ZIMBABWE SCHOOL EXAMINATIONS COUNCIL <br> General Certificate of Education Advanced Level
**MATHEMATICS**
PAPER 4
9164/4
3 hours
NOVEMBER 2008 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
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 | | | 0 | | 4 | 50 | | | | | | 3 <br> 0 | | 5 | 50 | | | | | | ， <br> 3 | | | | | | | | | |
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.
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?
Question 9
In each of seven successive weeks, the number \(N\), of road accidents between Harare and Chegutu, and the number \(P\), of police cars on patrol are recorded. The results are as shown in the table below:
| P | 2 | 3 | 4 | 4 | 5 | 6 | 6 | | :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | N | 45 | 40 | 36 | 42 | 30 | 25 | 24 |
(a)
Plot these results on a scatter diagram.
(b)
Write down the coordinates of one point through which the regression line of \(P\) on \(N\) must pass.
(c)
Calculate the regression line of \(P\) on \(N\) in the form \(\mathrm{P}=a+b \mathrm{~N}\). Draw this line on your graph and use it to estimate \(P\) when \(\mathrm{N}=35\).
(d)
#### (i)
Calculate the product-moment correlation coefficient for the given data.
#### (ii)
Interpret the result of this calculation in terms of your scatter diagram.
Question 10
In a seed viability test, 600 seeds were planted in 100 rows of 6. The number of seeds that germinated in each row was counted and the results are shown in the table below.
| No of seeds germinating per row | 0 | 1 | 2 | 3 | 4 | 5 | 6 | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | Observed number of rows | 1 | 4 | 7 | 29 | 33 | 18 | 8 |
(a)
Calculate
#### (i)
the mean number of seeds germinating per row,
#### (ii)
the expected frequencies corresponding to these observed values for a binomial distribution with the same mean as that in (i).
(b)
Carry out the appropriate \(\chi^{2}\)-test, at the \(5 \%\) level of significance, to determine whether the observed results confirm that the number of seeds germinating follows a binomial distribution.
**Section (b): Mechanics**
Question 11
A force of 2.4 newtons parallel to the wire on which the ring is threaded.
Calculate the coefficient of friction between the ring and the wire.
Question 12
A bus is travelling at \(72 \mathrm{~km} / \mathrm{hr}\) on a level road when it begins to ascend a hill of inclination \(\arcsin \left(\frac{1}{200}\right)\) to the horizontal. (See diagram).
The tractive force developed by the engine is maintained at a constant value \(\frac{\mathrm{Mg}}{100}\) and the resistance at \(\frac{\mathrm{Mg}}{10}\), where \(M\) is the mass of the bus. Find the distance moved up the hill before the bus comes to rest.
Question 13
(a)
A particle is projected with speed \(U\) at an angle of elevation \(\theta\) above the horizontal and moves freely under gravity. Prove that the range on a horizontal plane through the point of projection is
$$ \frac{\mathrm{U}^{2} \sin 2 \theta}{g}. $$
(b)
The point \(O\) is situated on the ground 10 m in front of a vertical wall. A particle is projected from \(O\), with speed \(\mathrm{U} \mathrm{ms}^{-1}\) at an angle of elevation \(30^{\circ}\) above the horizontal in the vertical plane through \(O\) perpendicular to the wall. The particle hits the wall while still moving upwards. Show that, \(\mathrm{U}>15.1\) correct to 3 significant figures.
Question 14
A bus travelling along a straight horizontal road passes through two garages \(P\) and \(Q\). The bus passes \(P\) at a speed of \(15 \mathrm{~ms}^{-1}\) and maintains this speed for \(t\) seconds, during which time it travels a distance of 900 m. The bus then decelerates uniformly at \(0.5 \mathrm{~ms}^{-2}\) over the next \(x\) metres to a speed of \(10 \mathrm{~ms}^{-1}\). The bus then accelerates uniformly at \(0.75 \mathrm{~ms}^{-2}\) for 20 seconds and reaches a speed of \(\mathrm{V} \mathrm{ms}^{-1}\). It maintains this speed for 45 seconds to reach \(Q\).
(i)
Calculate the values of \(\mathrm{t}, x\) and \(\mathrm{V}\).
(ii)
Sketch a \((t, v)\) graph for the motion of the bus between \(P\) and \(Q\).
(iii)
Find the distance between \(P\) and \(Q\).