Achieve Grade 7–9 Workbook
The ideal resource to help you achieve your potential at Higher tier mathematics; providing extra support and guidance where it is most needed. Written by experienced maths teachers and authors, the range of questions will help you focus on the key topics and skills required to succeed. Easy to use and revise with a write-in format Answers supplied at the back of the workbook; allowing you to check your own work Focus your learning with targeted, level-specific questions Test your problem-solving and reasoning skills with questions focused on these areas Assess your understanding by spotting the errors in questions designed to challenge you
AQA GCSE Maths Achieve Grade 7–9 Workbook
AQA GCSE Maths
AQA GCSE
Maths
Achieve Grade 7–9 Workbook
Su Nicholson ISBN 978-0-00-827126-8
9 780008 271268
Russell Timmins Greg Byrd
2 Algebra 2.1 Solving quadratic equations graphically 1
The diagram shows part of the graph of y = x2 + 2x − 3.
Hint: The roots of a quadratic equation ax2 + bx + c = 0 are the values of x that satisfy
the equation. They are the values of x where the graph of y = ax2 + bx + c crosses the x axis, i.e. where y = 0. If the graph does not cross the x axis, then the quadratic equation has no real roots. y
12 11 10
y = x2 + 2x − 3
9 8 7 6 5 4 3 2 1 –4 –3 –2 –1 0 –1
1
2
3
4x
–2 –3 –4
a Write down the roots of the equation x2 + 2x − 3 = 0.
b Write down the equation of the line of symmetry for the graph of y = x2 + 2x − 3.
c By drawing a suitable straight line, use your graph to find estimates for the solutions of x2 + x − 5 = 0.
Hint: Rearrange the equation as x2 + 2x − 3 = ax + b and find the values of a and b. The equation of the line to draw is y = ax + b.
2.1 Solving quadratic equations graphically
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P is the point on the graph where x = 0. d Calculate an estimate for the gradient of the graph at the point P.
Hint: The gradient of a curve at any point is the same as the gradient of the tangent to the curve at that point.
2
y
The diagram shows part of the graph of y = 5 + 2x − x2.
8 6
a Write down the coordinates of the maximum point.
4 2 –3 –2 –1 0 –2
1
2
3
4
5
x
b Write down estimates for the roots of the equation 5 + 2x − x2 = 0.
c Show that the equation 5 + 2x − x2 = 0 can be rearranged to give x =
d Use the iteration xn + 1 =
2 x + 5.
2 xn + 5, with x0 = 3, to find the positive root of the equation
5 + 2x − x = 0 to 3, decimal places. 2
Hint: Use the ANS key on your calculator: key in ‘3’ ‘=’ then '‘ 2 × ANS + 5’ ‘=’ ‘=’ … to find the values of x1, x2, x3 etc. Continue until two successive values agree to 3 decimal places.
2.2 Recognising shapes of graphs Hint: Match the obvious ones first. Match each graph with its equation.
14
1
y = 2x2 + 4 is graph
4
y = x3 + 4 is graph
2
y = x2 + 2x is graph
5
y = −x3 + 4 is graph
3
y = 2x + 4 is graph
6
y = x2 + 2x + 4 is graph
2 Algebra
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4 is graph x
7
y=
8
y = 3x is graph
A
B
y
D
G
E
x
C
x
y
F
x
y
x
0
I
y
x
0
y
x
0
y
0
y = sin x is graph
y
0
H
y
J
10
0
y
0
y = cos x is graph
y
x
0
9
0
x
0
x
2.2 Recognising shapes of graphs
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x
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2.3 Real-life graphs 1
The graph shows the velocity–time graph for a train journey. v 120 110 100 90
v (km/h)
80 70 60 50 40 30 20 10 0
0
1
2
3
4 5 6 t (hours)
7
8
9 10
t
Work out: a the acceleration in the first hour
b the deceleration in the last hour
c the acceleration between t = 2 and t = 4 hours
d the distance travelled in the first two hours
e the distance travelled in the last two hours.
2
Abby invested £1000 in an account that paid 9% compound interest per annum. The interest is added to the account at the end of each year. Abby left the money in the account for 15 years. She did not add or withdraw any money from the account in this time.
Hint: Remember that for compound interest the amount of interest will change each year.
16
2 Algebra
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a Explain why the amount of money, A, in the account at the end of n years can be written as A = 1000 × 1.09n.
b Draw the graph of A = 1000 × 1.09n for 0 n 15. 4000 3800 3600 3400 3200 3000 2800 2600
Amount, A (£)
2400 2200 2000 1800 1600 1400 1200 1000 800 600 400 200 0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Years, n
2.3 Real-life graphs
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c Use your graph to find the number of years it took for the initial amount Abby invested to triple in value.
Hint: You can check your answer by substituting into the formula for compound interest.
3
The velocity-time graph shows the velocity in m/s of a car in the first 10 seconds of its motion. y
50 40
v
30 20 10 0
1
2
3
4
5 t
6
7
8
9
10
x
a Describe how the acceleration of the car changes over the first 10 seconds.
b Estimate the acceleration of the car at time t = 4 seconds.
c Use five equal strips on the graph to estimate the distance travelled by the car in
the first 10 seconds.
Hint: Divide the area under the graph into a triangle and 4 trapezia and work out the total area.
Area of trapezium =
18
1 h(a + b) 2
2 Algebra
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d Is your answer to part c an overestimate or an underestimate? You must justify your answer.
Hint: Consider where the lines of the shapes lie in relation to the curve.
e The distance travelled in the first 10 seconds can be estimated using 10 equal strips on the graph. Comment on the effect this would have on your answer for the distance travelled. You must justify your answer.
2.4 Straight lines and their equations Hint: The equation of a straight line can be written in the form y = mx + c, where m is the gradient and c is the intercept on the y axis.
y2 − y1 . x2 − x1 The coordinates of the midpoint of the line joining the points (x1, y1) and (x2, y2) are
The gradient, m, of the line joining the points (x1, y1) and (x2, y2) is given by m = x1 + x2 y 1 + y 2 . , 2 2
1
For the line joining the points (−3, 1) and (2, −3), work out: a the gradient
b the coordinates of the midpoint.
2
a Work out the equation of the line with gradient 1 that passes through the point (6, −1). 2
Hint: The equation of a line with gradient, m, that passes through the point (x1, y1) is y − y1 = m(x − x1).
2.4 Straight lines and their equations
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b Work out the equation of the line that is perpendicular to the line 3x − 2y + 7 = 0 and passes through the point (−2, 1).
Hint: To find the gradient, m, of a straight line write it in the form y = mx + c.
Parallel lines have the same gradient. If two lines with gradients m1 and 1 m2 are perpendicular, then m2 = − . m1
3
For the points A (2, 1) and B (6, −7) find: a the exact length of AB
Hint: The length of the line joining the points (x1, y1) and (x2, y2) is ( x 2 − x 1 )2 + (y 2 − y 1 )2 .
b the equation of the line that passes through A and B
Hint: To find the equation of a line that passes through two given points, first
find the gradient, m, then use y − y1 = m(x − x1) using one of the points the line passes through.
c the equation of the perpendicular bisector of AB.
4
The coordinates of the vertices of a triangle are (0, 4), (2, 1) and (8, 5). Show that the triangle is right angled.
Hint: Use Pythagoras’ theorem.
20
2 Algebra
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