ZIMBABWE SCHOOL EXAMINATIONS COUNCIL
**General Certificate of Education Ordinary Level MATHEMATICS**
**4028/2**
PAPER 2
**NOVEMBER 2009 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, N2009.
**Section A [64 marks]**
Answer all the questions in this section.
Question 1
(a)
Simplify \(8-24 \div 6+3 \times 4\).
(b)
Expand \((1-2 x)(x+3)\).
(c)
#### (i)
Find the L.C.M. of
$$ 15 y^{2}, 25 x y^{3} \text { and }\left(x^{3}-x^{2}\right) . $$
#### (ii)
Evaluate \(\left(\log _{9} 81\right) \times\left(2 \log _{4} 8\right)\).
Question 2
(a)
Solve the equations
#### (i)
\(7(h+3)-2(h-4)=4\),
#### (ii)
\(3^{(m+4)}=9^{(m-1)}\).
(b)
Write down, in set notation, the set represented by the shaded region in the Venn diagram.
(c)
It is given that
$$ \begin{aligned} & \xi=\{x: 2 \leq x \leq 20, x \text { is an integer }\}, \\ & \mathrm{P}=\{x: x \text { is a prime number }\} \text { and } \\ & \mathrm{Q}=\{x: 4 \leq x<17\} . \end{aligned} $$
#### (i)
List the elements of P.
#### (ii)
Find \(n\left(Q^{1} \cap P\right)\).
Question 3
(a)
Simplify \(\frac{n-3}{6} \div \frac{n^{2}-9}{4}\).
(b)
Given that \(\mathbf{A}=\left(\begin{array}{ll}2 & 3\end{array}\right)\) and \(\mathbf{B}=\left(\begin{array}{cc}4 & -1 \\ 5 & 6\end{array}\right)\), find
#### (i)
AB ,
#### (ii)
\(\mathrm{B}^{-1}\).
(c)
If \(\left(\begin{array}{cc}-2 & p \\ p+3 & -4 p\end{array}\right)\) is singular, find the two possible values of \(p\).
(d)
A company invested money in a bank at \(270 \%\) simple interest per annum. Given that after 8 months, the total value of its investment was \(\$ 840\) million, calculate the amount invested.
Question 4
(a)
The diagram shows two squares ABCD and PQRS . Given that \(\mathrm{AB}=12 \mathrm{~cm}\), calculate
#### (i)
the perimeter of PQRS ,
#### (ii)
the area of \(\triangle Q R S\).
(b)
Sibongile's weekly wage W (in thousands of dollars), is partly constant and partly varies as the number of hours N of overtime she works per week.
#### (i)
Express W in terms of N and constants \(h\) and \(k\).
#### (ii)
Given that when \(\mathrm{W}=80, \mathrm{~N}=10\) and when \(\mathrm{W}=60, \mathrm{~N}=6\), find the value of \(h\) and the value of \(k\).
#### (iii)
Sibongile's normal working time is 44 hours in a week.
Find the total number of hours worked in a week in which she was paid \$90 thousand.
Question 5
(a)
The volume, \(V\), of material needed to make a cylindrical tube of internal radius \(r\), external radius \(R\) and length \(h\) is given by the formula
$$ \mathrm{V}=\pi\left(\mathrm{R}^{2}-r^{2}\right) h $$
#### (i)
Taking \(\pi\) to be \(\frac{22}{7}\), find the value of V when \(\mathrm{R}=4 \mathrm{~cm}, r=3 \mathrm{~cm}\) and \(h=150 \mathrm{~cm}\).
#### (ii)
Make R the subject of the formula.
(b)
A solid cuboid of density \(0,7 \mathrm{~g} / \mathrm{cm}^{3}\) measures 8 cm by 7 cm by \(x \mathrm{~cm}\) and has a total surface area of \(442 \mathrm{~cm}^{2}\).
Calculate
#### (i)
the value of \(x\),
#### (ii)
the mass of the solid.
Question 6
(a)
Factorise completely \(3 \mathrm{p}^{2}+7 \mathrm{p}-6\).
(b)
#### (i)
Using the graph, write down three inequalities other than \(3 y+x \leq 21\) which satisfy the region \(\boldsymbol{R}\).
#### (ii)
Find the maximum value of 5y-x for integer values of \(x\) and \(y\) in \(\boldsymbol{R}\).
**Section B [36 marks]**
Answer any three questions in this section.
**Each question carries 12 marks.**
Question 7
(a)
In the hexagon \(\mathbf{P Q R S T U}\), the lines \(\mathbf{P Q}\) and \(\mathbf{U T}\) are parallel. \(\mathbf{U} \hat{\mathbf{P Q}}=y^{\circ} \hat{\mathrm{PQR}}=130^{\circ}, \hat{\mathrm{RS}}=5 y^{\circ}, \hat{\mathrm{RST}}=155^{\circ}\) and \(\hat{\mathrm{STU}}=(180-2 y)^{\circ}\).
#### (i)
Write down an expression, in terms of \(y\), for PUTT.
#### (ii)
Using the sum of interior angles of the hexagon, form an equation in terms of \(y\) and solve it.
#### (iii)
Hence, write down the numerical value of QRS .
(b)
#### (i)
Show that the equation
$$ \frac{1}{2 x-5}+\frac{2}{3}=\frac{1}{x+3} $$
reduces to \(4 x^{2}-x-6=0\).
#### (ii)
Hence solve the equation \(4 x^{2}-x-6=0\). giving your answers correct to two decimal places.
Question 8
Answer the whole of this question on a sheet of plain paper.
Use ruler and compasses only for all constructions and show clearly all the Construction lines and arcs.
A landmine-infested area is in the form of a quadrilateral PQRS with \(\mathrm{PQ}=14 \mathrm{~km}, \mathrm{QR}=12 \mathrm{~km}, \mathrm{PS}=17 \mathrm{~km}, \mathrm{PQR}=90^{\circ}\) and \(\mathrm{QPS}=120^{\circ}\)
(a)
Using a scale of 1 cm to represent 2 km , construct quadrilateral PQRS.
(b)
For safety reasons, resettled families are to be at least 6 km from QR.
Construct the locus of points 6 km from QR .
(c)
Two landmines were located such that they were each equidistant from \(P S\) and \(S R\) and 10 km from \(P\).
#### (i)
Construct the locus of points equidistant from PS and SR.
#### (ii)
Construct the locus of points 10 km from P .
(d)
#### (i)
Label \(M_{1}\) and \(M_{2}\) the two positions of the landmines.
#### (ii)
Find the actual distance between the landmines.
Question 9
Use the diagram to answer the following questions.
(a)
\(\Delta \mathrm{B}\) is a reflection of \(\Delta \mathrm{A}\).
#### (i)
Write down the equation of the mirror line.
#### (ii)
Given that \((\mathrm{k} ; 8)\) is one of the invariant points under this reflection, find the value of \(k\).
(b)
Describe fully the single transformation which maps \(\triangle \mathrm{A}\) onto \(\triangle \mathrm{C}\).
(c)
\(\Delta \mathrm{D}\) is the image of \(\Delta \mathrm{A}\) under an enlargement, centre origin followed by a translation.
Write down
#### (i)
the scale factor of the enlargement,
#### (ii)
the translation vector.
(d)
Describe fully the single transformation which maps \(\triangle \mathrm{A}\) onto \(\triangle \mathrm{E}\).
Question 10
(a)
In the diagram, OR is parallel to PQ and \(\frac{\mathrm{PQ}}{\mathrm{OR}}=\frac{2}{3}\). OP and RQ produced meet at \(\mathrm{S} . \overrightarrow{\mathrm{OP}}=\boldsymbol{p}\) and \(\overrightarrow{\mathrm{PQ}}=\boldsymbol{q}\).
#### (i)
Express in terms of \(\boldsymbol{p}\) and/or \(\boldsymbol{q}\).
##### (a)
\(\quad \overrightarrow{\mathrm{OR}}\),
##### (b)
\(\quad \overrightarrow{\mathrm{RQ}}\).
#### (ii)
Write down, in its lowest terms, the ratio
##### (a)
\(\frac{\mathrm{QS}}{\mathrm{RS}}\),
##### (b)
\(\frac{\text { area of } \triangle \mathrm{PQS}}{\text { area of trapezium } \mathrm{OPQR}}\).
(b)
#### (i)
A shop that makes and sells curtains supplied the following price quotation to a customer:
| Curtaining material/metre............. | \(\$ 700000-00\) | | :--- | :--- | | Labour | \(10 \%\) of the total | | | cost of the material |