ZIMSEC A-Level Mathematics Paper 4 November 2004 Question Paper

About this question paper

This is the ZIMSEC A-Level Mathematics question paper for Paper 4 from November 2004. Use it for exam practice under timed conditions, then check your work against the marking scheme. Time allowed: 3 hours. See the marking scheme.

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Extracted text

# ZIMBABWE SCHOOL EXAMINATIONS COUNCIL

General Certificate of Education Advanced Level

**MATHEMATICS**

PAPER 4
Monday 8 NOVEMBER 2004
![](https://ocr.unopasa.com/p/ocr_assets/930f2c1ad03b99d564f8526776dd1fcbbd3df12048d0d5968bf0122833544359.jpg)

Morning

9164/4

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.

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 7 printed pages and 1 blank page.
Copyright: Zimbabwe School Examinations Council, N2004.
[Turn over

**Section (a): Statistics**

## Question 1
A number X is randomly selected from the interval \((-\pi, \pi)\).

Find the cumulative distribution function of X .

## Question 2
Transcription checkers have found out from experience that 1 in 20 marksheets have recording errors. A checker randomly draws a sample of 8 marksheets from a marker.

Calculate the probability that

### (i)

3 of the marksheets will have recording errors,

### (ii)

at most 2 of the 8 marksheets will have recording errors.

## Question 3
The function \(f(x)= \begin{cases}\frac{3}{2} x(2-x),  |  0 \leq x \leq 1  0,  |  \text { otherwise. }\end{cases}\)

### (i)

Verify that \(f(x)\) is a probability density function.

### (ii)

Find \(P\left(X<\frac{1}{2}\right)\).

### (iii)

Calculate the probability that 2 of 3 independent values of X observed will be less than \(\frac{1}{2}\).

## Question 4
### (a)

It is given that \(X \sim \operatorname{Geo}(0.2)\). Find

#### (i)

\(\mathrm{P}(\mathrm{X} \geq 3)\),

#### (ii)

\(\operatorname{Var}(\mathrm{X})\).

### (b)

A university student visits the National Free Library frequently. On each visit she records the number of people in the queue at the checkout point. The table below shows the records for 24 randomly chosen visits.

| No. of people | 4 | 5 | 6 | 7 | 8 | 9 |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| No. of visits | 2 | 3 | 7 | 6 | 4 | 2 |

Calculate the mean and the standard deviation of the number of people in the queue.

## Question 5
Three flower vendors X, Y and Z have equal chances of selling their flowers. X has 80 red and 20 white flowers, Y has 30 red and 40 white flowers and Z has 10 red and 60 white flowers. On Valentine's day, Kudzai wants to buy a flower.

### (a)

Find the probability that she picks a red flower.

### (b)

Given that she bought a red flower, find the probability that it came from Y .

## Question 6
A multiple choice test has 60 questions. Each question has three possible answers with only one correct. Using a suitable approximation find the probability that

### (i)

one passes given that a score of at least 25 is a pass mark,

### (ii)

guesswork yields 10 to 15 correct answers for 42 questions.

## Question 7
The table below shows the interruption of service per day due to a photocopying machine breakdown.

| \(x\) Interruptions per day | 0 | 1 | 2 | 3 | 4 or more |
| :--- | :---: | :---: | :---: | :---: | :---: |
| No. of days | 27 | 30 | 12 | 3 |  |

Test whether a Poisson distribution with parameter \(\lambda=1\) is a suitable model at the \(5\%\) significance level.

For this question answers must be given correct to 3 significant figures where appropriate.

## Question 8
The yield per hectare of a crop depends on the amount of rainfall in the growing season. The value of the yield, X , in tonnes per hectare and the rainfall, Y , in centimetres per nine successive growing seasons are given in the table below.

| X | 8 | 10 | 15 | 6 | 11 | 12 | 13 | 11 | 9 |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| Y | 14 | 10 | 18 | 13 | 14 | 13 | 16 | 11 | 12 |

$$
\begin{aligned}  |  \sum \mathrm{X}=95, \quad \sum \mathrm{X}^{2}=1061, \quad \sum \mathrm{XY}=1307,   |  \sum \mathrm{Y}=121 \text { and } \sum \mathrm{Y}^{2}=1675,
\end{aligned}
$$

### (i)

Find the product moment correlation coefficient.

What does the value of the product moment correlation coefficient indicate about the yield and rainfall amount?

### (ii)

Find the regression line of \(y\) on \(x\).

What does the value of the product moment correlation coefficient indicate about the relationship between the regression line of \(y\) on \(x\) and the regression line of \(x\) on \(y\) ?

## Question 9
The credit manager for a department store believes that the average monthly credit account balances have changed from the historical average of \($ 5870\). The internal auditor took a random sample of 35 credit account balances and calculated the unbiased estimates of the mean and variance to be \($ 5790\) and \($ 62500\) respectively.

### (a)

Explain whether a one-tailed test is appropriate.

### (b)

Stating the null and alternative hypotheses clearly, test at \(5\%\) significance level whether the sample evidence supports the credit manager's belief.

### (c)

Construct a 99\% confidence interval for the true mean monthly credit account balance.

## Question 10
### (a)

Mercy travels from her Harare office to her home by …