ZIMBABWE SCHOOL EXAMINATIONS COUNCIL
General Certificate of Education Ordinary Level **MATHEMATICS** 4028/2
**PAPER 2** **NOVEMBER 2010 SESSION**
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 calculator may be used to evaluate explicit numerical expressions.
This question paper consists of 12 printed pages. Copyright: Zimbabwe School Examinations Council, N2010. ©ZIMSEC N2010
**Section A [64 marks]**
Answer all questions in this section.
Question 1
(a)
Find the value of \(\frac{1}{3}+1 \frac{7}{9} \div 2 \frac{2}{3}\).
(b)
During a sale, the price of a camera was reduced from \(\$ 160\) to \(\$ 148,80\).
Calculate the percentage decrease in price.
(c)
Given that \(f(x)=x^{2}-4 x+3\), find all the values of \(x\) for which \(f(x)=0\).
Question 2
(a)
Express \(\frac{1}{x-1}+\frac{2}{x+1}\) as a single fraction in its simplest form.
Hence or otherwise, solve the equation
$$ \frac{1}{x-1}+\frac{2}{x+1}=\frac{3}{x} $$
(b)
Solve the inequality
\(y-4<3 y+2 \leq 6-y\).
Hence list the integral values of \(y\) that satisfy the inequality.
(c)
In an Olympiad test, there were 26 questions. Eight points were given for each correct answer and five points were deducted for each wrong answer.
Tamara answered all questions and scored zero. Find the number of questions she had got correct.
Question 3
(a)
It is given that \(s=u t-\frac{1}{2} g t^{2}\).
#### (i)
Find the value of \(s\) if \(g=9,8 ; u=20\) and \(t=2\).
#### (ii)
Make \(g\) the subject of the formula.
(b)
In the diagram, ADE is a triangle, B is a point on AD such that \(AB=2 cm\) and \(BD=8 cm\). C is a point on AE such that \(AC=4 cm\) and \(CE=1 cm\), \(DE=8 cm\).
#### (i)
Name the triangle that is similar to \(\triangle ABC\).
#### (ii)
Calculate the length of BC.
Question 4
(a)
It is given that P varies directly as T and inversely as V.
#### (i)
Write down an equation connecting P, V, T and a constant k.
#### (ii)
Given that \(P=2 \times 10^{5}\) when \(V=1 \times 10^{-3}\) and \(T=300\), calculate the value of \(k\).
#### (iii)
Calculate P if \(V=0,0025\) and \(T=300\).
(b)
Given that \(\mathbf{M}=\left(\begin{array}{rr}3 & -2 \\ -1 & 4\end{array}\right), \mathbf{N}=\binom{5}{7}\) and \(\mathbf{R}=\left(\begin{array}{ll}3 & -1\end{array}\right)\), find
#### (i)
MN
#### (ii)
\(\mathrm{M}^{-1}\)
#### (iii)
RN.
Question 5
(a)
In the diagram, A, B, C, D and E are points on the circumference of a circle centre O. BT is a tangent to the circle and TCD and AEF are straight lines. \(\angle CAE=68^\circ\), \(\angle CAB=36^\circ\) and BD is parallel to AE.
Find the size of
#### (i)
\(\angle CBO\),
#### (ii)
\(\angle BTC\),
#### (iii)
\(\angle DEF\),
#### (iv)
\(\angle ACB\).
(b)
In a recipe for an apple pie, 500 g of apples and 200 g of flour are needed in making an apple pie for 4 people.
#### (i)
If an apple pie was to be made for 6 people, calculate the quantity of apples needed.
#### (ii)
If the apple pie was to be made for 3 people, calculate the quantity of flour needed.
Question 6
Answer the whole of this question on a sheet of plain paper.
Use ruler and compasses only and show all construction lines and arcs. All constructions must be done on a single diagram.
A farmer has a plot in the shape of a quadrilateral \(ABCD\), in which \(AB=110 m\), \(BC=100 m\), \(CD=60 m\), \(AD=70 m\) and \(\angle ABC=60^\circ\).
(a)
Using a scale of \(1 cm : 10 m\), construct the quadrilateral \(ABCD\).
(b)
Draw the locus of points
#### (i)
30 m from AB,
#### (ii)
equidistant from A and B,
#### (iii)
inside the quadrilateral which are 60 m from B.
(c)
The farmer wishes to dig a well inside the plot such that it is at least 30 m from AB, at least 60 m from B and nearer to A than to B.
Shade the region in which the well must be.
**Section B [36 marks]**
Answer three questions in this section.
Question 7
(a)
The diagram shows a wine glass in the shape of a cone mounted on a stand. The depth of the cone is equal to its diameter at the top.
#### (i)
Write down an expression for the volume of the cone in terms of its radius \(r\) and \(\pi\).
#### (ii)
If the wine glass can hold 20 ml of wine when full, calculate the radius of the wine glass at the top.
#### (iii)
Wine is bought in bottles of volume 750 ml. Calculate the number of wine glasses that can be filled from one bottle.
[Volume of cone \(=\frac{1}{3}\) base area × height. \(\pi=\frac{22}{7}\)]
(b)
The base of a triangle is \(x cm\) and its height is \((x-7) cm\).
#### (i)
Write down an expression for the area of the triangle.
#### (ii)
If the area of the triangle is \(6 cm^2\), form an equation in \(x\) and show that it reduces to \(x^{2}-7x-12=0\).
(c)
Solve the equation \(x^{2}-7x-12=0\), giving your answers correct to 2 decimal places.
Question 8
Answer the whole of this question on a sheet of graph paper.
The following is a table of values for the graph of the function
$$ y=7-5x-x^{2} $$
| \(x\) | -7 | -6 | -5 | -4 | -3 | -2 | -1 | 0 | 1 | 2 | | :--- | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: | | \(y\) | -7 | 1 | 7 | 11 | 13 | 13 | 11 | 7 | 1 | -7 |
(a)
Using a scale of 2 cm to represent 1 unit on the horizontal axis and 2 cm to represent 5 units on the vertical axis, draw the graph of the function \(y=7-5x-x^{2}\) for \(-7 \leq x \leq 2\).
(b)
Use your graph to answer the following questions.
#### (i)
State the maximum value of the function \(y=7-5x-x^{2}\).
#### (ii)
Solve the equation \(7-5x-x^{2}=0\).
#### (iii)
Solve the equation \(-5x-x^{2}=2\).
#### (iv)
Find the gradient of the curve at the point where \(x=0\).
Question 9
(a)
The diagram shows the distance-time graph of a cyclist, Farai and a pedestrian, Tanya, who travelled from their home to the train station which was 5 km away. After sometime Farai came back home.
Use the diagram to answer the following questions.
#### (i)
Find Farai's speed on the outward journey.
#### (ii)
State
1. the time when Tanya arrived at the station, 2. the time when Farai overtook Tanya on the way to the station, 3. the distance that Tanya had covered when she was overtaken, 4. the total time that Farai was resting, 5. the distance that Tanya had left to cover when Farai met her the second time.
#### (iii)
Calculate Tanya's average speed for the whole journey.
(b)
Two cards were picked at random from a pack of 52 playing cards with replacement.
Find the probability that one was a Court card (i.e. J, K or Q) and the other was an Ace (A).
Question 10
In the diagram, A, B and C are three points on level ground. B is 12 km from A on a bearing of \(062^\circ\) and C is 15 km from A on a bearing of \(158^\circ\).
Calculate
#### (i)
the distance from B to C,
#### (ii)
\(\angle ACB\) to the nearest degree,
#### (iii)
the bearing of C from B.
Question 11
Answer the whole of this question on a sheet of graph paper.
A builder wishes to build houses and flats on \(6000 m^2\) plot of land.
(a)
The City Council insists that there must be more than 6 houses and that there must be more flats than houses.
Taking \(x\) to represent the number of houses and \(y\) to represent the number of flats, write down two inequalities, other than \(x>0\) and \(y>0\), which satisfy these conditions.
(b)
The builder allows \(300 m^2\) for each flat and \(400 m^2\) for each house. Write down another inequality which satisfies this condition and show that it reduces to \(4x+3y \leq 60\).
(c)
The point \((x ; y)\) represents \(x\) houses and \(y\) flats. Using a scale of 2 cm to represent 5 units on both axes, draw the \(x\) and \(y\) axes for \(0 \leq x \leq 20\) and \(0 \leq y \leq 20\).
Construct and show by shading the unwanted regions, the region in which \((x ; y)\) must lie.
(d)
Use your graph to find
#### (i)
the maximum number of flats that can be built,
#### (ii)
the maximum number of houses that can be built,