ZIMBABWE SCHOOL EXAMINATIONS COUNCIL <br> General Certificate of Education Ordinary Level
**MATHEMATICS**
PAPER 2
4008/2
NOVEMBER 2006 SESSION
2 hours 30 minutes
Additional materials:
Answer paper
Geometrical instruments
Graph paper (3 sheets)
Mathematical tables
Plain paper (1 sheet)
TIME 2 hours 30 minutes
INSTRUCTIONS TO CANDIDATES
Write your name, Centre number and candidate number in the spaces provided on the answer paper/answer booklet.
Answer all questions in Section A and any three questions from Section B.
Write your answers on the separate answer paper provided.
If you use more than one sheet of paper, fasten the sheets together.
**Electronic calculators must not be used.**
All working must be clearly shown. It should be done on the same sheet as the rest of the answer.
Omission of essential working will result in loss of marks.
If the degree of accuracy is not specified in the question and if the answer is not exact, the answer should be given to three significant figures. Answers in degrees should be given to one decimal place.
INFORMATION FOR CANDIDATES
The number of marks is given in brackets [ ] at the end of each question or part question. Mathematical tables may be used to evaluate explicit numerical expressions.
This question paper consists of 13 printed pages and 3 blank pages.
Copyright: Zimbabwe School Examinations Council, N2006.
CZIMSEC N2006
[Turn over]
**Section A: [64 marks]**
Answer all the questions in this section.
Question 1
(a)
Evaluate \(5 \frac{1}{4}-1 \frac{2}{3} \times 2 \frac{1}{10}\), giving your answer as a mixed number in its lowest terms.
(b)
Noma has \((4 x-3 y)\) dollars and Rudo has \((5 x-y)\) dollars. Find, in its simplest form, the amount of money
#### (i)
they have altogether,
#### (ii)
Rudo has more than Noma.
(c)
If \(f(x)=x^{3}-3 x^{2}+k x-4\), find \(k\) given that \(f(3)=11\).
Question 2
(a)
Factorise completely
#### (i)
\(3 m p+n p-6 m q-2 n q\),
#### (ii)
\(16-9 r^{2}\).
(b)
It is given that
\[ \mathrm{A}=\pi r\left(h^{2}-r^{2}\right) \]
#### (i)
Make \(h\) the subject of the formula.
#### (ii)
Find the value of \(h\) when \(\mathrm{A}=330, r=7\) and \(\pi=\frac{22}{7}\).
Question 3
(a)
In a class of 40, every pupil studies at least one of the subjects Mathematics, Geography and Accounts.
4 pupils study Mathematics and Geography,
5 study Mathematics and Accounts,
7 study Geography and Accounts,
15 study Mathematics only,
13 study Geography only and
4 study Accounts only.
Find the number of pupils who study all the three subjects.
(b)
Express \(\frac{b}{a^{2}-a b}+\frac{a}{b^{2}-a b}\) as a single fraction in its lowest terms.
(c)
Solve the equation
\[ \frac{1}{2-m}-\frac{3}{m-4}=0 \]
Question 4
Take \(\pi\) to be 3,142.
The diagram shows a clockface in the shape of two regular concentric octagons centre O. O, A and B are in a straight line, \(\mathrm{BO}=6 \mathrm{~cm}\) and \(\mathrm{AO}=5 \mathrm{~cm}\). The area between the octagons is decorated.
(a)
Determine the size of \(A \hat{O} D\).
(b)
Calculate
#### (i)
the area of \(\triangle A O D\),
#### (ii)
the area of \(\triangle \mathrm{BOC}\),
#### (iii)
the area that is decorated.
(c)
Calculate the angle through which the minute hand turns in 18 minutes.
(d)
If the length of the minute hand is 4 cm, calculate the distance the tip of the minute hand moves in 18 minutes.
Question 5
(a)
Given that \(\mathbf{M}=\left(\begin{array}{cc}4 & -9 \\ -2 & 5\end{array}\right), \quad \mathbf{N}=\left(\begin{array}{cc}1 & 3 \\ 0 & -1\end{array}\right)\) and \(\mathbf{L}=\left(\begin{array}{cc}2 d & 4 \\ 1 & 3\end{array}\right)\), find
#### (i)
\(\mathbf{M}+2 \mathbf{N}\),
#### (ii)
MN,
#### (iii)
the value of \(d\) which makes matrix L singular.
(b)
In the diagram, PRT and OQT are straight lines. \(\overrightarrow{\mathrm{OP}}=2 p, \overrightarrow{\mathrm{OQ}}=3 q\) and \(\overrightarrow{\mathrm{PR}}=3 \mathbf{p}-\mathbf{q}\).
#### (i)
Express \(\overrightarrow{\mathrm{RQ}}\) as simply as possible in terms of \(\mathbf{p}\) and/or \(\mathbf{q}\).
#### (ii)
Given that \(\mathrm{PT}=m \mathrm{PR}\), express \(\overrightarrow{\mathrm{PT}}\) in terms of \(\mathbf{p}, \mathbf{q}\) and \(m\).
#### (iii)
Given also that \(\mathrm{OT}=n \mathrm{OQ}\) form an equation connecting \(\mathbf{p}, \mathbf{q}, m\) and \(n\). Hence find the value of \(m\) and the value of \(n\).
Question 6
Answer the whole of this question on a sheet of plain paper.
Use ruler and compasses only for all constructions and clearly show all construction lines and arcs on a single diagram.
(a)
Construct a quadrilateral ABCD in which \(\mathrm{AB}=4 \mathrm{~cm}, \mathrm{BC}=6 \mathrm{~cm}\), \(\mathrm{CD}=5 \mathrm{~cm}, \hat{\mathrm{ABC}}=135^{\circ}\) and \(\hat{\mathrm{BCD}}=120^{\circ}\).
(b)
Measure and write down
#### (i)
the length of AD,
#### (ii)
\(\hat{B A D}\).
(c)
Construct the locus of points
#### (i)
equidistant from AB and BC,
#### (ii)
3 cm from BC and on the same side of BC as A,
#### (iii)
4 cm from B.
**Section B: [36 marks]**
Answer any three questions from this section.
Question 7
(a)
Solve the equation
\[ 2 x^{2}+6 x+1=0, \]
giving your answers to 2 decimal places.
(b)
A geographical globe has a diameter of 48 cm. A miniature model of the globe has a diameter of 8 cm.
#### (i)
Calculate the surface area of the model.
#### (ii)
On the globe, the map of Zimbabwe occupies an area of \(23,04 \mathrm{~cm}^{2}\). Calculate the corresponding area on the model.
(c)
If the globes are solid, calculate the volume of the model.
Volume of a sphere \(=\frac{4}{3} \pi r^{3}\)
Surface area of a sphere \(=4 \pi r^{2}\)
Take \(\pi\) to be 3,142
Question 8
(a)
In the diagram, R is the region bounded by lines \(m, n, l\) and p.
State the vertical scale.
(b)
Use the region R to answer the following questions.
#### (i)
Write down 3 inequalities other than \(y \leq-\frac{2}{5} x+32\) which define R.
#### (ii)
State the maximum value of \(y\).
#### (iii)
Given that \((x ; y)\) is a point inside the region R and that \(x\) and \(y\) are integers, write down the value of \(x\) and the value of \(y\) which make \((x+y)\) a maximum.
#### (iv)
Find the maximum value of \(40 x+20 y\).
Question 9
Answer the whole of this question on a sheet of graph paper.
The following is an incomplete table of values for the function \(y=\frac{12}{x}-1\).
| \(x\) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | \(y\) | 11 | 5 | 3 | 2 | \(p\) | 1 | 0,7 | \(q\) |
(a)
Calculate the value of \(p\) and the value of \(q\).
(b)
Using a scale of 2 cm to represent 1 unit on the \(x\)-axis and 2 cm to represent 2 units on the \(y\)-axis, draw the graph of \(y=\frac{12}{x}-1\) for \(1 \leq x \leq 8\).
(c)
Use the graph to estimate
#### (i)
the gradient of the curve at \(x=2\),
#### (ii)
the area of the region between the curve, the \(x\)-axis and the lines \(x=3\) and \(x=6\).
(d)
On the same axes, draw the graph of \(y=x+4\).
Hence solve the equation \(\frac{12}{x}-1=x+4\).
Question 10
(a)
In the diagram, A, B, C and D are points on the circumference of a circle centre O. ABT is a straight line and CT is a tangent to the circle at C. \(\mathrm{ABC}=71^{\circ}\) and \(\mathrm{B} \hat{\mathrm{CO}}=47^{\circ}\).
Calculate