ZIMBABWE SCHOOL EXAMINATIONS COUNCIL <br> General Certificate of Education Ordinary Level
**MATHEMATICS** 4028/2 PAPER 2
REPLACEMENT PAPER
NOVEMBER 2012 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, N2012.
**Section A [64 marks]**
Answer all the questions in this section.
Question 1
(b)
At a certain Secondary School, 400 pupils sat for a Form 1 entrance test. If 150 of them passed the test, find the percentage that failed.
(c)
Factorise completely
#### (i)
\(x^{2}+2 x-3\),
#### (ii)
\(x^{2}-1\).
Hence write down the lowest common multiple (L.C.M) of \(x^{2}+2 x-3\) and \(x^{2}-1\).
(d)
If \(m=2,6 \times 10^{-3}\) and \(n=4,0 \times 10^{7}\), calculate \(m n\), giving your answer in standard form.
Question 2
(a)
It is given that \(\xi=\{1 ; 2 ; 3 ; \ldots ; 8 ; 9 ; 10\}\), with subsets A and B such that \(A\) is a set of perfect squares and \(B\) is a set of multiples of 3.
#### (i)
Draw a Venn diagram to represent the sets above.
#### (ii)
Find \(n(A \cup B)\).
(b)
A salesman's salary is partly constant and partly varies directly as the total sales he makes for the month. If sales are \(\$ 10000\), his salary is \(\$ 550\) and if sales are \(\$ 15000\), his salary is \(\$ 600\).
Find his salary if sales are \(\$ 25000\).
Question 3
(a)
#### (i)
Simplify \(2 m^{3} \times 3 m^{0}\).
#### (ii)
Evaluate \(\sqrt{12 \frac{1}{4}}\).
(b)
#### (i)
Express \(\log _{5}(x+1)-\log _{5}(2 x)\) as a single logarithm.
#### (ii)
Solve the equation
$$ \log _{5}(x+1)-\log _{5}(2 x)=1 . $$
(c)
Solve the equations
#### (i)
\(1 \frac{1}{5}=x-4\),
#### (ii)
\(3^{2 x-1}=\frac{1}{9}\).
Question 4
(a)
In the diagram, ABCD is a circle centre O. FAE is a tangent, \(\mathrm{A} \hat{\mathrm{O} B}=106^{\circ}\) and the chords AC and BD intersect at T.
#### (i)
Name two angles which are equal to DAE.
#### (ii)
Name the triangle that is similar to triangle ADT.
#### (iii)
Calculate
1. \(\mathrm{A} \hat{\mathrm{D}} \mathrm{B}\),
2. \(\hat{\mathrm{ABO}}\),
3. \(\hat{\mathrm{B}} \hat{\mathrm{AF}}\).
(b)
#### (i)
Solve the inequality \(5 x-6<2 x-3 \leq 3 x+1\), giving your answer in the form \(a \leq x<b\), where \(a\) and \(b\) are integers.
#### (ii)
Illustrate your solution on a number line.
Question 5
Answer the whole of this question on a sheet of plain paper. Use ruler and compasses only and show clearly all construction lines and arcs.
(a)
On a single diagram, construct
#### (i)
triangle PQR with \(\mathrm{QR}=8 \mathrm{~cm}, \hat{\mathrm{PQR}}=45^{\circ}\) and \(\hat{\mathrm{QRP}}=60^{\circ}\),
#### (ii)
the bisector of \(P \hat{R} Q\).
#### (iii)
Measure and write the length of PR.
#### (iv)
Calculate the area of triangle PQR.
(b)
Describe fully the locus represented by the bisector of \(\mathrm{PR} \hat{\mathrm{R}}\) in (a).
Question 6
(a)
Solve the equation \(3 x^{2}-5 x-7=0\), giving your answers correct to 2 decimal places.
(b)
| Date | Account details | Amount (USD) | | :--- | :--- | :--- | | | Balance B/F | 529,74 | | | Interest at 2,5\% | \_\_\_\_ | | | SUBTOTAL (1) | \_\_\_\_ \_\_\_\_ | | | RENTAL FROM 01/07/09 TO 31/07/09 | 10,00 | | 26/06/09 | METERED PREVIOUS CURRENT UNITS READING READING 5577855926 | 31.08 | | | SUBTOTAL (2) | 41,08 | | | VAT AT 15\% | 6,16 | | | AMOUNT DUE | ----------------- |
The statement above shows the telephone bill for Mr Banda for the month of July 2009. Calculate
#### (i)
the interest on the balance B/F,
#### (ii)
the subtotal (1),
#### (iii)
the number of units used,
#### (iv)
the cost of one unit,
#### (v)
the total amount due.
**Section B [36 marks]**
Answer any three questions in this section.
Each question carries **12 marks**.
Question 7
Answer the whole of this question on a sheet of graph paper.
Triangle KLM has vertices \(\mathrm{K}(-5 ; 3), \mathrm{L}(-1 ; 2)\) and \(\mathrm{M}(-4 ; 4)\). Using a scale of 2 cm to represent 1 unit on each axis, draw the \(x\) and \(y\) axes for \(-5 \leq x \leq 4\) and \(-8 \leq y \leq 4\).
(a)
Draw and label the triangle KLM.
(b)
A single transformation maps triangle KLM onto triangle \(\mathrm{K}_{1} \mathrm{~L}_{1} \mathrm{M}_{1}\) with vertices \(\mathrm{K}_{1}(-1 ; 3), \mathrm{L}_{1}(-5 ; 2)\) and \(\mathrm{M}_{1}(-2 ; 4)\).
#### (i)
Draw and label the triangle \(\mathrm{K}_{1} \mathrm{~L}_{1} \mathrm{M}_{1}\).
#### (ii)
Describe this transformation fully.
(c)
\(\mathrm{M}_{2}(3 ;-1)\) is the image of M after translating triangle KLM.
#### (i)
State the translation vector.
#### (ii)
Write down the coordinates of the points \(\mathrm{K}_{2}\) and \(\mathrm{L}_{2}\), the images of K and L respectively, under this translation.
(d)
The triangle KLM is mapped onto the triangle \(\mathrm{K}_{3} \mathrm{~L}_{3} \mathrm{M}_{3}\) by a shear of factor 2 with the \(y\)-axis invariant.
#### (i)
Write down the matrix of the shear.
#### (ii)
Draw and label the triangle \(\mathrm{K}_{3} \mathrm{~L}_{3} \mathrm{M}_{3}\).
Question 8
Answer the whole of this question on a sheet of graph paper.
A newly constructed school wishes to buy desks and chairs for its pupils. Let \(x\) be the number of desks and \(y\) be the number of chairs.
(a)
#### (i)
The school wishes to buy at least 75 desks and at least 100 chairs.
Write down two inequalities which satisfy these conditions.
#### (ii)
The number of chairs should be more than the number of desks.
Write down an inequality which satisfies this condition.
#### (iii)
Desks cost \(\$ 25\) each and chairs cost \(\$ 17,50\) each. The school has only \(\$ 5000\) to spend on these items.
Write down an inequality and show that it reduces to \(10 x+7 y \leq 2000\)
(b)
Using a scale of 2 cm to represent 25 desks and 2 cm to represent 50 chairs, show, by shading the UNWANTED regions, the region in which \((x ; y)\) must lie.
(c)
Use your region to determine the number of desks and chairs that would use up the greatest possible amount.
Question 9
The diagram shows an athletics running track enclosing a rectangular field 90 m by 70 m, with semi-circular ends. The track is made up of 8 lanes each 1 m wide.
Use \(\frac{22}{7}\) for \(\pi\)
(a)
Calculate
#### (i)
the length of the inner boundary of the first lane,
#### (ii)
1. the length of the inner boundary of the second lane, 2. the distance between the starting points of lane 1 and lane 2 if competitors in lanes 1 and 2 are to run the same distance in one lap,
#### (iii)
the area covered by the 8-lane running track.
(b)
The track is to be covered by an artificial grass costing \(\$ 200\) per square metre.
Calculate the cost of covering the track.
Question 10
OABCD is a pentagon such that AB is parallel to OD and DC is parallel to OA. M is the mid-point of OC such that AM produced cuts OD at X.
$$ \overrightarrow{\mathrm{OA}}=\mathrm{a} \quad \overrightarrow{\mathrm{OD}}=\mathrm{b} \quad \text { and } \quad \overrightarrow{\mathrm{OD}}=3 \overrightarrow{\mathrm{XD}} . $$
(a)
Express the following in terms of \(\mathbf{a}\) and/or \(\mathbf{b}\),
#### (i)
\(\overrightarrow{\mathrm{AD}}\),
#### (ii)
\(\overrightarrow{\mathrm{OX}}\),
#### (iii)
\(\overrightarrow{\mathrm{AX}}\).
(b)
If \(\overrightarrow{\mathrm{MX}}=k \overrightarrow{\mathrm{AX}}\), express \(\overrightarrow{\mathrm{MX}}\) in terms of \(\mathbf{a}, \mathbf{b}\) and \(k\).
(c)
Given that \(\overrightarrow{\mathrm{DC}}=h \overrightarrow{\mathrm{OA}}\), express in terms of \(\mathbf{a}, \mathbf{b}\) and \(h\),
#### (i)
\(\overrightarrow{\mathrm{OM}}\),