ZIMBABWE SCHOOL EXAMINATIONS COUNCIL <br> General Certificate of Education Advanced Level
**MATHEMATICS** **PAPER 4**
9164/4
3 hours
JUNE 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 diameters of 25 steel rods are found to have a mean of 0.980 cm and a standard deviation of 0.015 cm .
Assuming the diameters of the steel rods are normally distributed with the same variance, find 99\% confidence limits for the population mean.
Question 2
The masses in grammes of 24 sweets in a bag are represented by the stem and leaf diagram shown below. The leaves are not ordered.
| 0.7 | 2,3,9 | | :--- | :--- | | 0.8 | 0,8 | | 0.9 | 1,9,1,8,4 | | 1.0 | 3,8,6,1 | | 1.1 | 3,3,9,3 | | 1.2 | 1,2 | | 1.3 | 9,3 | | 1.4 | 4,5 |
Key: \(0.7 \mid 2=0.72\)
(a)
Find
#### (i)
the median of the distribution,
#### (ii)
the mode of the distribution.
(b)
A sweet of mass more than 1.2 g is classified as large. Calculate the mean of large sweets that the bag contains.
Question 3
A fair die is tossed three times. Find the probability that
#### (i)
exactly one six is obtained,
#### (ii)
the first score is even, the second is odd and the third is either a one or a two.
Question 4
A dairy farmer claims that his milk bottles contain exactly one litre of milk. A consumer took a random sample of 20 bottles and found the average contents to be 0.980 litres with a standard deviation of 0.070 litres.
Test the farmer's claim at the 5\% significance level.
Question 5
Legal requirements specify that a pie manufacturer may only produce \(0.3 \%\) of pies below a mass of 80 g . The pie producing machine operates with a standard deviation of 0.8 g . The weekly output is 500000 pies. Pies with masses in excess of 83 g require additional packaging at a cost of \(\$ 10\) per pie. Assuming the weights are normally distributed, find the firm's weekly cost of extra packaging.
Question 6
A discrete random variable X takes the values 0,1 and 2 only, with probabilities \(P_{0}, P_{1}\) and \(P_{2}\) respectively. Find the values of \(P_{0}, P_{1}\) and \(P_{2}\) given \(E(X)=\frac{4}{3}\) and \(\operatorname{Var}(\mathrm{X})=\frac{5}{9}\).
Question 7
A random variable Y has the following probability distribution.
| Y | 4 | 8 | 9 | k | 16 | | :--- | :--- | :--- | :--- | :--- | :--- | | \(\mathrm{P}(\mathrm{Y}=\mathrm{y})\) | 0.14 | 0.17 | 0.19 | 0.29 | p |
Given that \(E(Y)=10.18\), calculate
#### (i)
the values of p and k ,
#### (ii)
\(\operatorname{Var}(\mathrm{Y})\).
If a random sample of 31 observations of Y is taken, calculate the probability that the sample mean is more than 9.78 correct to 2 decimal places.
Question 8
An air line experiences delays at an average rate of 1 per two weeks.
(a)
Calculate the probability that at least 2 delays will be experienced in a particular three-week period.
(b)
The air line's financial year lasts for 51 weeks. Taking their year to consist of 17 three-week periods, find the probability that in a year there are at most two three-week periods during which at least 2 delays occur.
(c)
Given that the probability of at least one delay occurring in a period of \(n\) weeks is greater than 0.875 , find the least possible value of \(n\).
Question 9
The personnel department of a company in Chegutu is doing a study about job satisfaction, classifying it as either high, medium or low. A random sample of 310 employees was given a test designed to diagnose the level of job satisfaction. Results were recorded according to salary levels.
| job satisfaction | Number of employees earning under \$10 million | Number of employees earning \$10 - \$20 million | Number of employees earning over \$20 million | | :--- | :--- | :--- | :--- | | High | 20 | 20 | 10 | | Medium | 100 | 65 | 35 | | Low | 40 | 15 | 5 |
Use a \(\chi^{2}\)-test to determine if salary and job satisfaction are independent at the \(5 \%\) level of significance.
Question 10
Independent continuous random variables \(X\) and \(Y\) have probability density functions
$$ f(x)= \begin{cases}\frac{1}{4} x, & 1 \leq x \leq 3 \\ 0, & \text { otherwise }\end{cases} $$
and
$$ g(y)= \begin{cases}\frac{1}{2} & , 2 \leq y \leq 4 \\ 0 & , \text { otherwise }\end{cases} $$
respectively.
#### (i)
Find the expectation and variance of X and Y .
#### (ii)
Hence calculate the expectation and variance of \(4 \mathrm{Y}-3 \mathrm{X}\) to 2 decimal places.
Question 11
Participants to a ZIMSEC workshop on syllabus interpretation were asked to report the distance \(d\), they drove in kilometres and the time \(t\), taken in minutes. The table below gives a random sample of the values reported.
| \(\mathrm{d}(\mathrm{km})\) | | 263 | 211 | 290 | 580 | 473 | 377 | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | \(t(\mathrm{~min})\) | 180 | 210 | 240 | 420 | 390 | 330 | |
$$ \begin{aligned} & \sum(d-300)=394, \quad \sum(d-300)^{2}=123648 \\ & \sum(t-200)=570, \quad \sum(t-200)^{2}=103500 \\ & \sum(d-300)(t-200)=103930 \end{aligned} $$
(a)
Plot these data on a scatter diagram. Use a scale of 2 cm to represent 50 km on horizontal axis and 2 cm to represent 50 minutes on vertical axis.
(b)
#### (i)
Obtain the equation of the estimated regression line of \(t\) on \(d\).
#### (ii)
Draw the regression line on your diagram.
#### (iii)
Use the regression line to estimate the time taken by a participant who travelled 350 km .
(c)
Find the product moment correlation coefficient between \(t\) and \(d\). Comment on the result.
**Section (b): Mechanics**
Question 12
The diagram shows a man pulling a crate of weight 20 N on a rough horizontal floor using a string inclined at \(60^{\circ}\) to the horizontal.
The crate is in limiting equilibrium when the man pulls with a force of magnitude 4 N .
Calculate the exact value of
#### (i)
the normal reaction between the crate and the floor,
#### (ii)
the coefficient of friction between the crate and the floor.
Question 13
A car is travelling at \(40 \mathrm{~ms}^{-1}\) when it begins to decelerate uniformly to \(25 \mathrm{~ms}^{-1}\) over 30 seconds. It travels at this speed of \(25 \mathrm{~ms}^{-1}\) for a distance of 500 m . On separate diagrams, draw fully labelled ( \(t, v\) ) and ( \(t, x\) ) graphs for the motion of the car.
Question 14
A stone is thrown from the top of a vertical wall 3 metres high with a velocity \(\mathrm{V} \mathrm{ms}^{-1}\) at an angle of depression \(\theta^{\circ}\) see diagram.
The stone hits the ground 0.5 seconds later at a point P which is 1.6 metres from the foot of the wall.
Find \(V\) and \(\theta\) by assuming that the air resistance is negligible.
Question 15
A toy car P of mass 4 kg is at rest on a rough horizontal plane. The coefficient of friction between P and the plane is \(\frac{1}{2} . \mathrm{P}\) is connected to another toy car Q of mass 8 kg by a light inextensible string passing over a smooth pulley \(\mathrm{X} . \mathrm{Q}\) is held at rest on a smooth plane inclined at \(30^{\circ}\) to the horizontal (see diagram).
When Q is released, calculate
#### (i)
the acceleration of the toy cars and the tension in the string, giving exact answers in terms of \(g\),
#### (ii)
the magnitude of the force exerted by X on the string, leaving your answer in the form \(k g \cos \alpha\), where \(k\) and \(\alpha\) are rational numbers.