ZIMBABWE SCHOOL EXAMINATIONS COUNCIL <br> General Certificate of Education Ordinary Level
**MATHEMATICS**
PAPER 2 NOVEMBER 2011 SESSION
Additional materials: Answer paper Geometrical instruments Graph paper (3 sheets) Mathematical tables Plain paper (1 sheet)
4028/2
2 hours 30 minutes
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, N2011.
**Section A [64 marks]**
Answer all the questions in this section.
Question 1
(a)
Simplify \(5 \frac{1}{6}-3 \frac{2}{3} \div 1 \frac{1}{4}\).
(b)
The virus that causes the common cold is \(5 \times 10^{-7} \mathrm{~m}\) long. Giving the answer in standard form, find the total length of 12000 such viruses.
(c)
Express as a single fraction in its lowest terms \(\frac{3}{x^{2}-x}-\frac{5}{x^{2}-1}\).
(d)
Calculate the Principal that earns \(\$ 300\) Simple Interest at \(5 \%\) per annum for 6 years.
Question 2
(a)
Solve the inequality \(15-3 x<2(x-5)\).
(b)
Factorise completely
#### (i)
\(2 x y-x-z+2 y z\),
#### (ii)
\(2 p^{2} q+p q^{2}\).
(c)
Assuming there is an equal chance of being born a boy or a girl, find the probability that
#### (i)
a child is born a boy,
#### (ii)
in a family of three children, there are two boys and one girl.
(d)
An office is \(4,35 \mathrm{~m}\) long and \(3,62 \mathrm{~m}\) wide.
Its floor is to be carpeted at a cost of \(\$ 15,99\) per square metre. Calculate
#### (i)
the area of the floor,
#### (ii)
the cost of carpeting the floor.
Question 3
(a)
Solve the equation \(x(x+1)+2 x(x-1)=3\left(x^{2}-1\right)\).
(b)
Given that \(b=\frac{1}{2} \sqrt{a^{2}-x^{2}}\), make \(x\) the subject of the formula.
(c)
A salt shaker is made up of a cylinder of height \(h\) and a hemisphere of internal diameter \(d\) as shown below.
#### (i)
Write down an expression for the volume of the salt shaker in terms of \(\pi, d\) and \(h\).
#### (ii)
Find the internal volume of the salt shaker if \(h=11 \mathrm{~cm}\) and \(d=3,5 \mathrm{~cm}\), leaving the answer in terms of \(\pi\).
\[ \left[\text { Volume of sphere }=\frac{4 \pi r^{3}}{3}\right] \]
Question 4
(a)
A compound is made up of potassium nitrate, sulphur and charcoal, mixed in the ratio \(33: 5: 7\) respectively.
#### (i)
Calculate the percentage of sulphur in the compound.
#### (ii)
Find the mass of charcoal needed to make 900 kg of the compound.
#### (iii)
Given that 10 kg of sulphur and 14 kg of charcoal are mixed, find the mass of potassium nitrate needed to make up the compound.
(b)
Solve the equation \(2 x^{2}-3 x-7=0\), giving your answers correct to 2 decimal places.
(c)
Given that \(M\) is directly proportional to \(t\), and that \(M=27,5\) when \(t=55\), find the value of \(t\) when \(m=43\).
Question 5
(a)
In the diagram, \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) and D are points on the circumference of a circle centre O . AT and CT are tangents to the circle, \(\mathrm{OAC}=30^{\circ}\) and \(A \hat{C} B=20^{\circ}\).
Find
#### (i)
\(\mathrm{A} \hat{\mathrm{O}} \mathrm{C}\),
#### (ii)
\(\mathrm{A} \hat{\mathrm{D}} \mathrm{C}\),
#### (iii)
\(B \hat{C} T\),
#### (iv)
\(\hat{\mathrm{CAB}}\),
#### (v)
\(\mathrm{A} \hat{\mathrm{T}} \mathrm{C}\).
(b)
Given that \(\mathbf{A}=\left(\begin{array}{ll}1 & 2 \\ 0 & 1\end{array}\right), \quad \mathbf{B}=\left(\begin{array}{ll}3 & 1 \\ 2 & 0\end{array}\right)\), find
#### (i)
\(2 \mathrm{~A}+\mathrm{B}\),
#### (ii)
BA ,
#### (iii)
\(\mathrm{B}^{-1}\)
Question 6
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)
quadrilateral ABCD in which \(\mathrm{AB}=9 \mathrm{~cm}, \mathrm{BC}=6 \mathrm{~cm}, \mathrm{AD}=7.3 \mathrm{~cm} \mathrm{DC}=5.5\) and \(\hat{\mathrm{BAD}}=45^{\circ}\),
#### (ii)
the locus of points equidistant from A and B ,
#### (iii)
the locus of points 5.7 cm from B .
(b)
Measure and write down the size of \(A \hat{D} C\).
(c)
Point P is inside the quadrilateral and is such that it is equidistant from \(A\) and \(B\) and is 5.7 cm from \(B\). Measure and write down the distance of \(P\) from \(D\).
**Section B [36 marks]**
Answer any three questions in this section.
Each question carries 12 marks.
Question 7
Take \(\pi\) to be \(\frac{22}{7}\)
The diagram shows a metal object made up of two identical cuboids and a cylinder riveted together. The cuboids have dimensions 10 cm by 6 cm by 4 cm . The cylinder has a height of 6 cm and a diameter of 4 cm .
(a)
Calculate
#### (i)
the exposed area of ABCD ,
#### (ii)
the surface area of cuboid ABCDEFGH ,
#### (iii)
the curved surface area of the cylinder,
#### (iv)
the total surface area of the object.
(b)
Given that the volume of the metal used is \(555,4 \mathrm{~cm}^{3}\) and that the density of the metal is \(9000 \mathrm{~kg} / \mathrm{m}^{3}\), calculate, giving your answer correct to the nearest kg , the mass of the object.
Question 8
Answer the whole of this question on a sheet of graph paper.
The table shows values for the graph of \(y=x^{3}\).
| \(x\) | -3 | -2 | -1 | 0 | 1 | 2 | 3 | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | \(y\) | -27 | -8 | -1 | 0 | 1 | 8 | 27 |
(a)
Using a scale of 2 cm to represent one unit on the \(x\)-axis and 2 cm to represent 5 units on the \(y\)-axis, draw the graph of \(y=x^{3}\) for \(-3 \leq x \leq 3\).
(b)
Use your graph to find
#### (i)
the gradient of the curve at \(x=1\),
#### (ii)
the area enclosed between the graph and the line \(y=2 x\) and the lines \(x=0\) and \(x=1,5\).
(c)
By drawing a suitable straight line, solve the equation \(x^{3}=3 x\).
Question 9
In the diagram, \(Y\) is 165 km from \(X\) on a bearing \(340^{\circ}\) and \(Z\) is 98 km from \(Y\) on a bearing of \(275^{\circ}\).
(a)
Calculate the distance between X and Z .
(b)
A helicopter took \(1 \frac{1}{2}\) hours to fly from X to Y direct. Find the speed of the helicopter.
(c)
#### (i)
Find \(Y \hat{X} Z\).
#### (ii)
State the bearing of Z from X .
Question 10
Answer the whole of this question on a sheet of graph paper.
The table shows the heights of 32 pupils measured to the nearest cm .
| Height \((\mathrm{h})\) | \(120 \leq \mathrm{h}<130\) | \(130 \leq \mathrm{h}<140\) | \(140 \leq \mathrm{h}<150\) | \(150 \leq \mathrm{h}<160\) | \(160 \leq \mathrm{h}<170\) | \(170 \leq \mathrm{h}<180\) | \(180 \leq \mathrm{h}<190\) | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | frequency | 1 | 3 | 6 | 10 | 7 | 4 | 1 |
Below is the cumulative frequency table for the information above.
| upper class <br> boundary | \(\mathrm{h}<130\) | \(\mathrm{~h}<140\) | \(\mathrm{~h}<150\) | \(\mathrm{~h}<160\) | \(\mathrm{~h}<170\) | \(\mathrm{~h}<180\) | \(\mathrm{~h}<190\) | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | cumulative <br> frequency | 1 | 4 | 10 | 20 | 27 | 31 | 32 |
(a)
Using a scale of 2 cm to represent 5 pupils on the vertical axis and 2 cm to represent 10 cm on the horizontal axis, draw the Cumulative frequency curve to illustrate this information.
(b)
Use your graph to find the median height of the pupils.
(c)
To get into a game park at half price, pupils have to be under 150 cm tall.
Find the number of pupils who can get in at half price.
(d)
If two pupils are chosen at random from the class, find the probability that the height of the first is less than 150 cm and that of the second is at least 160 cm .
(e)
Calculate an estimate of the mean height of the pupils.
Question 11
(a)
In the diagram, OABC is a quadrilateral and \(\mathrm{P}, \mathrm{Q}, \mathrm{R}\) and S are the midpoints of \(\mathrm{OA}, \mathrm{AB}, \mathrm{BC}\) and OC respectively. \(\overrightarrow{\mathrm{OA}}=\mathbf{a}, \overrightarrow{\mathrm{OC}}=\mathbf{b}\) and \(\overrightarrow{\mathrm{AB}}=\frac{3}{2} \mathbf{a}+\frac{5}{3} \mathbf{b}\).
Find in terms of \(\mathbf{a}\) and/or \(\mathbf{b}\)
#### (i)
\(\overrightarrow{\mathrm{PS}}\),