ZIMSEC A-Level Mathematics Paper 4 June 2013 Question Paper

About this question paper

This is the ZIMSEC A-Level Mathematics question paper for Paper 4 from June 2013. Use it for exam practice under timed conditions, then check your work against the marking scheme. The paper code is 9164/4. Time allowed: 3 hours. See the marking scheme.

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Extracted text

# ZIMBABWE SCHOOL EXAMINATIONS COUNCIL <br> General Certificate of Education Advanced Level

**MATHEMATICS <br> PAPER 4**

JUNE 2013 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 table shows the number of children below the age of 15 known to have suffered from measles in 2009 in a certain village.

| Age (in years) | Number of reported cases |
| :--- | :--- |
| Under 1 | 14 |
| 1-2 | 33 |
| 3-4 | 35 |
| 5-9 | 39 |
| 10-14 | 5 |

### (a)

Represent the above information by means of a histogram.

### (b)

Calculate the mean age of children who suffered from the disease.

## Question 2
During the 2010 World Cup in a certain city, the probability that there was electricity on any particular day was \(\frac{1}{3}\). In the case that there was no electricity, a generator would be switched on. Independently, the probability that John watched a soccer match being screened live was \(\frac{1}{4}\).

### (a)

Represent the above information by means of a tree diagram.

### (b)

Given that John watched a soccer match, find the probability that there was no electricity.

## Question 3
An ice-cream vendor records his daily takings ( \($ x\) ) over a period of 30 days. The results are summarised by \(\sum x=900\) and \(\sum x^{2}=34000\).

### (a)

Find the unbiased estimates of

#### (i)

the population mean,

#### (ii)

the population variance.

### (b)

Calculate at \(95 \%\) confidence interval the mean amount he receives assuming that his daily takings are normally distributed.

## Question 4
The probability that a learner driver passes his/her test at Vehicle Inspection Department is \(\frac{1}{4}\). A learner counts the number of attempts, \(n\), until he/she passes the driving test.

### (a)

State a suitable statistical distribution which can be used to model the above situation.

### (b)

Find the mean and variance of the distribution.

### (c)

Find the smallest value of \(n\), for which there is a probability of at least 0.7 , that the learner will need only \(n\) or fewer trials to pass the test.

## Question 5
A continuous random variable \(X\) has a probability density function \(f(x)\) given below

$$
f(x)= \begin{cases}\frac{1}{2}  |  0 \leq x<0.5  \frac{1}{5}(3-x),  |  0.5 \leq x \leq 3  0,  |  \text { otherwise. }\end{cases}
$$

### (a)

Sketch the graph of \(f(x)\).

### (b)

Find the median.

### (c)

Evaluate \(\mathrm{P}(x<1.2)\).

## Question 6
A sample of 10 items are taken from a production line to check if the machine is functioning properly. The components produced by the machine are set to have a mean diameter of 2 cm and a standard deviation of 0.03 cm . The ten items had their diameters measured and the results were:

| 2.17 | 1.93 | 2.02 | 1.97 | 2.00 |
| :--- | :--- | :--- | :--- | :--- |
| 2.01 | 2.02 | 1.89 | 1.99 | 2.01 |

Test at \(5 \%\) level of significance whether the components produced by the machine are of the required standard.

## Question 7
### (a)

Distinguish between a 1 -tailed test and a 2 -tailed test.

### (b)

A political party claims that it commands \(60 \%\) of the voters. To test this, a random sample of 300 potential voters were asked which party they would vote for. 160 confirmed that they would vote for that party.

Establish whether this sample supports the claim by the party. Test at \(10 \%\) level of significance.

## Question 8
Mangoes are boxed into cartons, each containing 500 mangoes. The probability that a mango is rotten is 0.002 . Buyers of these cartons of mangoes will return any carton that contains 4 or more rotten mangoes.

### (a)

#### (i)

Find the expected number of rotten mangoes per carton,

#### (ii)

State, giving a reason, the most appropriate statistical distribution which can be used to model the above situation,

### (b)

Find the probability that

#### (i)

a carton of mangoes is not returned,

#### (ii)

if two cartons of mangoes are chosen at random, they contain at least three rotten mangoes.

## Question 9
The following data shows the ownership of satellite dishes by different social classes in a randomly chosen sample of 150 households in a town.

| social class | number of people who own satellite dishes | number of those without a satellite dish. |
| :--- | :--- | :--- |
| Executive | 15 | 10 |
| Managerial | 23 | 8 |
| Working | 54 | 40 |

Test at \(5 \%\) level of significance to establish if there is an association between ownership of a satellite dish and social class in the town.

## Question 10
Oranges are sold in small pockets which are normally distributed with mean weight \(\mu\) and variance \(\sigma^{2}\). The probability that a randomly chosen pocket weighs more than 3.2 kg is 0.01 and that a randomly chosen pocket weighs less than 2.4 kg is 0.2 .

### (a)

Calculate \(\mu\) and show that \(\sigma^{2}=0.38\) correct to 2 decimal places.

### (b)

Potatoes are also sold in small …