ZIMBABWE SCHOOL EXAMINATIONS COUNCIL
General Certificate of Education Advanced Level
**MATHEMATICS**
PAPER 4 Monday 8 NOVEMBER 2004
Morning
9164/4
3 hours
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.
This question paper consists of 7 printed pages and 1 blank page. Copyright: Zimbabwe School Examinations Council, N2004. [Turn over
**Section (a): Statistics**
Question 1
A number X is randomly selected from the interval \((-\pi, \pi)\).
Find the cumulative distribution function of X .
Question 2
Transcription checkers have found out from experience that 1 in 20 marksheets have recording errors. A checker randomly draws a sample of 8 marksheets from a marker.
Calculate the probability that
(i)
3 of the marksheets will have recording errors,
(ii)
at most 2 of the 8 marksheets will have recording errors.
Question 3
The function \(f(x)= \begin{cases}\frac{3}{2} x(2-x), & 0 \leq x \leq 1 \\ 0, & \text { otherwise. }\end{cases}\)
(i)
Verify that \(f(x)\) is a probability density function.
(ii)
Find \(P\left(X<\frac{1}{2}\right)\).
(iii)
Calculate the probability that 2 of 3 independent values of X observed will be less than \(\frac{1}{2}\).
Question 4
(a)
It is given that \(X \sim \operatorname{Geo}(0.2)\). Find
#### (i)
\(\mathrm{P}(\mathrm{X} \geq 3)\),
#### (ii)
\(\operatorname{Var}(\mathrm{X})\).
(b)
A university student visits the National Free Library frequently. On each visit she records the number of people in the queue at the checkout point. The table below shows the records for 24 randomly chosen visits.
| No. of people | 4 | 5 | 6 | 7 | 8 | 9 | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | No. of visits | 2 | 3 | 7 | 6 | 4 | 2 |
Calculate the mean and the standard deviation of the number of people in the queue.
Question 5
Three flower vendors X, Y and Z have equal chances of selling their flowers. X has 80 red and 20 white flowers, Y has 30 red and 40 white flowers and Z has 10 red and 60 white flowers. On Valentine's day, Kudzai wants to buy a flower.
(a)
Find the probability that she picks a red flower.
(b)
Given that she bought a red flower, find the probability that it came from Y .
Question 6
A multiple choice test has 60 questions. Each question has three possible answers with only one correct. Using a suitable approximation find the probability that
(i)
one passes given that a score of at least 25 is a pass mark,
(ii)
guesswork yields 10 to 15 correct answers for 42 questions.
Question 7
The table below shows the interruption of service per day due to a photocopying machine breakdown.
| \(x\) Interruptions per day | 0 | 1 | 2 | 3 | 4 or more | | :--- | :---: | :---: | :---: | :---: | :---: | | No. of days | 27 | 30 | 12 | 3 | |
Test whether a Poisson distribution with parameter \(\lambda=1\) is a suitable model at the \(5\%\) significance level.
For this question answers must be given correct to 3 significant figures where appropriate.
Question 8
The yield per hectare of a crop depends on the amount of rainfall in the growing season. The value of the yield, X , in tonnes per hectare and the rainfall, Y , in centimetres per nine successive growing seasons are given in the table below.
| X | 8 | 10 | 15 | 6 | 11 | 12 | 13 | 11 | 9 | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | Y | 14 | 10 | 18 | 13 | 14 | 13 | 16 | 11 | 12 |
$$ \begin{aligned} & \sum \mathrm{X}=95, \quad \sum \mathrm{X}^{2}=1061, \quad \sum \mathrm{XY}=1307, \\ & \sum \mathrm{Y}=121 \text { and } \sum \mathrm{Y}^{2}=1675, \end{aligned} $$
(i)
Find the product moment correlation coefficient.
What does the value of the product moment correlation coefficient indicate about the yield and rainfall amount?
(ii)
Find the regression line of \(y\) on \(x\).
What does the value of the product moment correlation coefficient indicate about the relationship between the regression line of \(y\) on \(x\) and the regression line of \(x\) on \(y\) ?
Question 9
The credit manager for a department store believes that the average monthly credit account balances have changed from the historical average of \(\$ 5870\). The internal auditor took a random sample of 35 credit account balances and calculated the unbiased estimates of the mean and variance to be \(\$ 5790\) and \(\$ 62500\) respectively.
(a)
Explain whether a one-tailed test is appropriate.
(b)
Stating the null and alternative hypotheses clearly, test at \(5\%\) significance level whether the sample evidence supports the credit manager's belief.
(c)
Construct a 99\% confidence interval for the true mean monthly credit account balance.
Question 10
(a)
Mercy travels from her Harare office to her home by commuter omnibus from station A to station B. Her walking times to station A from the office and from station B to her home add up to 20 minutes. The variable factors measured in minutes are as shown in the table below.
| | Mean | Standard deviation | | :--- | :--- | :--- | | Waiting time | 30 | 5.4 | | Bus journey | 50 | 2.5 |
Assuming that these two factors are independent and normally distributed, find the probability that the whole journey takes
#### (i)
less than 88 minutes,
#### (ii)
between 94 and 102 minutes.
(b)
Tendai and Chipo make typographical errors on average at a rate of 0.4 and 0.6 per page respectively. The two are asked to type end of term examinations for English Language. Given that the examinations consists of 10 typewritten pages each, find the probability that
#### (i)
Tendai will make at most 2 errors,
#### (ii)
the total errors made by the two are more than 2.
Question 11
AIDS pandemic. In one study at an AIDS awareness campaign conference, 100 questionnaires were issued out randomly. Assuming that the population mean and standard deviation of the questionnaire scores are 38 and 5 respectively,
(i)
state the sampling distribution of the sample mean questionnaire score,
(ii)
calculate the probability that the sample mean score exceeds 39.1.
Given that the questionnaire mean score was 39.1, state and explain the nature of the sample.
(b)
A population of locusts has mean mass \(\mu \mathrm{g}\) and standard deviation 6 g. A random sample of size 100 is taken. State the distribution of the sample mean mass.
Given that the actual masses in the sample are summarised by \(\sum(x-50)=270\) and \(\sum(x-50)^{2}=2540\), where \(x \mathrm{~g}\) is the mass of a locust, find
#### (i)
unbiased estimates of \(\mu\) and \(\sigma\),
#### (ii)
a \(95\%\) confidence interval for the population mean mass.
Twenty different random samples are taken and a \(95\%\) confidence interval for \(\mu\) is calculated for each sample.
State the expectation of the number of these confidence intervals that will contain \(\mu\).
**Section (b) : Mechanics**
Question 12
The diagram below shows the \((t, x)\) graph for a particle which moves along a straight line.
(i)
Find the coordinates of the point A.
(ii)
Write down the final displacement of the particle.
(iii)
Find the average velocity of the particle.
Question 13
Two forces, P newtons and Q newtons are inclined at an angle of \(\theta^{\circ}\) to each other. When \(|\mathrm{P}|=9\) and \(|\mathrm{Q}|=4\), the resultant has the same magnitude, R newtons as the resultant in the case when \(|\mathrm{P}|=9\) and \(|\mathrm{Q}|=8\) with \(\theta\) unchanged.
Find the values of \(\cos \theta\) and R .
Question 14
The diagram above shows a particle at the point O being projected with speed \(8 \mathrm{~ms}^{-1}\) at an angle \(\theta^{\circ}\) above the horizontal. The particle passes through the point \(\mathrm{A}(8 ;-1.81)\).
(a)
Calculate the possible values of \(\theta\).
(b)
State the value of \(\theta\) corresponding to the minimum time taken to reach point A.
Question 15
A particle A of mass \(6m\) kg lies on a rough plane inclined at an angle of \(30^{\circ}\) to the horizontal. A light inextensible string is attached to A and passes over a smooth pulley P fixed at the top of the inclined plane. To the other end of the string is attached a particle B, of mass \(2m\) kg which hangs freely. The particles are released from rest with the string taut and the coefficient of friction between the plane and A is 0.25.
Calculate, in terms of \(m\) and \(g\),
(i)
the magnitude of the tension in the string,
(ii)
the acceleration of A down the plane,
(iii)
the magnitude of the force exerted by the string on P.
