Pre-course task for 2015 entry
Welcome to AS Use of Maths!
By now you are probably enjoying a well-earned rest after your GCSE exams. But we’d also like you to
give some thought to your start at College in August. Experience tells us that students who know what
to expect and who do some preparation during the holidays find adjusting to A Level study more
straightforward. This work comprises an element of the assessment used during induction.
As the name suggests. This course is about how Maths is used in a variety of commercial, industrial and
scientific contexts. You will study three modules:
Algebra
Here we study the techniques necessary for any advanced study of the subject. We concentrate on
four particular areas: linear modelling, using quadratics, exponential growth and decay, modelling using
trigonometric functions.
Data Analysis
This builds upon the statistics you have covered at GCSE level. Topics include: descriptive statistics,
graphical representation of data, correlation and regression, the normal distribution.
Decision Maths
This is a fairly modern branch of mathematics concerned with solving practical problems. We will
study Critical Path Analysis which is used to ensure that complex projects (such as the building of
Huddersfield’s new sports centre) run on time. We will also discover how a Sat-Nav device finds the
shortest route for a motorist.
Each module is examined by a one hour exam.
Graphical Calculators
You are expected to use one of these throughout the course and in the exam. Will are able to supply
them at a competitive rate (about £55). We will provide more details at the start of the year about the
particular model we need you to buy.
What follows is some work to help you prepare for the start of your course. This should ensure that you
keep your skills fresh and the questions we have set you should lead nicely into the new work that we
start in September.
You should have this task completed to the best of your ability by Monday 29th June which is
our Welcome Day and should hand it in to your subject workshop leader on this day. You will
shortly be receiving an invite to this event. Please ensure your name is written on the top of
every page of work you produce.
Finally, I’d like to wish you good luck with your exam results. I look forward to seeing you in September.
Michael Smith
Course Leader, A Level Use of Maths.
1
Section A. Data Analysis
1.
The number of goals scored in 15 hockey matches is shown in the table.
Number of goals
Number of matches
1
2
3
1
5
5
6
3
9
4
Calculate the mean number of goals scored.
…………..........…............……...................…………….....…………................................
…………..........…............……...................…………….....…………................................
…………..........…............……...................…………….....…………................................
…………..........…............……...................…………….....…………................................
Answer .......................................................... goals
(Total 3 marks)
2.
The table shows the results of a survey of children’s weights.
Weight, w, (kg)
Number of
children
w < 30
5
w < 40
9
w < 50
13
w < 60
8
w < 70
7
w < 80
8
Calculate an estimate of the mean weight of the children.
...............................................................................................................................................
...............................................................................................................................................
...............................................................................................................................................
Answer .................................................... kg
(Total 4 marks)
2
3.
(a)
The box plot shows the heights of a group of boys in a school.
130
(i)
140
150
160
Height (cm)
170
180
190
Write down the media height of these boys.
Answer …………………………………… cm
(1)
(ii)
Find the interquartile range of the heights of these boys.
...........................................................................................................................
Answer …………………………………… cm
(2)
(b)
15 girls in the school are chosen at random.
Their heights, in centimetres, are shown below.
142, 147, 152, 156, 156, 159, 164, 166, 166, 166, 167, 170, 171,
171, 175
There are a total of 450 girls in the school.
Use this sample to estimate how many girls in the school are less than 148 cm tall.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
Answer ...................................................................
(2)
(Total 5 marks)
3
4.
(a)
Adam and Betty take a mental arithmetic test each week for seven weeks.
Adam’s test scores are
(i)
What is the mode of Adam’s scores?
Answer ................................................................
(1)
(ii)
What is the median of Adam’s scores?
Answer ................................................................
(2)
(b)
Betty’s test scores are
3 6 7 8 8 4 6
Complete this table.
Range
Mean
Adam
3
8
Betty
5
.....................................................................................................................................
(3)
(c) Use the range and mean to compare their test scores.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(2)
(Total 8 marks)
4
5.
The weights of 80 bags of rice are measured. The table summarises the results.
(a)
Minimum
480 g
Lower quartile
500 g
Median
540 g
Upper quartile
620 g
Maximum
720 g
Draw a box plot to show this information.
500
550
600
Weight (g)
650
700
(3)
(b) Write down the interquartile range for these data.
Answer .............................................. g
(1)
(c) How many bags weigh
(i)
less than 480 g
Answer ..............................................
(1)
(ii)
less than 500 g?
Answer ..............................................
(1)
5
(d)
Draw a cumulative frequency diagram to show the information.
80
60
Cumulative
frequency
40
20
0
500
550
600
Weight (g)
650
700
(3)
(Total 6 marks)
6
6.
The table shows the number of petrol pumps and the number of cars queuing at midday
at six garages.
Number of petrol pumps
3
4
6
4
3
5
Number of cars queuing
6
5
3
4
5
4
(a)
Plot a scatter graph of these data on the axes below.
6
5
4
Number
of cars 3
queuing
2
1
0
0
1
2
5
3
4
6
Number of petrol pumps
7
8
9
(2)
b)
Draw a line of best fit on your scatter graph.
(1)
(c)
Use your line to estimate the number of cars queuing at a garage with 8 petrol pumps.
Answer ............................................
(1)
(d)
Explain why your answer in part (c) may be unreliable.
……………………………………………………………………………………….
(1)
(Total 5 marks)
7
Section B. Algebra.
7.
Solve the following equations.
(a)
2x + 5 = 3
.....................................................................................................................................
.....................................................................................................................................
Answer x = ....................................................
2)
(b) 4(y – 3) = 18
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
Answer y = ....................................................
(3)
(Total 5 marks)
8.
A company uses this formula to find the cost, in pounds, to hire out a car.
Cost = 25 × number of days hired + 20
a)
Calculate the cost of hiring a car for
(i)
two days,
Answer £ ....................................................................
(2)
(ii)
one week.
Answer £ ....................................................................
(2)
(b) Linda hires a car for her holiday.
She pays the company £270.
For how many days does she hire the car?
.....................................................................................................................................
Answer ................................................... days
(2)
(Total 6 marks)
8
9.
(a)
Complete this table of values for y = 2x – 1
x
–1
y
–3
0
1
2
3
1
5
.................................................................................................................................
(1)
(b) On the grid draw the graph of y = 2x – 1 for values of x from –1 to +3.
y
5
4
3
2
1
–1
O
1
2
3
x
–1
–2
–3
(2)
(c) Find the coordinates of the point where the line y = 2x – 1 crosses the line y = –2.
.................................................................................................................................
Answer (........................, ......................)
(2)
(Total 5 marks)
9
10.
The line l on the graph passes through the points A (0, 3) and B (–4, 11).
y
15
l
B
10
5
A
–10
(a)
O
–5
5
x
Calculate the gradient of the line l.
......................................................................................................................................
Answer ......................................................
(2)
(b) Write down the equation of the line l.
......................................................................................................................................
Answer ......................................................
(1)
(c) Write down the equation of the line which also passes through the point (0, 3) but is
perpendicular to line l.
......................................................................................................................................
......................................................................................................................................
Answer ......................................................
(2)
(Total 5 marks)
10
11.
(a)
On the grid below draw and label the lines y = –4
and
y = 2x + 1
...............……………………………………………………………………………..
...............……………………………………………………………………………..
...............……………………………………………………………………………..
y
7
6
5
4
3
2
1
–4
–3
–2
–1 O
1
2
3
4
x
–1
–2
–3
–4
–5
–6
–7
(4)
(b)
Write down the coordinates of the point where the lines y = –4 and y = 2x + 1 cross.
Answer ( ...................., ......................)
(1)
(Total 5 marks)
11
12.
(a)
Complete the table of values for y = 2x2 – 4x – 1
x
–2
y
15
–1
0
1
–1
2
3
–1
5
.............................................................................................................................
(2)
(b)
On the grid, draw the graph of y = 2x2 – 4x – 1 for values of x from – 2 to +3.
y
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
–2
–1
O
1
2
3
x
–1
–2
–3
(2)
12
(c)
An approximate solution of the equation 2x2 – 4x – 1 = 0 is x = 2.2
(i)
Explain how you can find this from the graph.
...................................................................................................................
(1)
(ii)
Use your graph to write down another solution of this equation.
Answer x = .................................................................
(1)
(Total 6 marks)
13