Contents 1 1.1 1.2 1.3 1.4
2 2.1 2.2 2.3 2.4 2.5
3 3.1 3.2 3.3
4 4.1 4.2 4.3 4.4 4.5
5 5.1 5.2 5.3
6 6.1 6.2 6.3 6.4
How to use this book
5
Percentages
6
Simple interest Percentage increases and decreases Calculating the original value Using percentages
6 7 9 11
Equations and formulae
13
Expansion Factorisation Equations with brackets Equations involving fractions Rearranging formulae
13 14 15 16 17
Polygons
19
Angles in polygons Angles in regular polygons Regular polygons and tessellations
19 21 23
Using data
25
Scatter graphs and correlation Time-series graphs Two-way tables Comparing two or more sets of data Statistical investigations
25 27 30 34 38
Applications of graphs
39
Step graphs Time graphs Exponential growth graphs
39 42 44
Pythagoras’ theorem
48
Introduction Calculating the hypotenuse Calculating the length of the shorter sides Solving problems using Pythagoras’ theorem
48 49 51
7 7.1 7.2 7.3 7.4
8 8.1 8.2 8.3
9 9.1 9.2 9.3 9.4 9.5
Fractions
55
Adding and subtracting Multiplying Multiplying with mixed numbers Dividing fractions and mixed numbers
55 56 57
Algebra
60
More about brackets Factorising expressions containing powers Expanding the product of two brackets
60
Decimal numbers
64
Powers of 10 Standard form Rounding appropriately Mental calculations Solving problems
64 65 66 68 70
61 62
10 Prisms and cylinders 10.1 10.2 10.3 10.4 10.5
72
Metric units for area and volume Volume of prisms Surface area of prisms Volume of a cylinder Surface area of cylinders
72 73 75 77 79
11 Solving equations graphically
81
11.1 11.2 11.3 11.4
Graphs from equations in the form ay ± bx = c Graphs from quadratic equations Solving quadratic equations by drawing graphs Solving simultaneous equations by using graphs
81 82 84 85
52
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12 Compound units 12.1 12.2 12.3
Speed More about proportion Unit costs
13 Right-angled triangles 13.1 13.2 13.3 13.4
4
Introducing trigonometric ratios How to find trigonometric ratios Using trigonometric ratios to find angles Using trigonometric ratios to find lengths
87 87 89 90 93 93 94 95
14 GCSE preparation 14.1 14.2 14.3 14.4 14.5 14.6
Number Algebra Ratio, proportion and rates of change Geometry and measures Probability Statistics Mixed GCSE-style questions
99 99 100 102 104 106 108 111
96
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11
Solving equations graphically
11.1 Graphs from equations in the form ay é bx = c 1
Solve the equations: a e
2
3
3x = 24 2y = 12
b 5x = 20 f −9y = −27
c g
−6x = 30 −5y = 45
d −2x = 22 h −4y = 8
Draw the graph of each equation. Use a grid that is numbered from −10 to +10 on both the x-axis and the y-axis. a
y=x+3
b y=x−4
a
Draw the graphs of all these equations on the same grid. Use a grid that is numbered from −10 to +10 on both the x-axis and the y-axis. i y = 12 x + 4 ii y = 12 x − 2 iii y = 12 x
c
y = 2x
d y = 3x + 1
b What do you notice about all these graphs? c Explain how you could now draw the graph of y = 12 x + 6. 4
Using a grid with axes numbered from −2 to 10, draw the graph of x + y = 6.
5
For each graph, find the coordinates of the two points where the graph intersects the x-axis and the y-axis. a
6
3x + 2y = 18
c
4y − 7x = −28
Draw the graph of each equation. Use a grid that is numbered from −10 to +10 on both the x-axis and the y-axis. a
3y + 2x = 12
d 6x − y = 6 7
b 5y − x = 15
b 4x + 5y = 40
c
3y + 7x = 21
3y − 4x = 24
f
2y − 5x = −20
e
Draw the graphs of all these equations on the same grid. Use a grid that is numbered from −10 to +10 on the x-axis and from −2 to +10 on the y-axis. a i 3y + x = 6 ii 4x − 5y = −10 iii x + y = 2 b What do you notice about all these graphs?
iv
2x − 9y = −18
8
a Using a grid with axes numbered from −2 to 12, draw the graph of y = 2x + 5. b Use the graph to solve these equations. i 2x + 5 = 4 ii 2x + 5 = 8 iii 2x + 5 = 10 iv 2x + 5 = 2
9
a
Complete the table for the graph y = 6x . x
1
2
3
4
5
6
y
b Why can you not find the value of y when x = 0? c Using a grid with axes numbered from 0 to 6, draw the graph of y = 6x . 11.1 Graphs from equations in the form ay ± bx = c
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11.2 Graphs from quadratic equations 1
Copy and complete this table of values for y = x2 + 3. −1
x
0
1
2
3
y
2
Copy and complete this table of values for y = x2 + x. −1
x x
0
1
2
3
2
x y
3
a
Copy and complete this table of values for y = x2 + 6x. −7
x
−6
−5
−4
−3
−2
−1
0
1
2
x
6x y
b Draw a grid with the x-axis numbered from −7 to 1 and the y-axis from −10 to 10. c Use the table to help you draw, on the grid, the graph of y = x2 + 6x. 4
a
Copy and complete this table of values for y = x2 + 2x − 4. x x
−4
−3
−2
−1
0
1
2
2
2x −4 y
b Draw a grid with the x-axis numbered from −4 to 2 and the y-axis from −6 to 5. c Use the table to help you draw, on the grid, the graph of y = x2 + 2x − 4. 5
a
Copy and complete this table of values for y = 2x2 + 1. x x
−2
−1
0
1
2
2
2 x2 y
b Draw a grid with the x-axis numbered from −2 to 2 and the y-axis from 0 to 10. c Use the table to help you draw, on the grid, the graph of y = 2x2 + 1.
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6
a
Construct a table of values for each equation. Then plot all their graphs on the same pair of axes. Number the x-axis from −2 to 2 and the y-axis from −3 to 16. i y = 3x2 − 2 ii y = 3x2 iii y = 3x2 + 1 iv y = 3x2 + 3
b Comment on your graphs. c Sketch onto your diagram the graph with the equation y = 3x2 + 2. 7
a
Copy and complete this table of values for y = 8 − 2x − x2. −5
x
−4
−3
−2
−1
0
1
2
3
8 −2x − x2 y
b Draw a grid with the x-axis numbered from −5 to 3 and the y-axis from −8 to 10. c Use the table to help you draw, on the grid, the graph of y = 8 − 2x − x2. d Comment on the shape of the graph. 8
a
Plot the graph of y =
( x − 5)( x + 5) 2
+
27 x
with the x-axis numbered from 1 to 7.
b Write down the coordinates of the minimum point on the graph of . y = ( x − 5)(2 x + 5) + 27 x 9
The length (L) of a pendulum is related to the period of its swing cycle (T ). The table shows the lengths of five pendulums and their periods. T (seconds)
2
4
6
8
10
L (metres)
1.0
4.0
8.9
15.9
24.8
a Draw a graph with T on the x-axis and L on the y-axis. b Use your graph to estimate the length of a pendulum with a period of 12.2 seconds. c Use your graph to estimate the period of a pendulum with a length of 5 m.
Brainteaser A circle has a circumference of 18 cm. a b c d
Find an estimate for the radius of the circle. Find an estimate for the area of the circle. Show that, if π = 3, the area of a circle, A, is given by the formula A = circumference. 2 Copy and complete this table of values for A = C12 . C
0
6
12
18
24
C2 12 ,
where C is the
30
A
e f
Draw a graph with C on the x-axis and A on the y-axis. Use your graph to estimate the area of a circle with a circumference of: i 5 cm ii 15 cm iii 25 cm.
g
Estimate the circumference of a circle with an area of 40 cm2.
11.2 Graphs from quadratic equations
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11.3 Solving quadratic equations by drawing graphs 1
a
Copy and complete the table for the equation y = x2 + 2x − 1. x
−5
−4
−3
−2
−1
0
1
2
3
2
x
2x −1 y
b Use the table to solve the equations: i x2 + 2x − 1 = 7 ii x2 + 2x − 1 = 2 2
a
iii x2 + 2x − 1 = −2
Copy and complete the table for the equation y = x2 + 3x. x
−5
−4
−3
−2
−1
0
1
2
2
x
3x y
b Use the table to solve the equations: ii x2 + 3x = 10 i x2 + 3x = 0 c Find the value of y when x = −3.2 3
a Draw the graph of y = x2 − 5x + 2 from x = −2 to 7. b Write down the value of y when x = 4.5 c Use the graph to find the solutions to the equations: ii x2 − 5x + 2 = 5 iii x2 − 5x + 2 = 10 i x2 − 5x + 2 = 0
4
a Draw the graph of y = x2 + x − 3 from x = −4 to 4. b Use the graph to find the solutions to the equations: ii x2 + x − 3 = 2 iii x2 + x − 3 = 5 i x2 + x − 3 = −2
5
a Draw the graph of y = x2 − 2x − 4 from x = −3 to 5. b Use the graph to find the solutions to the equations: ii x2 − 2x = 1 iii x2 − 2x = 10 i x2 − 2x − 4 = 0
6
Draw graphs to find the solutions of: a
PS
iii x2 + 3x = −2
x2 − 3x − 1 = 0
b x2 + 4x − 6 = 2
c
x2 − x − 4 = 5
7
Draw a graph to find the solutions of 2x2 − 3x − 1 = 0.
8
a Draw the graph of y = 2x2 + x − 3 from x = −3 to 3. b Write down the value of y when x = −1.7 c Use the graph to find the solutions to the equations: ii 2x2 + x = 7 iii 2x2 + x − 2 = 0 i 2x2 + x − 3 = 0 d Draw a straight line on your graph to solve the equations: i 2x2 + x − 3 = x + 5 ii 2x2 + 2x − 7 = 0
84
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11.4 Solving simultaneous equations by using graphs 1
The graph shows the lines with the equations x + y = 9 and y = 12 x + 3.
y 10 9
Use the graph to solve the simultaneous equations x + y = 9 and y = 12 x + 3.
8 7 6 5 4 3 2 1 0 –2 –1 0 –1
1
2
3
4
5
6
7
8
9
10
x
–2
2
The graph shows the lines with the equations 3y + x = 6 and y = 2x + 9. y 12 11 10 9 8 7 6 5 4 3 2 1 0 –6 –5 –4 –3 –2 –1 0 –1
1
2
3
4
5
6
7
8
9
10 11 12
x
–2
Use the graph to solve the simultaneous equations 3y + x = 6 and y = 2x + 9. 3
Pair up the simultaneous equations with the answers. a b c d
x + y = 10 and 2y + 3x = 23 2y − x = 20 and y + 4x = 1 3y + 4x = 14 and 3y − 4x = 22 y = 41 x + 3 and y = x − 3
i ii iii iv
(−1, 6) (3, 7) (8, 5) (−2, 9)
11.4 Solving simultaneous equations by using graphs
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4
Solve each pair of simultaneous equations by drawing their graphs. a b c d e
5
x+y=4 y = 2x + 5 3y + x = 15 4y − 3x = 6 y + 3x = 8
y = 3x + 8 y = 4x + 9 2y + 3x = 24 x + 2y = 8 y=3−1
Ben and Tiara are solving the simultaneous equations 3y + 5x = 45 and y + 2x = 16. Ben says that the answer is x = 6 and y = 5. Tiara says that the answer is x = 3 and y = 10. Who is correct? Explain your answer.
6
a
Draw the graphs of all these equations on the same grid. Use a grid that is numbered from −4 to + 10 on both the x-axis and the y-axis. i x+y=3 ii 2y = x + 3 iii y = 2x − 3
b Use your graph to solve the simultaneous equations x + y = 3 and 2y = x + 3. c Use your graph to solve the simultaneous equations x + y = 3 and y = 2x − 3. d Use your graph to solve the simultaneous equations 2y = x + 3 and y = 2x − 3. 7
Solve the simultaneous equations y = x2 − 2 and 3x + 2y = 6
8
Solve the simultaneous equations y=
16 x
and 2y + x = 12
Brainteaser The displacement (s) of a particle t seconds after being launched, at a velocity of u m/s with an acceleration of a m/s2, is given by s = ut + 12 at2. Take the acceleration to be −10 m/s2 in all questions.
86
a
A rocket is launched from the ground vertically upwards at 60 m/s. Find the two occasions it is 100 m above the ground.
b
A beachball is launched from the ground vertically upwards at 40 m/s. Find the time it takes to return to the ground.
c
A rocket is launched from the ground vertically upwards at 37.5 m/s. Find how long the rocket spends higher than 45 m above the ground.
d
A book is dropped from the top of the Leaning Tower of Pisa. Given that the tower is 56 m tall, find how long it takes the book to hit the ground.
e
A stone is thrown vertically upwards from the top of a cliff at 10 m/s. Given that the cliff is 75 m tall, find how long it takes the stone to splash into the water at the bottom of the cliff.
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