ZIMBABWE SCHOOL EXAMINATIONS COUNCIL <br> General Certificate of Education Advanced Level
**MATHEMATICS** 9164/4 **PAPER 4** JUNE 2012 SESSION 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.
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. Questions are printed in the order of their mark allocations and candidates are advised 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.
Question 1
Given that \(W \sim \operatorname{Geo}(p)\) and \(\operatorname{Var}(W)=30\), find \(E(W)\).
Question 2
\(X\) is a continuous random variable with probability density function \(\mathrm{f}(x)\) where
\[ f(x)= \begin{cases}2-2 x & 0 \leq x \leq 1 \\ 0 & \text { otherwise. }\end{cases} \]
(i)
Find the cumulative distribution function \(\mathrm{F}(x)\).
(ii)
Show that \(\mathrm{P}\left(x>\frac{1}{3}\right)=\frac{4}{9}\).
(iii)
Find the value of \(p\) such that \(\mathrm{P}(\mathrm{X}<\mathrm{p})=\frac{1}{5}\).
Question 3
In a certain court, there are only two verdicts on passing judgement, namely "convicted" or "discharged". Of all the cases that have been tried by this court, \(80\%\) of the verdicts were convictions. Suppose that when the court's verdict is "convicted" or "discharged", the respective probabilities of the accused person being innocent are 0.07 and 0.4 respectively.
By use of a tree diagram, find
(a)
the probability that a person tried by this court is innocent,
(b)
the conditional probability that an innocent person tried by this court is convicted.
Question 4
The performance of a group of 24 students who sat for their final 'O'-Level examinations in Biology and Integrated Science are shown below.
**Biology**
36, 45, 40, 60, 71, 66, 53, 42, 35, 54, 35, 43, 72, 37, 39, 34, 49, 43, 75, 58, 67, 59, 36, 67
**Integrated Science**
88, 89, 30, 34, 48, 49, 59, 65, 67, 78, 41, 70, 54, 66, 39, 49, 37, 59, 45, 63, 52, 75, 38, 38
(a)
Construct a back to back stem and leaf diagram to represent these two sets of data.
(b)
Using a scale of 2 cm to represent 10 marks, draw a pair of box and whisker plots to represent these two sets of data.
(c)
Comment briefly on the performance of the students in the tests.
Question 5
The table below shows the attitude of parents in various parts of the country towards the introduction of incentives for teachers.
| | \multicolumn{3}{|c|}{PROVINCES} | | :--- | :--- | :--- | :--- | | ATTITUDE | NORTH | MIDLANDS | SOUTH | | LIKE | 37 | 24 | 16 | | RESERVED | 33 | 50 | 38 | | DISLIKE | 20 | 36 | 46 |
Test at \(5\%\) level to find out whether there is an association between the geographical area of residence and attitude towards the introduction of incentives.
Question 6
The number of patients admitted into a clinic per day was recorded over a period of 30 days. The results are summarized below.
| Number of patients admitted | 0 | 1 | 2 | 3 | 4 | | :--- | :--- | :--- | :--- | :--- | :--- | | Number of days | 5 | 9 | 10 | 4 | 2 |
(a)
Show that the mean number of patients admitted per day is 1.63 correct to 2 decimal places.
(b)
Assuming that the above situation can be modelled by a Poisson distribution, calculate the probability that the clinic admits
#### (i)
exactly 2 patients on a particular day,
#### (ii)
at least 3 patients on two consecutive days.
(c)
If the clinic has only four beds available, calculate the probability that it will turn away some patients who need to be admitted on a particular day.
Question 7
A new vaccine claimed to cure AIDS is applied to 50 samples each of 5 monkeys and the number of living monkeys in each sample was counted after one year. The results are as follows:
| Number of living monkeys in a sample | 0 | 1 | 2 | 3 | 4 | 5 | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | Frequency | 17 | 20 | 9 | 2 | 1 | 1 |
Test at \(5\%\) level of significance to find out whether the number of living monkeys in each sample follows a Binomial distribution.
Question 8
(a)
A Form 5 class has 50 students and it is desired to choose 5 students to attend a seminar. The Form teacher gave the students small tickets numbered from 0 to 49. Those students with tickets numbered from 0 to 9 were asked to put their tickets in a hat and the Form teacher picked one at random. He noted the digit on the ticket. He then formed his sample with students whose tickets ended with the digit he had picked.
#### (i)
What do you understand by the term random sample?
#### (ii)
Explain whether or not the above method gives rise to a random sample.
(b)
A machine should be set up to cut pieces of planks which are 5.00 m. A random sample of 10 planks cut by the machine were measured and their lengths were as follows
| 4.94 | 4.93 | 5.00 | 4.76 | 5.00 | | :--- | :--- | :--- | :--- | :--- | | 4.73 | 4.63 | 5.01 | 4.65 | 5.03 |
Test at \(5\%\) level of significance to find out whether there is evidence that the machine requires resetting.
Question 9
The weight of broiler chickens are normally distributed with mean 2 kg and standard deviation 0.3 kg. The weights of layers chickens are also normally distributed with mean 1.5 kg and standard deviation 0.5 kg.
Determine correct to 2 significant figures the probability that
#### (i)
a broiler chicken weighs less than 1.8 kg,
#### (ii)
a random sample of 4 broiler chickens weigh more than 8.2 kg,
#### (iii)
a layers chicken weighs less than a broiler chicken,
#### (iv)
a random sample of 8 layers chickens weigh more than a sample of 6 broiler chickens.
Question 10
The marks obtained by 10 candidates in the two 'O'-level mathematics papers in a certain year are shown in the table below
| Paper 1 (x) | 74 | 46 | 30 | 60 | 80 | 52 | 67 | 20 | 64 | 73 | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | Paper 2 (y) | 70 | 40 | 18 | 42 | 81 | 35 | 40 | 08 | 72 | 68 |
(a)
Plot on a scatter diagram, the Paper 1 mark (\(x\)) against the Paper 2 mark (\(y\)) and comment on the relationship.
(b)
Find the appropriate regression line for estimating
#### (i)
the mark on Paper 2 by a candidate having a specific mark in Paper 1. Hence calculate the Paper 2 mark for a candidate who got \(65\%\) in Paper 1,
#### (ii)
the mark on Paper 1 by a candidate having a specific mark in Paper 2. Hence calculate the Paper 1 mark for a candidate who got \(50\%\) in Paper 2.
Question 11
On analysing the marks scored by a large number of candidates in an Olympiad Examination, the Director takes a random sample of 250 candidates and finds that the marks, \(x\), of these candidates gave
\[ \sum x=11872 \text { and } \sum x^{2}=646193 . \]
(a)
Calculate the unbiased estimate of
#### (i)
the population mean mark,
#### (ii)
its variance.
(b)
Calculate a \(90\%\) confidence interval for the population mean \(\mu\).
(c)
#### (i)
Test the hypothesis that \(\mu=49\), against an alternative that it is less than 49, at \(10\%\) significance level.
#### (ii)
Determine the values of \(\alpha\) for which the null hypothesis is rejected.
**Section (b) : Mechanics**
Question 12
A car travelling along a straight road passes through a fixed point A with a velocity of \(10 \mathrm{~ms}^{-1}\) and continues with this velocity for \(t_{1}\) seconds. The car then accelerates uniformly over the next \(t_{2}\) seconds till it attains a velocity of \(15 \mathrm{~ms}^{-1}\). The car subsequently decelerates to rest in a further \(t_{3}\) seconds.
(a)
Sketch a well labelled \((t ; v)\) graph to illustrate the motion of the car after passing A.
(b)
Given that the car's acceleration and deceleration are equal, show that \(t_{3}=3 t_{2}\).
Question 13
The diagram shows the trajectory of a ball that is projected from the top of a building 40 m high with initial velocity of \(20 \mathrm{~ms}^{-1}\) at an angle of \(30^{\circ}\) to the horizontal. The ball hits the ground at a point P (see diagram).
Find
#### (i)
the time of flight of the ball,
#### (ii)
the direction in which the ball is moving when it hits Point P.
Question 14
Two particles of mass \(m_{1} \mathrm{~kg}\) and \(m_{2} \mathrm{~kg}\) where \(m_{1}>m_{2}\) are connected by a light inelastic string which passes over a smooth fixed pulley.
(a)
Find in terms of \(\mathrm{g}, m_{1}\) and \(m_{2}\) the acceleration of the system.
(b)
Show that the tension in the string is given by \(m_{1} \mathrm{~g}\left(\frac{2 m_{2}}{m_{1}+m_{2}}\right)\).
Question 15
A particle of weight 20 N rests on a rough horizontal surface. A force \(P \mathrm{~N}\) inclined at an angle \(\theta\) to the horizontal, is applied to the particle until it is on the point of moving.
Given that the normal force \(\mathrm{R}=16 j\) and the frictional force \(\mathrm{F}=-9 i\) act on the particle where \(i\) and \(j\) are unit vectors (see diagram),
calculate
#### (i)
##### (a)
the coefficient of friction,
##### (b)
the magnitude of the constant force between the particle and the surface,
#### (ii)
##### (a)
\(\theta\),
##### (b)
\(P\).