ZIMBABWE SCHOOL EXAMINATIONS COUNCIL
General Certificate of Education Ordinary Level MATHEMATICS 4028/2
PAPER 2
JUNE 2010 SESSION Additional materials: 2 hours 30 minutes 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 12 printed pages.
Copyright: Zimbabwe School Examinations Council, J2010.
**Section A [64 marks]**
Answer all the questions in this section.
Question 1
(a)
Remove brackets and simplify the expression \(3(5-x)-2 x(x+3)\).
(b)
Integers \(x, y\) and \(z\) are such that \(x \leq 6, y \geq-2\) and \(-6 \leq z \leq 4\).
Find (i) the least possible value of \(y z^{2}\)
(ii) the greatest possible value of \(x-y\),
(c)
Factorise completely \(125 p^{3}-5 p\).
[Total :9]
Question 2
(a)
Express 252 as a product of its prime factors.
(b)
120 kg of a certain metal has a volume of \(0,4 \mathrm{~m}^{3}\). Find the density of the metal, giving your answer in \(\mathrm{g} / \mathrm{cm}^{3}\).
(c)
It is given that \(\xi=\{1 ; 2 ; 3 ; 5 ; 7 ; 8 ; 9 ;\}\), \(A=\{3 ; 5\}\), \(\mathrm{B}=\{1 ; 3 ; 7 ; 9\}\) and \(\mathrm{C}=\{1 ; 7 ; 9\}\)
(i) Draw a fully labelled Venn diagram to show all the elements in each subset.
(ii) Write down the elements of the following subsets
(a) \(\quad A \cap B \cap C^{\prime}\),
(b) \((A \cup B)^{1} U C\).
Question 3
(a)
In the diagram, \(A C E\) and \(A B D\) are straight lines, \(A B=B C=C D\) and \(B \hat{A} C=x^{0}\).
(i) Express in terms of \(x\)
(a) \(\quad C \hat{B} D\),
(b) \(\quad D \hat{C} E\),
(c) \(B \hat{C} D\).
(ii) If \(\mathrm{AC}=\mathrm{AD}\), find the numerical value of \(x\).
(b)
Given that \(\mathbf{M}=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right), \mathbf{N}=\left(\begin{array}{cc}4 & -2 \\ 3 & 0\end{array}\right)\) and \(3 \mathbf{M}+\mathbf{N}=\mathbf{M}\), find
(i) the matrix \(M\),
(ii) \(\quad \mathrm{N}^{2}\).
Question 4
Two variables R and V are connected by the equation \(\mathrm{R}=\mathrm{kV}+\mathrm{c}\), where k and c are constants.
(a)
Write down the type of variation between R and V .
(b)
If the graph of \(\mathrm{R}=\mathrm{kV}+\mathrm{c}\) is drawn with R on the vertical axis, write down, in terms of k and/or c the coordinates of the point where the graph crosses
(i) the vertical axis,
(ii) the horizontal axis.
(c)
Make V the subject of the equation \(\mathrm{R}=\mathrm{kV}+\mathrm{c}\).
(d)
Given that \(\mathrm{R}=14\) when \(\mathrm{V}=6\) and that \(\mathrm{R}=8\) when \(\mathrm{V}=2\),
(i) form a pair of simultaneous equations in k and c ,
(ii) hence find the numerical value of k and the numerical value of c .
Question 5
(a)
In a school with 1050 pupils, \(\frac{4}{7}\) of the pupils were boys. One quarter of the boys were suspended for misbehaviour.
(i) Find the number of boys suspended.
(ii) Express the number of girls as a fraction of the remaining pupils.
(iii) If two pupils were chosen at random from the remaining pupils to testify, find the probability that the two pupils were of the same sex.
(b)
In the diagram, \(\mathrm{P}, \mathrm{Q}, \mathrm{R}\) and S are points on the circumference of the circle with centre O . OS is parallel to QR and \(P \hat{R} Q=54^{\circ}\). POR is a straight line.
Calculate
(i) \(R O \hat{S}\),
(ii) \(R \hat{P} S\),
(iii) \(P \hat{Q} S\).
Question 6
Answer the whole of this question on a sheet of plain paper.
Use ruler and compasses only for all constructions and show all construction lines and arcs.
(a)
Construct on a single diagram
(i) quadrilateral ABCD in which \(\mathrm{AB}=8 \mathrm{~cm}, A \hat{B} C=90^{\circ}\); \(B \hat{C} D=120^{\circ}, \mathrm{BC}=10 \mathrm{~cm}\) and \(\mathrm{CD}=5 \mathrm{~cm}\),
(ii) the locus of points equidistant from B and C ,
(iii) the locus of points equidistant from DC and DA .
(b)
(i) Mark and label the point P which is equidistant from B and C and equidistant from DC and DA.
(ii) Draw a circle with centre P and radius PC .
(iii) Measure and write down the length of PC.
**Section B [36 marks]**
Answer any three questions in this section.
Question 7
In the diagram, PQ is parallel to \(\mathrm{OR}, P M=\frac{1}{3} P R, \overrightarrow{O P}=2 \mathbf{a}\) and \(\overrightarrow{O R}=3 \mathbf{b}\).
(a)
Express in terms of \(\mathbf{a}\) and/or \(\mathbf{b}\)
(i) \(\overrightarrow{P R}\),
(ii) \(\overrightarrow{P M}\),
(iii) \(\overrightarrow{O M}\).
(b)
(i) Given that \(\mathrm{PQ}=\mathrm{hOR}\), write down in terms of \(\mathrm{h}, \mathbf{a}\) and/or \(\mathbf{b}\) an expression for
(a) \(\overrightarrow{P Q}\),
(b) \(\overrightarrow{O Q}\).
(ii) Given also that \(\mathrm{OQ}=\mathrm{kOM}\), write down another expression for \(\overrightarrow{O Q}\) in terms of \(\mathbf{a}, \mathbf{b}\) and k .
(c)
Using the two expressions for \(\overrightarrow{O Q}\), form an equation and use it to find the value of \(k\) and the value of \(h\).
(d)
Write down \(\overrightarrow{O Q}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\) only.
(e)
Find the ratio \(\frac{\text { area of } \triangle \mathrm{OPQ}}{\text { area of trapezium } \mathrm{OPQR}}\).
Question 8
Answer the whole of this question on a single sheet of graph paper.
Using a scale of 2 cm to represent one unit on both axes, draw the \(x\) and \(y\) axes for \(-4 \leq x \leq 5\) and \(-5 \leq y \leq 5\).
(a)
The letter V has a vertex at \(\mathrm{A}(-2 ; 2)\) and the ends at \(\mathrm{B}(-3 ; 5)\) and \(C(-1 ; 5)\).
Draw and label the shape \(A B C\).
(b)
The shape ABC is mapped onto \(\mathrm{A}_{1} \mathrm{~B}_{1} \mathrm{C}_{1}\) with coordinates \(\mathrm{A}_{1}(1,2 ; 0,4)\), \(\mathrm{B}_{1}(4,2 ; 1,4)\) and \(\mathrm{C}_{1}(3 ; 3)\) by a certain transformation.
(i) Draw and label shape \(\mathrm{A}_{1} \mathrm{~B}_{1} \mathrm{C}_{1}\).
(ii) Describe completely the single transformation which maps shape ABC onto \(\mathrm{A}_{1} \mathrm{~B}_{1} \mathrm{C}_{1}\).
(c)
Shape ABC is enlarged by a scale factor \(-\frac{1}{2}\) with the origin as centre onto \(\mathrm{A}_{2} \mathrm{~B}_{2} \mathrm{C}_{2}\).
Draw and label \(\mathrm{A}_{2} \mathrm{~B}_{2} \mathrm{C}_{2}\).
(d)
A shear with y -axis invariant and scale factors 2 maps shape ABC onto \(\mathrm{A}_{3} \mathrm{~B}_{3} \mathrm{C}_{3}\).
Draw and label \(\mathrm{A}_{3} \mathrm{~B}_{3} \mathrm{C}_{3}\).
Question 9
Answer the whole of this question on a sheet of graph paper.
The table below shows the marks obtained by 40 students in a Mathematics test.
| \(\operatorname{Mark}(x)\) | \(8<x \leq 10\) | \(10<x \leq 11\) | \(11<x \leq 12\) | \(12<x \leq 14\) | \(14<x \leq 16\) | \(16<x \leq 19\) | | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | Frequency | 5 | 5 | 7 | 14 | 6 | 3 |
The following is a cumulative frequency table for this distribution.
| Mark (x) | \(x \leq 10\) | \(x \leq 11\) | \(x \leq 12\) | \(x \leq 14\) | \(x \leq 16\) | \(x \leq 19\) | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | Cumulative Frequency | 5 | 10 <br> q | 17 | 31 | 37 | 40 |
(a)
Find the value of \(\mathbf{q}\).
(b)
State the modal class.
(c)
Using a scale of 1 cm to represent 1 mark on the \(x\)-axis and 2 cm to represent 5 students on the \(y\)-axis, draw the cumulative frequency curve for the marks.
(d)
Use your graph to estimate
(i) the median mark,
(ii) the number of students who got 15 or more marks.
(e)
Calculate an estimate of the mean mark.
Question 10
(a)
(i) Solve the inequality \(-3<2 x-7 \leq 7\).
(ii) Illustrate your solutions on a number line.
(b)
Triangle ABC is such that \(\mathrm{BC}=x\) metres and the perpendicular distance of A from BC is \((3 x-5)\) metres. Given that the area of the triangle \(A B C\) is \(4 \mathrm{~m}^{2}\),
(i) form an equation in \(x\) and show that it reduces to \(3 x^{2}-5 x-8=0\),
(ii) solve this equation for \(x\), giving your answers correct to 2 decimal places,
(iii) hence write down the distance of A from BC .
Question 11
The diagram shows a swimming pool of uniform cross-section ABCDEF , of length \(50 m\) and breadth \(40 m\). \(\mathrm{AB}=50 m, \mathrm{BC}=3,5 m, \mathrm{DC}=\mathrm{FE}=20 m, \mathrm{AF}=1,5 m\) and \(B \hat{A} F=A \hat{F} E=B \hat{C} D=A \hat{B} C=90^{\circ}\)
(a)
Calculate
(i) the cross sectional area ABCDEF ,
(ii) the capacity of the swimming pool, giving your answer in kilolitres,
(iii) the length of DE .
(b)
The vertical walls of the pool are to be painted. Given that 7 litres of paint is needed to cover \(10 \mathrm{~m}^{2}\) of wall surface and that the paint is sold in 5 litre tins at a cost of \(\$ 27000\) per tin.
(i) the total area to be painted,
(ii) the number of tins of paint to be bought,
(iii) the amount of money needed to buy the paint.
Question 12
Answer the whole of this question on a sheet of graph paper.
(a)
The following is an incomplete table of values for the function
$$ y=\frac{3}{x+2} $$
| \(x\) | -6 | -5 | -4 | -3 | \(-2,5\) | -1 | 0 | 1 | 2 | | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | \(y\) | Sns <br> \(-\frac{3}{4}\) | -1 | \(\mathbf{p}\) | \(-\mathbf{y}\) | -3 | -6 | 3 | \(1 \frac{1}{2}^{\text {S }}\) | 1 |
(i) Calculation the value of \(\mathbf{p}\).
(ii) Using a scale of 2 cm to represent 1 unit on both axes draw the graph of \(y=\frac{3}{x+2}\) for \(-6 \leq x \leq 2\).
(b)
On the same axes draw the graph of the function \(y=2 x+3\) to intersect with the graph of \(y=\frac{3}{x+2}\).
(c)
Write down, in the form \(a x^{2}+b x+c=0\) (where \(\mathrm{a}, \mathrm{b}\) and c are constants), the equation whose roots are the \(x\) coordinates of the points of intersection of the two graphs.
(d)
By drawing a suitable tangent, find the gradient of the graph \(y=\frac{3}{x+2}\) at the point ( \(1 ; 1\) ).