x n + 1 = 7 – ----- find a solution of x = 7 – ---
xn x
correct to two decimal places.
3 3
3) Starting with x = 4 and using the iteration x n + 1 = 3 + ----- find a solution of x = 3 + ---
xn x
correct to two decimal places.
2 2
4) Show that x = 9 – --- can be re-arranged into the equation x – 9x + 2 = 0 .
x
2
Use the iterative formula x n + 1 = 9 – ----- together with a starting value of x 1 = 9 to obtain
xn
2
a root of the equation x – 9x + 2 = 0 .
1 2
5) Show that x = ------------ can be re-arranged into the equation x + x – 1 = 0 .
x+1
1
Use the iterative formula x n + 1 = -------------- together with a starting value of x 1 = 0.5 to
xn + 1
2
obtain a root of the equation x + x – 1 = 0 correct to 2 decimal places.
12 2
6) Show that x = ------------ – 4 can be re-arranged into the equation x + 6x – 4 = 0 .
x+2
12
Use the iterative formula x n + 1 = -------------- – 4 together with a starting value of x 1 = – 6 to
xn + 2
2
obtain a root of the equation x + 6x – 4 = 0 correct to 1 decimal place.
21
7) A sequence is given by the iteration x n + 1 = --------------
xn – 4
a) (i) The first term, x 1 , of the sequence is –2. Find the next 4 terms, correct to two decimal
places.
(ii) What do you think the value of x is as n approaches infinity?
2
b) Show that the equation this solves is x – 4x – 21 = 0 .
7
8) A sequence is given by the iteration x n + 1 = 6 + -----
xn
a) (i) The first term, x 1 , of the sequence is 5. Find the next 3 terms.
(ii) What do you think the value of x is as n approaches infinity?
2
b) Show that the equation this solves is x – 6x – 7 = 0 .
©Sum books 1998 Higher Level

17. Direct and Inverse Proportion

1) y varies directly as x and y = 4 when x = 6.

a) Find an expression for y in terms of x. b) Calculate the value of y when x = 8.
c) Calculate the value of x when y = 15.
3
2) If v varies directly as r and v = 24 when r = 2 find the value of v when r = 3.
3) y is directly proportional to √x and x = 9 when y = 2.
a) Find an expression for y in terms of x. b) Find the value of y when x = 4.
c) Calculate the value of x when y = 6.
3
4) a varies directly as b . If a = 32 when b = 2 find a when b = 4.
5) Given that y is inversely proportional to the square of x, and y is equal to 0.75 when x = 2.
a) Find an expression for y in terms of x.
1
b) Calculate y when x = 3. c) Calculate x when y = ------ .
12
6) y varies inversely as x. If y = 1.5 when x = 2 find y when x = 4.
7) y is proportional to the cube root of x . y = 10 when x = 8.
a) Find an expression for y in terms of x.
b) Calculate (i) y when x = 64. (ii) x when y = 25.
2
8) y varies inversely as x . Copy and complete the following table.

x 1 2 3 4
y 0.75

9) a varies directly as the square root of b. If a = 1 when b = 9 find a when b = 4.

10) y varies directly as the cube root of x. Copy and complete the following table.

x 0.125 1 8 64
y 1

11) The table below shows values of y for values of the variable x, which are linked by the
n
equation y = 8x .

x 0.5 1 2
y 1 8 64

Find (a) The value of n. (b) y when x = 0.25.

12) y varies inversely as the square of x. Copy and complete the table below.

x 0.5 0.75 1 1.5

y 6
©Sum books 1998 Higher Level

18. Solving Equations 1

Exercise 1
Solve the following equations
1) 7x – 3 = 60 2) 12x – 14 = 130
3) 24 – 3x = 6 4) 7 – 2x = – 1
x x
5) --- = 30 6) --- = 25
4 6
2 1
7) --- x = 8 8) --- x = 4
3 2
3 2
9) ------------ = 1 10) ----------- = 5
x+2 x–1
11) 2x + 3 = 3x + 2 12) 4x + 3 = 3x + 5
13) 4x + 5 = 3x + 3 14) 7x + 3 = 10x – 6
15) 5 ( x + 3 ) = 7x + 5 16) 3 ( 2x – 1 ) = 5 ( 3x – 15 )
17) 3 ( x + 2 ) – 2 ( 3x – 5 ) = 10 18) 4 ( 2x – 3 ) – 3 ( 3x – 10 ) = 11
x x x–2 x–2
19) --- + --- = 14 20) ----------- + ----------- = 25
3 4 3 2
2x – 1 x + 1 3 4
21) --------------- – ------------ = 4 22) --- = ------------
3 4 x x+2
2 3 2x – 1 3
23) ----------- = --------------- 24) --------------- = ---
x–1 2x + 4 x 4
1 3 1 2 1 1
25) --- = ------ + ------------ 26) ------ = --- – ------------
x 2x x + 1 3x x x+1

Exercise 2
Solve the following equations
2 2
1) x – 25 = 0 2) x – 81 = 0
2 2
3) 2x – 72 = 0 4) 3x – 27 = 0
5) ( x + 2 ) ( x – 3 ) = 0 6) ( x – 6 ) ( x + 5 ) = 0
7) ( 3x + 4 ) ( 2x – 1 ) = 0 8) x ( 3x – 2 ) = 0
9) x ( 4x + 3 ) = 0 10) 2x ( x – 4 ) = 0
2 2
11) 2x + 3x = 0 12) 6x – 4x = 0
2 2
13) 4x – x = 0 14) x – x – 6 = 0
2 2
15) 2x – x – 3 = 0 16) 6x + 4x – 2 = 0
2 2
17) x – 3x – 10 = 0 18) 3x + 6x = 24
2 2
19) 8x = 2x + 15 20) 4x – 25 = 0
4 1 6 1 3 1
21) --- + ------------ = --- 22) --- = ------ + ------------
x x+1 x x 2x x + 1
©Sum books 1998 Higher Level

19. Solving Equations 2

Exercise 1

Solve the following equations correct to 2 decimal places each time.

2 2
1) x + 4x – 6 = 0 2) x – 3x + 1 = 0
2 2
3) x – 4x – 3 = 0 4) x + 6x – 4 = 0
2 2
5) x + x – 1 = 0 6) 4x – 7x – 4 = 0
2 2
7) 6x + 3x – 1 = 0 8) 10x + 2x – 7 = 0
2 2
9) 4x + 5x – 10 = 0 10) 3x – 4x – 2 = 0
2 2
11) 3x + 4x = 5 12) 2x + 5x = 1
2
13) 4x + 3 = 7x 14) 3x ( 2x + 1 ) = 1
1 3 1 1
15) --- + ------------ = 7 16) ----------- – ------ = 4
x x+2 x – 3 2x

5 1 7 2
17) --- + ------------ = 7 18) ------ + ----------- = 4
x x+1 2x x – 1

2 2
19) x + 7x + 2 = 2x + 4 20) x + 13x = 4x + 5
2 2 2 2
21) 5x – 3x = x + 4 22) 4x – 3x = 2x + 7
2
23) x ( x + 4 ) = 3x + 2x – 6 24) 4 ( x + 2 ) = 3x ( x + 1 )
4 2x – 3 2x + 3 x
25) --- = --------------- 26) --------------- = ---
x x–1 x–1 4

Exercise 2

In each of the following calculate the value of a and the corresponding value of k.
2 2 2 2
1) 4x – 20x + k = ( 2x – a ) 2) ( 3x + a ) = 9x + 12x + k
2 2 2 2
3) ( 5x – a ) = 25x – 30x + k 4) x – 14x + k = ( x – a )
2 2
5) ( 3x + a ) = 9x ( x + 1 ) + k 6) 16x ( x + 1 ) + k = ( 4x + a )
2 2 2
7) 9x – 30x + k = ( 3x – a ) 8) ( 4x – a ) = 16x ( x – 4 ) + k
2 2 2 2
9) ( 2x + a ) = 4x + 24x + k 10) 36x + 48x + k = ( 6x + a )
©Sum books 1998 Higher Level

20. Problems Involving Equations

1. In each of the triangles below, calculate the value of x and hence the sizes of the angles of
the triangles.
a) b) 3x° 2x – 20°
2x°

2x – 10°
x° x + 44°
2. It takes an aeroplane 6 1--2- hours to travel from London to the USA, a distance of 3,500 miles.
a) What was the average speed?
b) If the same aeroplane travels from London to Spain, a distance of x miles, write down in
terms of x the time taken.
c) If the same aeroplane travels from London to Italy in y minutes, write down an expression
in terms of y for the distance travelled.
3. The dimensions of a square and a rectangle are given in the two diagrams. If their areas are
equal,
a) calculate the value of x
b) calculate their areas.

(x + 1)cm (x – 2)cm
(x + 7.6)cm
4. Three consecutive numbers add up to 156.
a) If the middle number is x, what are the values of the other two numbers, in terms of x.
b) Write down an equation in terms of x and solve it to find x.
5. A wine merchant has x bottles of wine in her shop and y bottles in her cellar. She transfers a
quarter of the bottles from the cellar to the shop.
a) How many bottles does she now have (i) in the shop (ii) in the cellar?
She now finds that she has twice as many bottles in the cellar as she has in the shop.
b) Write down an equation linking x and y and simplify it.
c) If she originally had 2,000 bottles in the shop, how many has she altogether?
6. For the annual village fete, the vicar orders 250 bottles of drinks. He orders x bottles of
lemonade and the remainder of cola.
Bottles of lemonade cost 35p each and bottles of cola cost 38p each. He spends £91.70
altogether.
a) Write down in terms of x the number of bottles of cola he bought.
b) Write down an equation for the total cost of the bottles and from it calculate the value of
x.
c) How many bottles of cola did he buy?
7. Three numbers are added together. The second number is 6 more than the first number and
the third number is 15 less than the first number.
a) If the second number is x, write down in terms of x, the value of each of the other two
numbers.
b) The three numbers are added together. Write down a simplified expression for the total of
the three numbers.
c) If the sum of the numbers is 93, what are the three numbers?
©Sum books 1998 Higher Level

21. Simultaneous Equations

1) 2x + 2y = 10 2) 3x + y = 18 3) 4x + 2y = 2 4) 5x + 3y = 18
x + 2y = 6 2x + y = 13 2x + 2y = 0 5x + y = 16
5) x + y = 1 6) 3x + 4y = 29 7) 3x – 2y = 10 8) 3x + 4y = 18
x–y = 5 x – 4y = – 17 – 3x + y = – 11 3x – 4y = – 6

9) 4x + 3y = 11 10) 5x + 2y = 33 11) 6x + 2y = 10 12) 3x – 2y = 13

2x + y = 7 2x + y = 14 4x + y = 7 x–y = 5

13) 2x + 3y = 28 14) 2x + 3y = 15 15) 4x + 3y = 13 16) 5x + 3y = 14

3x – y = 9 5x – y = 46 6x – 2y = 13 2x + 2y = 4

17) A family of 2 adults and 2 children go to the cinema. Their tickets cost a total of £14.00.
Another family of 1 adult and 4 children go to the same cinema and their bill is £13.60.
a) Letting x represent the cost of an adults ticket and y the cost of a childs ticket, write down
two equations connecting x and y
b) Solve for x and y.
18) The sum of two numbers is 39 and their difference is 9.
a) Letting x and y be the two numbers, write down two equations connecting x and y.
b) Solve the equations.
19) A rectangle has a perimeter of 42cm. Another rectangle has a length double that of the first
and a width one third of that of the first. The perimeter of the second is 57cm.
Letting x and y represent the dimensions of the first rectangle, write down two equations
containing x and y. Solve the equations and write down the dimensions of the second
rectangle.
20) 4 oranges and 3 apples weigh 720 grams. 3 oranges and 4 apples weigh 750 grams. Let x
and y represent their weights. Write down two equations containing x and y. Calculate the
weights of each piece of fruit.
21) Three mugs and two plates cost £7.20, but four mugs and one plate cost £7.90. Let x
represent the cost of a mug and y the cost of a plate. Write down two equations involving x
and y. Solve these equations and calculate the cost of seven mugs and six plates.
22) Sandra withdrew £400 from the bank. She was given £20 and £10 notes, a total of 23
altogether. Let x represent the number of £20 notes and y the number of £10 notes. Write
down two equations and solve them.
23) A quiz game has two types of question, hard (h) and easy (e). Team A answers 7 hard
questions and 13 easy questions. Team B answers 13 hard questions and 3 easy questions. If
they both score 74 points, find how many points were given for each of the two types of
question.
24) A man stays at a hotel. He has bed and breakfast (b) for three nights and two dinners (d).
A second man has four nights bed and breakfast and three dinners. If the first man’s bill is
£90 and the second man’s bill is £124, calculate the cost of a dinner.
25) Four large buckets and two small buckets hold 58 litres. Three large buckets and five small
buckets hold 68 litres. How much does each bucket hold?
26) Caroline buys three first class stamps and five second class stamps for £1.94. Jeremy buys
five first class stamps and three second class stamps for £2.06. Calculate the cost of each
type of stamp.
©Sum books 1998 Higher Level

22. Problems Involving Quadratic Equations

1. A rectangle has a length of ( x + 4 ) centimetres ( x + 4 )cm

and a width of ( 2x – 7 ) centimetres.
a) If the perimeter is 36cm, what is the value of x?
2
( 2x – 7 )cm
b) If the area of a similar rectangle is 63 cm ,
2
show that 2x + x – 91 = 0 and calculate the
value of x.

2. The areas of the two triangles on the right are

equal.
a) Write down an equation in x and simplify it.
b) Solve this equation and calculate the area of
one of the triangles. 3x–9
2x–5
3. Bill enters a 30 kilometre race. He runs the first
20km at a speed of x kilometres per hour and the
x+4 x
last 10km at (x –5) kph. His total time for the
race was 4 hours.
a) Write down an equation in terms of x and solve it.
b) What are his speeds for the two parts of the run?

4. A piece of wire is cut into 2 parts. The first part is bent into the shape of a square. The
second part is bent into the shape of a rectangle with one side 4cm long and the other side
twice the length of the square’s side. Let x represent one side of the square.
a) Write down two expressions in x for the areas of the two shapes.
2
b) If the sum of the two areas is 105cm2, show that x + 8x – 105 = 0 .
c) Calculate the length of the original wire.

5. A small rectangular lawn is twice as long as it is wide. It has a path around it which is 2
metres wide. The area of the path is twice the area of the lawn
a) If the small side of the lawn is x metres, write down the dimensions of the outside edge of
the path.
b) By writing down the area of the lawn in terms of x and using the answer to part a), form
an equation in x.
2
c) Simplify this equation so that it can be written as 4 ( x – 3x – 4 ) = 0 and solve it.
d) Write down the dimensions of the lawn.

6. Bill and Dan take part in a fun run. Bill’s average speed is (x + 4) kph and Dan’s is (3x) kph.
Bill completes his run in (x – 1) hours and Dan in (x – 2) hours.
a) Write down two expressions representing the distance travelled by both runners.
b) Combine these expressions to find the value of x.
c) What was Bill’s speed?
©Sum books 1998 Higher Level

23. Straight Line Graphs

Exercise 1
In each of the following questions say what the gradient of the line is.
1) y = x + 2 2) y = 3x + 3 3) y = – 5x – 2 4) y = 3 – 4x
5) 2y = 4x + 3 6) 3y = 6x – 5 7) 3y = – 5x + 2 8) 4y = 6 – x
9) y – 2x + 3 = 0 10) 2x – y – 3 = 0 11) y + x + 3 = 0 12) 2y + 2x – 2 = 0
13) 2y + 2x = 1 14) x + y = 3 15) 4x – 3y = 4 16) x – 3y = 2

Exercise 2
1) What is the equation of the line parallel to y = x which goes through the point (3,0)?
2) What is the equation of the line parallel to y = 2x which goes through the point (5,0)?
3) What is the equation of the line parallel to y = 3x + 2 which goes through the point (0,0)?
4) What is the equation of the line parallel to y = – x – 6 which goes through the point (0,2)?
5) What is the equation of the line parallel to y = – 3x + 2 which goes through the point
(1,1)?
6) What is the equation of the line parallel to y = 4x – 7 which goes through the point (4,2)?

Exercise 3
1) The table below shows the relationship between x and y.

x 10 20 30 40 50

y 14 34 54 74 94
a) On graph paper plot the values of x and y and show that this graph is of the form
y = mx + c b) What are the values of m and c? c) What is the value of y when x = 65?
d) What is the value of x when y = 234?
2) The table below shows the relationship between two sets of values x and y .

x 2 4 6 8 10

y 30 40 50 60 70
a) On graph paper plot the values of x and y and show that this graph is of the form
y = mx + c b) What are the values of m and c? c) What is the value of y when x = 15?
d) What is the value of x when y = 105?
3) A coach company charges for the hire 140
of their coaches according to the
120
graph shown.
a) Write down the equation to the line 100
b) What is the cost of hiring a coach Cost 80
for 5 hours? £ 60
c) What is the cost of hiring a coach (C)
40
for 6 1--2- hours? 20
d) For how many hours is a coach
hired for if it costs £100? 2 4 6 8 10
Number of hours (H)
©Sum books 1998 Higher Level

24. Inequalities

Exercise 1
In each of the following inequalities, solve them in the form x > n, x < n, x ≥ n or x ≤ n where n
is a number.
1) x + 1 > 7 2) x + 4 > 6 3) x + 12 > 20
4) x + 6 > 3 5) x + 5 > 2 6) x + 7 > 9
7) 2x > 12 8) 3x > 12 9) 4x > 20
10) x + 3 < 5 – x 11) x + 4 ≤ 10 – x 12) x + 3 ≤ 13 – x
13) 3x – 9 ≥ x + 7 14) 5x – 6 < 2x + 3 15) 6x – 8 < 4x + 2
16) 2 ( x – 3 ) > 3 ( 2 – x ) 17) 3 ( x + 1 ) ≤ 2 ( x + 7 ) 18) 5 ( x + 3 ) ≤ 2 ( x – 4 )
19) 5 ( 2x – 4 ) ≤ 3 ( 3x – 7 ) 20) 3 ( 2x – 7 ) ≤ 4 ( 2x + 8 ) 21) 3 ( 2x – 6 ) ≥ 2 ( 4x – 7 )

Exercise 2
In questions 1 to 3, copy the number line into your book. Mark on it the integer values of x which
satisfy the inequality.
1. 2x + 3 > 3x + 4

–4 –3 –2 –1 0 1 2 3 4
2. 3x + 7 > 4x + 2

0 1 2 3 4 5 6 7 8
3. 4x + 7 < 2x – 6

–8 –7 –6 –5 –4 –3 –2 –1 0

In questions 4 and 5, copy the number line into your book. Mark on the number line the
range of values which satisfy the inequality
4. 4x + 3 > 2x – 4

–4 –3 –2 –1 0 1 2 3 4
5. 5x – 6 < 2x + 4

–1 0 1 2 3 4 5 6 7
6. If 3x + 4 > x + 7 , what is the least whole number that satisfies this inequality?
7. If 3x + 4 < x + 7 , what is the greatest whole number that satisfies this inequality?
8. A social club want to hire a bus to take them out for the day. Company A charge £20 per
hour. Company B charge £70 plus £10 per hour. Let x be the number of hours they want to
hire the bus for.
a) Write down the inequality satisfied by x where the cost of the number of hours charged
by company A is less than the cost charged by company B.
b) Solve the inequality and explain what the solution tells you.
©Sum books 1998 Higher Level

25. Linear Inequalities

x+8
1. Plot the graphs of x = 6, y = 10 – x and y = ------------ for values of x from 0 to 6 and values of
2
y from 0 to 10. Clearly show the area satisfied by the inequalities x ≤ 6, y ≥ 10 – x and
x+8
y ≤ ------------ . Write down the integer points within this area.
2
2. Plot the graphs of y = 6, y = 6 – 2x and y = 4x – 2 for values of x from 0 to 4 and values of y
from –3 to 7. Clearly show the area satisfied by the inequalities y ≤ 6, y ≥ 6 – 2x and y ≥ 4x
– 2. Write down the integer points within this area.
x
3. For values of 0 to 5 on the x axis and 0 to 12 on the y axis, plot the graphs of y + --- = 4 ,
3
1
y = --- x + 3 and y = 12 – 3x . On your diagram clearly indicate the area satisfied by the
2
x 1
inequalities y + --- ≥ 4 , y ≤ --- x + 3 and y ≤ 12 – 3x . List the integer points lying within
3 2
this area.
1
4. Plot the graphs of y = x , y = 12 – 3x and y = 7 – --- x for values of x from 0 to 6 and
2
values of y from 0 to 12. Clearly indicate the area satisfied by the inequalities y > x ,
1
y > 12 – 3x and y < 7 – --- x . Write down the integer points lying within this area.
2
1 1
5. Plot the graphs representing the equations x + y = 10 , y + --- x = 5 , y = --- x + 4 and
2 2
y = 2x – 3 for values of x from 0 to 6 and values of y from –3 to 10. On your diagram
1 1
shade in the region representing the inequalities x + y < 10 , y + --- x > 5 , y ≤ --- x + 4
2 2
and y > 2x – 3 . Write down the integer points lying within this region.
6. On graph paper draw lines and shade in the area representing the inequalities y ≤ 8 – 2x ,
1
y ≥ 6 – 3x , y ≤ --- x – 1 and y ≥ x – 4 for values of x from 0 to 6 and y from –4 to 8. Write
2
down the integer points lying within this region.
1
7. Plot lines and shade in the area representing the inequalities y + --- x < 1 , y ≥ x – 3 ,
2
1
y < 2x – 2 and y > – --- ( x + 5 ) for values of x from –5 to 4 and y from –3 to 2. Write down
3
the integer points lying within this region.
8. Plot lines and shade in the area representing the inequalities y + 2x < 8 , y < x + 3 ,
1
y + --- x > 2 and y > 2x – 2 for values of x from –1 to 5 and values of y from –2 to 8. Write
2
down the integer points lying within this area.
9. Plot lines and shade in the area representing the inequalities y > x + 1 , y + 3x < 7 ,
y < 2x + 6 and y > 1 – x for values of x from –3 to 4 and values of y from –3 to 8. Write
down the integer points lying within this area.
©Sum books 1998 Higher Level

26. Linear Programming

1) A van can carry a maximum load of 400 kg. It carries boxes weighing 20 kg and 40 kg. It
carries at least 7 boxes weighing 40 kg. The number of boxes weighing 40 kg is not more
than twice the number of 20 kg boxes.
Let x represent the number of 20 kg boxes and y the number of 40 kg boxes.
a) Write down three inequalities involving x and y .
b) Illustrate the three inequalities by a suitable diagram on graph paper. Let 2 cm represent 1
box on both axes.
c) From the diagram determine the least weight the van carries.
d) What combinations give the greatest weight?
2) Orange is produced by mixing together red and yellow in certain ratios. David wants to
make up to 10 litres of orange paint. He mixes together 200 ml tins of red and yellow. To be
classified as orange there must not be more than twice as much of one colour than the other.
Let x represent the number of red tins and y the number of yellow tins.
a) Write down 3 inequalities in x and y.
b) On graph paper, illustrate these inequalities using a scale of 2 cm to represent 10 tins on
each axis, showing clearly the area representing the orange mix.
c) How much of each colour would he use to make 10 litres if he wanted;
(i) the reddest possible shade of orange.
(ii) the yellowest possible shade of orange.
d) What is the maximum amount of orange paint he can make with 14 tins of red?
3) A factory produces curtains for large and small windows. Each large curtain requires 10 m2
of fabric and each small curtain requires 5 m2 of fabric. There is a total of 500 m2 of fabric
available each day.
For each of the large curtains the factory makes a profit of £5 and for each small curtain it
makes a profit of £8. To cover costs the factory needs to make a profit of at least £400 each
day. Due to the type of demand for the curtains, they never make more than twice as many
small curtains as large ones.
Let x represent the number of large curtains and y the number of small curtains.
a) Write down three inequalities involving x and y .
b) Represent these inequalities on a graph and clearly indicate the area which satisfies these
inequalities. Use a scale of 2 cm to represent 10 curtains on each axis.
c) From the diagram find the values of x and y which will satisfy all the three conditions and
give the greatest profit.
4) The dimensions of a rectangle are such that its perimeter is greater than 20 metres and less
than 30 metres. One side must be greater than the other. The larger side must be less than
twice the size of the smaller side.
Let x represent the length of the smaller side and y the length of the larger one.
a) Write down four inequalities involving x and y.
b) On graph paper, illustrate these inequalities using a scale of 2 cm to represent 2 metres on
each axis, clearly showing the area containing the solution.
c) What whole number dimensions will satisfy these three inequalities?
©Sum books 1998 Higher Level

27. Recognising Graphs.

1. Water runs into a conical container. The height (h) of the water is Water
plotted against the time (t) it takes for the water to flow into the
container. Which of the following sketches represents this?
h

a) b) c) d)

h h h h

t t t t

2. A weight is suspended from the bottom end of a piece of wire. The top end is fixed. The
weight makes the wire extend at a constant rate. Which of the following diagrams shows
this?
a) b) c) d)

length length length length

time time time time

3. David buys an antique table. He estimates that each year it’s value will increase by 20% of
it’s value at the beginning of that year. Which of the following diagrams represents this?
a) b) c) d)

value value value value

time time time time

4. A car sets out from town A and drives to town B. The car is slowed down by traffic at the
beginning and the end of the journey, but speeds up in the middle section. Which of the fol-
lowing diagrams shows this?
a) b) c) d)

dist- dist- dist- dist-

ance ance ance ance

time time time time

©Sum books 1998 Higher Level

28. Graphs 1

2
1. a) Complete this table for values of y = x + 2x – 3 .
x= –4 –3 –2 –1 0 1 2
2
y = x + 2x – 3
2
b) Draw the graph of y = x + 2x – 3 using a scale of 2cm to represent 1 unit on the x axis
and 2cm to represent 1 unit on the y axis.
c) Using the same axis, draw the graph of y = x + 2 .
d) Write down the two values of x where the two graphs cross.
e) Write down a simplified equation which satisfies these two values of x.
3
2. a) Complete this table for the values of y = x – 2x + 2 .
x= –2.5 –2 –1 0 1 2 2.5
3
y = x – 2x + 2 –8.625 3 6
3
b) Plot the graph of y = x – 2x + 2 using a scale of 2cm to represent 1 unit on the x axis
and 2cm to represent 2 units on the y axis.
c) On the same axes, draw the graph of y = 3x + 2 .
3
d) Show on the graph that the equation x – 5x = 0 has three solutions. From the graphs
give approximate values of these solutions.
2
3. a) Complete the table of values for y = 2x – 4x – 3 .
x= –2 –1 0 1 2 3 4
2
y = 2x – 4x – 3
2
b) Plot the graph of y = 2x – 4x – 3 using a scale of 2cm to represent 1 unit on the x axis
and 2cm to represent 2 units on the y axis.
2
c) From the graph write down the solution to the equation 2x – 4x – 3 = 0 .
d) On the graph draw in the line y = 4 and from the graph write down the approximate
2
solution to the equation 2x – 4x – 7 = 0 .
4. a) Complete the table of values for y = ( x + 3 ) ( x – 1 )
x= –4 –3 –2 –1 0 1 2
x+3
x–1
y = ( x + 3)( x – 1)
b) Using a scale of 2cm to represent 1 unit on both axes, draw the graph of
y = ( x + 3)( x – 1) .
c) On the same axes, draw the graph of y = x + 1 .
d) From your graph, estimate the x co-ordinates of the points of intersection of the two
graphs and write down the quadratic equation which these values of x satisfy.
©Sum books 1998 Higher Level

29. Graphs 2

8
1. a) Complete the table of values of y = 2x + --- for the values of x from 0.5 to 6.
x

x 0.5 1.0 1.5 2.0 2.5 3 4 6 7

8
y = 2x + ---
x

b) Using a scale of 2cm to represent 1 unit on the x axis and 1cm to represent 1 unit on the y
8
axis plot the graph of y = 2x + --- .
x
x
c) Using the same axes draw the lines representing y = 14 and y = 12 – --- .
2
d) By considering the points of intersection of two graphs write down the approximate
8
solutions to the equation 2x + --- – 14 = 0 .
x
8 x
e) Show that the intersection of the graphs y = 2x + --- and y = 12 – --- gives a solution to
x 2
2
the equation 5x – 24x + 16 = 0 . What are the approximate solutions to this equation?
8
f) What is the gradient of the curve y = 2x + --- when x = 4?
x
2
2. a) Complete the table of values of y = 4 + 3x – x for values of x from –4 to +4.

x= –4 –3 –2 –1 0 1 2 3 4 5
2
y = 4 + 3x – x

2
b) Draw the graph of y = 4 + 3x – x using a scale of 2cm to represent 1 unit on the x axis
and 2cm to represent 4 units on the y axis.
c) On the same axes draw the line y = 3 and write down the approximate co-ordinates of the
point of intersection of the two graphs.
d) Show that the x co-ordinates at this point are an approximate solution to the equation
2
3x – x + 1 = 0 .
2
e) What is the solution to the equation 4 + 3x – x = 0 ?
2
f) By drawing a straight line, find an approximate solution to the equation 8 + 3x – x = 0 .
3x + 5
3. Draw the graphs of y = ( x + 3 ) ( 3 – 2x ) and y = --------------- for values of x from –4 to +2
2
using a scale of 2cm to 1 unit on the x axis and 1 cm to 2 unit on the y axis. From your graph
estimate the solutions to the equations;
2
a) 13 – 9x – 4x = 0 and
2
b) 3 – 3x – 2x = 0
©Sum books 1998 Higher Level

30. Graphs 3

2
1. The table below shows values of y = x + c . What is the value of c?
x 1 2 3 4 5
y 2.5 5.5 10.5 17.5 26.5

2
2. The table below shows values of y which are approximately equal to ax + b , where a and b
are constants.
x 1 2 4 6 8
y 53 62 98 160 240
a) Plot the values of y against x, using a scale of 2cm to represent 1 unit on the x axis and
2cm to represent 20 units on the y axis.
b) From your graph, determine the approximate values of a and b.
c) What is the approximate value of y when x = 7?
x
3. The table shows the approximate values of y which satisfy the equation y = pq , where p
and q are constants.
x 0 0.5 1 1.5 2 2.5 3
y 3 4.24 6 8.5 12 17 24
a) Plot the values of y against x, using a scale of 4cm to represent 1 unit on the x axis and
2cm to represent 2 units on the y axis.
b) Use your graph to help you estimate the values of p and q.
c) What is the approximate value of y when x = 2.25?
4. The table below shows the approximate values of y which satisfy the equation
y = a sin x + b where a and b are constants.
x° 0 30 60 90 120 150 180
y –0.5 0.25 0.8 1.0 0.8 0.25 –0.5
a) Using a scale of 2cm to represent 30° on the x axis and 10cm to represent 1 unit on the y
axis, plot the graph of y = a sin x + b
b) From your graph estimate the values of a and b.
c) What is the approximate value of y when x = 45° ?
x
5. The table below shows the approximate values of y which satisfy the equation y = a + b
where a and b are constants.
x 0 0.5 1.0 1.5 2.0 2.5
y 3.0 3.73 5.0 7.2 11 17.6
x
a) Draw the graph of y = a + b . Allow 4cm to represent 1 unit in the x axis and 2cm to
represent 2 units on the y axis.
b) From your graph estimate the values of a and b.
c) What is the approximate value of y when x = 1.3?
©Sum books 1998 Higher Level

31. Growth and Decay

–x
1. The relationship between x and y is given by the equation y = 1.5 .
(a) Complete the table, giving y correct to 3 decimal places where necessary.

x 0 1 2 3 4 5 6
y 0.667 0.198

–x
(b) Draw the graph of y = 1.5 , allowing 2cm to represent 1 unit on the x axis and 2cm to
represent 0.1 on the y axis.
(c) From your graph, estimate the following, showing clearly where your readings are taken.
(i) the value of x when y = 0.7.
(ii) the value of y when x = 3.5.

2. The population of a country grows over a period of 7 years according to the equation
t
P = P 0 × 1.1 where t is the time in years, P is the population after time t and P 0 is the
initial population.
a) If P 0 = 10 million, complete the table below giving your values correct to 2 decimal
places where necessary.

t 0 1 2 3 4 5 6 7
P (million) 10 14.64 19.49

b) Plot P against t. Allow 2cm to represent 1 year on the horizontal axis and 2cm to
represent 1 million on the vertical axis (Begin the vertical axis at 8 million).
c) From your diagram estimate the population after 5 1--2- years.
d) How long will the population take to reach 15 million?

3. The percentage of the nuclei remaining in a sample of radioactive material after time t is
–t
given by the formula P = 100 × a , where P is the percentage of the nuclei remaining
after t days and a is a constant.
a) Copy and complete the table below for a = 3.

t 0 0.5 1 1.5 2 2.5 3 3.5 4

P 100 19.3 3.7

b) Draw a graph showing P vertically and t horizontally. Use a scale of 4cm to represent 1
day on the horizontal axis and 2cm to represent 10% on the vertical axis.
c) From the graph, estimate the following, showing clearly where your readings are taken.
(i) The half life of the material (ie when 50% of the nuclei remain) correct to the nearest
hour.
(ii) The percentage of the sample remaining after 2.25 days.
(iii) The time at which three times as much remains as has decayed.
©Sum books 1998 Higher Level

32. Distance - Time Diagrams

1. The graph shows a journey

undertaken by a group of
10
walkers. From the graph
C
determine
8

Distance (miles)
a) their average speed
between points A and B.
6
b) their average speed
between points B and C.
4
c) their approximate speed B
at 2pm.
2
d) At C they rest for half an
hour and then return to A at A
a constant speed. If they 11:00 13:00 15:00 17:00 19:00
arrive home at 8.00pm, what Time (hours)
is their average speed?
2. An object is projected vertically upwards so that its height above the ground h in time t is
given in the following table.

Time t seconds 0 0.5 1 1.5 2 2.5 3 3.5 4

Height h metres 0 7 12 15 16 15 12 7 0
Draw a graph to show this information using a scale of 4cm to represent 1 second on the
horizontal axis and 2cm to represent 2 metres on the vertical axis. From your graph find
a) the time, to the nearest 0.1 second, it takes for the object to reach 10 metres.
b) the velocity of the object when t = 1.7secs.
3. The curve shows the distance
200
Distance travelled (s metres)

travelled (s) by a car in time (t).

a) Find its approximate speed
160
when t = 14 seconds.
Explain what is happening to 120

the car between 80

b) t = 0 and t = 8 secs
40
c) t = 8 secs and t = 12 secs
d) t = 12 secs and t = 20 secs
0 10 20
Time (t seconds)
4. An object is projected vertically upwards. Its height h above the ground after time t is given
2
by the formula h = 30t – 6t where h is measured in metres and t is in seconds. Draw a
graph to show this relationship for values of t from 0 to 5 seconds. From your graph find
a) the height when t = 1.4 seconds.
b) the approximate speed of the object when t = 2 seconds.
c) the distance travelled in the fourth second.
d) the maximum height gained by the object.
©Sum books 1998 Higher Level

33. Velocity-Time Graphs

1. The velocity of a vehicle

after time t seconds is given 30
by the graph on the right.
25
a) Find the area underneath

–1
Velocity v ms
the graph between 20
0 < t < 20 seconds.
15
b) What is the approximate
distance travelled by the 10
vehicle in this time? 5
c) What is the acceleration
of the vehicle when the 0 2 4 6 8 10 12 14 16 18 20 22 24
–1 Time t seconds
velocity is 10 ms ?

2. The diagram below

shows a velocity-time 20
graph for the journey a
car makes. Use it to –1 15
Velocity v ms
calculate
a) the total distance 10
travelled in 180
seconds 5
b) the average
acceleration in the first 0
40 seconds. 40 80 120 160
Time t seconds
3. The diagram on the right
shows the journey a car makes 40
–1

between two sets of traffic 30

Velocity v ms

lights. Use it to find the

20
approximate distance between
10
the traffic lights

–1 0 10 20 30 40
4. The velocity v ms of a Time t seconds
particle over the first 5 seconds
of its motion is represented by the equation v = t ( 5 + 2t ) where t is in seconds.
a) Copy and complete the table below and from it draw the graph of v = t ( 5 + 2t ) . Use a
Time t 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Velocity v
–1
scale of 2cm to represent 1 second on the horizontal axis and 1cm to represent 5 ms on
the vertical axis.
b) Estimate the distance travelled by the particle in the first 3 seconds.
c) What is the approximate acceleration of the particle when t = 3 seconds?
©Sum books 1998 Higher Level

34. Angles and Triangles

In each of the following questions write down the sizes of the unknown angles.
1) 2)
x° y°

x°
z°

84° 79°
49° z° y°

3) 4)
y° x°

x°

z° y° 42° z°

5) 6)
w° x° y° 57° y°

z°
z° x°

7) 8) C
19° B
x° z° 44°
y° A
46°
y° x°
18° AC=AD
z° E
D

9) 10)
30° x°
20°
y°
110°
y°

x° 40°
z° z°
©Sum books 1998 Higher Level

35. Regular Polygons

1. a) Calculate the value of the angle x in this regular octagon.

b) What is the size of angle y?
c) Calculate the size of angle z.
In each, clearly explain how you arrive at your answer.
z
2. What is the sum of all the exterior angles of a polygon? y

x
3. Regular hexagons and squares are put together in
a row, as shown in the diagram on the right. x
Calculate the size of the angle marked x. Explain
clearly how you arrive at your answer.

4. A five pointed regular star shape is to be cut from

y
a piece of card. In order to draw it accurately it is
necessary to calculate the angles x and y. Calculate
these angles, explaining clearly how you arrive at the x

answers.

5. What regular shape can be made from a regular

hexagon and three regular triangles, all with side
lengths of x centimetres? What will be the length of
one side of the new shape?

6. In the regular heptagon shown on the right AB and H

DC are produced to meet at H. B
a) Calculate the sizes of angles BCD and BHC, in
A C
each case explaining how you arrive at your
answer.
b) Prove that BC is parallel to AD.
G D
7. Three regular polygons fit together around a point.
One is a triangle and one is an octagon. Calculate the
number of sides the third shape has. F E
©Sum books 1998 Higher Level

36. Congruent Triangles

In questions 1 to 4, say whether all, two or none of the triangles are congruent.
1.
a) b) 3.5cm c)
3cm 4cm
4cm 4cm 3.5cm
3cm
3.5cm 3cm
2.
a) b) c)

4cm 3cm 4cm 60° 3cm

3cm 4cm
60° 60°
3.
a) b) c)

7cm 7cm 7cm

6cm
45°
6cm
4.
a) b) c) 6cm
70°
7cm 6cm
60°
60° 70° 70° 60°
In questions 5 and 6 the triangles indicated are congruent. In each case explain why they are
congruent.
5. ∆ABC and ∆ADC A B

D C
6. ∆ABX and ∆CDX A B

D C
7. Which, if any, of the following
a) b) c)
statements are true for the triangles
on the right 50°
(i) Triangles a and b are congruent 5cm 5cm 5cm
(ii) Triangles a and c are congruent
(iii) Triangles b and c are congruent 65° 65° 65° 65°
(iv) All three triangles are congruent 5cm
(v) None of the triangles are congruent.
©Sum books 1998 Higher Level

37. Geometry of a circle 1

x
y

O O
o
84 o
y 33
x z z

1) Calculate the sizes of the 2) Calculate the sizes of the

angles x, y and z. angles x, y and z.

P A
o
o 48
40
D B
O
y .O
o
S z 36
Q
x
R C

3) O is the centre of the circle. Lines 4) Calculate the sizes of angles ADC
OQ, and OS are equal in length. ABC and AOD.
Calculate the sizes of angles x, y
and z.

z x
y
x Oy .
O
.
o
45 o
35
z 33
o

5) Calculate the sizes of angles x, y 6) Calculate the values of the

and z. angles x, y and z.
©Sum books 1998 Higher Level

38. Geometry of a circle 2

A
45°
x y
o
47
o D
86

B o
o
36 z
42
C

1) If AB=BC, calculate the sizes of 2) Find the values of the angles x, y

the angles ABC, ACB, ACD and and z.
DAC.

o
60 C
D
O.
c E
bO o
110
a o
68 B
A
d
4) Calculate the angles DEB, BCD,
3) O is the centre of the circle. Calculate and DBO.
the sizes of the angles a, b, c and d.

o
22 B
z o C
62
x
.O
y O . A

D
E
5) Calculate the angles x, y and z. 6) Calculate the angles BDA, BOD,
BAD and DBO.
©Sum books 1998 Higher Level

39. Vectors 1

C
1. In the quadrilateral, AB, BC, and CD are c
represented by the vectors b, c, and d. Find, in B
d
terms of b, c, and d AC , AD , and BD
2. AB =   and BC =   calculate AC in
2 3 b
 3  5 D
bracket form.
A
3. Express AB + BC + CD in its simplest form.
A B
4. OABC is a parallelogram. If OA = 4a and
OC = 4c. Find, in terms of a and c (a) AC 4a
(b) AP where P is the mid point of AC
(c) OP (d) If X is the mid point of CB find O 4c C
PX . (e) What is the geometrical relationship
between PX and OC ?
5. In triangle OAB, OA = 6a and OB = 9b. Point P is on AB such that AP = 1--2- PB . Find, in

terms of a and b (a) AB (b) AP (c) OP .

A
6. OABC is a triangle with point C half way along OB. OA = a
and AB = b. Find, in terms of a and b a b
(a) OB (b) OC (c) AC
O C B
7. ABCDEF is a regular hexagon with centre O . OA = a and F A

OB = b . Express in terms of a and b the vectors

OD, OE, DA, BC, OC and CF .
E .O B

8. (a) If AB = 3a and CD = 6a write down the geometrical

D C
relationship between AB and CD .
(b) What is the geometrical relationship between vectors 4a + 2b and 6a + 3b?
9. OAB is a triangle with point X half way along OA and Y O
half way along OB. If OA = 2a and OB = 2b. Find, in
terms of a and b, (a) AB (b) XY and (c) show that XY
X. .Y
is parallel to AB.
10. In a rectangle ABCD, AB = a and BC = b. Find, in A B
terms of a and b the vectors AC and BD .
©Sum books 1998 Higher Level

40. Vectors 2

A D B
1. OABC is a square. OA = a and OC = b. Point D cuts the
line AB in the ratio 2:1.
(a) Find, in terms of a and b the vector that represents AD .
(b) If point E is halfway along BC, find the vector that a E
represents DE .
(c) Using your answers to parts a and b show that the vector
OE is represented by b + 1--2- a.
O b C
A B
2. OABC is a rhombus with OA represented by b
and OC represented by a.
b
(a) What is the vector OB represented by? P. .Q
(b) What vector represents OP if point P O a C
divides the line OB in the ratio 1:2?
(c) Point Q divides the line CB in the ratio 1:2. What vector represents CQ ?
(d) Show that the line PQ is parallel to the line AB.
3. In the parallelogram OABC, point Y cuts the line A B
OB in the ratio 5:1. Point X cuts the line AC in .Y
the ratio 5:1 also. Write down in terms of a and b a
(a) OB (b) YB (c) AC (d) XC (e) XY . .X
(f) What is the geometric relationship between O C
b
lines XY and CB?
4. OABC is a rhombus. A B
CB = a and OC = b.
X.
Point X is halfway along OA. P
. a
Point P cuts XC in the ratio 1:2. Write
down, in terms of a and b vectors for O C
b
(a) OX (b) XC (c) OB

(d) XP (e) OP (f) How far along OB is point P?

5. ABCD is a trapezium with BC parallel to
B C
AD and 3--4- of its length.
E.
AD = 12b DC = 6a. 6a
Point E cuts the line AC in the ratio 5:1.
Find, in terms of a and b
A D
(a) AC (b) DE (c) AB 12b
©Sum books 1998 Higher Level

41. Similar Shapes

1. Two hollow cylinders are similar in shape. One is

12cm tall and the other is 18cm tall.
Calculate the ratio of
a) the areas of their ends
b) their volumes.
3
c) If the smaller one has a volume of 100 cm , what is
the volume of the larger one?

2. A cone of height 3h is cut into two parts to make a smaller cone of

height 2h.
a) What is the ratio of the volumes of the two parts? 2h
2
b) If the surface area of the top part is 200 cm , what was the
surface area of the original cone?

3. Two similar blocks of metal made from the same material have a
2 2
total surface area of 25 cm and 64 cm . If the smaller one has a h
mass of 30 grams, what is the mass of the larger one?

4. A ball has a diameter of 8cm and weighs 200 grams. Calculate the weight of a ball of 10cm
diameter made from the same material.

5. A cylindrical can of height 15cm holds one litre of orange juice. What height, to the nearest
mm, must a similar can be if it holds 500ml?

6. Two similar cones have heights of 100cm and

50cm. If the volume of the smaller one is

3
1000 cm , calculate the volume of the larger one.

7. Two similar boxes have corresponding

sides of 6cm and 4cm.
a) If the surface area of the smaller box
2
is 50 cm , what is the surface area of
4cm
the larger box?
b) If the volume of the larger box is 6cm
3
50 cm , what is the volume of the
smaller box?

8. Two similar cornflakes packets have widths of 25cm and 20cm. Cornflakes are sold in
packets of 750g, 500g and 350g. If the 25cm packet holds 750g, how much does the 20cm
packet hold?
©Sum books 1998 Higher Level

42. Similarity

1. In the triangle ABCDE, BE = 3cm, CD = 5cm , angle CDE = 105° and angle BAE = 28° .
Line BE is parallel to line CD.
C

B
5cm
3cm

A 28° 105°
E 4cm D
a) (i) Calculate the size of angle DEB
(ii) Explain your answer.
b) Calculate the size of AE.
2. In the diagram, AB = 12cm, DE = 18cm, AC = 5cm, angle ACB = 116° and angle
CDE = 25° . Line AB is parallel to line DE.
A 12cm B

5cm 116°
C

25°
D 18cm E
a) (i) Calculate the size of angle DEC.
(ii) Explain how you get your answer.
b) Calculate the length of CE
3. In the triangle, AC = 20cm, DC = 10cm and A
FB = 6cm.
a) Calculate the length of AB. F B

b) If AF = 4.5 cm, calculate the length of FE.

E D C
4. In the triangle shown below, AD = 1cm, A
DC = 3cm, ED = 2cm and BE = 4.5cm. 1cm
Line AB is parallel to line DF. E 2cm D
a) Calculate the lengths of FC and AE. 4.5cm
3cm
b) What is the ratio of the areas of triangle
AED to that of triangle ABC.
B F C
©Sum books 1998 Higher Level

43. Bearings

1. An aeroplane sets off on a bearing of N 28° E ( 028° ) but after some time has to turn back to
the airport it came from. On what bearing must it travel?
2. A ship sets sail on a bearing of S 17° E ( 163° ). It then turns through an angle of 90°
anticlockwise. What is its new bearing?
3. ABC is an equilateral triangle with line BC pointing due east. A
Write down the following bearings:
a) A from B
b) C from A
c) A from C B C
4. The diagram shows the approximate Belfast
relative positions of Belfast, Dublin
N
and Liverpool. Calculate the
bearings of:
a) Belfast from Dublin
b) Belfast from Liverpool 6°
81° 38°
Liverpool
c) Liverpool from Belfast. Dublin

5. The diagram shows a regular N

hexagon ABCDEF with one side pointing due north. Calculate A
the bearings of:
F B
a) A from F
b) C from B
c) D from C
E C
d) E from B
D
6. The diagram shows the approximate
N
relative positions of Bardsey Island, Bardsey Island
33° 84°
Barmouth and Aberystwyth. Barmouth
Calculate the bearings of:
a) Bardsey Island from Aberystwyth
b) Barmouth from Aberystwyth
c) Bardsey Island from Barmouth.
55°

Aberystwyth
©Sum books 1998 Higher Level

44. Constructions

In the questions below, use only a ruler and a compass, not a protractor.
1. Construct an angle of 60° and bisect it to make an angle of 30° .
2. Construct an angle of 90° and bisect it to make an angle of 45° .
3. Construct this equilateral triangle, with sides of 5cm.
60°

60° 60°
4. Construct this rectangle
6cm

8cm
5. Draw a line 12cm long and bisect it by construction.
6. Joanna wants to measure the height of a tower. She A
measures out a distance from the bottom of the
tower (BC) until the angle between the ground and
the top of the tower is 30° . She measures BC to be
37 metres. By making a scale drawing and letting
30°
1cm represent 5 metres, estimate the height of the C B
37 metres
tower.

7. The diagram shows the sketch of an 8cm C

square metal plate with a hole at the centre.
x
The shape is symmetric about the lines AB
and CD. A 4cm B 6cm dia
Make an accurate drawing of the hole.
What is the length of dimension x?

D
N
8. A ship is sailing due south when it detects a light
Direction house at an angle of 30° from its direction of travel.
of ship It carries on for a further 1200 metres and now finds
30°
that the lighthouse is at an angle of 45° to its direction

45°
of travel. Draw an accurate diagram to a scale of 1cm
to represent 200 metres. By measuring on your
diagram, estimate the closest distance the ship will be
Lighthouse to the lighthouse if it continues on this course.
©Sum books 1998 Higher Level

45. Loci

1. A wheel of diameter 3cm rotates around

the inside of a rectangle measuring 8cm
by 11cm. Draw accurately the locus of the
centre of the wheel.

2. A wheel of diameter 3cm rotates around

the outside of a rectangle measuring 10cm
by 4cm. Draw the locus of the centre of the
wheel.

3. A rectangular field ABCD is to have a pipe A 170 metres B

line laid underneath it. The pipe enters the field
through the side AD, is parallel to the side AB 110
metres
and 40 metres from it. After some distance its
route turns at a point X towards corner C.
D C
Between X and C it is always equidistant from
the sides BC and CD. It finally leaves the field by the corner C.
Copy the plan of the field, using a scale of 1cm to represent 20 metres. On the plan clearly
indicate the path of the pipe line. Measure and record the distance BX.

4. The diagram shows the plan of a garden lawn. Draw an 10m

accurate diagram using the scale of 1cm to represent 1
5m
metre. Point X represents an electrical socket on the Lawn
side of the house. An electric lawn mower has a cable 10m
5m
attached to it allowing it to travel up to 9 metres from
5m
the socket. Show clearly on the diagram the area of the
2m
lawn which can be mown. House X

5. A ship sails between two rocks X and Y. The ship is at North

all times equidistant from both rocks. At point P, Z
when it is the same distance from rocks Y and Z, it
Y
alters course, keeping to a path which is equidistant 1km 1.2km
from Y and Z. Draw an accurate diagram to a scale of 30°
Direction 45°
5cm to represent 1km to show the route of the ship. of ship
X
What is the distance XP?
©Sum books 1998 Higher Level

46. Transformations 1

1. a) The triangle S is reflected across the line y

y = 0 to triangle T. Copy the diagram and 5

show triangle T. What are the co-ordinates 4

of triangle T? S
3

2
b) Triangle T is reflected across the line x = 0
1
to triangle U. Draw triangle U and write
x
–5 –4 –3 –2 –1 0 1 2 3 4 5
down its co-ordinates. –1

c) What is the single transformation that –2

transforms triangle U onto triangle S? –3

–4

–5

2. Triangle A has vertices at (–2,2), (–4,2) and y

(–2,4). Copy the diagram and draw each of the 4
following transformations.
3
In each case write down their co-ordinates. A
2
a) A rotation of triangle A of 90° clockwise
about the origin. Label this B. 1

b) A reflection of A about the line y = x. Label -4 -3 -2 -1 0 1 2 3 4 x

this C. -1
c) An enlargement of A with a scale factor of 1--2-
-2
through the origin (0,0). Label this D.
-3
d) What single transformation will transform D
back into A? -4

3. Triangle A, B, C is transformed into triangle y

2 5 A B
A 1 ,B 1 ,C 1 by an enlargement of scale factor ---
5
through the origin (0,0). 4
a) Draw triangles A, B, C and A 1 ,B 1 ,C 1 .
3
b) Write down the co-ordinates of triangle
C
A 1 ,B 1 ,C 1 . 2
c) A 1 ,B 1 ,C 1 is transformed onto A, B, C. What 1
single transformation will do this?
0 1 2 3 4 5 x
©Sum books 1998 Higher Level

47. Transformations 2

1. Triangle X is rotated 180° anticlockwise y

5
about the origin to triangle Y.
4
a) Draw triangles X and Y and write
3
down the co-ordinates of Y. 2
X
b) Triangle Y is translated by the vector 1

 0 to triangle Z. Draw triangle Z x

 4 –5 –4 –3 –2 –1 0 1 2 3 4 5
–1
and write down its co-ordinates.
–2
c) What single transformation will –3

transform triangle X onto Z? –4

–5
2. a) Triangle ABC is transformed onto y
triangle XYZ. What single transformation Z Y 4

will do this? 3
b) Triangle XYZ is transformed onto triangle 2
X
PQR. What single transformation will do
1
this?
-4 -3 -2 -1 0 1 2 3 4 x
c) If triangle PQR is rotated through 180°
-1
about the point (–1,0) to triangle STU,
R Q -2 A
what are the co-ordinates of the new shape?
-3
d) What single transformation will transform
triangle XYZ onto triangle STU? -4 C
P B

3. a) Triangle A is reflected about the line y = 2 to y

6
triangle B. Draw triangles A and B and write
5
down the co-ordinates of triangle B.
4 A
b) Triangle B is rotated 90° clockwise about
3
the point (1,2) onto triangle C. Draw triangle
2
C and write down its co-ordinates.
1
c) What single transformation will transform
-3 -2 -1 0 1 2 3 4 5 x
triangle A onto triangle C? -1

-2
©Sum books 1998 Higher Level

48. Transformations of Graphs

2
1. Diagram ‘a’ below shows a sketch of the graph y = x . The other graphs show
2 2 2 2 2
(i) y = x + 1 , (ii) y = x – 1 , (iii) y = ( x – 1 ) , (iv) y = ( 2x ) , and (v) y = ( x + 2 )
but not in that order. Match the diagram to the equation.
a) y b) y c) y

x x x
d) y e) y f) y

x
x x
2. Explain clearly each of the transformations in question 1.
3. The diagram on the right shows the graph of the f(x)
3 2
function f ( x ) = 2x + 3x .
a) If this graph is reflected in the x axis what will its
equation be? 1
b) Sketch the graph.
c) If f(x) is transformed into f(–x) write down its
equation and explain the transformation.
4. The function f(x) is defined for 0 < x < 3 in the diagram
below. Sketch the functions
a) y = f ( x – 2 ) and b) y = f ( 2x )
y
-1 0 x
3

0 1 2 3 x
©Sum books 1998 Higher Level

49. Matrix Transformations

1. By first drawing triangle A(1,1), B(3,1), C(3,4) use each of the following matrices to trans-
form it. State the transformation each represents.
       
a)  – 1 0  b)  2 0  c)  – 1 0  d)  0 1 
 0 –1   0 2  0 1  1 0

 
2. A transformation is given by the matrix P =  – 1 1  . Find P , the inverse of P and use it
–1
 1 1
to find the co-ordinates of the point whose image is (1,3).
3. A' is the image of triangle A after it has been reflected about the y axis. A'' is the image of
the triangle A' after it has been rotated 180° about the origin (0,0).
a) If triangle A has co-ordinates (1,1), (2,1), and (1,3) draw triangle A and the images A'
and A'' .
b) Find the 2 × 2 matrix M associated with the transformation A to A' .
c) Find the 2 × 2 matrix N associated with the transformation A' to A'' .
d) Calculate the matrix product NM and state the single transformation which is defined by
this matrix.
4. Triangle T has co-ordinates (1,1), (3,2), and (1,2). Triangle T' is the image of triangle T
when rotated through 180° about the origin.
a) What is the matrix M associated with the transformation of T to T' ?
 
Triangle T' is further transformed to T'' by the matrix N =  0 – 1 
 1 0 
b) What transformation does matrix N perform?
c) State the single transformation which is defined by the matrix product NM and write
down this matrix.
5. Triangle A(1,1), B(2,2), C(1,4) is transformed onto triangle A 1 B 1 C 1 by the matrix M

 
where M =  1 0  .
 0 –1 

 
Triangle A 1 B 1 C 1 is transformed by the matrix N where N =  0 – 1  onto the triangle
 –1 0 
A1 B1 C2 .
a) Find the co-ordinates of A 1 , B 1 , C 1 and C 2 and draw the three triangles.
b) What single transformation will transform triangle ABC onto triangle A 1 B 1 C 2 ?
c) By calculating the matrix W = NM show that this matrix represents the transformation
found in part ‘b’.
–1
d) Calculate W , the inverse of W. Use it to find the co-ordinates of the point whose image
is (3,–5).
©Sum books 1998 Higher Level

50. Area and Perimeter

1. Calculate the area of the trapezium on the right. 6cm

2. Calculate the area of the shaded part of the ring below.
8cm

9cm
9cm dia.

6cm dia.

3. Calculate the area and arc length of the following sectors where r is the radius of the circle
and θ is the angle at the centre of the circle.

a) r = 6cm b) r = 12cm c) r = 16.5cm d) r = 16.75cm

θ = 30° θ = 140° θ = 200° θ = 195°

4. The shaded part of the diagram shows a flower garden in

the shape of an arc. It has an inside radius of 4 metres and
an outside radius of 9 metres. a) Calculate the area of the
garden. b) If it is completely surrounded by an edging strip,
calculate its length. 75°
4m
5. A conical lamp shade is to be made from the piece of fabric
shown in the diagram. a) Calculate its area. b) If a fringe is 9m
sewn onto it’s bottom edge, what length will it be?
25cm 20cm
170°

6. A water tank is in the form of a cylinder of 4 metres diameter and 4 4m

metres high, open at the top. It is to be made from sheet steel and
painted with three coats of paint, both inside and outside. Calculate
a) the total area of steel needed and b) the number of tins of paint
needed if one tin is sufficient to cover 28m2 4m
7. The ceiling of a church is in the shape of a hemisphere. If its radius
is 8 metres, calculate its surface area.
8. The diameter of the earth is approximately 12734 kilometres.
Calculate its surface area.
©Sum books 1998 Higher Level

51. Volume

1. Calculate the volume of a 12cm square based pyramid with a height of 20cm.
2. Calculate the height of a 4cm square based pyramid whose volume is 40cm3
3. A rolling pin is in the shape of a cylinder with hemispherical ends. Its total length is 40cm

and its diameter is 5cm. Calculate a) its volume and b) its weight if 1cm3 of wood weighs
0.75 grammes.
4. A marble paperweight is in the shape of a hemisphere of radius 4cm. Calculate its volume.
A supplier packs them into boxes of 50. Calculate their weight, correct to the 0.1kg, if 1cm3
of marble weighs 2.7 grammes.
5. A metal sphere of diameter 10cm is lowered into a cylindrical jar of 16cm height and 12cm
diameter which contains water to a depth of 10cm. How far up the side of the jar will the
water rise?
6. The diagram shows a swimming pool. It 22m
measures 22 metres long by 10 metres wide. 10m
It is 1 metre deep at the shallow end and 2 1m
metres deep at the other end. Calculate the 2m
amount of water it will hold, in litres.
7. A bucket is made by cutting a cone into two parts, as shown in the
diagram. If the rim of the bucket measures 30cm diameter, calculate Bucket
the amount of water it will hold, correct to the nearest litre. 40cm
8. A piece of metal, in the shape of a square based pyramid of height
10cm and base sides of 5cm, is melted down and re-cast into spheres 40cm
of diameter 3mm. How many spheres can be made?

9. Plastic plant pots are made in the shape of inverted 5cm

truncated pyramids. The top is a square of 5cm sides, the
bottom a square of 3cm sides and its height is 6cm.
a) Calculate its volume b) How many can be filled from
a bag containing 2 litres of compost.? 6cm
10. Tennis balls of 6cm diameter are packed into cubic boxes
which hold 27 balls. Calculate a) the volume of the box
3cm
b) the percentage of the box filled by the tennis balls.
11. A measuring cylinder with a base diameter of 3.8cm contains water. 500
spheres of diameter 5.3mm are dropped into it and are completely covered
by water. Calculate the height by which the water rises. 14
12. A hollow metal sphere has an outside diameter of 10cm and is 1cm thick. cm
Calculate the volume of the metal. 6cm
13. The diagram shows a wine glass. The bowl is in the shape of a hemisphere
with part of a cone on top of it. The diameter of the hemisphere is 7cm and
the height of the cone is 14cm. The diameter of the rim of the glass is 6cm
and is 2cm above the hemisphere. Calculate the amount of wine the glass
will hold, correct to the nearest millilitre.
©Sum books 1998 Higher Level

52. Ratios and Scales

1. Divide £450 between Albert, Bob and Colin in the ratio 4:5:6.
2. An amount of money is divided between three people in the ratio 5:6:7. If the first gets
£125, what do the others get?
3. £560 is shared among three people. The second gets twice as much as the first who receives
twice as much as the third. a) Into what ratio is it divided? b) How much do they each get?
4. Mary makes 1.5 litres of lemon squash. The instructions say that the juice should be mixed
with water in the ratio three parts juice to seven parts water. How much juice is needed?
5. An architect draws the plans of a house to a scale of 1:20. Complete the table below.

Dimensions on drawing Actual dimensions

Width of house cm 10 metres

Height of door 11cm m

Area of dining room floor 2 2

cm 20m
Angle of staircase to floor ° 40°

6. The plans of a small housing estate are drawn to a scale of 1:500.

Copy and complete the following table.

Dimensions on drawing Actual dimensions

Length of street cm 0.5km

Angle between two streets ° 80°

Area of school field 2 2

cm 3000m
Width of street 2.5cm m
7. Mr Jones makes a doll’s house for his grand-daughter. He makes it a model of his own
house to a scale of 1:8. Copy and complete the following table.

Dimensions on doll’s house Actual dimensions

Total height cm 12m

Area of hall floor 2 2

1200cm m
Volume of dining room 3 3
cm 48m

Mr Jones sees some model tables to put into the house. There are three designs. They are
5cm, 10cm and 15cm high. Which one should he buy?
8. A model aeroplane is made to the scale of 1:40.
a) If the wingspan is 20 metres, what will it be on the model?
b) If the wing area is 500 square centimetres on the model, what is its actual area?
c) The volume of the fuselage is 240 cubic metres. What is its volume on the model?
©Sum books 1998 Higher Level

53. Degree of Accuracy

1. A length of wood measures 3 metres long by 10cm wide. Smaller pieces of wood are to be
cut from it, each measuring 10cm by 35mm, the width correct to the nearest millimetre.
Calculate the maximum and minimum number of pieces that can be cut from it.
2. Mugs are made in the shape of a cylinder, with internal dimensions of 9.3cm tall and 7.5cm
diameter, both measurements correct to the nearest millimetre.
a) What is the maximum amount of liquid it will hold?
b) What is the minimum amount of liquid it will hold?
3. A garden is 15 metres long, correct to the nearest metre. A path is made with cobbles
measuring 22cm long by 8.5cm wide, both dimensions correct to the nearest millimetre.
The cobbles are laid side by side without any space between them, in the way shown in the

15 metres
diagram with the long sides touching. How many cobbles must be ordered to ensure the
path is completed?
4. Tiles are made to the dimensions shown in the diagram, correct to the nearest millimetre.
8.4cm

4.1cm x cm gap

They are fitted together in the way shown with two shorter sides against a longer side, and
the top and bottom edges level. Calculate the maximum and minimum values of the gap, x .
5. Erasers are made in the shape of a cubiod measuring 1cm by 2cm by 4.5cm. All dimensions
correct to the nearest millimetre.
a) Calculate the maximum volume an eraser can be.
b) What is the minimum volume it can be?
c) If the manufacturer produces all the erasers to the lower dimensions, what is their
percentage saving in rubber, over the maximum dimension?
6. The plans of a new warehouse show that the floor measures 50 metres by 26 metres, correct
2
to the nearest metre. The concrete floor is to be covered with paint . The paint covers 15m
per litre.
a) What amount of paint needs to be ordered to ensure that the floor is covered?
b) What is the maximum amount of paint that will be left over if this is bought?
7. Floor tiles measure 20cm square, correct to the nearest millimetre. A rectangular floor
measures 4.21 metres long by 3.84m wide, both measurements correct to the nearest
centimetre.
a) What is the maximum number of tiles needed to fit on one row along the length of the
floor?
b) What is the maximum number of tiles needed to fit along the width of the room?
c) What is the maximum number of tiles needed for the whole of the room?
d) Ian buys the maximum number of tiles needed, but finds that they have been made to
their maximum size and the room measured to its minimum size. How many tiles does he
have left over?
©Sum books 1998 Higher Level

54. Formulae

Exercise 1
In each of the questions below, taking l, b, h, r and d to be length, classify each of the
following formulae into length, area, volume or those making no sense.
2 2
1) 4 ( r + lb ) 2) 3 ( l + d )b 3) π ( l + b )
4) 3bd + c 5) 3 ( l × b ) 6) 3b ( l + h )
2 2
3 ( l + 2b ) l +b 2 2
7) ---------------------- 8) ---------------- 9) 5 r + h
2 r
2 2
10) 2lbh 11) 3lr 12) 5b ( l + h )
2 2
13) 4b + r 14) 5b ( d + c ) 15) 6 ( l + b )
1 1 bdh
16) --- ( l + b ) 17) --- ( bh ) 18) ---------
2 2 3
2 2
19) a bd + e 20) d ab + lr 21) a b + lrd

Exercise 2
1) For the diagram shown on the right, choose one of the d
formulae listed below that you think best suits a) its area
and b) its perimeter
1 2
(i) a + b + c + 1.3d (ii) --- a b
3
b(c + a) c
(iii) -------------------- (iv) abcd
2 a
1 (a + b + c)
(v) --- abc (vi) --------------------------
2 2
b

2) For the diagram shown below, choose one of the formulae listed underneath you think best
suits a) the area of the shaded end and b) its volume.

c
b
d

e
a

1 1 1 2
(i) --- abc (ii) --- ( ac – de ) (iii) --- ( ab c )
2 2 2
(v)  ------------------ b
2 ac – de
(iv) a b – bc (vi) abc
 2 
©Sum books 1998 Higher Level

55. Pythagoras Theorem

1. Calculate the length of the unknown side in each of the following diagrams
a) b) c) d)
5.2cm
5cm
6cm 5.5cm
9cm 7.1cm
7.3cm

4cm

2. The diagram shows a rectangle ABCD A B

with CD = 42cm, and the triangle ADE E
with AE = 10.5cm and DE = 14cm.
Calculate the length of AC.

42cm C
D
3. In the diagram on the right, AOC and BOD A
are diameters of a circle of radius 4.5cm.
If DC = 7cm, calculate the lengths of BC and OE.

4. A rhombus has diagonals measuring 10cm and E D

8cm. Calculate the lengths of its sides.
.O
5. A window cleaner positions the base of his ladder 2
metres from the bottom of a block of flats. He opens B
his ladder up to 4 metres to clean the first floor
window and 6.5 metres to clean the second floor
window. On both occasions the ladder just touches
the window sill. Calculate the distance between the
two window sills. C

6. A Radio mast AD is held in a vertical position by four wires, AF, AE, AC and AB. AF = AB
and AE = AC. If AF = 50 metres, AE = 35 metres and AD = 25 metres, calculate the distance
CB.
A

F E D C B
©Sum books 1998 Higher Level

56. Sine, Cosine and Tangent Ratios

1. Calculate the length of the unknown side, x, in each of the following triangles
a) b) c) d)

22cm 33°
x 15cm
73°
18cm x
44° x
64° 12.3cm
x
2. Calculate the sizes of the unknown angle, x in each of the following diagrams.
a) b) c) d)
4.9cm
2.8cm x
3.2cm 4.1cm
5.4cm
3.5cm 5.2cm
6.0cm
x
x x

3. In the diagram below, AE = 10cm, BD = 7cm and ∠ ACE = 22° . Calculate the sizes of the
lines CD, AC and EB. Calculate also the size of angle DEB.
A

B
10cm
7cm

22° C
E D
4. The diagram shows the positions of three towns, A, B and C. A N
The bearing of town B from town C is 235° . The distance
between towns A and B is 10.5km, and the distance between
towns B and C is 9km. Calculate a) the bearing of town A 10.5km
from town B and b) the distance from town A to town C.
C
5. The diagram below shows the roof truss of a house. Its
height in the middle is 2 metres, and the angle between the 55°
horizontal and the roof is 28° . It is symmetrical about the
9km
line AC. Calculate a) the width of the truss and b) the angle x
if point D is half way along AB.
A B

D
2m
x 28°
C B
©Sum books 1998 Higher Level

57. Sine and Cosine Rules

1. Calculate the sizes of the unknown value of x in each of the following triangles.
a) 12cm b) c)
5.2m
58°
11cm 16cm
9cm 11cm
x 4.5m
x x 37°

d) e) f)
6.3m 87°
81°
43°
x
12cm 14m
7.7m 17cm
x
x 72°

2. In the diagram on the right, B

calculate the length of the A 34°
line AB.
3. In a parallelogram ABCD,
AB = DC, and AD = BC. 12.5cm
The diagonal BD = 20cm
and the diagonal AC = 14cm.
If angle ACB = 85° , 71°
calculate the sizes of the 47°
D 11cm C
two interior angles of the
parallelogram.
4. The bearing of a beacon from an aeroplane is 160° (S 20° E). The aeroplane flies due south
for 5km and now finds that the bearing is 135° (S 45° E).What is now the distance of the
beacon from the aeroplane?
5. In the diagram below, calculate the length of CB
A

17m

35° 18°
D C B
6. The diagram shows a series of congruent triangles used
in the making of a patchwork design. If the sides of the 8cm 9cm
triangle measure 8cm, 9cm and 10cm, calculate its
10cm
angles.
©Sum books 1998 Higher Level

58. Areas of Triangles

In questions 1 to 5 calculate the areas of these triangles

1. In triangle ABC, ∠ ABC = 61° , AB = 6cm and BC = 8cm.
2. In triangle XYZ, ∠XYZ = 77° , XY = 8.5cm and YZ = 13cm.
3. In triangle CDE, ∠CDE = 110° , CD = 5.6cm and DE = 8.5cm.
4. In triangle PQR, ∠PQR = 119° , PQ = 7.2cm and QR = 8.7cm.
5. In triangle RST, ∠RST = 47.7° , RS = 22.3cm and ST = 16cm.
6. In the diagram below both AB and AC are equal to 9cm. Show that the area of triangle ACD
is equal to the area of triangle ABD.
B C

20° 20°
A D
7. By calculating the area of triangle ADC, calculate the area of the parallelogram ABCD.
A B
5cm

65°
D C
11cm
8. A patchwork bed cover is to be made from 300 pieces of A

fabric cut into the shape of the triangle shown. Calculate

12cm
the area of the bed cover, in square metres, correct to one
decimal place.
60°
B 15cm C

9. A regular pentagon lies inside a circle of diameter 20cm

with its vertices touching the circle. Calculate the area
of the pentagon.

10. A field is in the shape of a triangle. Calculate

its area in hectares, correct to 1 decimal place.
307m
(10,000 sq. metres is equal to 1 hectare)

57°
265metres
©Sum books 1998 Higher Level

59. Trigonometry Mixed Exercise

1. Two circles with diameters AO1B and BO2C intersect at A

B and C. If AB = 11cm and BC = 9cm, calculate the
dimensions of AC and CD.
2. BD, the diameter of the circle below, is 12cm in length. D C
A .O 1

.O
B 2

. B
O

D C

AD = 11cm and BC = 8cm. Calculate the length of the line AC.

3. The diagram shows a river with a post P on its bank. When viewed from two points, A and
B, on the other side of the bank, the angles between the lines of sight and the bank are 32°
and 58° . If the distance between A and B is 9 metres calculate the width of the river.
P.

58° 32°
B 9 metres A
4. The diagram on the right shows a A F B
pentagon ABCDE which is symmetrical
about the line FD. The lengths of AE and ED
are 10cm. If AB = 6cm, calculate the sizes of
angles ABC and CDE.

5. In the triangle ABC, AB = AC and angle E C

ABC = 70° . If BC = 10cm, calculate the
length of AC.
D
6. A building AB, is observed across a street A
from the window of a house 7 metres above the
ground. If the angle of elevation of the top of the 23°
building is 23° and the angle of depression of the foot 26°
of the building is 26° , calculate the height of the 7m
building. B
©Sum books 1998 Higher Level

60. Graphs of Sines, Cosines and Tangents

1. The diagram shows the graph of y = sin x

a) Sketch this into your book and mark on it the 1.0
approximate solutions to the equation
sin x° = – 0.5 where x lies between 0° and
360° . 0
0° 180° 360°
b) Calculate accurately the solution to the
equation sin x° = – 0.5 where x lies between
0° and 360° . –1.0

2. a) Draw a graph of y = 2 sin x° + 1 for

0° ≤ x ≤ 180° using a scale of 2cm for 1 unit on the y axis and 2cm for 30° on the x axis.
b) From your diagram, calculate the values of x which satisfy the equation
2 sin x° + 1 = 2.5 for 0° ≤ x ≤ 180° .

3. a) Sketch the graph of y = cos x for 0° ≤ x ≤ 360° .

b) Show on the diagram the approximate locations of the solutions to the equation
cos x = – 0.5 for 0° ≤ x ≤ 360° .

4. a) Draw the graph of y = 3 cos x + 2 for 0° ≤ x ≤ 360° . Use a scale of 2cm for 1 unit on
the y axis and 4cm for 90° on the x axis.
b) From the graph, calculate the solutions to the equation 3 cos x° + 2 = 3 for
0° ≤ x ≤ 360° .

5. The diagram on the right shows a sketch of

y = tan x for 0° ≤ x ≤ 360° . From the graph 10
determine the approximate solutions to the equa-
tion tan x = 4 for 0° ≤ x ≤ 360° . 5

6. a) Sketch the graph of y = tan x + 2 , for

0
0° ≤ x ≤ 360° . Indicate on the graph the 0˚ 180˚ 360˚
approximate location of the solution to the
equation tan x + 2 = 0 for 0° ≤ x ≤ 360° . –5
b) Sketch the graph of y = 3 tan x on the same
axis as y = tan x for 0° ≤ x ≤ 180° clearly –10
showing the difference between the two
graphs.

7. a) Sketch the graph of y = 2 tan x – 1 for 0° ≤ x ≤ 360° .

b) Indicate on the graph the approximate location of the solution to the equation
2 tan x – 1 = 2 for 0° ≤ x ≤ 360° .
©Sum books 1998 Higher Level

61. Three Dimensional Trigonometry

1. A flag pole stands in the middle of a

square field. The angle of elevation of the
top of the flag pole from one corner is 27° .
27°
If the flag pole is 12 metres high, calculate
the lengths of the sides of the field.

2. The diagram shows a square based right V

pyramid.
a) If the base edges are 20cm, calculate the
length of the base diagonal AB.
b) If the angle VBA is 55° , calculate the 55° B
height VC. C
A

3. The diagram shows a cuboid with vertices B

ABCDEFGH. Its height is 2 metres and its C
width is 3 metres. Angle GHF measures 50° E 2m
Calculate: A
F
a) the diagonal of the base, HF D

b) angle CHF
H 50°
c) the diagonal of the box HC. 3m
G
4. The roof of a building is in the E
shape of a triangular prism and is
shown in the diagram. B D
Calculate:
F
a) the length of BC
A 50°
b) the angle BFC. 40° 10m
5m
C

5. A radio mast is due east of an observer A and 150 metres away. Another observer, B, stands
due north of A on level ground. If the angle of elevation of the top of the mast from B is 28°
and the bearing of the mast from B is 132° , calculate the height of the mast.
©Sum books 1998 Higher Level

62. Questionnaires

1. Meg is following a course in tourism. As part of the course she has to do a statistical survey.
She decides to find out where most tourists in Britain come from. She thinks that the USA
will be her answer.
Her home town lies more than 300 miles from London and is very historical. She carries out
a survey on the main street of her town at 12:00 midday each day during the month of July.
a) Will her survey be biased? Explain your answer.
b) How could she improve the survey?
c) Write down three questions she might ask.
2. Jim lives at Ayton-on-sea. He works for the local newspaper who want to begin a regular
feature for people over the age of 60. To help decide whether the venture is worth
undertaking, his editor asks him to do a survey of the over 60’s.
Jim decides to place a questionnaire in the newspaper and invite people to fill it in and send
it back to him. This was the questionnaire.
‘The Ayton-on-sea Observer are thinking of starting a feature for the over 60’s. If you are in
this age range we would be grateful if you would fill in the following questionnaire and send
it back to us.
Would you regularly read a feature for the over 60’s? Please tick the
appropriate box.

Yes No
What types of articles would you like included? Please list them in
the space below.

a) What other information do you think is needed?

b) Is the survey biased in any way?

3. Jane believes that the life expectancy of a car is 15 years. She carries out a survey by
observing cars travelling down her high street and noting their registration number (this tells
her how old they are).
a) Explain why this will only give her an idea of the life expectancy and not the full answer.
b) In order to add to this data, she did a further survey by questioning people leaving a
supermarket. What questions do you think she needs to ask?
©Sum books 1998 Higher Level

63. Sampling

1) The table shows the number of pupils in each year at Clivedenbrook Community College.

Year Group 7 8 9 10 11 12 13
Number of Pupils 173 147 166 144 140 49 21

Emma does a survey to find out what types of TV programmes people like to watch. She decides
to interview 70 pupils.
a) Use a stratified sampling technique to decide how many pupils should be asked from each
year.
b) Without using the names of any specific TV programmes, write down three relevant
questions she could ask with a choice of at least 2 replies in each case.

2) The table shows the salaries of the 180 employees at the ACME sausage factory.

Amount Earned <£10,000 £10,000 to £15,000 to £20,000 to £30,000

<£15,000 <£20,000 <£30,000 and over
Number of Employees 45 65 44 16 10

In order to understand whether they have satisfactory working conditions the management plan
to interview 40 employees. They are to be chosen at random using a stratified sampling
technique.
a) Calculate how many employees from each group should be interviewed.
b) Write down two questions which could be asked, each with a choice of two responses.
c) What other type of stratification could be used? Give one example.

3) The manager of a fitness centre wants to find out whether the members are satisfied with the
amenities available. She would like ideas to improve the facilities. The table shows the age
ranges for all the members.

Age Range Under 25 25 to 40 40 to 60 Over 60

Number of Members 237 463 414 136

She decides to interview 100 members using a stratified sampling technique.

a) Calculate the number of people from each group that should be interviewed.
b) Write down two relevant questions, each with a choice from at least two responses.
©Sum books 1998 Higher Level

64. Scatter Diagrams

1. The table shows the sales of cans of drink from a vending machine, over a period of 12 days
in June

Day S M T W T F S S M T W T

Temp. °C 15 17 20 25 21 18 15 20 23 27 30 29

Cans sold 104 113 188 275 212 150 90 205 251 330 425 404

a) Construct a scatter diagram of the data and draw on it a line of best fit.
b) Approximately how many cans would be sold on a day when the average temperature is
22 °C ?
2. Timothy’s old freezer registers –4 °C , which is higher than it should be so he buys a new
one. The diagram shows the temperature inside the new freezer after he has started it up.

20
x
x
10 x
x
Temperature°C

Time
0 x
1:00 02:00 03:00 04:00 x 05:00 06:00
x
x
–10 x

–20
The freezer is switched on at 1:00pm and he takes the temperature every 30 minutes.
a) At what time would you expect it to reach its minimum temperature of –20 °C ?
b) He missed taking the temperature at 03:30. What was the approximate temperature?
c) Tim transfers food from his old freezer to the new one when both the temperatures are
equal. At approximately what time does he do this?
3. A lorry can carry up to 20 tonnes of sand. The lorry moves sand from the quarry to a
collection point 2km away. The times it takes for 10 deliveries are shown in the table below.
Draw a scatter graph of the data showing a line of best fit.

Time taken (minutes) 5.0 4.5 3.5 3.7 4.8 4.3 4.1 4.0 3.6 4.2
Amount of sand (tonnes) 10 14 20 18 13 15 17 16 19 17

a) What weight of sand would take approximately 4 minutes 20 seconds to deliver?

b) How long would a load of 16 tonnes take to deliver?
©Sum books 1998 Higher Level

65. Pie charts

1. The diagram shows the way in which a small

company spends its money. Other Business

costs stock
During a year it spends £50,000.
Advertising
By measuring the angles on the diagram, Employee
General costs
calculate the approximate amount of money administration
spent in each area.
Premises
costs

2. An Italian restaurant sells 6 different types of pizzas.

Type of pizza Number sold
During one week the manager keeps
Margherita 256
records of the pizzas sold. Seafood 97
The table shows the results. Hot and spicy 83
From the table draw a pie chart to show the data, Vegetarian 132

clearly indicating how the angles are calculated. Chef’s special 185
Hawaiian 47

3. Jake asked a number of people how they intended to

vote in the forthcoming election, for the Left, Right,

Centre or Reform Party.

He began to draw a pie chart for the data. The angle he

has drawn on this pie chart represents the 210 people

who voted for the Centre Party.

a) Measure the angle and calculate how many people

were interviewed altogether.

b) If 392 people voted for the Right Party, what angle would be used to show this?

c) If 100° was used to represent the Left Party, show this information on a pie chart, and

complete the pie chart.

d) How many people intended to vote for the Reform Party?

©Sum books 1998 Higher Level

66. Flowcharts 1

1. This flowchart can be used to calculate how

Start
many numbers (n) there are in the sequence
Input
2 T, n
n + 3 smaller than or equal to T. For
2 Evaluate
example, the first number is 1 + 3 = 4 , 2
S = n +3
2
the second is 2 + 3 = 7 and so on.
Is No Increase
Starting with n = 40 and T = 2000, list each S > T? n by 1
successive value of n and each corresponding Yes
Print
value of S. Find how many numbers there are n–1

in the sequence which are less than 2000 Stop

Start
2. This flowchart can be used to
Input name
print out a list of students
Input score
names and their examination
Is Is Is
score N score N score N results. If the following are
>79%? >64%? >49%?
inputted, write down what you
Y Y Y
Output Output Output would expect to be outputted.
name and name and name and
‘Distinction’ ‘Merit’ ‘Pass’
W Jones 67

J Connah 56
Any
Y more C Smith 83
names?
N R Rogers 24
Stop
H Patel 52
©Sum books 1998 Higher Level

67. Flowcharts 2

Start
1. The flow diagram can be used to calculate the Input x

first 20 elements of a number sequence. Use it Print x

to calculate the first 20 numbers of the sequence. Input y

Print y
Explain how each number in the sequence

is formed. Let z = x + y
Let y = z
Start with x = 0 and y = 1. Print z

Let x = y
Have
20 numbers No
been
printed?

Yes
Stop

Start
2. This flow chart is used to process the
Input T

Input x following values of Z.

10, 7, 9, 11, 6, 3, 6, 13, 5, 10, 8.

Input Z
Carry out the process.
Let T = T + Z
Let x = x + 1 Write down the value of M and explain

Any what the flowchart does.

more values Yes
Start the process with T = 0 and x = 1.
of Z?

No
Let M = T ÷ x

Print M
Stop
©Sum books 1998 Higher Level

68. Histograms 1- Bar Charts

1. The time taken for the pupils in year 11 to get to school on Monday morning are shown
below in the diagram.

35

30

25

Frequency
20

15

10

0
0 5 10
20 25 30 15 35 40 45 50
Time (minutes)
a) Copy and complete this frequency chart.

Time 0- 5- 10- 15- 20- 25- 30- 35- 40- 45-50

Freq
Use the data to calculate
b) the mean time
c) the percentage of pupils who took 25 minutes or more.

2. The members of a fitness centre were weighed and their masses noted in the table below.
51 53 61 77 82 93 107 67 73 70 69
51 81 99 105 47 64 77 69 82 41 65
79 62 108 98 80 75 61 65 52 41 43
69 73 78 81 73 63 74 86 42 56 58
64 81 76 63 84 92 103 94 85 72 63

a) Copy and complete this frequency table.

Mass 40- 50- 60- 70- 80- 90- 100-110
Frequency

b) Show this data on a bar chart

c) What is the modal class?
©Sum books 1998 Higher Level

69. Histograms 2

1. The histogram shows the range of ages of members of a sports centre.

6.0

Frequency density
4.0

2.0

0
15 25 30 35 45 60 70
Age (years)
a) Complete this frequency table
Age 15- 25- 30- 35- 45- 60-70
Number of people

b) Calculate an estimate for the mean age of the members.

2. David carries out a survey of the speeds of vehicles passing a point on a motorway. From
the data he draws this histogram.

12.0
Frequency density

8.0

4.0

0 20 30 40 60 70 100
Speed (miles per hour)
a) Complete this table

Speed 20 < x ≤ 30 30 < x ≤ 40 40 < x ≤ 60 60 < x ≤ 70 70 < x ≤ 100

Number of cars

b) Which range of speeds is the modal range?

©Sum books 1998 Higher Level

70. Histograms 3

1. The following list shows the heights of the tomato plants in a greenhouse.
94 124 113 103 127 106 131 132 112 118
106 117 117 123 102 114 133 117 118 101
127 109 119 93 110 126 139 108 97 113
109 119 114 128 101 129 111 121 126 110
91 122 116 99 125 116 108 125 114 107
a) Complete the following frequency table.
Height in centimetres Number of plants Frequency density
90 < x ≤ 100 5 0.5
100 < x ≤ 105
105 < x ≤ 115
115 < x ≤ 120
120 < x ≤ 130
130 < x ≤ 140
b) Draw a histogram of the data.
c) From your histogram, find the percentage of tomato plants which are greater than 115cm.

2. A battery manufacturer tests the life of a sample of batteries by putting them into electric
toys and timing them. The results she gets, in minutes, from 58 tests are shown below.
726 945 863 673 876 842 645 942 621 833
1042 526 735 893 621 773 531 733 635 998
954 763 1073 550 725 1084 747 849 716 1032
721 962 683 768 872 632 787 641 752 800
1063 794 1062 613 714 867 590 749 854 943
1021 681 943 842 841 730 961 982
a) Complete the following frequency table
Time - Minutes Frequency Frequency density
500 < x ≤ 600 4 0.04
600 < x ≤ 700
700 < x ≤ 750
750 < x ≤ 900
900 < x ≤ 1100
b) Draw a histogram to show this data
c) From the frequency table, estimate the percentage of batteries whose life expectancy is
less than 800 minutes.
©Sum books 1998 Higher Level

71. Mean

1. The table below shows the wages paid to a number of people working in a factory.
Complete the table and calculate the mean wage.

Wages, £ Frequency Mid value Frequency × Mid value

60 ≤ £ < 100 4
100 ≤ £ <140 19
140 ≤ £ < 170 24
170 ≤ £ < 200 11
200 ≤ £ < 220 6

2. The table below shows the heights of a number of rose trees at a garden centre. Copy and
complete the table of results. Calculate the approximate mean height of the roses.

Height of plant, h, centimetres Frequency

50 ≤ h < 70 5
70 ≤ h < 90 14
90 ≤ h < 100 16
100 ≤ h < 110 24
110 ≤ h < 120 21
120 ≤ h < 140 23
140 ≤ h < 160 17

3. The table below shows the speeds of 60 vehicles passing a certain point on a motorway.

27.6 58.5 80.5 64.8 54.8 46.6 77.9 84.1 54.9 59.6
64.1 45.8 43.6 30.6 73.9 28.5 43.1 43.9 39.5 49.6
40.4 76.0 24.7 48.6 45.8 75.6 22.5 58.9 45.5 60.8
37.4 42.8 54.8 35.9 45.2 32.6 83.5 43.9 39.4 42.4
51.6 47.9 33.7 57.8 33.6 57.2 54.9 64.5 61.0 73.6
32.1 67.9 57.8 75.7 23.6 52.0 38.6 54.2 27.3 55.8

Make a frequency table from the values and hence calculate the approximate mean speed of
the traffic, in miles per hour.
©Sum books 1998 Higher Level

72. Mean, Median and Mode.

1. 30 pupils in a class were asked to keep a record of the number of pints of milk their family
bought during one week. The results are given below.
18 21 15 21 22 14 14 28 21 14
14 17 21 22 14 15 21 21 16 24
15 13 14 21 25 21 14 18 12 22
a) What was the modal number of pints bought per week?
b) What was the mean amount of milk bought, correct to the nearest 0.1 pint?
c) What was the median amount of milk bought?
2. The length of the words in the first two sentences of No. of letters Frequency
Pride and Prejudice by Jane Austin are given in the table 1 6
2 21
on the right. 3 9
a) Calculate the mean length of the words (correct to 1 4 10
5 9
decimal place) 6 2
b) Calculate the median number of letters per word. 7 2
8 5
c) What is the modal number of letters in a word? 9 1
d) Do you think that these values are a fair indication of 10 2
11 2
the words in the whole book? Explain. 12 1
13 1
3. The table on the right shows the weights
Weight w Frequency Mid Value
of 100 packages brought into a post
office during one day. 60g < w ≤ 100g 21
a) Complete the table for the mid values. 100g < w ≤ 300g 44
b) Calculate an estimate for the mean
weight of a package brought in that 300g < w ≤ 600g 17
day. 600g < w ≤ 1kg 9
c) In which class interval does the
1kg < w ≤ 2kg 5
median value lie?
d) What is the modal class interval? 2kg < w ≤ 3kg 4

4. 84 packs of orange juice were sampled

from a days production at a drinks 40
factory. The contents of each pack were
measured. The amounts are shown in 30
Frequency

the diagram on the right.

From the graph:
20
a) Write down the modal class.
b) In which class interval does the median
10
value lie?
c) Calculate an estimate for the mean 0
contents of a pack. 1.00 1.01 1.02 1.03 1.04 1.05 1.06
Contents in litres
©Sum books 1998 Higher Level

73. Mean and Standard Deviation

1. Nigel weighs 10 bags of potatoes. His readings are:

5.01kg, 5.03kg, 5.07kg, 5.05kg, 5.08kg, 5.04kg, 5.03kg, 5.01kg, 5.04kg, 5.08kg.
a) Calculate the mean and standard deviation of the readings.
b) Nigel notices a sign on the scales saying that all measurements are undersize by 0.02kg.
Explain the effect this has on (i) the mean (ii) the standard deviation.
2. The heights of the children in year 10 were measured by the school nurse. Her results are
shown below.
Height of child (cm) Frequency f Mid point x
120 < h ≤ 130 3
130 < h ≤ 140 7
140 < h ≤ 150 20
150 < h ≤ 160 54
160 < h ≤ 170 32
170 < h ≤ 180 9
a) Complete the table for the mid point values.
b) Use the mid point values to calculate (i) the mean (ii) the standard deviation, for the
heights of the children, giving your answer correct to one decimal place.
3. The weekly wages of the employees at a supermarket are given in the table below.
Wage £ Frequency f Mid Value x
80 < £ ≤ 120 5
120 < £ ≤ 160 8
160 < £ ≤ 200 9
200 < £ ≤ 240 11
240 < £ ≤ 280 14
280 < £ ≤ 320 17
320 < £ ≤ 360 3
a) Complete the table for the mid values.
b) Using the mid point values, calculate the mean and standard deviation of the distribution,
giving your answers correct to two decimal places.
c) It is decided to give each person in the store a bonus of £10 for one week. What effect will
this have on the mean and the standard deviation for that week?
4. The list below shows the ages of the 30 members of a tennis club.
21 65 33 29 32 29 53 61 39 48
42 45 46 27 48 35 46 25 43 59
24 37 44 63 39 22 38 33 55 51
a) Complete this grouped frequency table.
Age (years) Frequency Mid value
20 ≤ a < 30
etc
b) Use this table of values to calculate an estimate of the mean and standard deviation of the
distribution.
c) At the Badminton club, the mean age is 48 and the standard deviation is 8.4. Comment on
the differences between these figures and those for the tennis club.
©Sum books 1998 Higher Level

74. Frequency Polygons

1. The data below shows the weights of 30 cats treated by a vet over a period of a week.

3.1 4.9 4.1 3.7 4.3 3.6 4.3 4.8 4.2 4.2
3.7 4.1 5.4 4.6 3.1 4.2 3.3 4.4 4.4 3.7
3.3 4.3 4.7 5.3 4.1 4.6 5.1 3.6 4.1 3.6

a) Use the data to complete this grouped frequency table.

Weight w kg Frequency Mid value

3.0 < w ≤ 3.5
3.5 < w ≤ 4.0
4.0 < w ≤ 4.5
4.5 < w ≤ 5.0
5.0 < w ≤ 5.5

b) From your table construct a frequency polygon for the data. Use a scale of 4cm to
represent 1kg on the horizontal axis and 2cm to represent 2 cats on the vertical axis.
2. The histogram below shows the heights of 112 greenhouse plants

40

30
Frequency

20

10

0
100 120 140 160 180 200 220
Height of plants
a) Use the histogram to complete this grouped frequency table.
Height h cm Frequency Mid value
80 <h ≤ 100
100 < h ≤ 120
120 < h ≤ 140
140 < h ≤ 160
etc.
b) Use the frequency table to plot a frequency polygon for the data. Use a scale of 1cm to
represent 10 cm on the horizontal axis and 2cm to represent 4 plants on the vertical axis.
©Sum books 1998 Higher Level

75. Cumulative Frequency (1)

1. The table below shows the lengths (l) of 100 engineering components taken at random from
one days production in a light engineering works.

Length l l ≤9.7 9.7<l ≤9.8 9.8<l≤9.9 9.9<l≤10.0 10.0<l≤10.1 10.1<l≤10.2

Frequency 4 13 24 36 17 6

a) Copy and complete this cumulative frequency table for this data.

Length l 9.7 9.8 9.9 10.0 10.1 10.2

Cumulative Frequency 4 17

b) Draw a cumulative frequency diagram. Use the scale of 2cm for 10 components on the
vertical axis and 2cm for 0.1 unit on the horizontal axis.
From your diagram find:
c) The median length.
d) The upper and lower quartiles and hence the interquartile range.
e) The components measuring 9.75cm or less are under size and will be scrapped.
Approximately what percentage will be scrapped?

2. The table shows the ages of the 250 employees at the headquarters of ‘MoneyBankPlc’.

Age 16- 20- 30- 40- 50- 60-65

Frequency 17 65 78 51 30 9

a) Draw up a cumulative frequency table for this data.

b) From the table draw a cumulative frequency diagram. Use a scale of 1cm to represent 10
employees on the vertical axis and 1cm to represent 5 years on the horizontal axis.
c) The company decide to offer early retirement to those who are 45 or over. From the
diagram estimate approximately what percentage of employees this will involve.

3. The table shows the marks gained by 200 students taking a mathematics examination.

Mark ≤10 11- 21- 31- 41- 51- 61- 71- 81- 91-
20 30 40 50 60 70 80 90 100
Frequency 3 7 22 31 34 37 28 19 13 6

a) Draw up a cumulative frequency table from this data.

b) From the table draw a cumulative frequency diagram. Use the scale of 1cm to represent
10 pupils on the vertical axis and 2cm to represent 20 marks on the horizontal axis.
c) If 60% of the students passed the examination, what was the approximate pass mark?
d) Those who get a mark of 75 or more are awarded a distinction. What percentage of the
students are awarded a distinction?
©Sum books 1998 Higher Level

76. Cumulative Frequency (2)

1. ‘Shiny’ long life bulbs are guaranteed to last at least 7,500 hours before breaking down. In
order to make this guarantee, ‘Shiny’ did trials on 300 bulbs. These are the results:
Life of bulb (hrs) 5000 to 6000 to 7000 to 8000 to 9000 to 10,000 to
5999 6999 7999 8999 9999 11,000
Frequency of breakdown 5 20 42 95 80 58
a) Make a cumulative frequency table for this data.
b) From the table, draw a cumulative frequency diagram. Use a scale of 2cm to 1000 hours
on the horizontal axis and 2cm to 40 bulbs on the vertical axis.
c) From the graph, estimate the percentage of bulbs that fulfil the guarantee.
2. A flying school gives 1 hour lessons. Some of that time is spent on the ground having
instruction and some in the air. The time spent in the air for 100 lessons were as follows.
Time spent 20-30 30-35 35-40 40-45 45-50 50-55
in the air mins mins mins mins mins mins
Frequency 6 10 12 34 23 15
a) Make a cumulative frequency table for this data. From the table, draw a cumulative
frequency diagram. Use a scale of 1cm to 2 mins horizontally and 1cm to 5 people
vertically.
c) From the diagram, estimate the median and interquartile range of the data.
3. The times that trains arrive at a station are recorded by the station staff. They record whether
a train is early or late and by how much. The table below shows their data for one day.

Timing 10-5 mins 5-0 mins 0-5 mins 5-10 mins 10-15 mins 15-20mins
early early late late late late
Frequency 33 66 22 17 10 2
a) Make a cumulative frequency table for this data.
b) From the table, draw a cumulative frequency diagram. Use a scale of 2cm to represent 5
minutes on the horizontal axis and 1cm to represent 10 trains on the vertical axis.
c) The train company guarantees that its services will not be more than 7 minutes late. Use
your graph to check what percentage were outside the guarantee.
4. A large electrical retailer keeps stocks of television sets in its main warehouse for delivery
to its shops. Over a period of 1 year (365 days) the number of TV’s in the warehouse are
given in the table below.
No. of TV’s in warehouse 50-100 101-150 151-200 201-250 251-300 301-350
No of days (frequency) 5 27 77 145 83 28
a) Make a cumulative frequency table for this data.
b) Draw a cumulative frequency diagram using a scale of 2cm to represent 40 TV’s on the
horizontal axis, and 2cm to represent 40 days on the vertical axis.
It is the policy of the company to keep a stock of between 130 and 260 TV’s.
c) For approximately how many days was the stock within these limits?
d) A stock of 110 TV’s is regarded as the minimum quantity that the warehouse should
have. For approximately how many days was the stock below the minimum requirement?
©Sum books 1998 Higher Level

77. Probability 1

1. A bag contains 4 red discs and 5 green discs. A disc is selected at random from the bag, its
colour noted and then replaced. This is carried out 3 times. Calculate the probability of
getting:
a) a green disc on the first draw
b) a green disc followed by a red disc
c) three green discs
d) a green disc and two red discs in any order.

2. A game is played using a four sided

1
spinner and a three sided spinner. The two

4
spinners are spun together and their two

2
3+2 = 5

3
1
values added together in the way shown in 3 2
the diagram.
Calculate the probability of getting an outcome of a) 3 b) 5 c) not 6

3. The probability that a parcel will be delivered the next day is 0.4. Two parcels are sent,
independently of each other, on the same day. What is the probability that:
a) both will be delivered the next day?
b) only one will be delivered the next day?
c) neither will be delivered the next day?

4. A biased dice has the numbers 1 to 6 on its faces. The probability of throwing a 1 is 0.1 and
the probability of throwing a 2 is also 0.1. The probability of throwing each of the
remaining 4 numbers is 0.2. Calculate the probability of throwing:
a) a 1 followed by a 3
b) a 2 followed by a 4
c) two 2’s.

5. The probability that the post arrives before 8.00 am is 0.2 and the probability that the daily
newspaper arrives before 8.00 am is 0.7. Calculate the probability that:
a) both newspaper and post arrive before 8.00
b) the post arrives before 8.00 but the newspaper arrives after 8.00
c) both arrive after 8.00.
d) one or both of them don’t arrive before 8.00.

6. A fair dice having the numbers 1 to 6 on it is rolled and its value noted. It is rolled a second
time and its value added to the first one. For example, if 3 is rolled the first time and 2 the
second time, the total after 2 rolls will be 5.
a) What is the probability that the score will be greater than 12?
b) What is the probability of scoring between 6 and 9 inclusive with two throws of the dice?
©Sum books 1998 Higher Level

78. Probability 2

1. A bag contains 3 black discs and 7 white discs. A disc is taken at random from the bag and
not replaced. This is carried out three times. Calculate the probability of getting:
a) a black disc on the first draw
b) a black disc followed by a white disc
c) a black disc followed by two white discs.
2. It is known from past experience that of every 350 students entered for a typing examination
280 will pass the first time. Of those that fail, 80% of them pass on their second attempt.
What is the probability that a student, chosen at random from a group of students sitting the
examination for the first time, will pass on the first or second attempt?
3. The probability that there will be passengers waiting to be picked up at a bus stop is 0.85. A
long distance bus has 3 stops to make. Calculate the probability that it will have to pick up
passengers one or more times on its journey.
4. In a game of chance, two people each have a dice numbered 1, 1, 1, 2, 2, 2. They both throw
the dice together. If both the outcomes are the same, then they have a second try. If they are
the same again, then they have a third try, and so on until someone gets a 2 while the other
gets a 1. The winner is the person who gets the 2.
a) What is the probability that the game is over on the first pair of throws?
b) What is the probability that someone wins on the second pair of throws?
5. In a class of 30 pupils there are 14 boys and 16 girls. Two names are chosen at random.
Calculate the probability that:
a) the first name selected is a girls’
b) both the names selected are girls’.
6. In a bag there are 5 discs, 3 red and 2 blue. The red discs have the numbers 1, 2 and 3
written on them and the blue discs have the numbers 1 and 2 on them.

1 2 3 1 2
Red Discs Blue discs
a) What is the probability of withdrawing a disc with a 2 on it from the bag?
b) Two discs are drawn from the bag. What is the probability that one disc has a 2 on it and
one disc is red?
7. An engineering works produces metal rods. The rods are 6.2cm long and 0.5mm in
diameter, both dimensions to the nearest 0.1mm. When the rods are inspected there can be
four different outcomes:
either one or both dimensions are under size, in which case the rod is scrapped,
or one dimension is oversize and one is correct, in which case it is reworked,
or both dimensions are oversize, in which case the rod is reworked,
or both dimensions are correct, in which case the rod is acceptable.
It is known from past experience that 1.2% are under size in length and 1.5% are under size
in diameter. Also 2.5% are oversize in length and 2.2% are oversize in diameter.
A rod is selected at random. What is the probability that it is:
a) scrapped b) reworked c) accepted?
©Sum books 1998 Higher Level

79. Relative Probability

1. A biased coin is tossed 50 times. The number of times it comes down heads is 31 and the
number of times it comes down tails is 19. If it is now tossed 1000 times, how many times
would you expect it to show heads?
2. In an opinion poll for the general election, 500 people in a constituency were asked what
party they would vote for. 220 said they would vote for the Left party, 170 for the Right
party and 110 for the Centre party. The 500 people can be taken as representative of the
population of the constituency.
a) What is the probability that a person chosen at random will vote for:
(i) the Left party (ii) the Right party (iii) the Centre party?
b) It is expected that 45,000 people will vote in the election. Estimate the number of people
who vote for each party.
3. A chocolate company make sugar coated chocolate sweets in red, yellow and green. It is
known that more people prefer red sweets than any other colour so the company mix them
in the ratio 5 red to 3 yellow to 2 green.
a) In a bag of 50 sweets, how many of each colour would you expect?
b) David takes a sweet from the bag without looking at it. What is the probability that it will
be:
(i) red (ii) yellow (iii) green?
Sarah only likes the red sweets. She eats them all from her bag without eating any of the
others. She gives the bag to David who now takes a sweet. What is the probability that it is:
(i) green (ii) yellow?
During one days production of sweets at the factory, the company make 100,000 green
sweets. How many red and yellow sweets do they make?
4. An unfair triangular spinner has the numbers 1, 2 and 3 on it. It is spun 100 times. Number
1 is obtained 25 times, number 2 occurs 30 times and number 3 occurs 45 times.
a) Write down the probability of each number occurring.
b) If it is spun 550 times, how often would you expect each number to occur?
5. A survey is carried out over a period of six days to determine whether the voters are for or
against having a new by-pass outside their town. Each day 50 people were asked their
opinion. The results were put into the following table.

Number of people 50 100 150 200 250 300

Number ‘for’ 30 45 55 83 96 122
Number ‘against’ 20 55
Probability ‘for’ 0.60 0.45
Probability ‘against’ 0.40 0.55

a) Copy and complete the table.

b) On graph paper plot ‘Number of people’ against ‘Probability for’. Use a scale of 2cm to
represent 40 people on the horizontal axis and 2cm to represent 0.1 on the vertical axis.
c) From your graph estimate the probability that a voter chosen at random would be for the
by-pass.
d) There are 70,000 voters in the town. If the people asked were a representative sample of
the voting population, how many would you expect to be against the by-pass?
©Sum books 1998 Higher Level

80. Tree Diagrams

1. The probability that Sarah will win the long jump 100 metres
is 0.1 and, independently, the probability that she Long jump
W
will win the 100 metres is 0.15.
a) Complete the tree diagram to show all the
Win n/w
possible outcomes.
b) What is the probability that she wins one or
both events?
W
c) What is the probability that she wins neither
Not win
event?
n/w
2. William waits at the bus stop
each morning in order to Late 0.7 × 0.05 = 0.035
catch the bus to work. Some Bus
mornings his friend arrives in
his car and gives him a ride to
work. From past experience
the probabilities of William
getting the bus is 0.7 and the Car
probability that he gets a ride
with his friend is 0.3. When
he gets the bus, the chance of him being late for work is 0.05 and when he gets a ride the
chance of him being late is 0.15.
a) Complete the tree diagram to show all the possible outcomes.
b) From the diagram calculate the probability that on a day chosen at random he will be late.
3. A bag contains 5 green
1st sweet 2nd sweet Red 3--- × 2--- = -----
3
-
sweets and 3 red sweets. 8 7 28
Two sweets are withdrawn
at random from the bag. Red
sweet
a) Complete the diagram to
show all the possible
outcomes
b) What is the probability
of getting:
(i) a sweet of each colour
(ii) at least one red sweet? Green
sweet
4. A bag contains 8 discs, 4
black and 4 white. 2 discs
are removed at random
from the bag. By drawing a tree diagram calculate the probability that:
a) both discs will be black b) at least one of the discs will be white.
5. In class 7B, two pupils are chosen at random from the class register to be the monitors.
There are 12 boys and 16 girls in the class. By drawing a tree diagram calculate the
probability that:
a) the two people chosen will be girls b) the two people chosen will be boys
c) one person of each gender will be chosen.
Higher Page 81
Answers 30 × 0.9
27) a) 2.099 b) ------------------- c) 2.7
10
1. Estimations and Calculation 2
30 × 0.4
28) a) 16.89 b) ---------------------- c) 18
2×5 20
1) a) 1.857 b) ------------ c) 2
5 40 × 10
29) a) 1231 b) ------------------ c) 1000
10 × 4 0.4
2) a) 20.12 b) --------------- c) 20
2 2
30 × 20
9×9 30) a) 639.9 b) -------------------- c) 600
3) a) 18.02 b) ------------ c) 20 3 × 10
4 2
30 × 20
0.5 × 7 31) a) 7009 b) -------------------- c) 10,000
4) a) 1.002 b) ---------------- c) 1 0.3 × 4
3 2 2
3 × 0.4 30 × 0.8
5) a) 0.1074 b) ---------------- c) 0.1 32) a) 1372 b) ------------------------ c) 1440
9 0.4
2
8 × 0.3 40 × 4
6) a) 0.3274 b) ---------------- c) 0.3 33) a) 213.1 b) ------------------- c) 160
8 0.3 × 10
0.7 × 0.8 10 × 30
7) a) 0.1099 b) --------------------- c) 0.1 34) a) 8.471 b) ------------------ c) 7.5
5 40
0.2 × 0.6 40 × 0.02
2
8) a) 0.01814 b) --------------------- c) 0.02 35) a) 2455 b) ------------------------- c) 4000
6 0.8 × 0.01
0.6 × 0.8 2
0.5 × 0.006
9) a) 0.06096 b) --------------------- c) 0.06 36) a) 0.1352 b) ----------------------------- c) 0.09
8 0.8 × 0.02
2 × 0.2 100
10) a) 1.035 b) ---------------- c) 1 37) a) 6784 b) ---------- c) 5000
0.4 0.02
8 × 0.3 27000 × 0.008
11) a) 12.22 b) ---------------- c) 12 38) a) 5.866 b) ---------------------------------- c) 7
0.2 32
20 × 0.2
12) a) 4.332 b) ------------------- c) 5
0.8
10 × 30 2. Standard Form
13) a) 616.3 b) ------------------ c) 500 Exercise 1
0.6
30 × 100 1) 4.57 × 10
2
2) 1.427 × 10
3
3) 9.431 × 10
3
14) a) 6500 b) --------------------- c) 7500
0.4
5 7 –1
200 × 30 4) 1.56321 × 10 5) 1.7 × 10 6) 2.813 × 10
15) a) 21080 b) --------------------- c) 20000
0.3 –2 –4 –6
7) 8.142 × 10 8) 4.86 × 10 9) 9.7 × 10
20 × 0.02
16) a) 0.9753 b) ---------------------- c)1
0.3
Exercise 2
30 × 0.8 1) 280,000 2) 64,000,000 3) 93,000
17) a) 1445 b) ------------------- c) 1200
0.02
70 × 30 4) 4,315,000 5) 8,614,000,000 6) 0.0431
18) a) 1,411,000 b) ------------------ c) 2,100,000
0.001 7) 0.0000032 8) 0.000000684 9) 0.00000000438
2
( 30 ) × 10
19) a) 395.9 b) ------------------------- c) 300
30 Exercise 3
7 9 –4
5 × ( 0.8 )
2 1) 2.432 × 10 2) 4.482 × 10 3) 1.564 × 10
20) a) 5.303 b) ------------------------ c) 5 –1 7 –3
0.6 4) 3.784 × 10 5) 5.312 × 10 6) 2.438 × 10
2
20 × ( 20 ) 7) 1.676 × 10
–4
8) 1.537 × 10
–9
9) 3.5 × 10
8
21) a) 58.88 b) ------------------------- c) 80
100
–5 10 4
( 20 ) × 2
2 10) 1.25 × 10 11) 1.37 × 10 12) 8.123 × 10
22) a) 246.9 b) ---------------------- c) 400
0.6 × 3 13) 1.316 × 10
8
14) 2.400 × 10
–4
15) 1.207 × 10
4
2
100 × ( 0.3 ) 16) 2.093 × 10
2
17) 9.928 × 10
5
18) 1.963 × 10
– 14
23) a) 0.5510 b) ------------------------------ c) 0.4
0.8 × 30 2 –3
2 19) 2 × 10 20) 3 × 10
( 0.3 ) × 30
24) a) 0.7783 b) --------------------------- c) 0.8
0.5 × 7
Exercise 4
40 × 7 12
25) a) 13.36 b) ---------------
20
c) 14 1) a) 9.4608 × 10 km b) 8.456 years
21 19
10 × 20 2) a) 5.976 × 10 and 7.35 × 10 b) 81 times
26) a) 6.883 b) ------------------ c) 4 30
50 3) a) 1,839 b) 1.098 × 10 electrons
Higher Page 82

3. Rational & Irrational Numbers 6 2 3

Exercise 1 14) 2 × 3 × 11 15) 2 × 3 × 5 × 7 × 11
23 14 49 Exercise 2
1) ------ 2) ------ 3) ------ 2
1) 2 × 3 × 5 × 11 165
2
2) 2 × 3 × 5
2
3
90 45 90
2 2 2 2
28 2 11 3) 2 × 3 × 5 2 4) 2 × 5 × 7 7
4) ------ 5) ------ 6) ------
45 45 15 2
5) 3 × 5 × 7
2
7
2
6) 3 × 5 × 7
2
5
7 4 5 3 2 2 2
7) --------- 8) --------- 9) ------ 7) 2 × 3 × 5 6 8) 2 × 3 × 11 11
900 225 33
3 2 4 2
32 53 4
9) 2 × 3 × 5 6 10) 2 × 3 × 13 13
10) ------ 11) ------ 12) ------ 2 2 8
99 99 55 11) 2 × 3 × 5 × 11 22 12) 2 × 3 × 5 15
Exercise 2 2 2 2 4 2
13) 2 × 3 × 5 × 7 5 14) 2 × 3 × 5 × 7 15
1, 2, 7, 9, 10, 12, 13, 15 2 4
Exercise 3 15) 2 × 3 × 5 × 7 35
2, 3, 4, 5, 6, 8, 9, 14, 15, 16 Exercise 3
Exercise 4 1) 15 2) 105 3) 27
1, 3, 4, 5, 6, 8, 11 4) 13 5) 55 6) 11
Exercise 5 7) 143 8) 75 9) 115
c, e 10) 231 11) 77 12) 33
13) 735 14) 4725 15) 1155
4. Surds Exercise 4
Exercise 1 1) a) 21 b) 2.1cm × 2.1cm c) 1260
1) 2 2 2) 2 3 3) 2 6 4) 2 7 2) a) 15 b) 15
3) a) 18 b) 180 boxes c) 55 and 42
5) 6 3 6) 2 10 7) 5 2 8) 3 2
9) 4 3 10) 4 2 11) 2 5 12) 5 5 6. Percentages
13) 10 2 14) 6 6 15) 8 3 16) 8 5 1) £2000
Exercise 2 2) 15%
1) 2 2) 3 3) 2 4) 3 2 3) a) £282 b) £24 c) £850
4) a) £97.50 b) £21.45
5) 2 7 6) 4 2 7) 3 3 8) 4 3
5) a) £24,725 b) 10.4%
9) 7 2 10) 10 2 11) 10 3 12) 10 5 6) 61,252
13) 14 5 14) 13 3 15) 7 7 16) 3 21 7) 20.5%
Exercise 3 8) £5222.25
1) 3 2 2) 4 3 3) 5 2 4) 3 2 9) a) 800ml b) Yes - 20% increase in
cola but a 33% increase in price.
5) 2 6) 3 7) 5 8) 2 2
10) a) 39% b) 4096
9) 5 10) 5 11) 2 7 12) 7 5 11) 57%
Exercise 4 12) 6 years
2
1) 2 2) ------- 3) 3 3 4) 8 2 13) 546,250
2
14) 6 years
5) 8 3 6) 7 7) 7 5 8) 3
15) A by £315
9) 4 7 10) 2 5 11) 9 5 12) 18 2 16) £1116.25
Exercise 5
1) 3 2 2) 5 2 3) 6 4) 6 2 7. Conversion Graphs
1) a) 2,880 b) 62,000 c) £49
5) 14 6) 24 7) 16 6 8) 60 2
2) a) £17.20 b) 1m 60cm
9) 48 10) 20 5 11) 42 6 12) 112 6 3) b) 42cm c) £27

5. Prime Factors 8. Fractions

Exercise 1 Exercise 1
2 2 2 2 2 2
1) 2 × 3 × 5 2) 2 × 3 × 5 3) 2 × 3 × 5 × 7 3 2
1) --- 2) 1 3) 1 ---
2
4) 2 × 5 × 7
2 3
5) 2 × 3 × 11
2 3
6) 3 × 5 × 7 5 5
3 2 2 3 3 8 15 1
7) 2 × 5 × 7 8) 3 × 5 × 11 9) 2 × 3 × 7 4) ------
15
5) ------
56
6) ------
12
2 4
10) 2 × 3 × 5 × 11 × 13 11) 2 × 3 × 5 × 11 4 18 13
7) 1 ------ 8) ------ 9) 1 ------
2
12) 2 × 3 × 5 × 7 × 13
5
13) 2 × 3 × 13
2 15 35 36
Higher Page 83
3 23 22
10) 1 ------ 11) 1 ------ 12) 3 ------ 3 4 2 5
20 24 35 9) a 10) b 11) y 12) y
3a 6a 6 18 28 12
13) ------ 14) ------ 15) a 13) a 14) c 15) x 16) x
7 5
3 3 7 6 4 8 7 7
8x 12x 3b 17) x y 18) a b 19) x y 20) a b
16) ------ 17) --------- 18) ------ 2 4 6 8
15 35 10 21) 16x 22) 9x 23) 16y 24) 16a
x 29x 67x 2 3 3 3
19) --- 20) --------- 21) --------- 25) 3a 26) 18x 27) 4y 28) 4a
8 70 90
2 6 2
9a 37x 117x 29) 4a 30) 6b 31) 4a b 32) 3xy
22) ------ 23) --------- 24) ------------ 4 8 4 8
4 14 44 33) x 34) 6a 35) 14y 36) 3x
Exercise 2 Exercise 2
5 1 15 0 –2 –1 1
1) --- 2) --- 3) ------ 1) x or 1 2) a 3) 2y 4) --------2
x a x 2b
5 4 11 3
4) ------ 5) --- 6) --------- 1 –1 6 12 6a
2x x 20x 5) --- or y 6) ----3- 7) -----3- 8) --------
y x a b
13 9 a+b 3 1 2
7) ------ 8) --------- 9) ------------ 9) x 10) ----3- 11) 6x 12) 24
6x 20a ab
y 1
---
2y – x 3y + 5x b+5 2 –1
10) --------------- 11) ------------------ 12) ------------ 13) a 14) 1 15) b 16) y
xy xy 5b
2 1 3 1
--- --- ---
2b + 15 4+x 16 – x 17) 1 18) y 4 19) a 4 20) b 2
13) ------------------ 14) ------------ 15) -----------------
5b 5 4x 3 1
--- ---
2
x – 12 21) x 2 22) a 2 23) 8b
16) -----------------
4x 3
---
2 3
Exercise 3 24) 5y or 5y
3a + 11 5x – 3 Exercise 3
1) -------------------- 2) -------------------- 1 1
4(a + 1) 8( x + 1) 1) 5 2) ---
5
3) 2 4) ---
3
3x – 23 7x – 3
3) ----------------------- 4) --------------- 5) 4 6) 8 7) 8 8) 27
10 ( x – 1 ) 12
9) 27 10) 16 11) 8 12) 125
3a – 4 ( x + 1 ) 8b + 15a
5) -------------------------------- 6) ---------------------- 1 1 5 25
12 20 13) --------- 14) ------ 15) --- 16) ------
125 32 2 4
53x + 6 33a + 12
7) ------------------ 8) --------------------- Exercise 4
18 20
1) x = 2 2) x = 2 3) x = 4 4) x = 3
41x + 32 2–x
9) --------------------- 10) -----------
24 6 5) x = 1--- 6) x = – 1--- 7) x = 2 8) x = 3
14y + 15 61a + 20 5 2
11) --------------------- 12) ---------------------
30 28 9) x = 5 10) x = 81 11) 16 12) x = – 1---
2
9( x – 3) 18x + 1
13) --------------------------------- 14) ------------------------------------- 10. Simplifying
( x – 1)( x – 4) ( x + 2 ) ( 2x – 1 )
Exercise 1
14x + 1 15 + 2xy
15) ----------------------------------------- 16) --------------------- 1) 4x 2) – 2y 3) – 5y
( 3x + 2 ) ( 4x + 1 ) 6x
4) – 13x 5) 9x + 9y 6) 3x – 2y
19x 7x
17) --------- 18) ------ 7) 5y – 10x 8) – 4x – 10y 9) 3b – 4a – 4ab
5a 3a
2 2
1 1 10) 12b – 11a + 3ab 11) 2x + 3x 12) y – 4y
19) ------ ( x – 9 ) 20) ------ ( 9x + 73 ) 2 2 2
30 40 13) 4y 14) 3y – 10x 15) 3x + 4y
1 7x + 9 16) 10x – 3y 17) 7x + 4y 18) 4x – 6y
21) ------ ( 36a – 57 ) 22) ---------------
35 4 19) 6y – 2x 20) 8y – 2x 21) 5x + y
a + 30 3 ( x + 11 ) 22) 7x + 10y 23) 8x – 14y 24) –y
23) --------------- 24) -----------------------
4 4 2
2 25) 3x + 3 26) – 5x – 9y 27) 2x – 24x
x +2 13a + 1 17x – 3
25) -------------- 26) ------------------ 27) ------------------ 2 2
x 3 4 28) 18y – 14x – 5xy 29) 8y – 6xy – 2x
2 2 7x
30) 4xy + 12x + 9y 31) ------
9. Indices 12
2
x y y
Exercise 1 32) --- 33) --- 34) -----
6 6 6
7 11 7 12
1) x 2) y 3) a 4) b Exercise 2
2 7 5 11 2 2
5) 12x 6) 5a 7) 24y 8) 35x 1) x – 2x – 3 2) x – 2x – 8
Higher Page 84

2 2
3) 2x – 11x – 21 4) 6x + 11x – 10 12. Number Sequences
2
5) 6x – 17x – 14
2
6) 6x – 7x – 20 Exercise 1
2 2 1) 20, 23 2) 25, 29 3) 25, 32
7) 35x – x – 6 8) 12x + 16x – 16 4) 17, 23 5) 44, 52 6) – 1, – 8
2 2 2
9) 8x – 4x – 4 10) 6x + 13xy + 6y 7) – 2, – 4 8) – 21, – 34 9) – 3, – 8
2
11) 4x + 13xy + 3y
2
12) 2x – 14x + 24
2 10) 128, 256 11) – 22, – 29 12) – 21, – 27
2 2 13) 29, 35 14) – 6, – 10 15) 127, 255
13) x + 2x + 1 14) 9x + 12x + 4 16) 49, 64 17) – 5, – 7 18) 343, 512
2 2
15) 25x – 20x + 4 16) 9x + 24x + 16 19) 335, 504 20) 46, 61 21) 64, 128
2
17) 25x – 60x + 36 18) 49x + 28x + 4
2 22) 146, 191 23) 8, 13 24) 86, 139
2 2 25) 28, 36 26) 49, 60
19) 9x – 12x + 4 20) 25x + 70x + 49 Exercise 2
2 2 2
21) 16x + 48x + 36 22) x + 2xy + y 1) 22, 25 3n + 1 2) 27, 32 5n – 8
2
23) 4x + 12xy + 9y
2
24) 16x – 16xy + 4y
2 2 3) – 26, – 30 2 – 4n 4) – 22, – 27 13 – 5n
5) – 9, – 17 47 – 8n 6) 36, 41 5n + 1
2
11. Re-arranging Formulae 7) 23, 30 7n – 26 8) 49, 64 n
2 2
Exercise 1 9) 51, 66 n +2 10) 42, 57 n –7
v–u d+c
1) v – at 2) ----------- 3) ------------ 11) 36, 49 (n – 1)
2
12) 42, 56 n –n
2
t 3
14 16 2n 7 8 n
4) c – pd 5) 7y – x
a–c
6) ----------- 13) ------, ------ --------------- 14) ---, --- ------------
3 15 17 2n + 1 8 9 n+1
2
3w – u 49 64 n 21 24 3n
7) ----------------
4
8) 4x – 5y 9) x 15) ------, ------ ------------ 16) ------, --------- -----------------
2
10 11 n+3 97 127 2n – 1
8y
10) ------ 11) 2 ( p – 3b ) 12) w u
---- – --- or --12- ( w – --14- u ) Exercise 3
3 2 8 1)a) n + 1
a+b 2r – q b) ( n + 1 ) ( y + 1 ) = ny + n + y + 1 = ( ny + n + y ) + 1
13) ------------ 14) -------------- 15) --2x- – y
c 3p which is a term in the sequence
--- ( --- – 3b )
1 a
16) 4 3 17) 2x – z 18) 9a – 2b 2) a) ( 2n + 1 )
19) y – 12x
3v – 15w
20) ---------------------- 21) 1411------ y b) ( 2n + 1 ) ( 2y + 1 ) = 4ny + 2n + 2y + 1
2 = 2 ( 2ny + n + y ) + 1 which is a term.
3
22) – -----
19
-b
2 2 2 2
Exercise 2 3) a) n b) n y = ( ny ) which is a term.
1) ac 2) --y 4) a) 4n + 1 b) 4 ( 4ny + n + y ) + 1 is a term.
x
bc 3 13. Substitution
3) ---------- 4) ---
2 v Exercise 1
5) y = – --x- 6) y = – 8x 1) 38 2) 0 3) – 16 4) – 8
6 5) 50 6) 116 7) 52 8) 105
7) x = --y- 8) b = – 27
------ c 9) 2116 10) 156 11) –100 12) –589
6 7 Exercise 2
ab 2yz 1) a) 22 b) x = 5
9) x = -----------
- 10) x = -------------------------
-
b+a 3 ( 2z + 3y ) 2) a) – 27 b) b = 20
2 2
11) x =  ------------------- 12) x =  ---------------
6ab 4y 3) a) – 39 b) x = – 12
3b + 2a 3 + by 4) a) 700 cm b) R = 26
3
6x xy 5) a) 23.565cm b) 7cm
13) z = -----------------
- 14) z = ----------------------
-
3xy – 4 3 ( 2x – y ) 6) a) £71.25 b) £476.19
b 2 7) a) – 29° b) F = 14°
15) a = --------------
- 16) a = --------------------------
xb – 1 3 ( 4y – 3b ) 8) a) 55 minutes b) 1 hour 19 minutes
y c) Decrease the numbers that D and S are divided
17) b = 2c 18) x = ---
by.
3
z b 14. Factorising
19) y = --------------
- 20) c = ---------------
Exercise 1
3x – 2 4a + 3
x–4 1) 4 ( x + 2 ) 2) 3 ( 2y – 3 ) 3) 7 ( b – 2a )
21) x = 1 – 3y 22) y = -----------
3 4) x ( y – 1 ) 5) x ( 3 + y ) 6) 2y ( 2 + 5x )
2
7) 2 ( 3x + 1 ) 8) x ( 5x – 1 ) 9) 3x ( 3x – 1 )
10) ab ( a + b ) 11) ab ( 4 – a ) 12) 2ab ( 4 + 3b )
Higher Page 85

2 17. Direct and Inverse Proportion

13) a ( 1 + b – a ) 14) a ( 3b – c + a ) 15) y ( 5x – 4y – 3x ) 1) a) y = --23- x b) y = 5 --13- c) x = 22 --12-
2
x y x 2) v = 81
16) ----- ( 2 – x ) 17) --- ( 2y – x ) 18) --- ( 5x – 4 )
4 6 6 3) a) y = --23- x b) y = --43- c) x = 81
Exercise 2
1) ( m – n ) ( m + n ) 2) ( a – 2 ) ( a + 2 ) 4) a = 256
3) ( xy – z ) ( xy + z ) 4) ( ab – 3 ) ( ab + 3 ) 5) a) y = ----3- b) y = --13-
2 c) x = 6
x
3
5) ( xy – 2 ) ( xy + 2 ) 6) ( vw – 5 ) ( vw + 5 ) 6) y = 4--
-

7) ( ab – 3c ) ( ab + 3c ) 8) ( 5a – 3b ) ( 5a + 3b ) 7) a) y = 5 3 x b) i) y = 20 ii) x = 125
9) ( b – 1 ) ( b + 1 ) 10) 2 ( a – 5 ) ( a + 5 ) 8)
11) 2 ( 2a – 5 ) ( 2a + 5 ) 12) 3 ( 2x – 3y ) ( 2x + 3y ) x 1 2 3 4
13) x ( y – 2x ) ( y + 2x ) 14) 2x ( y – 2x ) ( y + 2x )
2 2 2 y 3 0.75 0.3̇ 0.1875
15) x ( 2y – 3 ) ( 2y + 3 ) 16) ( x – y ) ( x + y ) ( x + y )
2 2 2
17) ( 2x – 3y ) ( 2x + 3y ) ( 4x + 9y ) 9) a = ---
3
2 2
18) 3 ( a – 2b ) ( a + 2b ) 10)
Exercise 3 x 0.125 1 8 64
1) ( x + 3 ) ( x + 1 ) 2) ( x + 2 ) ( x + 2 ) y 0.25 0.5 1 2
3) ( x + 7 ) ( x + 1 ) 4) ( x + 5 ) ( x + 2 )
5) ( x + 4 ) ( x + 3 ) 6) ( x + 6 ) ( x + 5 ) 11) a) n = 3 b) y = 0.125
7) ( x – 1 ) ( x + 3 ) 8) ( x – 3 ) ( x + 1 ) 12)
9) ( x + 5 ) ( x – 1 ) 10) ( x + 2 ) ( x – 4 ) x 0.5 0.75 1 1.5
11) ( x – 5 ) ( x + 3 ) 12) ( x + 3 ) ( x – 4 )
y 6 2.6̇ 1.5 0.6̇
13) ( x – 6 ) ( x – 4 ) 14) ( x – 3 ) ( x – 5 )
15) ( x – 4 ) ( x – 7 )
Exercise 4 18. Solving Equations 1
1) ( 2x + 1 ) ( x + 1 ) 2) ( 2x + 1 ) ( x + 4 ) Exercise 1
3) ( 2x + 1 ) ( x + 3 ) 4) ( 2x + 2 ) ( x + 3 ) 1) x = 9 2) x = 12 3) x = 6
5) ( 2x – 3 ) ( x + 2 ) 6) ( 3x + 2 ) ( x – 3 ) 4) x = 4 5) x = 120 6) x = 150
7) ( 2x – 1 ) ( x – 4 ) 8) ( 3x – 1 ) ( x – 3 ) 7) x = 12 8) x = 8 9) x = 1
9) ( 3x – 2 ) ( x – 4 ) 10) ( 3x + 7 ) ( x – 2 ) 10) x = 7--5- 11) x = 1 12) x = 2
11) ( 3x + 4 ) ( x + 5 ) 12) ( 3x – 6 ) ( x – 2 ) 13) x = – 2 14) x = 3 15) x = 5
13) ( 2x + 2 ) ( 2x + 3 ) 14) 2 ( x – 1 ) ( 2x – 3 ) 16) x = 8 17) x = 2 18) x = 7
15) ( 4x + 1 ) ( x + 3 ) 16) ( x + 5 ) ( 4x + 1 ) 19) x = 24 20) x = 32 21) x = 11
17) ( 5x – 2 ) ( x + 3 ) 18) ( 3x + 2 ) ( 2x – 3 ) 22) x = 6 23) x = – 11 24) x = 4
---
5
19) ( 3x + 1 ) ( 2x + 1 ) 20) ( 3x + 2 ) ( 3x + 2 ) 25) x = – 1--3- 26) x = 1
---
2
21) ( x + 1 ) ( 8x + 3 ) 22) ( x – 5 ) ( 4x – 3 ) Exercise 2
23) ( 5x + 2 ) ( x – 3 ) 24) ( 4x – 3 ) ( 3x – 1 ) 1) x = 5 or – 5 2) x = 9 or – 9
3) 6 or – 6 4) 3 or – 3
15. Trial and Improvement 5) – 2 or 3 6) 6 or – 5
Exercise 1 7) – 4--3- or 1--2- 8) 0 or 2--3-
1) 3.8 2) 4.3 3) 4.7
9) 0 or – 3--4- 10) 0 or 4
4) 5.5 5) 4.3 6) 6.5
7) 5.2 8) 3.8 9) 2.4 11) 0 or – 3--2- 12) 0 or --23-
10) 2.1 11) 4.8 12) 2.9 13) 0 or 1--4- 14) – 2 or 3
13) 3.5 14) 1.9 15) 1.9 15) 32--- or – 1 16) – 1 or 13---
Exercise 2 17) 5 or – 2 18) 2 or – 4
1) 6.7 2) 8.2 3) 9.4 19) – 5--4- or --32- 20) 2.5 or – 2.5
4) 3.3 5) 3.5 6) 1.5
21) – 2 22) – 1--3-
7) 2.4 8) 3.3
19. Solving Equations 2
16. Iteration
Exercise 2
1) 3.73 2) 5.30 3) 3.79
1) 1.16 or – 5.16 2) 2.62 or 0.38
4) 8.77 5) 0.62 6) – 6.6
3) 4.65 or – 0.65 4) 0.61 or – 6.61
7) a) (i) –3.50, –2.80, –3.09, –2.96 (ii) –3
5) 0.62 or – 1.62 6) 2.20 or –0.45
(iii) 9+12–21 = 0
7) –0.73 or 0.23 8) 0.74 or – 0.94
8) a) (i) 7.4, 6.946, 7.008 (ii) 7.0 b) 49–42–7 = 0
9) 1.08 or – 2.33 10) 1.72 or – 0.39
Higher Page 86

11) 0.79 or – 2.12 12) 0.19 or – 2.69 23. Straight Line Graphs
13) 1 or 0.75 14) 0.23 or – 0.73 Exercise 1
15) 0.18 or – 1.61 16) 3.24 or 0.12 1) 1 2) 3 3) – 5 4) – 4
17) 0.78 or – 0.92 18) 1.92 or 0.46 5) 2 6) 2 7) – 5--3- 8) – 1--4-
19) 0.37 or – 5.37 20) 0.52 or – 9.52
21) 1.44 or – 0.69 22) 2.77 or – 1.27 9) 2 10) 2 11) – 1 12) – 1
4 1
23) 2.30 or –1.30 24) 1.81 or – 1.47 13) – 1 14) – 1 15) ---
3
16) ---
3
25) 2.78 or 0.72 26) 10.18 or – 1.18 Exercise 2
Exercise 2 1) y = x – 3 2) y = 2x – 10 3) y = 3x
1) a = 5 k = 25 2) a = 2 k = 4 4) y = – x + 2 5) y = – 3x + 4 6) y = 4x – 14
3) a = 3 k = 9 4) a = 7 k = 49 Exercise 3
5) a = 1.5 k = 2.25 6) a = 2 k = 4 1) b) m = 2 , c = – 6 c) 124 d) 120
7) a = 5 k = 25 8) a = 8 k = 64 2) b) m = 5 , c = 20 c) 95 d) 17
9) a = 6 k = 36 10) a = 4 k = 16 3) a) C = 10H + 40 b) £90 c) £105 d) 6 hours
20. Problems Involving Equations 24. Inequalities
1) a) x = 34° 34°, 68°, 78° Exercise 1
b) x = 30° 90°, 40°, 50° 1) x > 6 2) x > 2 3) x > 8 4) x > – 3
2) a) 538.46 mph 5) x > – 3 6) x > 2 7) x > 6 8) x > 4
x 538.46y
b) ---------------- hours c) ------------------- miles 9) x > 5 10) x < 1 11) x ≤ 3 12) x ≤ 5
538.46 60
2 13) x ≥ 8 14) x < 3 15) x < 5 16) x > 2 --25-
3) a) x = 4.5 cm b) 30.25cm 17) x ≤ 11 18) x ≤ – 7 --23- 19) x ≤ – 1 20) x ≥ – 26 --12-
4) a) x – 1 and x + 1 b) x = 52 21) x ≤ – 2
3y
5) a) i) x + --y- ii) ------ Exercise 2
4 4 1) x x x
b) y = 8x c) 18,000 bottles –4 –3 –2 –1 0 1 2 3 4
6) a) 250 – x b) x = 110 c) 140 2) x x x x x
7) a) x – 6 and x – 21 b) 3x – 27 c) x = 40 0 1 2 3 4 5 6 7 8
34, 40 and 19 3) x x
–8 –7 –6 –5 –4 –3 –2 –1 0
21. Simultaneous Equations 4)
1) 4,1 2) 5,3 3) 1,–1 – 3 1--2-
4) 3,1 5) 3,–2 6) 3,5 –4 –3 –2 –1 0 1 2 3 4
7) 4,1 8) 2,3 9) 5,–3 5)
10) 5,4 11) 2,–1 12) 3,–2 3 1--3-
13) 5,6 14) 9,–1 15) 2.5,1 –1 0 1 2 3 4 5 6 7
16) 4,–2 6) 2
17) a) 2x + 2y = 14 x + 4y = 13.60 b) x = 4.80 7) 1
c) Adults £4.80 Children £2.20 8) a) 20x < 10x + 70
18) a) x + y = 39 x–y = 9 b) 24,15 b) x < 7 Company A is cheaper when the number
19) 25.8 by 2.7 20) 90g and 120g of hours is less than 7
21) £1.72 and £1.02 22) 17 and 6 Company A and B offer the same price when the
23) 5, 3 24) £22, £12 number of hours is 7
25) 11, 7 26) 28p and 22p Company B is cheaper when the number of
hours is greater than 7
22. Problems Involving Quadratic Equations
1) a) x = 7 b) x = 6.5 cm 25. Linear Inequalities
2
2) a) x = 10 b) 105cm 1) (4,6) (5,5) (5,6) (6,4) (6,5) (6,6) (6,7)
10
3) a) 20
------ + ----------- = 4 x = 2.5 or 10 2) (0,6) (1,4) (1,5) (1,6) (2,6)
x x–5 3) (2,4) (3,3)
b) 10 kph and 5 kph 4) (3,4) (3,5)
2 2
4) a) x and 8x b) x + 8x = 105 5) (3,4) (3,5) (2,5)
2
so x + 8x – 105 = 0 c) 64 cm 6) (2,0) (3,0) (4,0) ( 3, – 1 )
5) a) 2x + 4 and x + 4 d) x = 4 m 8m by 4m. 7) ( 1, – 1 ) ( 2, – 1 )
6) a) ( x + 4 ) ( x – 1 ) and 3x ( x – 2 ) 8) (1,2) (1,3) (2,3)
b) x = 4 c) 8 km/hour 9) (0,2) ( – 1, 3 ) (0,3) (1,3) (0,4) (0,5)
Higher Page 87

26. Linear Programming 31. Growth and Decay

1) a) y ≥ 7 y ≤ 2x x + 2y ≤ 20 c) 360kg d) 6,7 and 1) a)
4,8 0 1 2 3 4 5 6
2) a) x + y ≤ 50 y ≤ 2x x ≤ 2y 1 0.667 0.444 0.296 0.198 0.132 0.088
c) i) 33 --13- tins of red, 16 --23- tins of orange
ii) 33 --13- tins of yellow, 16 --23- tins of orange c) i) 0.85 ii) 0.24
d) 8.4 litres 2) a)
3) a) i) 2x + y ≤ 100 ii) 5x + 8y ≥ 400 iii) y ≤ 2x 0 1 2 3 4 5 6 7
c) 25 large, 50 small. 10 11 12.1 13.31 14.64 16.11 17.72 19.49
4) a) i) x + y < 15 ii) x + y > 10 iii) y > x iv) y < 2x
c) (4,7) (5,6) (5,7) (5,8) (6,7) (6,8) (5,9) c) 16.9 million d) 4 --14- years
3) a)
27. Recognising Graphs 0 0.5 1 1.5 2 2.5 3 3.5 4
1) c 2) a 3) c 4) a 100 57.7 33.3 19.3 11.1 6.4 3.7 2.1 1.2
c) i) 0.63 days (15 hours 7 minutes)
28. Graphs 1
ii) 8.44% iii) 0.26 days
1) a)
5 0 –3 –4 –3 0 5 32. Distance - Time Diagrams
d) – 2.8 and 1.8 1) a) 3.2 miles per hour b) 1.61
2 c) 1.7 miles per hour d) 2.6 miles per hour
e) x + x – 5 = 0 –1
2) a) 0.8 b) 2.3ms
2) a) –1
3) a) 8.8ms b) Speed is increasing (accelerating)
– 8.625 –2 3 2 1 6 12.625 c) Speed is approximately constant
d) Speed is decreasing
b) x = – 2.25 , 0 and 2.25 –1
4) a) 30 metres b) 6ms c) 12m d) 37.5m
3) a)
13 3 –3 –5 –3 3 13 33. Velocity - Time Diagrams
–2
c) x = – 0.6 and 2.6 1) a) Area = 363 b) 363 metres c) 2.4ms
–2
d) x = – 1.1 and 3.1 2) a) 2880 metres b) 0.5ms
4) a) 3) 995 metres
4) a)
–1 0 1 2 3 4 5
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
–5 –4 –3 –2 –1 0 1 0 3 7 12 18 25 33 42 52 63 75
5 0 –3 –4 –3 0 5 –2
b) 40.75 metres c) 17ms
2
d) – 2.55 and 1.6 x +x–4 = 0
34. Angles and Triangles
29. Graphs 2 1) 49°, 131°, 49° 2) 79°, 84°, 17°
1) a) 3) 120°, 60°, 30° 4) 42°, 45°, 93°
5) 60°, 60°, 60°, 30° 6) 57°, 57°, 33°
17 10 8.3̇ 8 8.2 8.6̇ 10 13.3̇ 15.1 7) 45°, 27°, 45° 8) 44°, 69°, 67°
d) 0.6 and 6.4 e) 0.8 and 4.0 f) 1.5 9) 150°, 150°, 30° 10) 40°, 70°, 30°
2) a)
35. Regular Polygons
– 24 – 14 –6 0 4 6 6 4 0 –6
1) a) 135° b) 67.5° c) 45° 2) 360°
c) x = – 0.25 and 3.25 3) 360° minus the two interior angles of the shapes
e) x = – 1 and 4 4) x = 108° (Interior angle of a regular pentagon)
f) y = – 4 . x = – 1.7 or 4.7 y = 36° ( 180° – 2 × 72° )
3) a) x = – 3.25 and 1 b) x = – 2.2 and 0.7 5) Regular triangle with side 3x
6) a) 128.6°, 77.1° b) ∠HCB = ∠HDA
30. Graphs 3 7) 24 sides
1) c) c = 1.5
2) b) a = 3 , b = 50 c) 197 36. Congruent Triangles
3) b) p = 3 , q = 2 c) 14.3 1) All 2) a and b 3) a and c 4) b and c
4) b) a = 1.5 , b = – 0.5 c) 0.56 5) Two sides and the included angle are equal on both
5) b) a = 3 , b = 2 c) 6.2 triangles.
Higher Page 88

6) Two angles and a side are equal on both triangles. 43. Bearings
7) (i) True - rest are false. 1) S 28° W ( 208° ) 2) N 73° E ( 073° )
37. Geometry of a Circle 1 3) a) N 30° E ( 030° ) b) S 30° E ( 150° )
1) 48°, 42°, 42° 2) 90°, 33°, 57° c) N 30° W ( 330° )
3) 140°, 80°, 50° 4) 96°, 84°, 72° 4) a) N 6° E ( 006° ) b) N 55° W ( 305° )
5) 12°, 57°, 78° 6) 27.5°, 125°, 97.5° c) S 55° E ( 125° )
5) a) N 60° E ( 060° ) b) Due South ( 180° )
38. Geometry of a Circle 2 c) S 60° W ( 240° ) d) S 60° W ( 240° )
1) 94°, 43°, 52°, 42° 2) 47°, 88°, 9° 6) a) N 51° W ( 309° ) b) N 4° E ( 004° )
3) 50°, 130°, 120°, 70° 4) 124°, 56°, 34° c) N 84° W ( 276° )
5) 124°, 68°, 56° 6) 62°, 124°, 56°, 28°
44. Construction
39. Vectors 1 6) 21 metres 7) 4.5 cm 8) 1630 m
1) b + c, b + c + d, c + d 2)  5 3) AD
8
4) a) 4c – 4a b) 2c – 2a c) 4a + 2c – 2a = 2c + 2a 45. Loci
d) 2c e) PX is parallel to OC and half its length 1)
5) a) 9b – 6a b) 3b – 2a c) 6a + 3b – 2a = 4a + 3b
6) a) a + b b) 12--- (a + b) c) 12--- (b – a)
7) –a, –b, 2a, –a, b–a, 2a–2b
8) a) AB is parallel to CD, CD is twice as long as AB
b) Parallel. One is 23--- the length of the other 2)
9) a) 2b – 2a b) b – a c) XY is parallel to AB
because AB = 2 × XY 3)81metres
10) AC = a + b BD = b – a 4)

40. Vectors 2
1) a) 2--3- b b) 1--3- b – 1--2- a c) b + 1--2- a
2) a) a + b b) 13--- (a + b) c) 13--- b
d) --23- a PQ is parallel to OC and --23- of its length 5) 700 metres
3) a) (a + b) b) 1--6- (a + b) c) b – a d) 1--6- (b – a)
46. Transformations 1
e) 2--3- a f) They are parallel and XY is 2--3- the length
1) a) (–1, –2) (–3,–2) (–4,–5)
of CB
b) (1,–2) (3,–2) (4,–5)
4) a) 1--2- a b) b – 1--2- a c) a + b d) 1--3- b – 1--6- a c) Rotation of 180˚ about origin (0,0)
e) 1--3- a + 1--3- b f) 1--3- 2) a) (2,2) (2,4) (4,2)
5) a) 12b + 6a b) 5a – 2b c) 3b + 6a b) (2,–2) (2,–4) (4,–2)
c) (–1,1) (–2,1) (–1,2)
41. Similar Shapes d) An enlargement with a scale factor of 2 through
3
1) a) 4:9 b) 8:27 c) 337.5 cm the origin (0,0)
2 3) b) (1,2) (2,2) (2,1)
2) a) 8:19 b) 450 cm
3) 122.88 grams 4) 390.625 grams c) An enlargement with a scale factor of 2 12--- through
3 the origin (0,0)
5) 11.9 cm 6) 8,000 cm
2 3
7) a) 112.5 cm b) 14.81 cm
8) 350 grams 47. Transformations 2
1) a) (–1,–1) (–4,–4) (–3,–1)
42. Similarity b) (–1,3) (–3,3) (–4,0)
1) a) i) 75° ii) Since DC and BE are parallel then c) Rotation of 180˚ (clockwise or anticlockwise)
angle CDE and BEA are equal about the point (0,2)
b) AE = 6 cm 2) a) Rotation of 180˚ about the origin (0,0)
2) a) i) 39° ii) Angle DCE = 116° (vertically b) Translation  0  c) (–1,4) (–1,2) (1,2)
–6
opposite) Angle DEC = 180 – (116 + 25) = 39° d) Rotation of 180˚ about (–1,3)
(3 angles of a triangle add up to 180° ) 3) a) (2,1) (5,1) (4,–2) b) (0,1) (0,–2) (–3,–1)
b) 7.5 cm c) Reflection in the line y = 3 – x
3) a) 7.5 cm b) 7.5cm 4) a) 6 cm 1.5cm b) 1:16
Higher Page 89

48. Transformations of Graphs b) Reflection in the x axis (y = 0)

1) b) iii c) v d) iv e) ii f) i  
d)  0 – 1 
2) b) Translation  1 c) Translation  – 2
(5,3)
0 0  1 0 

d) y dimensions x 4 e) Translation of  0  50. Area and Perimeter

–1
f) Translation of  0 1) 60 cm
2
2) 35.3475 cm
2
1
3 2 2
3) a) – 2x – 3x 3) a) 9.426 cm 3.142 cm
2
b) y b) 175.952 cm 29.3 cm (to 1 d.p.)
2
x c) 475.2 cm (to 1 d.p.) 57.6 cm (to 1 d.p.)
-1 0 2
d) 477.5 cm (to 1 d.p.) 57.0 cm (to 1 d.p.)
2
4) a) 42.55 m (to 2 d.p.) b) 27.02 m (to 2 d.p.)
2
5) a) 333.8 cm (to 2 d.p.) b) 74.2 cm (to 1 d.p.)
–1 2 2
6) a) 62.84 m b) 14 tins 7) 402.2 m (to 1 d.p.)
2
8) 509,500,000 km (to 4 sig. figures)
3 2
c) – 2x + 3x . Reflection in the y axis.
4) a) y 51. Volume
1 3
1) 960 cm 2) 7.5 cm
x 3
0 1 2 3 3) a) 752.8 cm (to 4 sig. figures)
–1 b) 564.6 grammes (to 4 sig. figures)
3
–2 4) a) 134.06 cm b) 18.1 kg
5) 4.63 cm (to 2 d.p.) 6) 330,000 litres
7) 16 litres 8) 5893 spheres
3
b) y
9) a) 98 cm b) 20
4 3
10) a) 5832 cm b) 52.4%
3 11) 3.4 cm (to 1 d.p.)
3
2 12) 255.55 cm (to 2 d.p.) 13) 156 ml
1
x 52. Ratios and Scales
0 1 2 3
1) 120, 150, 180 2) 150, 175
49. Matrix Transformations 3) a) 2:4:1 b) 160, 320, 80
2
1) a) Rotation of 180˚ 4) 450 ml 5) 50 cm, 2.2 m, 500 cm , 40˚
2
b) Enlargement of x 2 through origin (0,0) 6) 100 cm, 80˚, 120 cm , 12.5 m
2 3
c) Reflection in the y axis 7) 150 cm, 7.68 m , 93,750 cm 10 cm table
2 3
d) Reflection in the line y = x 8) a) 50 cm b) 80 m c) 3,750 cm
 1
– ---
1
--- 
2)  2 2
 (1,2) 53. Degree of Accuracy
 1
---
2
1
---
2  1) 84, 86 2) a) 418.65 cm
3
b) 403.27 cm
3

3) a) Points are (–1,1) (–2,1) (–1,3) and 3) 368 4) 0.35 cm, 0.05 cm
3 3
(1,–1) (2,–1) (1,–3) 5) a) 9.79 cm b) 8.24 cm c) 15.8%
    6) a) 89.22 litres b) 5.07 litres
b)  – 1 0  c)  – 1 0  7) a) 22 b) 20 c) 440 d) 20
 0 1  0 –1 
  54. Formulae
d)  1 0  Reflection in the x axis (y = 0)
 0 –1  Exercise 1
1) area 2) area 3) area
 
4) a)  – 1 0  b) Rotation of 90˚ anticlockwise 4) none 5) area 6) area
 0 –1  7) length 8) length 9) length
about (0,0) 10) volume 11) volume 12) volume
13) none 14) none 15) length
 
c) Rotation of 90˚ clockwise about (0,0)  0 1  16) length 17) area 18) volume
 –1 0  19) none 20) area 21) volume
5) a) (1,–1) (2,–2) (1,–4) (4,–1)
Higher Page 90

Exercise 2 6) a) 10
1) a) iii b) i 2) a) ii b) v
5

55. Pythagoras Theorem

0
All to 4 significant figures 0˚ 180˚ 360˚

1) a) 7.211 b) 7.483 c) 4.8 d) 8.801 –5

2) 45.5 cm 3) 5.657 cm, 2.828 cm
–10
4) 6.403 cm 5) 2.721 metres 6) 18.81 metres
b) 10

56. Sine, Cosine and Tangent Ratios

5
1) (to 4 sig. figures) a) 15.28 cm b) 6.576 cm
c) 5.503 cm d) 22.58 cm 0
0˚ 180˚
2) a) 41.2˚ b) 46.9˚ c) 42.3˚ d) 42.2˚
3) CD = 17.33 cm AC = 26.69 cm –5

EB = 10.20 cm DEB = 43.3˚ –10

4) a) N24˚E (024˚) b) 5.41 km (to 2 d.p.)
5) a) 7.523 metres (to 4 sig. figures) b) 28˚ 7) Approx 56˚ and 236˚

57. Sine and Cosine Rules 61. Three Dimensional Trigonometry

1) a) 72.97˚ b) 61.09˚ c) 4.74˚ d) 54.79˚ 1) 33.31 m (to 4 sig. figures)
e) 54.80˚ f) 10.54 cm 2) a) 28.28 cm b) 20.20 cm
2) 16.01 cm 3) 115.18˚ and 64.82˚ 3) a) 4.667 m b) 23.2˚ c) HC = 5.077 m
4) 4.05 km 5) 16.08 m 4) a) 3.830 m b) 21.0˚
6) 58.75˚, 71.79˚, 49.46˚ 5) 107.3 m

58. Areas of Triangles 62. Questionnaires

2 2 2
1) 20.99 cm 2) 53.83 cm 3) 22.36 cm The following answers are for guidance only. There
2 2
4) 27.39 cm 5) 131.95 cm are other solutions.
6) The perpendicular height from C to AD is 9 sin 20˚. 1) a) The survey could be biased for the following
The perpendicular height from B to DA produced reasons:
is also 9 sin 20˚. Triangles ACD and ABD have the i) It is done just in the month of July only.
same base ie. AD. Therefore their areas are equal. ii) It is carried out at 12.00 only and not at other
2 2
7) 49.85 cm 8) 2.3 m times of the day
2
9) 237.8 cm 10) 3.4 hectares iii) It is carried out in just one particular place. To
get information about the whole of Britain she
59. Trigonometry Mixed Exercise needs information from other towns.
1) AC = 6.325 cm CD = 5.175cm iv) Town is of one particular type ie historical.
2) 10.91 cm 3) 9.226 m b) The survey could be improved by including the
4) 122.75˚, 114.5˚ 5) 14.62 cm 6) 13.09 m points raised in part a)
c) Are you a tourist ?
Which country do you come from ?
60. Graphs of Sines, Cosines and Tangents What is your gender m/f ?
1) 210˚ and 330˚ 2) Approx 49˚ and 131˚ 2) a) How old are you ?
3) What is your gender, male or female ?
1.0
What other newspapers do you read?
b) i) It only asks those who already read the
0 180˚ 360˚ newspaper. It does not get to potential readers.
3) a) It tells her the age of the cars on the road now.
–1.0
She could, for example, calculate the average age
of cars on the road but not the age at which they
4) Approx 71˚ and 289˚ 5) Approx 76˚ and 256˚ are taken off the road.
b) i) Was your last car sold to someone to use again
or was it scrapped ?
ii) How old was the car when it was scrapped ?
Higher Page 91

63. Sampling 68. Histograms 1

1) a) 14, 12, 14, 12, 12, 4, 2 1) a)
b) Do you watch TV ? Yes/no Time 0– 5– 10– 15– 20– 25– 30– 35– 40– 45–50
What is your gender ? Male/female Freq 7 12 22 28 33 24 14 8 6 3
Which channels do you watch mostly ? 1, 2, 3 etc.
b) mean = 21.83 minutes
Which types of programmes do you mainly
c) 35%
watch ? Plays, sport, news etc.
2) a)
2) a) 10, 14, 10, 4, 2
b) Are you happy with your working conditions ? Mass 40– 50– 60– 70– 80– 90– 100–110
What areas of your working environment need Freq. 5 6 14 12 9 5 4
to be improved? b)
What type of work do you do ?
How old are you ? 15
Gender, male/female. 10
c) Age range, type of work undertaken, gender etc.
3) a) 19, 37, 33, 11 5
b) i) What would you like to see improved in the
centre ? c) 60–70
ii) Which facilities do you use most often?
iii) Would you use other facilities if they were 69. Histograms 2
introduced ? Yes/no 1) a)
iv) What other facilities do you think ought to be
Age 15– 25– 30– 35– 45– 60–70
introduced ? (give a choice)
64. Scatter Diagrams People 18 21 28 32 12 6
1) b) 235 b) mean = 35.45
2) a) 7:30 pm b) 3˚C c) 4:45 pm 2) a)
3) a) 15.5 tonnes (approximately) b) 4.8 minutes Speed 20 < x ≤ 30 30 < x ≤ 40 40 < x ≤ 60 60 < x ≤ 70 70 < x ≤ 100
No.of 36 80 208 112 60
65. Pie Charts cars
1) Business stock £5972 Employee costs £14722 b) 40 < x ≤ 60 is the modal range
Premises £7361 General administration £10278
Advertising £3889 Other costs £7778 70. Histograms 3
2) Angles 115.2˚, 43.65˚, 37.35˚, 59.4˚, 83.25˚, 21.15˚ 1) a)
3) a) 1008 b) 140˚ d) 126
Height in cm No. of plants Frequency density
66. Flow Charts 1 90 < x ≤ 100 5 0.5
1) n S 100 < x ≤ 105 4 0.8
40 1603
105 < x ≤ 115 16 1.6
41 1684
42 1767 115 < x ≤ 120 9 1.8
43 1852 120 < x ≤ 130 12 1.2
44 1939
130 < x ≤ 140 4 0.4
45 2028
There are 44 numbers less than 2000 b)
2) W. Jones - Merit 0.2
J. Connah - Pass
0.15
C. Smith - Distinction
H. Patel - Pass 0.1

0.05
67. Flow Charts 2
1) 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377,
610, 987, 1597, 2584, 4181. 90 110 130
Each number is obtained by adding together the c) 50%
previous two numbers.
2) M = 8 It calculates the mean of a list of numbers.
Higher Page 92

2) a) 3)
Time Frequency Frequency density Speed Frequency Mid Frequency x
500 < x ≤ 600 4 0.04 value mid value
600 < x ≤ 700 10 0.10 20 ≤ s < 30 6 25 150

700 < x ≤ 750 10 0.20 30 ≤ s < 40 10 35 350

750 < x ≤ 900 18 0.12 40 ≤ s < 50 15 45 675

900 < x ≤ 1100 16 0.08 50 ≤ s < 60 14 55 770

60 ≤ s < 70 6 65 390
b)
0.2 70 ≤ s < 80 6 75 450

0.15 80 ≤ s < 90 3 85 255

Totals 60 3040
0.1
mean = 50.67
0.05
72. Mean, Median and Mode
500 700 900 1100 1) a) 21 pints b) 18.3 pints c) 18 pints
2) a) 4.3 letters b) 3 c) 2
c) 52% d) No - because the sample is very small compared
with the whole book and it is only taken from one
71. Mean part of the book.
1) 3) b) 428.3 kg c) 100 – 300 d) 100 – 300
Wages, £ Frequency Mid Frequency x 4) a) 1.02 – 1.03 litres b) 1.02 – 1.03 litres
value mid value c) 1.025 ml (to 4 sig. figures)
60 - 100 4 80 320
73. Mean and Standard Deviation
100 - 140 19 120 2280
1) a) 5.044, 0.0246
140 - 170 24 155 3720 b) i) The mean is increased by 0.02 kg
170 - 200 11 185 2035 ii) There is no change to the standard deviation
200 - 220 6 210 1260 2) b) i) 155.6 ii) 10.7
3) b) £230.15, 67.48 c) Mean is increased by £10,
mean = £150.23
standard deviation remains the same.
2)
4) b) 41 and 12.54
Weight Frequency Mid Frequency c) The mean age at the badminton club is higher,
value x mid value indicating that the members tend to be older.
50 - 70 5 60 300 The S.D. at the badminton club is lower,
70 - 90 14 80 1120 indicating that a greater proportion of the ages
90 - 100 16 95 1520 are closer to the mean than at the tennis club.
100 - 110 24 105 2520
74. Frequency Polygons
110 - 120 21 115 2415
1) a)
120 - 140 23 130 2990
Weight Frequency Mid value
140 - 160 17 150 2550
3.0 - 3.5 4 3.25
mean = 111.8 cm 3.5 - 4.0 6 3.75
4.0 - 4.5 12 4.25
4.5 - 5.0 5 4.75
5.0 - 5.5 3 5.25
Higher Page 93

2) c) 246 days d) 10 days

Height Frequency Mid value
77. Probability 1
80 - 100 8 90
1) a) 59--- b) 2081------ c) 125
---------
729
d) 80
---------
243
100 - 120 14 110 2) a) 6---
1
b) 4--- 1
c) 56---
120 - 140 26 130
3) a) 0.16 b) 0.48 c) 0.36
140 - 160 34 150 4) a) 0.02 b) 0.02 c) 0.01
160 - 180 18 170 5) a) 0.14 b) 0.06 c) 0.24 d) 0.86
180 - 200 8 190 6) a) 0 b) 59---
200 - 220 4 210
78. Probability 2
1) a) 10-----3- b) 30-----7- c) 7
------ 2) 0.96 3) 0.996625
75. Cumulative Frequency 1 40

1) a) 4) a) 1
---
2
b) 1
---
4
5) a) 8
------
15
b) 8
------
29
6) a) 25--- b) 20-----3-
Length l 9.7 9.8 9.9 10.0 10.1 10.2 7) a) 0.02682 b) 0.045811 c) 0.927369
Cumulative
Frequency 4 17 41 77 94 100 79. Relative Probability
c) 9.92 cm d) 9.99, 9.83, 0.16 e) 10% 1) 620
2) a) 2) a) i) 0.44 ii) 0.34 iii) 0.22
b) 19,800; 15,300; 9,900
Age 20 30 40 50 60 65 3) a) 25, 15, 10 b) i) 0.5 ii) 0.3 iii) 0.2
Cumulative c) i) 0.6 ii) 0.4 d) 250,000 and 150,000
Frequency 17 82 160 211 241 250 4) a) 0.25, 0.3, 0.45 b) 137, 165, 247
c) 26%
5) a)
3) a)
Mark 10 20 30 40 50 60 70 80 90 100 No. of 50 100 150 200 250 300
Cumulative 3 10 32 63 97 134 162 181 194 200 people
Frequency No. ‘for’ 30 45 55 83 96 122
c) 44 d) 14% No. 20 55 95 117 154 178
‘against’
76. Cumulative Frequency 2 Probability 0.60 0.45 0.37 0.415 0.384 0.407
1) a) ‘for’
Life of bulb 6000 7000 8000 9000 10,000 11,000 Probability 0.40 0.55 0.63 0.585 0.616 0.593
Cumulative ‘against’
Frequency 5 25 67 162 242 300

c) 86% c) 0.4 d) 42,000

2) a)
Time spent 30 35 40 45 50 55 80. Tree Diagrams
in air 1) a) 0.015, 0.085, 0.135, 0.765
Cumulative 6 16 28 62 85 100 b) 0.235 c) 0.765
Frequency 2) a) 0.035, 0.665, 0.045, 0.255
b) 0.08
c) 43 minutes, 10 minutes 3) a) -----283- , 15-----56- , 15-----56- , -----145- b) i) 15-----28- ii) 9
------
14
3) a)
4) a) 3
------
14
b) 11
------
14
5) a) 20
------
63
b) 11
------
63
c) 32
------
63
Timing 5 early 0 5 late 10 late 15 late 20 late
(minutes)
Cumulative 33 99 121 138 148 150
Frequency
c) 15%
4) a)
No. of T.V.’s 100 150 200 250 300 350
Cumulative
Frequency 5 32 109 254 337 365
.