ZIMBABWE SCHOOL EXAMINATIONS COUNCIL <br> General Certificate of Education Advanced Level
**MATHEMATICS** **PAPER 4**
**JUNE 2010 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.
**This question paper consists of 8 printed pages.**
**Section (a): Statistics**
Question 1
The diagram shows two cumulative frequency graphs for a test, marked out of 100 , given to two classes each of 40 pupils. Calculate the difference between the medians of the two classes.
Question 2
In a particular survey involving 7076 households, \(0.0248 \%\) were in favour of amending city council by-laws. By using a suitable approximation, find the probability that at least 8 households favour the amendment.
Question 3
X is a continuous random variable with probability density function \(f(x)\). The \(f(x)\) is defined by a curve \(y=x^{2} ; 0 \leq x \leq 1\) and a straight line as indicated on the diagram.
(i)
Find the value of \(h\).
(ii)
Write down the mode.
(iii)
Find the lower quartile.
Question 4
A local bookshop reported that 11\% of books sold in a year are romance novels. If the bookshop sells 316 books on a given day, find the probability that less than 40 are romance novels.
Question 5
A car is never kept in a garage at night. The probability that a night is wet in summer is 0,80 . On the morning following a wet night the probability that the car does not start is 0,25 . On the morning following a dry night this probability is 0,04 . The starter performance is independent of each morning.
Find the probability that
(i)
the car does not start in a given morning in summer,
(ii)
the night was wet given that the car did not start.
Question 6
A random variable X is defined by \(X \sim \operatorname{Bin}(n, p)\) Given that \(E(X)=\frac{3}{2}\) and \(\operatorname{Var}(X)=\frac{9}{8}\), find
(i)
the value of \(p\) and the value of \(n\),
(ii)
\(P(X \geq 3)\).
Question 7
The following table shows the observed frequency distribution on the number of matches per box in a random sample of 100 boxes.
| Number of matches | 45 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | | :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | Number of boxes | 5 | 4 | 10 | 20 | 25 | 25 | 9 | 2 |
(i)
Calculate the unbiased estimates of the mean and variance of the number of matches per box.
(ii)
Determine a \(95 \%\) confidence interval for the mean number of matches per box.
Question 8
A Harare commuter notes the departure time of the commuter train each morning. She records the duration of delays in minutes and seconds and her results for a given month are as follows: \(1: 15,12: 17,5: 49,2: 09,8: 54,6 ; 22,2: 43,2: 39,6: 24,4: 21,3: 13\), \(0: 28,1: 47,10: 37,1: 25,3: 10,4: 58,3: 16,5: 28,1: 32\)
(a)
Draw a frequency table in 2-minute intervals.
(b)
State the interval with the highest frequency.
(c)
By using the number of minutes as the stem and the number of seconds as the leaf, produce a stem and leaf plot.
(d)
A railway spokesman claims that \(80 \%\) of trains leave within 5 minutes of the scheduled time of departure. On the basis of the above information, what do you think of this claim?
Question 9
(a)
John bought a packet of 15 tomato seeds for his garden. If the probability of a seed not germinating is 0.01 , find the probability that he gets at most 13 tomato plants from his seeds.
(b)
In a competition, the title holder has a probability of 0.97 of hitting the target.
Find the probability that he hits the target at least 190 times in 200 shots.
(i)
During a practice session a gunman fires at a target from the same place until he succeeds. Given that independently for each attempt the probability of hitting the target is 0,8 , find the probability that he will need at least 3 attempts to his first success.
(ii)
If X is the number of attempts to his first success and Y is time in minutes he will take over his practice session, find the mean and variance of \(Y=3 X+2\).
Question 11
(i)
In a certain country it is known that \(34 \%\) of the adult population has some knowledge of a foreign language. If 6 adults from this country are chosen at random, find the probability that at least one of those chosen will have some knowledge of a foreign language.
(ii)
For a particular foreign language, only a very small proportion \(r \%\) of the adult population have some knowledge of it. It is required to select \(n\) adults at random, where \(n\) is chosen so that the probability of obtaining at least one adult having some knowledge of the language is to be 0.98 . Use a Poisson approximation to show that \(n=\frac{391.2}{r}\)
(iii)
For the case where \(r=\frac{1}{3}\) and \(n=917\), find the probability that precisely 5 adults in the sample will have some knowledge of the language.
Question 12
Two schools entered their pupils for the Mathematics Olympiad examination and the numbers of pupils falling within each grade are shown in the table below.
| Grade A | | Grade B | Grade C | | :--- | :---: | :---: | :---: | | School \(X\) | 23 | 19 | 18 | | School \(Y\) | 15 | 13 | 32 |
Use a \(X^{2}\) test at \(5 \%\) level of significance to determine whether proportion of pupils in the three grades are independent of the school.
Question 13
The following figures relate to the carriage of goods by road in Zimbabwe from 1988 to 1995.
| Year | Goods carried (Y) <br> (million tonnes) | Number of goods vehicles <br> registered (X) (thousands) | | :--- | :---: | :---: | | 1988 | 85 | 1630 | | 1989 | 86 | 1632 | | 1996 | 88 | 1660 | | 1991 | 90 | 1736 | | 1992 | 90 | 1778 | | 1993 | 95 | 1791 | | 1994 | 96 | 1773 | | 1995 | 98 | 1712 |
(i)
Draw a scatter diagram.
(ii)
Calculate the regression line of goods vehicles registered on goods carried and draw this line on your scatter diagram.
(iii)
Use this equation to estimate the number of goods vehicles registered if the amount of goods carried is 120 .
(iv)
Calculate the product moment correlation coefficient and comment
**Section (b): Mechanics**
Question 14
Four coplanar forces act at the point O with magnitudes and directions as shown in the diagram. The forces \(\boldsymbol{i}\) and \(\boldsymbol{j}\) are of 1 N magnitude in the directions of the \(x\) and \(y\) axes respectively. Find the resultant of the four forces giving your answer in the form \(\mathbf{a} \boldsymbol{i}+\mathbf{b} \boldsymbol{j}\), where \(a\) and \(b\) are exact constants.
Question 15
The diagram above shows three particles \(\mathrm{A}, \mathrm{B}\) and C . Particles A and B are Connected by a light inextensible string which passes over a smooth, light fixed pulley. Particle A has mass 2 kg and Particle B has mass 5 kg . The two particles are released from rest with string taut.
(a)
Find in terms of \(g\) the acceleration of this system.
(b)
At the same instant Particle C is released from rest 14 m above the horizontal ground. Given that Particle B and Particle C reach the ground at the same time, find
#### (i)
the time taken for the particles to reach the ground,
#### (ii)
the initial vertical distance of B above the ground.
Question 16
A parcel of mass 5 kg is released from rest on a rough ramp of inclination \(\theta=\sin ^{-1}\left(\frac{3}{5}\right)\) and slides down the ramp (see diagram).
Find the reactional force, R , in terms of \(g\).
Given that after 3 seconds the parcel has a speed of \(4,9 \mathrm{~m} / \mathrm{s}\) and treating the parcel as a particle, find the coefficient of friction between the particle and the ramp.
Question 17
An archer shoots an arrow from a point 1.5 m above level ground, at an angle \(30^{\circ}\) above the horizontal. The speed of projection is \(20 \mathrm{~ms}^{-1}\). The arrow strikes a tree which is 30 m horizontally from the point where the archer is standing. Calculate the magnitude and direction of the velocity of the arrow when it strikes the tree.