ZIMBABWE SCHOOL EXAMINATIONS COUNCIL <br> General Certificate of Education Ordinary Level
**MATHEMATICS**
4028/2 PAPER 2 NOVEMBER 2013 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.
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 or electronic calculators 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, N2013.
**Section A [48 marks]**
Answer all the questions in this section.
Question 1
(a)
Simplify \(15,6-3 \times 4,6\).
(b)
Express \(\frac{8}{x+1}-3\) as a single fraction in its simplest form.
(c)
Solve the equation \(\frac{2}{3}(p+1)=2 \frac{1}{5}\).
(d)
Factorise completely
#### (i)
\(2 a x+3 a y+4 x+6 y\),
#### (ii)
\(8 x^{2}-18\).
Question 2
(a)
Given that \(x\) is an odd number, find the possible values of \(x\), which satisfy the inequalities \(x \geq 3\) and \(5 x-10<35\).
(b)
If \(f(x)=x^{2}-2 x+k\), where \(k\) is a constant, and \(f(3)=-32\), find the value of \(k\).
Hence, find the values of \(x\) for which \(f(x)=0\).
(c)
Albert can weed the family garden in 4 hours. His sister, Biddy, takes 6 hours to complete the same job.
If they decide to work together, assuming they maintain their working rates, calculate
#### (i)
the fraction of the garden that they can weed in 1 hour,
#### (ii)
the total time that they can take to weed the whole garden.
Question 3
(a)
Given that \(m=2 p+1\) and \(n=p-2\), express \(mn\) in terms of \(p\) in its simplest form.
(b)
In the diagram, ABC is an isosceles triangle with \(\mathrm{AB}=\mathrm{BC}\). \(\mathrm{AB}=(x+9) \mathrm{cm}, \mathrm{BC}=(2 x+5) \mathrm{cm}\) and the base, \(\mathrm{AC}=10 \mathrm{~cm}\).
#### (i)
Form an equation in terms of \(x\) and solve it.
#### (ii)
Write down the length of \(BC\).
#### (iii)
Calculate the area of triangle ABC .
#### (iv)
Given that all the lengths of the sides of \(\triangle \mathrm{ABC}\) were given to the nearest centimetre, calculate the least possible perimeter of the triangle.
Question 4
(a)
If \(d x=r+q x\),
#### (i)
find the value of \(d\) when \(q=3, r=-1\) and \(x=2\),
#### (ii)
express \(x\) in terms of \(d, q\) and \(r\).
(b)
Fifty five people attending a workshop were asked to indicate which food item they liked, choosing from 'sadza', rice or potatoes. \(S\) is the set of people who liked 'sadza', where \(n(S)=23\), \(R\) is the set of people who liked rice, where \(n(R)=19\) and \(P\) is the set of people who liked potatoes, where \(n(P)=12\). The information was displayed in a Venn diagram and the numbers in each region represent the numbers of people who liked at least one of those food items.
#### (i)
Find the value of \(x\).
#### (ii)
Calculate the number of people who liked 'sadza' only.
#### (iii)
Find the number of people who did not like any of the food items on offer.
Question 5
(a)
Given that \(\mathbf{P}=\left(\begin{array}{cc}3 & -4 \\ 5 & 1\end{array}\right)\) and \(\mathbf{Q}=\binom{2}{8}\),
find
#### (i)
the value of \(n\) if \(\mathbf{P Q}=\binom{4 n}{18}\),
#### (ii)
\(\mathrm{P}^{-1}\).
(b)
In the diagram, TUVW is a circle centre O. TOV is a diameter. STX and XUY are tangents to the circle at T and U respectively. SWV is parallel to TU and \(\mathrm{UTA}=37^{\circ}\)
#### (i)
Find
1. \(\mathrm{T} \hat{\mathrm{S}} \mathrm{V}\),
2. \(\hat{\mathrm{SVT}}\),
3. \(\mathrm{UVT}\),
4. \(\quad T \hat{X} U\).
#### (ii)
Name the triangle that is similar to \(\triangle \mathrm{UVT}\).
#### (iii)
Given that \(\mathrm{TU}=4,7 \mathrm{~cm}\), calculate the radius of the circle.
Question 6
Answer the whole of this question on a sheet of plain paper provided. Use ruler and compasses only and show clearly all construction lines and arcs.
(a)
Construct on a single diagram,
#### (i)
quadrilateral ABCD in which \(\mathrm{AB}=6 \mathrm{~cm}, \mathrm{ABC}=120^{\circ}, \mathrm{BC}=7 \mathrm{~cm}\), \(\mathrm{CD}=9 \mathrm{~cm}\) and \(\mathrm{AD}=7 \mathrm{~cm}\),
#### (ii)
the locus of points equidistant from AB and AD ,
#### (iii)
the locus of points, inside the quadrilateral ABCD , that are 3 cm from BC ,
#### (iv)
the shortest line from point C to AB produced.
(b)
Mark and label clearly the point P , inside the quadrilateral, which is equidistant from AB and AD and 3 cm from BC .
**Section B [36 marks]**
Answer any three questions in this section.
Each question carries 12 marks.
Question 7
(a)
Express \(2-2 \log 50\) as a logarithm of a single number.
(b)
In the diagram, AB and CD intersect at E . the lines \(\mathrm{AD}, \mathrm{GF}\) and CB are parallel to each other. It is given that \(\mathrm{BC}=\mathrm{GF}=4,2 \mathrm{~cm}\) and \(\mathrm{AD}=10,5 \mathrm{~cm}\).
#### (i)
Describe fully the single transformation that maps \(\triangle \mathrm{BCE}\) onto \(\triangle \mathrm{GFE}\).
#### (ii)
\(\triangle \mathrm{ADE}\) is the image of \(\triangle \mathrm{BCE}\) under an enlargement.
State the centre and the scale factor of the enlargement.
(c)
The diagram shows \(\triangle \mathrm{UVW}, \Delta \mathrm{U}_{1} \mathrm{~V}_{1} \mathrm{~W}_{1}\) and point \(\mathrm{U}_{2}\).
#### (i)
Describe completely the single transformation that maps \(\triangle \mathrm{UVW}\) onto \(\triangle \mathrm{U}_{1} \mathrm{~V}_{1} \mathrm{~W}_{1}\).
#### (ii)
Point \(\mathrm{U}_{2}\) is the image of point U under a two-way stretch.
Find
1. the stretch factor with the \(y\)-axis invariant,
2. the co-ordinates of \(\mathrm{V}_{2}\), the image of V under this two-way stretch.
Question 8
(a)
If \(\mathbf{g}=\binom{-5}{2}\) and \(\mathbf{h}=\binom{3}{4}\), express \(\mathbf{g}+2 \mathbf{h}\) in the form \(\binom{x}{y}\).
(b)
In the diagram, OABC is a parallelogram in which \(\overrightarrow{\mathrm{OA}}=4 \mathbf{p}\) and \(\overrightarrow{\mathrm{OC}}=5 \mathbf{q}\). X is a point on AC such that \(\mathrm{AX}: \mathrm{XC}=2: 3\).
#### (i)
Express in terms of \(\mathbf{p}\) and/or \(\mathbf{q}\)
1. \(\overrightarrow{\mathrm{AC}}\),
2. \(\quad \overrightarrow{\mathrm{OX}}\) in its simplest terms.
#### (ii)
Y is a point on AB such that \(\frac{\mathrm{AY}}{\mathrm{AB}}=k\), where \(k\) is a constant.
Express \(\overrightarrow{\mathrm{OY}}\) in terms of \(\mathbf{p}, \mathbf{q}\) and k .
#### (iii)
Given that \(\mathrm{OY}=h \mathrm{OX}\), where \(h\) is a constant, write down another expression for \(\overrightarrow{\mathrm{OY}}\) in terms of \(\mathbf{p}, \mathbf{q}\) and \(h\).
#### (iv)
Using results in (ii) and (iii), find the value of \(h\) and the value of \(k\).
#### (v)
Express \(\frac{\text { the area of } \triangle \mathrm{OAY}}{\text { the area of parallelogram } \mathrm{OABC}}\), as a fraction in its simplest form.
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 \(y=2+\frac{5}{x}\).
| \(x\) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | \(y\) | 7 | \(p\) | 3,7 | 3,3 | 3 | \(q\) | 2,7 | 2,6 |
(a)
Calculate the value of \(p\) and the value of \(q\).
(b)
Using a scale of 2 cm to represent 1 unit on each axis, draw the graph of \(y=2+\frac{5}{x}\) for \(1 \leq x \leq 8\).
(c)
Use the graph to estimate
#### (i)
the gradient of the graph of \(y=2+\frac{5}{x}\) when \(x=2\),
#### (ii)
the area enclosed by the graphs \(y=2+\frac{5}{x}, x=1, x=2\) and the \(x\)-axis.
(d)
On the same axes, draw the graph of \(y=x\).
Hence solve the equation \(2+\frac{5}{x}=x\).
Question 10
(a)
Solve the equation \(2 x^{2}-4 x-3=0\), giving your answers correct to one decimal place.
(b)
The diagram shows three points, \(\mathrm{P}, \mathrm{R}\) and T which are on level ground and \(\mathrm{TR}=4 \mathrm{~km}\). From T , the bearing of P is \(\mathrm{N} 56^{\circ} \mathrm{E}\), the bearing of R is \(\mathrm{S} 60^{\circ} \mathrm{E}\) and P is due north of R .
#### (i)
Calculate
1. the shortest distance between line PR and T,
2. PR .
#### (ii)
From an acroplane flying directly above point \(T\) the angle of depression of R is \(20,3^{\circ}\).
Calculate the height of the aeroplane above the ground at T.