ZIMBABWE SCHOOL EXAMINATIONS COUNCIL General Certificate of Education Ordinary Level
**MATHEMATICS**
PAPER 2
4004/2
2 hours 30 minutes
**NOVEMBER 2024 SESSION**
Additional materials:
Mathematical Instruments
Mathematical Tables
Non programmable Electronic Calculator
Answer booklet
INSTRUCTIONS TO CANDIDATES
Write your Name, Centre number and Candidate number in the spaces provided on the answer booklet.
Answer all questions in Section A and any four questions from Section B.
If you use more than one booklet, fasten them together.
All working must be clearly shown 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 correct to three significant figures. Answers in degrees should be given correct to one decimal place.
Mathematical tables and Non-programmable electronic calculators may be used to evaluate explicit numerical expressions.
INFORMATION FOR CANDIDATES
The number of marks is given in brackets [ ] at the end of each question or part question.
**Section A [52 Marks]**
Answer all questions in this section
Question 1
(a)
Find the Highest Common Factor (H.C.F) of 96 and 120.
(b)
On a certain day the temperature rose from \(-5^\circ \mathrm{c}\) to \(+27^\circ \mathrm{c}\).
Calculate the rise in temperature.
Question 2
(a)
Matrix \(A=\left(\begin{array}{cc}5 & 0 \\ -8 & 9\end{array}\right)\) and matrix \(B=\left(\begin{array}{cc}6 & -4 \\ 0 & 2\end{array}\right)\).
Evaluate,
#### (i)
\(A-\frac{1}{2} B\)
#### (ii)
\(A B\)
(b)
The matrix \(\left(\begin{array}{cc}x-1 & 6 \\ 2 & x+3\end{array}\right)\) is singular.
#### (i)
Form an equation in \(x\) and show that it reduces to
$$ x^{2}+2 x-15=0 . $$
#### (ii)
Solve the equation \(x^{2}+2 x-15=0\).
Question 3
(a)
The selling price of a certain model of a mobile phone is \(\$ 2760,00\) including a Value Added Tax (VAT) of 15\%.
#### (i)
Calculate the price of the mobile phone excluding Value Added Tax.
#### (ii)
The exchange rate of Zimbabwean dollars ZWL(\(\$\)) to Botswana Pula (P) on a certain day was ZWL \(\$ 9,00\) to 1 Pula. Find the cost of the mobile phone in Pula.
#### (iii)
A Sales person is paid \(3,5 \%\) commission for each same model of mobile phone sold. Calculate the total commission the sales person gets when 23 such mobile phones are sold.
(b)
Electricity consumed by a household in a month is charged using the following consumption bands.
| Consumption Band (Units in Kwh) | Cost Per Unit (\$/Kwh) | | :--- | :--- | | First 50 units | 2,25 | | 51-100 | 4,51 | | 101-300 | 11,26 | | 301 and above | 12,94 |
Calculate the cost of 375 units bought by a household.
Question 4
The diagram shows an athletics field. It is made up of a rectangle ABDE and two semi - circles each of radius \(31,8 \mathrm{~m}\).
\(A B=100 \mathrm{~m}\).
(In this question take \(\pi\) to be 3,142)
(a)
Calculate the,
#### (i)
length of \(\operatorname{arc} \mathrm{BCD}\),
#### (ii)
perimeter of the athletics field,
#### (iii)
total area of the athletics field to the nearest \(\mathrm{m}^{2}\),
#### (iv)
cost of covering the whole athletics field with grass, if it costs \(\$ 620 / \mathrm{m}^{2}\).
(b)
On a map the length AB is represented by a line 4 cm long.
Find the scale used to draw the athletics field in the form \(1: n\), where \(n\) is an integer.
Question 5
ANSWER THE WHOLE OF THIS QUESTION ON A SHEET OF PLAIN PAPER
Use ruler and compasses only for all constructions and show clearly all construction lines and arcs.
All constructions should be done on a single diagram.
(a)
Construct a triangle ABC in which \(B C=10,5 \mathrm{~cm}, A \hat{B} C=90^{\circ}\) and \(B \widehat{C} A=30^{\circ}\).
(b)
Construct locus of points,
#### (i)
equidistant from AB and AC ,
#### (ii)
3 cm from BC .
(c)
Measure and write down the length of AB .
(d)
A point R is inside the triangle, nearer to AB than AC and less than \(3 cm\) from BC . Shade the region in which R lies.
**Section B [48 Marks]**
Answer any four questions from this section.
Each question carries 12 marks
Question 6
(a)
The length of each side of a regular hexagon is \(\frac{3 x-5}{2} \mathrm{~cm}\)
The perimeter of the hexagon is 15 cm .
Find the value of \(x\).
(b)
Given the formula \(V=k r^{2} h\)
#### (i)
find \(V\) when \(k=\frac{22}{7}, r=3.5\) and \(h=15\),
#### (ii)
express \(r\) in terms of \(k, V\) and \(h\).
Question 7
ANSWER THE WHOLE OF THIS QUESTION ON A SHEET OF GRAPH PAPER.
A particle moves such that the velocity, \(v \mathrm{~m} / \mathrm{s}\), after t seconds is given by \(v=t^{2}-5 t+10\).
The following is an incomplete table of corresponding values of \(t\) and \(v\).
| \(t\) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | \(v\) | 10 | 6 | 4 | 4 | 6 | \(m\) | 16 |
(a)
Calculate the value of \(m\).
(b)
Using a scale of 2 cm to represent 1 unit on the \(t\)-axis and 2 cm to represent 2 units on the \(v\)-axis, draw the graph of \(v=t^{2}-5 t+10\) for \(0 \leq t \leq 6\).
(c)
Use the graph to find the
#### (i)
minimum velocity reached by the object
#### (ii)
acceleration at time \(t=1\)
#### (iii)
range of values of \(t\) when velocity was less than \(7 \mathrm{~m} / \mathrm{s}\),
#### (iv)
distance travelled by the particle from \(t=2\) to \(t=4\)
Question 8
The table gives the frequency of marks obtained in a Mathematics test by a class of 100 learners.
| Mark | \(0 \leq x<10\) | \(10 \leq x<20\) | \(20 \leq x<30\) | \(30 \leq x<40\) | \(40 \leq x<50\) | | :--- | :---: | :---: | :---: | :---: | :---: | | Frequency | 7 | 13 | 31 | 37 | 12 |
(a)
Answer this part of the question on a sheet of graph paper.
Using a scale of 2 cm to represent 10 units on both axes, draw a frequency polygon for the distribution.
(b)
State the modal class.
(c)
Calculate an estimate mean mark of the distribution.
(d)
State the class interval in which the median mark is found.
(e)
One learner is picked at random from the class.
Find the probability that the learner's mark is greater than or equal to 10 but less than 40 .
Question 9
A dressmaker specialises in making trousers and skirts for school uniforms. He has 18 m of cloth. Let \(x\) be the number of trousers and \(y\) the number of skirts produced.
(a)
Each skirt requires \(0,5 \mathrm{~m}\) of cloth and a pair of trousers requires \(0,75 \mathrm{~m}\) of cloth. Form an inequality in \(x\) and \(y\) and show that it reduces to \(2 y+3 x \leq 72\).
(b)
To meet the demand of customers the dressmaker should make at least 6 pairs of trousers and at least 10 skirts.
Write down two inequalities that satisfy these conditions.
(c)
The number of skirts should be more than the number of trousers. Form another inequality that satisfy this condition.
(d)
Answer this part of the question on a sheet of graph paper.
Point \((x, y)\) satisfies all the conditions given. Show by shading the unwanted region the region where ( \(x, y\) ) lies.
Use a scale of 2 cm to represent 5 units on both axes for \(0 \leq x \leq 30\) and \(0 \leq y \leq 40\).
(e)
Each skirt makes a profit of \(\$ 120\) and each pair of trousers makes a profit of \(\$ 300\).
Calculate the maximum possible profit the dressmaker can get.
Question 10
Answer the whole of this question on a sheet of graph paper.
Use a scale of \(\mathbf{2} \mathbf{~ c m}\) to represent \(\mathbf{2}\) units on both axes for \(-8 \leq x \leq 10\) and \(-6 \leq y \leq 10\).
(a)
Triangle ABC has vertices at \(\mathrm{A}(4 ; 2), \mathrm{B}(4 ; 4)\) and \(\mathrm{C}(7 ; 4)\).
Draw and label triangle \(A B C\).
(b)
Triangle \(A_{1} B_{1} C_{1}\) is the image of triangle ABC after a translation by a vector \(\binom{2}{-5}\).
Draw and label triangle \(A_{1} B_{1} C_{1}\).
(c)
Triangle \(A_{2} B_{2} C_{2}\) is the image of triangle ABC under a reflection in the line \(y=x+2\).
#### (i)
Draw line \(y=x+2\).
#### (ii)
Draw and label triangle \(A_{2} B_{2} C_{2}\).
(d)
A transformation S represented by matrix \(\left(\begin{array}{cc}1 & -2 \frac{1}{2} \\ 0 & 1\end{array}\right)\) maps triangle ABC onto triangle \(A_{3} B_{3} C_{3}\).
#### (i)
Draw and label triangle \(A_{3} B_{3} C_{3}\).
#### (ii)
Describe fully the single transformation, S , that maps triangle ABC onto \(A_{3} B_{3} C_{3}\).
Question 11
PQRS is a parallelogram, \(\overrightarrow{P Q}=a ; \overrightarrow{P S}=b\).
M is the midpoint of SR .