A Strand 4 Unit 5 Band h
Problem solving
Exam-style
1 Pete wants to hire a car. He wants to spend as little as possible.
He can hire it from one of two companies: U hire and Cars 2 go.
U hire
50p a mile
Cars 2 go
£60 plus
20p a mile
a Write down the equations for the cost £ C of hiring a car for m miles from each company.
b On the same graph draw lines for your equations.
Use the vertical axis for C with a scale from 0 to 200 and the horizontal axis for m with a
scale from 0 to 400.
Exam-style
c Pete expects to drive 250 miles. Which company would you advise him to use?
2 Ros makes an accurate drawing of this diamond shape on graph paper.
The equations of the four lines are
y = 3x – 3
y = 3x – 9
y = 3x + 15
y = 3x + 9
R
Q
a Make tables of values for these lines, taking values of x from 0 to 5.
b Draw a graph showing the parts of the lines that make the shape.
S
P
c Write down the co-ordinates of the four vertices, P, Q, R and S.
d Show how Ros could have used algebra to find the co-ordinates of the vertices.
Use this method to check your answers to part c.
e What is the mathematical name for this shape?
3 Catherine draws a triangle which is bounded by the lines x + 2y = 12, 2y = x + 4 and y = x −3.
a Draw the three lines on the same graph.
b i Write the coordinates of the vertices of the triangle.
ii Use your diagram to solve the following pairs of simultaneous equations.
A x + 2y = 12, 2y = x + 4
B x + 2y = 12, y = x − 3
C 2y = x + 4, y = x − 3
Catherine adds 2 horizontal and 2 vertical lines to her diagram to form a rectangle around her
triangle. Each line passes through one of the vertices of the triangle.
c i Add these lines to your diagram.
ii State the equations of the lines that Catherine adds.
iii Find the area of the rectangle bounded by these 4 lines.
d Find the area of Catherine’s triangle.
© Hodder & Stoughton Ltd 2015
1
A Strand 4 Unit 5 Band h
4 Look at these pay-as-you-go tariffs for 2 mobile phone companies.
Q-Mobile
30p per minute for calls
8p per text
Pear
20p per minute for calls
12p per text
Let m stand for the number of call minutes and t stand for the number of texts.
a Write down an equation for m and t for a bill of £5 with
i Q-Mobile ii Pear.
b Draw a graph of your equations.
Chloe uses Q-Mobile and Daisy uses Pear.
c One week they use the same number of call minutes as each other, and they send the
same number of texts. They both use exactly £5 of credit.
How many call minutes and texts do they each use?
d Another week both girls make 4 minutes of calls and sent 40 texts.
i Mark a point on your graph to show this.
Both girls have £5 credit.
ii Who has to buy more credit? How much more credit does she need?
y
iii Who has credit left over? How much?
3
5 Here is the graph of the circle x + y = 4.
The line y = 2x + 1 cuts the circle at 2 points.
Find the co-ordinates of these 2 points.
2
2
2
−3
−2
−1
0
1
2
3
1
2
3
x
−1
Higher Tier only
1
−2
−3
y
6 Here is the graph of the ellipse
3
x2 + y2 = 1.
9
4
The line x + y = 1 cuts the ellipse at two points.
Find the co-ordinates of these two points.
2
−3
−2
−1
0
−1
−2
−3
2
© Hodder & Stoughton Ltd 2015
x
Higher Tier only
1