ZIMBABWE SCHOOL EXAMINATIONS COUNCIL <br> General Certificate of Education Ordinary Level
**MATHEMATICS**
PAPER 2 JUNE 2007 SESSION Additional materials: Answer paper Geometrical instruments Graph paper (3 sheets) Mathematical tables Plain paper (1 sheet)
4008/2
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, J2007.
**Section A [64 marks]** Answer all the questions in this section.
Question 1
(a)
Find the value of \(4-2 \frac{2}{3} \times 1 \frac{1}{5}\).
(b)
P, Q and R are subsets of \(\xi\) where \(n(\mathrm{P} \cap \mathrm{Q})=0\), \(\mathrm{R} \subset \mathrm{Q}\), \(\mathrm{P} \cap \mathrm{R}=\phi\). Draw and label the Venn diagram that represents these sets.
(c)
Factorise completely
#### (i)
\(3 d f-d^{2}-3 e f+d e\),
#### (ii)
\(3 m^{3}-27 m\).
Question 2
(a)
Given that \(\mathbf{A}=\left(\begin{array}{ll}3 & -2 \\ 4 & -5\end{array}\right), \quad \mathbf{B}=\binom{2}{3}\) and \(\mathbf{C}=\left(\begin{array}{cc}-1 & 5 \\ 3 & 0\end{array}\right) \quad\).
Find
#### (i)
\(2 \mathrm{~A}+\mathrm{C}\),
#### (ii)
AB,
#### (iii)
\(\mathrm{A}^{-1}\).
(b)
Given that \(V=\sqrt{\frac{l+b}{l-b}}\),
#### (i)
make \(l\) the subject of the formula,
#### (ii)
find \(l\) when \(b=16\) and \(V=\frac{1}{3}\).
Question 3
(a)
#### (i)
Solve the inequalities
$$ y-4<3 y+2 \leq 6, $$
leaving your answer in the form \(a<y \leq b\).
#### (ii)
Illustrate your answer on a number line.
(b)
Solve the equation
$$ \frac{4}{x-3}=\frac{2}{x}+1 $$
(c)
A tourist exchanged 4200 South African Rands (R) for 157500 Zimbabwean dollars (Z$).
Calculate
#### (i)
the equivalent of R1 in Z$,
#### (ii)
the amount in Z$ he would get if he exchanged R250,
#### (iii)
the equivalent cost, in Rands, of a gift marked Z$ 46200.
Question 4
(a)
In the diagram, OACB is a sector of a circle of radius 15 cm. \(A \hat{D} O=90^{\circ}\) and \(O D=12 \mathrm{~cm}\).
Calculate
#### (i)
the size of \(A \hat{O} B\) to the nearest degree,
#### (ii)
the maximum number of such sectors which can be drawn in the circle without overlapping,
#### (iii)
the angle of the sector which remains.
(b)
In the diagram, A, B, C and D are points on the circumference of a circle centre O. AT and CT are tangents to the circle. \(\hat{\mathrm{BAT}}=30^{\circ}\) and \(\hat{\mathrm{AOC}}=100^{\circ}\).
Find
#### (i)
\(\mathrm{OAB}\),
#### (ii)
\(\mathrm{A} \hat{\mathrm{D}} \mathrm{B}\),
#### (iii)
\(\mathrm{A} \hat{\mathrm{T}} \mathrm{C}\),
#### (iv)
\(B \hat{D} C\).
Question 5
(a)
The distances of three planets from the sun are as follows.
| Planet | Distance \(/ \mathrm{km}\) | | :--- | :--- | | B | \(5,97 \times 10^{7}\) | | C | \(7,78 \times 10^{8}\) | | D | \(2,87 \times 10^{9}\) |
#### (i)
Find the ratio of the distance of planet D from the sun to the distance of planet B from the sun, in the form \(k: 1\), giving \(k\) to the nearest whole number.
#### (ii)
Light travels at a speed of \(3 \times 10^{8}\) metres per second. Calculate the time, correct to the nearest minute, that light takes to travel from the sun to planet C.
(b)
A packet contains 60 identical sweets of which 40 are red and the rest are yellow.
#### (i)
If one sweet is picked at random from the packet, find the probability that it is red.
#### (ii)
Two sweets are picked from the packet without replacement. Draw a probability tree diagram to illustrate all the possible outcomes.
#### (iii)
Hence or otherwise find the probability that the two sweets picked are of the same colour.
Question 6
Answer the whole of this question on a sheet of plain paper.
**Use ruler and compasses only and show all construction lines and arcs.**
(a)
Construct, on a single diagram,
#### (i)
a quadrilateral ABCD in which \(\mathrm{AB}=10 \mathrm{~cm}, \mathrm{AD}=6 \mathrm{~cm}\). \(B C=13 \mathrm{~cm}, A \hat{B} C=30^{\circ}\) and \(\hat{D A B}=120^{\circ}\),
#### (ii)
the locus of points equidistant from AB and AD,
#### (iii)
the locus of points 5 cm from A,
#### (iv)
the locus of points 3 cm from AD and on the same side of AD as B.
(b)
Measure and write down the size of \(B \hat{C} D\).
**Section B [36 marks]** Answer any three questions in this section.
Question 7
(a)
The cash price of an electric stove is $3400000. The stove may be bought on hire purchase by paying a deposit of \(35 \%\) of the cash price and 12 equal monthly instalment of $300000 each.
Calculate
#### (i)
the deposit,
#### (ii)
the total amount to be paid if the stove is bought by hire purchase,
#### (iii)
the difference between the cash price and the hire purchase price.
(b)
A bus was travelling at \(120 \mathrm{~km} / \mathrm{h}\) when the driver spotted a police roadblock ahead. He reduced speed uniformly until the bus stopped at the roadblock after 30 seconds.
#### (i)
Convert \(120 \mathrm{~km} / \mathrm{h}\) to a speed in \(\mathrm{m} / \mathrm{s}\).
#### (ii)
Calculate the deceleration in \(\mathrm{m} / \mathrm{s}^{2}\).
#### (iii)
Calculate the distance, in metres, covered by the bus while decelerating.
Question 8
(a)
Two points P and Q have position vectors \(\mathbf{p}\) and \(\mathbf{q}\) respectively relative to the origin O. Given that \(\mathbf{p}=\binom{2}{5}\) and \(\overrightarrow{\mathrm{PQ}}=\binom{3}{1}\), find
#### (i)
\(\mathbf{q}\),
#### (ii)
\(|\overrightarrow{\mathrm{PQ}}|\).
(b)
Given also that \(\mathbf{r}=\binom{-3}{2}, \mathbf{t}=\binom{1}{6}\) and \(m \mathbf{p}+n \mathbf{r}=\mathbf{t}\), write down two equations in \(m\) and \(n\) and solve them simultaneously.
(c)
Solve the equation \(2 x^{2}+4 x-3=0\), giving your answers correct to 2 decimal places.
Question 9
Answer the whole of this question on a single sheet of graph paper.
Below is an incomplete table of values for the graph of \(y=\frac{8}{x^{2}}\).
| \(x\) | 1 | 1,5 | 2 | 2,5 | 3 | 4 | 5 | 6 | | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | \(y\) | 8 | 3,5 | 2 | 1,3 | 0,9 | \(p\) | 0,3 | 0,2 |
(a)
Find the value of \(p\).
(b)
Using a scale of 2 cm to represent 1 unit on both axes, draw the graph of \(y=\frac{8}{x^{2}}\) for \(1 \leq x \leq 6\).
(c)
Use the graph to find the gradient of the curve at \(x=2\).
(d)
On the same axes, draw the straight line passing through \((2 ; 3)\) with gradient \(-\frac{5}{4}\).
(e)
Write down the \(x\)-coordinate of each point of intersection of the two graphs.
Question 10
Answer the whole of this question on a sheet of graph paper.
Using a scale of 2 cm to represent 1 unit on both axes, draw the \(x\) and \(y\) axes for \(-6 \leq x \leq 3\) and \(-5 \leq y \leq 5\).
(a)
A kite K has vertices at \((1 ; 3),(2 ; 1),(3 ; 3)\) and \((2 ; 4)\). Draw and label the kite K.
(b)
The kite K is reflected in the line \(x=-1\). Draw and label the image \(K_{1}\) of the kite under this reflection.
(c)
Another kite \(\mathrm{K}_{2}\) has vertices at \((-1 ; 2),(-3 ; 1),(-4 ; 2)\) and \((-3 ; 3)\). Draw and label the kite \(\mathrm{K}_{2}\). Describe completely the single transformation which maps K onto \(\mathrm{K}_{2}\).
(d)
\(\mathrm{K}_{3}\) is the image of K under an enlargement of scale factor \(-\frac{1}{2}\) with the origin as centre. Draw and label \(\mathrm{K}_{3}\).
(e)
\(\mathrm{K}_{4}\) is the image of K under a translation of vector \(\binom{1}{-4}\). Draw and label \(\mathrm{K}_{4}\).
Question 11
In this question take \(\pi\) to be 3,142.
The diagram above shows a fuel tank, 5 metres long, lying horizontally on its side. ABCD is the surface area of the liquid fuel in the tank. Safety standards require that when the fuel level is above half, the surface area ABCD must not be less than 2 square metres.
(a)
Calculate the minimum value of AB.
The diagram shows the cross-section of the tank with centre O and diameter 2 metres. A dipstick PQ, passing through the centre of the cross-section, is used to measure the depth of the fuel in the tank. The depth of the fuel was found to be 1,8 metres.