Download and practise with this ZIMSEC A-Level Mathematics Paper 4 June 2011 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 June 2011. Use it for exam practice under timed conditions. The paper code is 9164/4. Time allowed: 3 hours.
# ZIMBABWE SCHOOL EXAMINATIONS COUNCIL <br> General Certificate of Education Advanced Level
# MATHEMATICS
PAPER 4
JUNE 2011 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
Given that the discrete random variable Y follows a geometric distribution with variance 12, find the probability that Y exceeds 3.
## Question 2
Out of 50 patients being treated at a clinic for a severe skin disease, 15 are selected at random to receive a new dietary treatment as opposed to the standard drug treatment. It has been established that the probability of a cure with the standard drug treatment is 0.65 whereas a cure with the new treatment is 0.95. After a week, one of the patients treated was selected at random.
Find the probability that
### (i)
the patient was cured,
### (ii)
the patient received the new dietary treatment given that she was cured.
## Question 3
An unbiased tetrahedral die has faces numbered \(1, 2, 3, 4\). Two such dice are thrown and the score X is found by adding together the numbers which show on the dice.
### (a)
Obtain the probability distribution of X.
### (b)
Calculate
#### (i)
\(\mathrm{E}(\mathrm{X})\),
#### (ii)
\(\operatorname{Var}(X)\).
## Question 4
X is a continuous random variable which follows a rectangular distribution over the interval \([a, b]\) where \(a < b\). Given that \(\mathrm{P}(\mathrm{X} > 4) = 0.5\) and that \(\mathrm{P}(\mathrm{X} > -5) = 0.9\), find the values of \(a\) and \(b\).
Hence calculate the value of \(\operatorname{Var}(\mathrm{X})\).
## Question 5
### (a)
If a large number of samples, size \(n\), are taken from a population which follows a normal distribution with mean 70 and standard deviation 5, find \(n\) if the probability that the sample mean exceeds 68 is 0.9254.
### (b)
A random sample of 100 adults in Muzarabani drank 7100 ml of water in one week with a standard deviation of 400 ml.
Calculate the \(95\%\) confidence interval that the mean weekly consumption of water will lie.
## Question 6
On average \(45\%\) of those taking a driving test will pass. In one particular week, 100 examinees took a test and 40 passed. One disgruntled examinee complained that these figures showed that the examiners were too harsh during that particular week. Examine the validity of the disgruntled examinee's statement at \(5\%\) level of significance.
## Question 7
Kizito High School has 1500 students who come to school every day. The probability that a student is late on a particular day is 0.002. Find, correct to 3 decimal places,
### (a)
the probability that on any given day, at least one student will be late,
## Question 8
In Bhudilana, 3 parties A, B and C contested an election. To facilitate this, the country was divided into 3 regions: North, Central and South. The results of the poll were as follows:
| | REGION | | |
| :---- | :----- | :---- | :---- |
| PARTY | NORTH | SOUTH | CENTRAL |
| A | 35 | 40 | 25 |
| B | 33 | 30 | 42 |
| C | 37 | 30 | 28 |
Test at \(5\%\) level of significance whether there is an association between the Party one voted for and the Region one resides in.
## Question 9
The masses of professional soccer players are normally distributed with a mean of 66 kg and a standard deviation \(\sigma\).
### (a)
Given that \(10\%\) of the players have masses which exceed 72 kg, find the value of \(\sigma\) correct to 3 significant figures.
### (b)
Show that the probability that the mass of a randomly chosen professional soccer player is at most 63 kg is 0.2608.
### (c)
Eleven professional soccer players are randomly selected for a particular soccer match. Find the probability that at least 3 of them weigh at most 63 kg.
## Question 10
A continuous random variable X has a probability density function \(f(x)\) given by
$$
f(x)= \begin{cases}k | \text { if } 0 \leq x \leq 3 k(4-x) | \text { if } 3 < x \leq 4 0 | \text { otherwise }\end{cases}
$$
### (a)
Show that \(k = \frac{2}{7}\).
### (b)
Find
#### (i)
\(\mathrm{E}(\mathrm{X})\),
#### (ii)
the median value.
### (c)
Show that \(\sigma\), the standard deviation of X, is 1.03 correct to 2 decimal places.
## Question 11
The mass of a toffee sweet has a normal distribution with mean 3.9 g and standard deviation 0.1 g. The mass of a mint has a normal distribution with mean 5 g and standard deviation 0.16 g.
Find the probability that
### (i)
a randomly chosen toffee sweet weighs more than 4 g,
### (ii)
of two randomly chosen toffee sweets, one weighs more than 4 g and the other one weighs less than 4 g,
### (iii)
five randomly chosen toffee sweets weigh a total of more than 20 g,
### (iv)
the total mass of five randomly chosen toffee sweets is more than the total mass of four randomly chosen mints.
## Question 12
There is a general conception that students who …