Download and practise with this ZIMSEC A-Level Mathematics Paper 4 November 2012 question paper. Useful for exam revision, past paper practice, and preparation in Zimbabwe.
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This is the ZIMSEC A-Level Mathematics question paper for Paper 4 from November 2012. Use it for exam practice under timed conditions. The paper code is 9164/4. Time allowed: 3 hours.
# ZIMBABWE SCHOOL EXAMINATIONS COUNCIL
General Certificate of Education Advanced Level
MATHEMATICS
PAPER 4
NOVEMBER 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.
**Section (a): Statistics**
## Question 1
A group of 50 children raised money for charity. The amount raised by each child to the nearest dollar ($) are recorded in the table.
| amount ($) | \(1-5\) | \(6-10\) | \(11-15\) | \(16-20\) | \(21-25\) | \(26-30\) |
| :--- | :--- | :---: | :---: | :---: | :---: | :---: |
| number of <br> children | 20 | 10 | 5 | 6 | 8 | 1 |
### (i)
State the smallest possible amount which may have been raised by one child.
### (ii)
Find
#### (a)
the mean amount,
#### (b)
standard deviation.
## Question 2
A discrete random variable \(X\), has a probability distribution given in the table.
| \(x\) | 0 | 1 | 2 |
| :---: | :---: | :---: | :---: |
| \(\mathrm{P}(X=x)\) | \(\mathrm{P}_{0}\) | \(\mathrm{P}_{1}\) | \(\mathrm{P}_{2}\) |
Given that \(\mathrm{E}(X)=1.2\) and \(\operatorname{Var}(X)=0.36\).
find the values of \(P_{0}, P_{1}\) and \(P_{2}\).
## Question 3
A student wishes to be enrolled at one of the three Zimbabwean universities BUSE, NUST and MSU. It is equally likely that the student will apply to any one of these three universities. The probability that the student will be admitted at BUSE is \(60 \%\) while the probability that the student will be admitted at NUST and MSU are \(45 \%\) and \(35 \%\) respectively.
Find the probability that
### (i)
the student will not be admitted at any one of the universities.
### (ii)
the student will apply to NUST given that the student will not be admitted into any one of the universities.
## Question 4
The masses of passengers on a flight are normally distributed with mean \(\mu\) and standard deviation \(\sigma\). \(1.7 \%\) of the passengers had masses greater than 75 kg and \(98 \%\) had masses more than 45 kg .
Find the value of \(\mu\) and the value of \(\sigma\).
## Question 5
A continuous variable \(X\) has a probability density function given by
$$
f(x)=\left\{\begin{array}{cc}
a x^{2} | 0 \leq x \leq 1
\frac{1}{4}(7-3 x) | \text { for } 1 \leq x \leq \frac{7}{3}
0 | \text { otherwise }
\end{array}\right.
$$
where \(a\) is a constant.
### (a)
Show that \(a=1\).
### (b)
Find the
#### (i)
mean,
#### (ii)
median.
## Question 6
\(20 \%\) of the employees of an organisation joined a Housing Scheme.
### (a)
From a random sample of 8 employees of the organisation, find the probability that
#### (i)
only 2 employees joined the scheme,
#### (ii)
less than 3 employees joined the scheme.
### (b)
A larger random sample of 100 employees of the organisation was taken, calculate the probability that less than 25 employees joined the scheme.
## Question 7
A transport company has 2 vans available for daily hire to ferry vegetables to a market. The daily demand for these vans may be assumed to have a Poisson distribution with mean 2.
### (a)
Show that demands for one and 2 vans on any day are equally probable.
### (b)
Calculate the probability that
#### (i)
on a particular day, the transporter may not be able to meet the demand for these vans,
#### (ii)
exactly 4 hirings will occur in two consecutive days.
## Question 8
A truck is carrying boxes of bathing soap and washing powder. The average mass of a box of bathing soap is 65 kg with variance 8 kg. A box of washing powder has an average mass of 50 kg with variance 7 kg.
Find the probability that
### (i)
a box of bathing soap weighs less than 62 kg,
### (ii)
2 randomly chosen boxes of bathing soap weigh a total of more than 140 kg,
### (iii)
the mass of a box of bath soap is greater than \(\frac{4}{3}\) of the mass of a box of washing powder.
## Question 9
The table shows a random sample of employees classified by educational qualifications and earnings.
| qualifications | earnings | | | |
| :--- | :---: | :---: | :---: | :--- |
| | low | medium | high | 650 |
| 'A'level | 115 | 293 | 242 | 950 |
| diploma | 171 | 415 | 364 | |
| degree | 684 | 992 | 864 | | 2540
Test, at \(5 \%\) level of significance, if there is any association between qualification and earnings.
## Question 10
In a region, the best ' \(O\) ' level mathematics student can score between \(70 \%\) to \(100 \%\). In experiments carried out to test this claim the percentage scored, \(x\), was recorded for 250 students after writing a Mathematics examination and the data is summarised by
$$
\begin{aligned} | \sum(x-80)=700 \text { and } | \sum(x-80)^{2}=25142 .
\end{aligned}
$$
The population mean and variance of X are denoted by \(\mu\) and \(\sigma^{2}\) respectively.
### (i)
Show that the unbiased estimate of \(\sigma^{2}\) is 93.1 , correct to one decimal place.
### (ii)
Test the hypothesis that \(\mu=85\) against the alternative \(\mu<85\) using a \(5 \%\) significant level.
### (iii)
Calculate a symmetric \(95 \%\) confidence interval for the mean score.
## Question 11
A farmer supplies potatoes daily to a …