ZIMBABWE SCHOOL EXAMINATIONS COUNCIL <br> General Certificate of Education Ordinary Level
**MATHEMATICS**
PAPER 2 June 2012 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 11 printed pages and 1 blank pages. Copyright: Zimbabwe School Examinations Council, J2012.
**Section A [64 marks]**
Answer all the questions in this section.
Question 1
(a)
Express as a single fraction in its simplest form
#### (i)
\(3 \frac{3}{4}-2 \frac{1}{2}\),
#### (ii)
\(\frac{2 a-5}{a-4}-\frac{1}{2}\).
(b)
Find the numerical value of \(d\left(n-d^{2}\right)\) when \(n=4\) and \(d=\frac{1}{2}\).
(c)
By selling an article for \(\$ 11,40\), a shop owner made a loss of \(5\%\).
Calculate the cost price of the article.
Question 2
(a)
Factorise completely
#### (i)
\(2 k^{2}-7 k-15\),
#### (ii)
\(2 a m^{2}-2 a n^{2}-b m^{2}+b n^{2}\).
(b)
Mary has \(\$(3 x-4 y)\) and Diana has \(\$(2 y-x)\).
#### (i)
Write down in terms of \(x\) and \(y\), the simplified expression for the amount of money that Mary has more than Diana.
#### (ii)
Given that Mary has \(\$ 12\) and Diana has \(\$ 8\), find the value of \(x\) and the value of \(y\).
Question 3
(a)
Given that \(T=\frac{11 v}{6}+20\),
#### (i)
make \(v\) the subject of the formula,
#### (ii)
find \(v\) when \(T=v\).
(b)
In the diagram, \(\mathrm{AM}=7 \mathrm{~cm}\) and \(\mathrm{MC}=8 \mathrm{~cm}\). If the area of \(\triangle \mathrm{AMC}=25 \mathrm{~cm}^{2}\), calculate \(\mathrm{AMC}\).
(c)
It is given that \(\mathbf{P}=\left(\begin{array}{rr}4 & -3 \\ 5 & 2\end{array}\right)\) and \(\mathbf{Q}=\left(\begin{array}{ll}7 & 3 \\ 4 & 6\end{array}\right)\).
Find
#### (i)
the inverse of \(\mathbf{Q}\),
#### (ii)
the matrix \(\mathbf{R}\) such that \(\mathbf{P}+\mathbf{R}=\mathbf{Q}\).
Question 4
Answer the whole of this question on a sheet of plain paper. Use ruler and pair of compasses only for all constructions and show clearly all the construction lines and arcs.
(a)
On a single diagram, construct
#### (i)
trapezium ABCD in which \(\mathrm{AB}=8,5 \mathrm{~cm}, \mathrm{~A} \hat{\mathrm{~B}} \mathrm{C}=120^{\circ}\), \(\mathrm{BC}=6 \mathrm{~cm}, \mathrm{CD}=12 \mathrm{~cm}\) and AB is parallel to DC,
#### (ii)
the locus of points which are equidistant from C and D,
#### (iii)
the circle of which CD is the diameter.
(b)
Measure and write down \(\hat{\mathrm{CB}}\).
(c)
CD satisfies a certain locus. Describe this locus fully.
Question 5
(a)
In the diagram, K is 5 km due east of \(\mathrm{M}\), \(\mathrm{D}\) is 8 km due south of K and \(C\) is 10 km due east of \(D\).
Calculate
#### (i)
the length of straight line MC,
#### (ii)
the bearing of C from M correct to the nearest degree.
(b)
In the diagram, \(\mathrm{P}, \mathrm{Q}, \mathrm{R}\) and S are points on the circle centre O. AT is a tangent to the circle at P and QO is parallel to RS.
Given that \(\mathrm{PRQ}=32^{\circ}\), calculate
#### (i)
\(\mathrm{QPT}\),
#### (ii)
QPS,
#### (iii)
\(P \hat{S} R\),
#### (iv)
PRO.
Question 6
(a)
Find the equation of a straight line which passes through \(\mathrm{P}(3 ;-4)\) and \(\mathrm{Q}(-1 ; 2)\).
(b)
Solve the equation
$$ \frac{3}{2 x-5}-\frac{4}{x-3}=0 $$
(c)
In the diagram, \(A B C D\) is a cyclic quadrilateral in which \(A C\) and \(B D\) intersect at \(\mathrm{P}\). \(\mathrm{AP}=3 \mathrm{~cm}, \mathrm{PC}=4 \mathrm{~cm}\) and \(\mathrm{BP}=6 \mathrm{~cm}\).
#### (i)
Name in correct order, the triangle that is similar to triangle APD.
#### (ii)
Find the ratio of \(\frac{\text { area of } \triangle \mathrm{APD}}{\text { area of } \triangle \mathrm{CPD}}\).
#### (iii)
Given that the area of \(\triangle \mathrm{CPD}=8 \mathrm{~cm}^{2}\), calculate the area of the quadrilateral \(A B C D\).
**Section B [36 marks]**
Answer any three questions in this section.
Question 7
In the diagram, the points \(\mathrm{H}, \mathrm{G}\) and D are in a straight line on level ground. DE is a tree 6 m high and the angle of elevation of E from G is \(52^{\circ}\). The angle of depression of E from the top of a tower HF, is \(24^{\circ}\).
Given that \(\mathrm{EF}=9 \mathrm{~m}\). calculate
(a)
GEF,
(b)
the length of
#### (i)
GE,
#### (ii)
FG.
(c)
the angle of depression of G from F.
Question 8
Answer the whole of this question on a sheet of graph paper. Using a scale of 2 cm to represent 2 units on both axes, draw the \(x\) and \(y\) axes for \(-6 \leq x \leq 10\) and \(-10 \leq y \leq 8\).
(a)
Triangle \(A\) has vertices at \((3 ; 1),(1 ; 2)\) and \((2 ; 4)\). Draw and label clearly the triangle.
(b)
Triangle B is the image of triangle A under an anticlockwise rotation of \(90^{\circ}\) about \((-2 ; 2)\).
Draw and label clearly the triangle B.
(c)
A single transformation \(P\) maps triangle A onto triangle C with vertices at \((9 ; 1),(3 ; 2)\) and \((6 ; 4)\).
#### (i)
Draw and label clearly triangle C.
#### (ii)
Find and write down, the matrix which represents the transformation \(P\).
#### (iii)
Describe fully the single transformation \(P\).
(d)
Triangle \(A\) is mapped onto triangle \(D\) by an enlargement of scale factor -2 with the origin as the centre.
Draw and label clearly triangle D.
Question 9
(a)
The resistance, \(R\) newtons, to a train travelling at \(v \mathrm{~km} / \mathrm{h}\), is given by the formula \(R=\mathrm{c}+\mathrm{d} v^{2}\) where c and d are constants.
Given that \(R=4\) when \(v=20\) and that \(R=10\) when \(v=40\), find
#### (i)
the value of \(c\) and the value of d,
#### (ii)
\(v\) when \(R=3\).
(b)
In the diagram, OAB is a triangle in which H and K are points on OA and BA respectively such that the ratio \(\mathrm{OH}: \mathrm{HA}=1: 2\) and the ratio \(\mathrm{BK}: \mathrm{KA}=1: 2\), \(\overrightarrow{\mathrm{OA}}=3 \mathbf{a}\) and \(\overrightarrow{\mathrm{OB}}=3 \mathbf{b}\).
Express in terms of \(\mathbf{a}\) and/ or \(\mathbf{b}\)
#### (i)
1. \(\overrightarrow{\mathrm{OH}}\), 2. \(\overrightarrow{\mathrm{BA}}\), 3. \(\overrightarrow{\mathrm{BK}}\), 4. \(\overrightarrow{\mathrm{KA}}\), 5. \(\overrightarrow{\mathrm{HK}}\).
#### (ii)
Write down the ratio \(\frac{\mathrm{HK}}{\mathrm{OB}}\).
Question 10
(a)
Solve the equation \(3 q^{2}-5 q-5=0\) giving your answers correct to two decimal places.
(b)
$$ \text { use } \pi=\frac{22}{7} $$
In the diagram, ABCDE is a composite solid which is made up of a cylinder and a cone with a common radius \(r\) metres. \(\mathrm{AE}=4,81 \mathrm{~m}\) and \(\mathrm{DE}=2,2 \mathrm{~m}\).
Calculate
#### (i)
the common radius of the solid,
#### (ii)
the surface area of the solid excluding the shaded base,
#### (iii)
the volume of the solid.
[Curved surface area of a cone \(=\pi r l\), volume of a cone \(=\frac{1}{3} \pi r^{2} h\)]
Question 11
Answer the whole of this question on a sheet of graph paper.
Below is an incomplete table of values for \(y=x^{3}-5 x+3\).
| \(x\) | -3 | -2 | -1 | 0 | 1 | 2 | 3 | | :--- | ---: | ---: | ---: | ---: | :--- | :--- | :--- | | \(y\) | -9 | \(m\) | 7 | 3 | -1 | 1 | \(n\) |
(a)
Find the value of \(m\) and the value of \(n\).
(b)
Using scale of 2 cm to represent 1 unit on the \(x\)-axis and 2 cm to represent 5 units on the \(y\)-axis, draw the graph of \(y=x^{3}-5 x+3\).