Mathematics: Paper Iii Preliminary Examination 2008

MATHEMATICS: PAPER III PRELIMINARY EXAMINATION 2008

SECTION A

QUESTION 1

(a)An alarm code consists of a combination of 2 letters followed by 3 single-digit numbers.

(1) How many possible outcomes are there for the threedigits if repititions are allowed?

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(2)How many possible outcomes are there for the two letters if repititions are not

allowed?

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(3)What is the probability of the combined outcomeSV 232 occurring if no repititions

are allowed for the two letters?

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(4)What is the probability that the first letter is not a ‘t’?

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(b) The probability that a certain rugby team has all its players fit to play is 70%.

The probability that they will win a game if all their players are fit is 90%.

When they are not fit the probability of them winning becomes 45%.

(1)Sketch a tree diagram to represent this situation.

(3)

(2)Use the tree diagram to determine the probability of them winning their next

game. (Round off to one decimal place)

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QUESTION 2

Provide reasons for all your statements

(a)The diagram below shows a clockface. 12 is joined to the 4 and 2 to the 7 with straight

lines AB and CD respectively. O is the centre of the clockface.

(1)If and are joined, show that . (No reasons required)

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(2) If and are joined calculate

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(3) Hence, calculate , providing reasons.

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(b) Refer to the given diagram.

In // and //.

(1)Determine the value of , with reasons.

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(2) Determine the value of , without reasons. (Round off to one decimal place)

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Total for Section A: 28 marks

SECTION B

QUESTION 3

(a)The following two-way contingency table is given to show the numbers of males and

females suffering from high blood pressure:

Male / Female / Total
High blood pressure / 393 / 15 / 408
Normal blood pressure / 4517 / 5075 / 9592
Total / 4910 / 5090 / 10000

(1)Does the evidence support the statement that high blood pressure is independent

of gender? Show your working out.

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(2) Determine the numbers in the cells for the events to be perfectly independent?

Male / Female / Total
High blood pressure
Normal blood pressure
Total

(5)

(3)On the strength of the differences between the actual and expected numbers

in the various cells, decide whether the events in the rows are independent of

the events in the columns and give reasons for your decision.

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QUESTION 4

(a)Courtney decided to collect data on the hourly earnings of 100 grade 12 learners during the

month of August for various holiday jobs. She organized the data into the following table.

Hourly earnings (rand) / Frequency
/ 5
/ 16
/ 25
/ 30
/ 24

(1)Calculate the mean for the hourly earnings of the grade 12 learners. Complete and

use the table below to calculate this. (Round off to one decimal where necessary)

Hourly earnings (rand) / Midpoint of
Interval ( / Frequency
( / Total

/ 5
/ 16
/ 25
/ 30
/ 24
Sum

(4)

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(2) Calculate the standard deviation for the hourly earnings of the grade 12 learners.

Complete and use the table below to calculate this.

(Round off to two decimal places where necessary)

Hourly earnings (rand) / Midpoint of
Interval ( / Frequency
( / / /

/ 5
/ 16
/ 25
/ 30
/ 24
Sum

(4)

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(3) Use the table below to draw an ogive (cumulative frequency curve) for this data on

the axes provided on the following page.

Hourly earnings (rand) / Frequency
/ Cumulative
frequency
/ 5
/ 16
/ 25
/ 30
/ 24

(2)

Hourly earnings of learners

(2)

(4) Estimate the median hourly earnings using your graph.

______(1)

(5) By referring to the relationship between the mean and the median, state

whether thedistribution of the data is normal, positively skewed or

negatively skewed.

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QUESTION 5

The percentages achieved by 10 learners studying Mathematics Core in Grade 11 were recorded. They then changed to Mathematical Literacy in Grade 12 and their results were recorded in the table below.

Candidate number / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10
Mathematics Core / 17 / 19 / 32 / 39 / 35 / 27 / 26 / 29 / 37 / 40
Mathematical Literacy / 72 / 74 / 79 / 84 / 83 / 68 / 78 / 82 / 60 / 90

(1)Draw a scatter plot of the data on the axes given and draw in a line of best fit.

(3)

(2)Calculate the equation of the regression line of best fit using the table below

Candi-date
Nr /
Math.
Core /
Math.
Literacy / / / /
1 / 17 / 72
2 / 19 / 74
3 / 32 / 79
4 / 39 / 84
5 / 35 / 83
6 / 27 / 68
7 / 26 / 78
8 / 29 / 82
9 / 37 / 60
10 / 40 / 90
Sum
Mean

(5)

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(3) Use your calculator to calculate the correlation coefficient for the data. (Round

off to two decimal places)

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(4)Describe the strength of the correlation.

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(5)Which candidate appears to be an outlier?

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(6)David wants to study a course at University that requires a Level 3 (40% - 50%)

for Mathematics Core or a Level 7 (80% - 100%) for Mathematical Literacy.

He is currently in Grade 11 and achieved a Level 2 (38%) on his latest report.

Would you advise him to continue with Mathematics Core? Show all your

calculations and motivate your answer.

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QUESTION 6

(a)In the diagram is a tangent to the circle with centre. and are on the

circumference of the circle and is produced to so that .

cuts at . Let

(1) Name, with reasons, two angles equal to .

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(2)Prove that is a diameter of the circle .

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(3)Determine the size of in terms of .

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(4)If bisects , find the value of .

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(b)In the diagram, is bisected by with on . and intersect at with

on so that

(1) Prove that ///

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(2) Hence, show that

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QUESTION 7

A piece of string is wound symmetrically around a circular rod. The string goes exactly

four times around the rod. The circumference of the rod is 4cm and its length is 12cm.

Find the length of the piece of string. Show all your working out.

______(5)

Total for Section B: 72 marks

Grand Total: 100 marks

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