ZIMBABWE SCHOOL EXAMINATIONS COUNCIL <br> General Certificate of Education Ordinary Level
**MATHEMATICS** **4008/2** **PAPER 2** **JUNE 2008 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, J2008.
**Section A [64 marks]**
Answer all the questions in this section.
Question 1
(a)
Solve the equation
$$ \frac{2}{3}(x+4)=x-1 $$
(b)
Factorise completely
$$ 6 y^{2}-y-12 . $$
(c)
Express \(\frac{2}{2 x-1}-\frac{3}{x}\) as a single fraction in its lowest terms.
(d)
Given that \(z=r \sqrt{n-1}\),
#### (i)
find \(z\) when \(r=0,3\) and \(n=50\),
#### (ii)
express \(n\) in terms of \(z\) and \(r\).
Question 2
(a)
In the diagram, TR and TS are tangents to the circle centre O . SR is parallel to OQ and \(\mathrm{SOR}=116^{\circ}\).
Calculate
#### (i)
\(\quad \mathrm{S} \hat{\mathrm{Q}} \mathrm{R}\),
#### (ii)
\(\quad \mathrm{R} \hat{\mathrm{S}} \mathrm{Q}\),
#### (iii)
\(\mathrm{R} \hat{\mathrm{TS}}\).
(b)
In the diagram, ABCD is a rectangle and BCEF is a square. Given that \(\mathrm{DA}=x \mathrm{~cm}, \mathrm{AF}=y \mathrm{~cm}, x+y=15\) and \(x-y=7\),
Calculate
#### (i)
the value of \(x\),
#### (ii)
the area of ABCD .
(c)
Six angles of an octagon are \(140^{\circ}\) each. The remaining angles are equal.
Find the size of each of the remaining angles.
Question 3
(a)
Given that \(\left(\begin{array}{ll}x & 2\end{array}\right)\left(\begin{array}{ll}3 & 1 \\ 0 & y\end{array}\right)=\left(\begin{array}{ll}15 & -7\end{array}\right)\),
find the value of
#### (i)
\(\quad x\),
#### (ii)
\(y\).
(b)
It is given that \(\xi \quad=\quad\{\) all triangles \(\}\), \(\mathrm{A}=\quad\) \{all equilateral triangles \}, \(\mathrm{B} \quad=\quad\) \{all isosceles triangles \} and \(\mathrm{C}=\quad\{\) all right angled triangles \(\}\).
Draw a clearly labelled Venn diagram to illustrate the relationship between the sets.
(c)
Study the number patterns shown in the table.
| Column 1 | Column 2 | Column 3 | | :--- | :--- | :--- | | 1 | 1 | 1 | | 3 | 3 | 9 | | 4 | 6 | 36 | | 4 | 10 | 100 | | 5 | | \(p\) | | 6 | \(q\) | | | . | • | - | | | 66 | | | \(r\) | 78 | | | . | . | . | | \(n\) | \(w\) | \(v\) |
#### (i)
Write down the numerical value of
##### (a)
\(p\),
##### (b)
\(q\),
##### (c)
\(\quad r\).
#### (ii)
Express \(v\) in terms of \(w\).
Question 4
(a)
In this question take \(\pi\) to be \(\frac{22}{7}\).
The diagram ABCDEF represents a metal drinking trough made from a closed cyclindrical drum that was bisected lengthwise.
The trough has a diameter of 56 cm and a capacity of 110 litres.
#### (i)
Calculate
##### (a)
the area of the cross-section ADE,
##### (b)
AB .
#### (ii)
The whole drum was bought for \(\$ 25500\) and this represents a \(70 \%\) increase in price of such a drum in the previous year.
Calculate the price of such a drum in the previous year.
(b)
Solve the equation
$$ 4 x^{2}-2 x-3=0, $$
giving your answers correct to 2 significant figures.
Question 5
(a)
In the diagram \(\overrightarrow{\mathrm{PQ}}=\mathbf{p}\) and \(\overrightarrow{\mathrm{PR}}=\mathbf{q} . \mathbf{M}\) is a midpoint of \(\mathrm{PR}, \mathrm{QL}: \mathrm{LR}=2: 1\) and MN is parallel to RQ .
#### (i)
Express in terms of \(\mathbf{p}\) and/or \(\mathbf{q}\)
##### (a)
\(\overrightarrow{\mathrm{QR}}\),
##### (b)
\(\quad \overrightarrow{\mathrm{LR}}\),
##### (c)
\(\quad \overrightarrow{\mathrm{MR}}\).
#### (ii)
Given that \(\overrightarrow{\mathrm{NM}}=k \overrightarrow{\mathrm{QR}}\), find the scalar \(k\).
(b)
In this question take \(\pi\) to be \(\frac{22}{7}\).
A hemispherical bowl is made of wood 2 cm thick.
#### (i)
Given that the bowl has an internal diameter of 14 cm , calculate the capacity of the bowl in litres.
#### (ii)
Calculate the mass of the bowl given that the density of the wood is \(0,8 \mathrm{~g} / \mathrm{cm}^{3}\).
$$ \left[\text { Volume of a sphere }=\frac{4}{3} \pi r^{3}\right] $$
Question 6
In the diagram, ABCD is a quadrilateral with AB parallel to DC . Diagonals AC and BD meet at \(\mathrm{E} . \mathrm{AB}=6 \mathrm{~cm}, \mathrm{BE}=3 \mathrm{~cm}\) and \(\mathrm{DC}=15 \mathrm{~cm}\).
(a)
Name, in the correct order, the triangle that is similar to \(\triangle \mathrm{ABE}\).
(b)
Calculate DE.
(c)
If the area of \(\triangle \mathrm{BEC}\) is \(22,5 \mathrm{~cm}^{2}\), calculate
#### (i)
the area of \(\triangle D E C\),
#### (ii)
the ratio \(\frac{\text { area of } \triangle \mathrm{ABE}}{\text { area of } \triangle \mathrm{ADC}}\) in its simplest form.
**Section B [36 marks]**
Answer any three questions in this section.
Question 7
Answer the whole of this question on a single sheet of graph paper.
Triangle \(A\) has vertices \((6 ; 4),(8 ; 6)\) and \(4 ; 6)\).
Using a scale of 2 cm to represent 2 units on each axis, draw the \(x\) and \(y\) axes for \(-8 \leq x \leq 8\) and \(-8 \leq y \leq 12\).
(a)
Draw and label triangle A.
(b)
A reflection in the line \(y=x+2\) maps triangle \(A\) onto triangle \(B\).
#### (i)
Draw the line \(y=x+2\).
#### (ii)
Draw and label triangle \(B\).
(c)
Triangle C has vertices at \((-6 ;-2),(0 ;-2)\) and \((-3 ;-6)\).
#### (i)
Draw and label triangle C .
#### (ii)
Describe fully the single transformation which maps triangle A into triangle \(C\).
(d)
A transformation P represented by the matrix \(\left(\begin{array}{cc}1 & 0 \\ -1 \frac{1}{2} & 1\end{array}\right)\) maps triangle A onto triangle D.
#### (i)
Draw and label triangle D.
#### (ii)
State the name of the transformation represented by P .
(e)
A clockwise rotation of \(90^{\circ}\), centre \((0 ; 10)\) maps triangle A onto triangle E .
Draw and label triangle E .
Question 8
Answer the whole of this question on a sheet of plain paper.
Use ruler and compasses only. All construction arcs and lines must be clearly shown.
Three schools, \(P, Q\) and \(R\) are such that the bearing of \(Q\) from \(P\) is \(045^{\circ}\) and that of \(R\) from \(P\) is \(300^{\circ}\). The distance between \(P\) and \(R\) is 18 km and \(Q\) is due east of \(R\).
(a)
#### (i)
Using a scale of 1 cm to represent 2 km , construct a single diagram to show the relative positions of the 3 schools, \(\mathrm{P}, \mathrm{Q}\) and R .
#### (ii)
Use the diagram to find the actual distance between P and Q .
#### (iii)
Construct the perpendicular from R to QP produced.
(b)
Calculate the area of the triangular region PQR , giving your answer in \(\mathrm{km}^{2}\).
Question 9
Answer the whole of this question on a single sheet of graph paper.
The following is an incomplete table of values for the graph of \(y=x^{2}+\frac{1}{x}\).
| \(x\) | 0,25 | 0,5 | 0,8 | 1 | 1,5 | 2 | 2,5 | 3 | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | \(y\) | 4,1 | 2,3 | 1,9 | \(p\) | 2,9 | 4,5 | \(q\) | 9,3 |
(a)
Calculate the value of \(p\) and the value of \(q\).
(b)
Taking 4 cm to represent 1 unit on the \(x\)-axis and 2 cm to represent 1 unit on the \(y\)-axis, draw the graph of
$$ y=x^{2}+\frac{1}{x} \text { for } 0,25 \leq x \leq 3 . $$
(c)
On the same axes draw the graph of \(2 y=5 x+2\).