Mathematics, Computing and Technology

MU123 Discovering mathematics

MU123

TMA 02 2015B

Covers Units 3, 4 and 5. Cut-o date: 21 April 2015

Submission instructions

You will nd instructions for completing TMAs in the Assessmentresources area of the MU123 website. Please read these instructions beforebeginning work on this TMA.

Reviewing your tutors comments on your previous TMA will help you asyou work on this one.

Special instructions

Remember that you need to explain your reasoning and communicate yourideas clearly, as described in Subsection 5.3 of Unit 1. This includes:

explaining your mathematics in the context of the question the correct use of notation and units appropriate rounding.

Copyright c 2014 The Open University WEB 04181 211.1

MU123 TMA 02 Cut-o date 21 April 2015

Question 1 30 marks

This question is based on your work on MU123 up to and including Unit 3.

(a) (i) Use a factor tree to write 1530 as a product of prime factors. Youshould display your factor tree in your answer. [4]

(ii) Calculate

79 16 + 812 ,

leaving your answer as a fraction in its simplest form, showing allyour working. [4]

(iii) Simplify the surd

31442

by writing it as a surd in its simplest form, showing your working. [3]

(b) A group of 255 students are required to take a test. Of these,225 actually take the test, and 30 are absent that day.

(i) Write down the ratio of the number of students who take the testto the number of students who are absent. Simplify this as far aspossible. [3]

(ii) Of the 225 students who take the test, the numbers of studentsobtaining grades A, B, C and D are in the ratio 2 : 9 : 3 : 1,respectively. Calculate the number of students obtaining eachgrade. Explain how you could check that your answers arereasonable. [6]

(c) A group of 285 teenagers each send an average of 37 text messages perday.

(i) Calculate the number of text messages that this group would sendover a period of a year (365 days).

Give your answer in ordinary notation unrounded, then write itin scientic notation correct to three signicant gures. [4]

(ii) If each text message sent costs 12.3p, calculate the total cost(in ) of the messages sent by the group over a period of a year.

Give your answer in ordinary notation correct to two signicantgures. [2]

(iii) One of the teenagers pays 5 per month to send text messages.Calculate the average cost (in ) per text for sending 37 textseach day for a month (30 days).

Give your answer in ordinary notation correct to two signicantgures. [4]

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Question 2 30 marks

This question is based on your work on MU123 up to and including Unit 4.

After a remarkably good summer, John decides to investigate whether itreally does rain more in summer than in winter. He looks up some guresfor his local area on the climate summaries pages of the Met Oce website.

John records the gures for his local area, East Anglia, in Table 1.

Table 1 Seasonal rainfall (mm)

Year Summer Winter

2001 189 2022002 159 1272003 124 1972004 226 1822005 152 992006 157 852007 28 1862008 192 1352009 165 1342010 182 2122011 179 1322012 254 1192013 105 176

[Source: www.metoce.gov.uk/climate/uk]

(a) (i) Is this investigation comparing or seeking a relationship?

(ii) Are rainfall amounts continuous or discrete?

(iii) Are these primary or secondary data, from Johns point of view?

Explain your answers briey. [6]

(b) Before analysing these datasets on a computer, spend a few minutesscanning them by eye. Identify any outlier that appears to be anerror, briey justifying your choice. [2]

John doesnt remember any particularly dry summer recently, so hedecides that the outlier must be a typing error and should not be includedin his analysis.

(c) Open Dataplotter and create a new dataset in each column by clickingon New. Enter the data for summer in the rst column and thedata for winter in the second column. (Do not include the outlier;you will now have 12 values for summer and 13 for winter.)

Copy and complete the following table. Where values need to berounded, give them correct to one decimal place. The means (roundedto one decimal place) are given for you as a check that you haveentered the data correctly.

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Seasonal rainfall (mm)

Summer Winter

Minimum (Min)

Lower quartile (Q1)

Median

Upper quartile (Q3)

Maximum (Max)

Mean 173.7 152.8

Standard deviation (SD)

Interquartile range (IQR)

Range

Size of dataset (n) 12 13[8]

(d) (i) Identify the two measures of location from the table in part (c).Use both of these measures to determine which of the twodatasets has the higher location. [3]

(ii) Identify the three measures of spread from the table in part (c).Which of the two datasets has the wider spread, as measured byeach of these three measures? [4]

(e) John concludes that it rains more in summer than in winter. Is this areasonable conclusion? Explain your answer briey. [2]

(f) John discovers that the correct value for the outlier is 228mm, andwonders how the error aects his investigation.

Use Dataplotter to nd the mean and median of the summer data:

including the incorrect outlier; including the correct value of 228 instead of the incorrect outlier.(Your dataset will have 13 values each time.)

Copy and complete the following table, rounding values for the meanto one decimal place.

Summer

with incorrect outlier with correct value

Mean

Median

Size of dataset 13 13

What is the eect on the mean and on the median of including theoutlier instead of the correct value? Explain why this happens. [4]

(g) Would including the correct gure aect Johns conclusion inpart (e)? Explain your answer briey. [1]

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Question 3 30 marks

This question is based on your work on MU123 up to and including Unit 5.

(a) Simplify each of the following expressions as far as possible; multiplyout any brackets, expand any algebraic fractions, and collect liketerms together. Show your working.

(i) 10a 12 12a+ 17 [2](ii) 4(9 5r) [2](iii) 3 + q(8 4q) 8q [3](iv) 9x 3(7y 4x) [3](v) 7(2x2 9x) + 5x(9 2x+ 3x2) [5]

(vi)p2 6p+ 4

p[4]

(b) Solve the following equations. Show your working and check that youranswers are correct.

(i) 5x+ 12 = 2x 3 [5](ii)

a

3 1 = 2(a 13) [6]

Question 4 10 marks

This question is based on your work on MU123 up to and including Unit 5.

In this question, you are asked to comment critically on a studentsincorrect attempt at solving the equation

37(x 3) = 3.

The attempt is shown below.

D bY%[hk ] ?

fY[e; hY[ _%[hk ?

fY[e; hY[ [ ?

v[ _ h [ ?

W "; ?

D ]hY[hk ] ?

(a) Substitute the students solution x = 14 into the left-hand side of theequation 37(x 3) = 3. Explain why this shows that the studentssolution is incorrect. [2]

(b) Write out your own full and correct attempt at solving the equationand checking your solution. [4]

(c) Identify the two lines in the students attempt where a mistake hasbeen made. Explain, as if directly to the student, why their working isincorrect. [4]

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