ZIMBABWE SCHOOL EXAMINATIONS COUNCIL <br> General Certificate of Education Advanced Level
MATHEMATICS
PAPER 4
NOVEMBER 2009 SESSION
Additional materials: Answer paper Graph paper List of Formulae
TIME: 3 hours
INSTRUCTIONS TO CANDIDATES
Write your name, Centre number and candidate number in the spaces provided on the answer paper/answer booklet.
There is no restriction on the number of questions which you may attempt. If a numerical answer cannot be given exactly, and the accuracy required is not specified in the question, then in the case of an angle it should be given to the nearest degree, and in other cases it should be given correct to 2 significant figures. If a numerical value for \(g\) is necessary, take \(g=9.81 \mathrm{~ms}^{-2}\).
INFORMATION FOR CANDIDATES
The number of marks is given in brackets [] at the end of each question or part question. The total number of marks for this paper is 120 . Within each section of the paper, questions are printed in the order of their mark allocations and candidates are advised, within each section, to attempt questions sequentially. The use of an electronic calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers.
**Section (a): Statistics**
Question 1
A random variable \(X\) has \(E(X)=10\) and \(\operatorname{Var}(X)=9\). Find the expected value and variance of \(Y=2 X-3\).
Question 2
The owners of a motel in Mutare have noticed that in the long run \(40 \%\) of the people who stop and inquire about a room for the night, actually book a room.
How many inquiries must the owners answer to be \(99 \%\) sure of at least one booking?
Question 3
Three tickets for a musical show are sent to a high school musical club. Fifteen girls and ten boys would like a ticket. If the three people to receive a ticket are chosen at random, find the probability that they will be
(i)
exactly 2 boys,
(ii)
at least 2 girls.
Question 4
(a)
#### (i)
Give two advantages of using stem and leaf diagrams in analysing data.
#### (ii)
Describe with an example, a statistical situation in which it would be appropriate to use the mode as a measure of central tendency.
(b)
The mean and standard deviation of the masses of a group of adult males are 65 kg and 10 kg respectively. Males are considered overweight if they are in the top \(5 \%\) of the group by mass. Assuming that the masses of this group are normally distributed, find the least mass to be considered overweight.
Question 5
A random sample of 400 students was asked to indicate their view on infusion of environmental issues in their college curriculum. The results are summarised in the following table.
| | In favour | Opposed | Undecided | | :--- | :--- | :--- | :--- | | Females | 115 | 60 | 36 | | Males | 90 | 85 | 14 |
Test at \(5 \%\) level of significance the hypothesis that there is no difference in opinion between males and females.
Question 6
The duration \(X\) minutes of a telephone call by a school head to the Provincial Education Director is a continuous random variable with a probability density function defined by
$$ f(x)= \begin{cases}x^{-2}, & x \geq 1 \\ 0, & \text { otherwise }\end{cases} $$
Given that a call has already lasted for 5 minutes, find the conditional probability that its total duration will be less than 7 minutes.
Question 7
The table shows the bus fares in thousands of dollars paid by 19 football fans selected at random from a football crowd.
| 73 | 85 | (48) | 80 | 53 | 75 | (55) | | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | 58 | (62) | 69 | 63 | 64 | 73 | 65 | | 55 | 54 | 55 | 45 | 55 | | |
(a)
Construct a stem and leaf diagram representing this data.
(b)
Calculate the median and the interquartile range.
Question 8
A manufacturer of an item used for the production of metal rods claims that the new machine that he has acquired has resulted in an improved product. The old machine is known to have given \(20 \%\) defectives per output. Test at \(5 \%\) significance level the validity of the claim if out of a sample of 20 items 2 were found to be defective. Use the binomial test.
Question 9
The diagram above shows a triangular prism with two equilateral triangular faces and three rectangular faces. The rectangular faces are numbered 1,2 and 3 whilst the triangular faces are numbered 4 and 5 .
When the prism is tossed, the probability that it lands on each rectangular face is \(2 k\) and the probability that it lands on each triangular face is \(k\).
(a)
Calculate the value of \(k\),
(b)
Define X as the random variable "the number on which the prism lands".
#### (i)
Show that \(\mathrm{E}(\mathrm{X})=2 \frac{5}{8}\).
#### (ii)
Find \(\operatorname{Var}(\mathrm{X})\).
Question 10
Two judges, A and B , independently awarded marks, \(x\) and \(y\) respectively to the architectural designers. The table below summarises the marks awarded.
| Design | Judge A(x) | Judge B(y) | | :--- | :--- | :--- | | 1 | 55 | 56 | | 2 | 40 | 37 | | 3 | 60 | 54 | | 4 | 65 | 49 | | 5 | 90 | 77 | | 6 | 30 | 33 | | 7 | 70 | 55 | | 8 | 95 | 75 | | 9 | 50 | 43 | | 10 | 45 | 47 |
$$ \begin{array}{ll} \sum x=600, & \sum x^{2}=39900 \\ \sum y=526, & \sum y^{2}=29548, \sum x y=34145 \end{array} $$
(a)
Calculate the product-moment correlation coefficient between the marks awarded by the two judges and comment.
(b)
Find the equation of the regression line of \(y\) on \(x\).
Question 11
(a)
A random sample of size 40 is selected from a particular population of fish in a fishery pond. The random variable X denotes the length of a fish in centimetres. Given that the actual lengths in the sample are summarised by \(\sum(x-20)=19\) and \(\sum(x-20)^{2}=68\), find the unbiased estimates of
#### (i)
the population mean.
#### (ii)
the population variance.
(b)
The \(95 \%\) confidence interval for the mean life of light bulbs constructed from a sample of size 36 is ( \(1023.3 \mathrm{hrs} ; 1161.7 \mathrm{hrs}\) ).
Assuming that the life of light bulbs is normally distributed, find the 99\% confidence interval for the mean life of this brand of light bulbs.
Question 12
Data from the Consumer Council of Zimbabwe shows that \(42 \%\) of Zimbabweans eat breakfast everyday. Find the probability that in a random sample of 300 Zimbabweans, the number who eat breakfast is
(i)
at most 100
(ii)
from 130 to 140
Question 13
The number of computer system breakdowns per month at a University were observed over a period of 100 months and summarised in the table below.
| Breakdowns (X) | 0 | 1 | 2 | 3 | 4 | 5 or more | | :--- | :---: | :---: | :---: | :---: | :---: | :---: | | Frequency | 15 | 25 | 30 | 21 | 9 | 0 |
Test the hypothesis that X has a Poisson distribution.
**Section (b): Mechanics**
Question 14
A block of mass 3.5 kg is released from rest at point A on a plane inclined at angle \(\propto\) to the horizontal, where \(\tan \propto=\frac{3}{4}\). The block slides down until it reaches point B at the base of the plane (see diagram).
Given that the coefficient of friction between the block and the plane is \(\frac{1}{4}\), calculate
(i)
the velocity of the block at the instant it reaches point B ,
(ii)
the time it takes the block to slide from point A to point B .
Question 15
The diagram above shows a ring of mass \(m \mathrm{~kg}\) accelerating at \(1 \mathrm{~ms}^{-2}\) along a rough horizontal wire. The accelerating force of 6 N is at an angle of \(60^{\circ}\) in the same vertical plane with the wire.
The coefficient of friction between the ring and the wire is \(\frac{1}{4}\). Find in terms of \(m\) and or \(g\), the exact value of
(i)
the normal reaction between the ring and the wire,
(ii)
the frictional force.
Hence find the value of \(m\), giving your answer to 3 decimal places.
Question 16
A particle is projected from a point O on the ground with a speed of \(\mathrm{V} \mathrm{ms}^{-1}\) at an angle of \(60^{\circ}\) to the horizontal and passes through the points A and B , where \(\mathrm{A}(\sqrt{3} ; 2)\) is a point before it reaches its maximum height above O at \(B\) (see diagram).
(a)
Express \(v^{2}\) in terms of \(g\).
(b)
Find the angle that AB makes with the horizontal.
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ZIMBABWE SCHOOL EXAMINATIONS COUNCIL <br> General Certificate of Education Advanced Level
MATHEMATICS
PAPER 4
NOVEMBER 2008 SESSION
Additional materials:
TIME: 3 hours
INSTRUCTIONS TO CANDIDATES
Write your name, Centre number and candidate number in the spaces provided on the answer paper/answer booklet. There is no restriction on the number of questions which you may attempt. If a numerical answer cannot be given exactly, and the accuracy required is not specified in the question, then in the case of an angle it should be given to the nearest degree, and in other cases it should be given correct to 2 significant figures. If a numerical value for \(g\) is necessary, take \(g=9.81 \mathrm{~ms}^{-2}\).
INFORMATION FOR CANDIDATES
The number of marks is given in brackets [] at the end of each question or part question. The total number of marks for this paper is 120 . Within each section of the paper, questions are printed in the order of their mark allocations and candidates are advised, within each section, to attempt questions sequentially. The use of an electronic calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers.
**This question paper consists of 6 printed pages and 2 blank pages.**
EZMSEC N2008 Copyright: Zimbabwe School Examinations Council, N2008.
**Section (a): Statistics**
Question 1
The stem and leaf diagram below shows the pocket money received by a group of girls in the year 1980.
| stem | leaf | | | | | | :--- | :--- | :--- | :--- | :--- | :--- | | 0 | 50 | 50 | 50 | 75 | | | 1 | 00 | 00 | 00 | 50 | 75 | | 2 | 00 | 00 | 00 | 50 | 50 | | 3 | 00 | 25 | 30 | 75 | | | 4 | 50 | | | | | | 5 | 50 | | | | |
KEY \(3 \mid 30=\$ 3.30\)
Find the mean and the standard deviation of the distribution of the pocket money received by the girls.
Question 2
A school selects \(55 \%\) of its lower sixth pupils from its own O-level pupils and the remainder comes from other schools. It is established that \(90 \%\) of accepted A-level pupils who did their O-level outside the school pass their A-level studies, and that \(70 \%\) of those who did their O-level at the school pass their A-level studies.
A pupil is selected at random from the recent A-level graduates of the school.
Find the probability that the pupil
(i)
passed A-level studies,
(ii)
did O-level outside the school, given that the pupil passed A-level studies. [2]
Question 3
A fairly constructed die has three of its sides numbered 0 each, two sides numbered 3 and one side numbered 6 . A boy pays \(\$ 5\) in order to toss the die twice and win an amount equal to the product of the two scores shown on the die.
Let Y be the random variable "the product of the two scores shown."
(i)
Construct a probability distribution table for Y .
(ii)
Find the expected value of Y and hence write down the expected profit or loss in a single game. Comment on the fairness of the game.
Question 4
A library contains a very large number of books of which \(60 \%\) are fiction and the remainder are non-fiction.
(a)
Determine correct to three decimal places, the probability that a random collection of 6 books from the library contains 5 or more fiction books.
(b)
Using a suitable approximation determine the probability that a random collection of 200 books from the library contains exactly 80 non-fiction books.
Question 5
A company in Gweru uses two machines, A and B , to manufacture steel rods whose lengths are normally distributed. The table below gives the distributions of the steel rods from the two machines.
| Machine | Mean | Variance | | :--- | :---: | :---: | | A | 15 | 0.5 | | B | 16 | 0.1 |
If two rods are selected at random from the production of each machine, find
(i)
the mean and variance of their combined length for each machine,
(ii)
the probability that the total lengths of rods from machine B is greater than the total length of rods from machine A .
Question 6
Let X be the number of claims for severe medical conditions requiring hospitalisation received by a medical insurance company in a year. Such medical conditions are estimated to affect 1 in 1000 of the population in a year.
(a)
Given that the medical insurance company receives \(n\) claims in a year, state the distribution of X.
(b)
This medical insurance company deals with two manufacturing companies A and B with 500 and 750 employees respectively. Find the probability that the number of claims received from
#### (i)
company A is 2 or more,
#### (ii)
both companies is 2 .
Question 7
(a)
A large number of samples of size \(n\) are taken from \(\mathrm{N}(100,225)\). Given that \(95 \%\) of the sample means are less than 105 , estimate the value of \(n\).
(b)
The random variables X and Y are independent and normally distributed, \(X\) being \(N(4,9)\) and \(Y\) being \(N(5,16)\). Given that a sample of 20 observations is taken from the distribution of X and a sample of 25 from the distribution of Y , find \(\mathrm{P}(\overline{\mathrm{Y}}>\overline{\mathrm{X}})\).
Question 8
(a)
The distribution of a population is known to have mean 9.27 and standard deviation 1.40. A sample of 36 was taken from this population and it gave a mean of 8.39 . Test whether there is evidence at the \(1 \%\) level that the distribution mean has decreased.
(b)
An animal breeder claims that the length of a certain species of animal is distributed normally with mean 44 cm . In order to test the truth of his claim, a sample of 21 such animals was taken and it was found that \(\bar{x}=42 \mathrm{~cm}\) and \(\mathrm{s}=6 \mathrm{~cm}\). Is there evidence at the \(5 \%\) level to refute the animal breeder's claim?