ZIMBABWE SCHOOL EXAMINATIONS COUNCIL
**General Certificate of Education Ordinary Level**
**MATHEMATICS** **PAPER 1**
4004/1 NOVEMBER 2023 SESSION 2 hours 30 minutes Candidates answer on the question paper. Additional materials: Geometrical instruments Allow candidates 5 minutes to count pages before the examination. This booklet should not be punched or stapled and pages should not be removed.
TIME 2 hours 30 minutes
INSTRUCTIONS TO CANDIDATES
Write your name, Centre number and candidate number in the spaces at the top of this page and your Centre number and Candidate number on the top right corner of every page of this paper.
Answer all questions. Check that all the pages are in the booklet and ask the invigilator for a replacement if there are duplicate or missing pages. Write your answers in the spaces provided on the question paper using black or blue pens. If working is needed for any question it must be shown in the space below that question. Omission of essential working will result in loss of marks. Decimal answers which are not exact should be given correct to three significant figures unless stated otherwise.
Mathematical tables, slide rules and calculators should not be brought into the examination room.
INFORMATION FOR CANDIDATES
The number of marks is given in brackets | | at the end of each question or part question. This question paper consists of 28 printed pages. Copyright: Zimbabwe School Examinations Council. N2023.
Question 7
\(A B C D\) is a parallelogram. \(X\) is the point on \(B C\) such that \(B X: X C=2: 1\). \(\overrightarrow{A B}=2 \mathbf{p}\) and \(\overrightarrow{A D}=3 \mathbf{q}\)
Find, in terms of \(\mathbf{p}\) and \(\mathbf{q}\).
(a)
\(\overrightarrow{A C}\).
Answer \(\overrightarrow{A C}=\)
(b)
\(\overrightarrow{A X}\).
Answer \(\overrightarrow{A X}=\)
(c)
\(\overrightarrow{X D}\).
Answer \(\overrightarrow{X D}=\)
Question 8
A group of 80 students took a physics test. This table shows the distribution of their marks.
| Mark \((m)\) | \(0<m \leqslant 10\) | \(10<m \leqslant 20\) | \(20<m \leqslant 30\) | \(30<m \leqslant 40\) | \(40<m \leqslant 50\) | \(50<m \leqslant 60\) | | :--- | :---: | :---: | :---: | :---: | :---: | :---: | | Frequency | 4 | 12 | 14 | 22 | 18 | 10 |
(a)
Complete the cumulative frequency table.
| Mark \((m)\) | \(m \leqslant 10\) | \(m \leqslant 20\) | \(m \leqslant 30\) | \(m \leqslant 40\) | \(m \leqslant 50\) | \(m \leqslant 60\) | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | Cumulative <br> frequency | | | | | | |
(b)
Draw a cumulative frequency curve for this information.
(c)
The pass mark for the test is 45.
Use your cumulative frequency curve to estimate the number of students who passed.
Question 9
| 0.2 | 2 | \(\sqrt{2}\) | \(\frac{1}{3}\) | 0.83 | 8 | 81 | | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
From the numbers listed above, write down
(a)
a square number,
(b)
a cube number,
(c)
an irrational number
Answer \(\_\_\_\_\)
Question 10
\(A=\left(\begin{array}{rr}4 & -2 \\ -1 & 1\end{array}\right) \quad B=\left(\begin{array}{ll}-3 & 2 \\ -1 & 4\end{array}\right)\)
(a)
Find \(2 A-B\)
Answer ( )[2]
(b)
Find \(A^{-1}\).
Answer ( )[2]
Question 11
Write these numbers in order, starting with the smallest.
\[ \begin{array}{lllll} \frac{3}{4} & 0 & -1 & -\frac{17}{20} & -\frac{4}{5} \end{array} \]
Answer \(\_\_\_\_\)
Question 12
The diagram shows the regions \(A\) to \(I\).
Give the letter of the region defined by each set of inequalities.
(a)
\(x>0, y>0, y<1\) and \(y<4-2x\)
\(\_\_\_\_\)
(b)
\(y>1, y<x-2\) and \(y<5-x\)
\(\_\_\_\_\)
Question 13
The two triangles below are similar. The lengths are in centimetres.
Calculate \(a\) and \(b\).
\[ \begin{aligned} \text{Answer } a & = \\ b & = \end{aligned} \]
Question 14
\(f(x)=\frac{7-3x}{2x}\)
(a)
Find \(f(4)\).
\(\_\_\_\_\)
Answer
(b)
Find \(f^{-1}(x)\).
Answer \(f^{-1}(x)=\) \(\_\_\_\_\)
Question 15
The table shows part of Ms Dube's payslip for a particular month.
| Earnings | \$ | Deductions | \$ | | :--- | :--- | :--- | :--- | | transport allowance | 100.00 | pension contribution | 6.00 | | housing allowance | 129.00 | union subscription | 10.00 | | | | medical aid | 8.00 | | | | Insurance | 17.50 | | basic salary | 275.00 | total deductions | - | | net salary | - | | |
(a)
Calculate the
#### (i)
total deductions.
#### (ii)
net salary.
(b)
Express the pension contribution as a percentage of her basic salary.
Answer (a)
(i) \$ \(\_\_\_\_\) ..... [1]
(ii) \$ \(\_\_\_\_\)[1]
(b) \(\_\_\_\_\)[2]
Question 16
The times of buses from Aytown to Deetown are shown.
| Aytown | 0704 | 0804 | 0856 | 0900 | 0932 | 1056 | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | Beetown | - | - | 0905 | - | 0941 | 1105 | | Ceetown | 0718 | 0818 | 0914 | - | - | 1114 | | Deetown | 0735 | 0835 | 0931 | 0928 | 1005 | 1131 |
(a)
Maryam lives in Ceetown and has to be in Deetown by 0930.
What time is the latest bus from Ceetown that she can catch?
Answer
(b)
Aadil catches the 0932 from Aytown to Deetown.
How long does his journey take?
Answer minutes [1]
Question 17
The first four terms \(u_{1}, u_{2}, u_{3}\) and \(u_{4}\), in a sequence of numbers are given by
\[ \begin{aligned} & u_{1}=1 \times 2+3^{2}=11 \\ & u_{2}=2 \times 3+4^{2}=22 \\ & u_{3}=3 \times 4+5^{2}=37 \\ & u_{4}=4 \times 5+6^{2}=56 . \end{aligned} \]
(a)
Evaluate \(u_{5}\).
Answer \(\_\_\_\_\)
(b)
The \(n\)th term of the sequence is \(u_{n}\).
Write down an expression for \(u_{n}\) in terms of \(n\).
Answer \(\_\_\_\_\)
(c)
Given that \(u_{n}=A n^{2}+B n+C\), find the values of \(A, B\) and \(C\).
Answer \(A=\) \(\_\_\_\_\) \(B=\) \(\_\_\_\_\) \(C=\) \(\_\_\_\_\)
Question 18
(a)
Evaluate \(\left(\frac{5}{3}\right)^{-2}\).
\(\_\_\_\_\)
Answer
(b)
Simplify \(\left(\frac{9}{t^{6}}\right)^{\frac{1}{2}}\).
\(\_\_\_\_\)
Answer
(c)
Simplify \(\frac{2 x^{3} y}{6 x y^{2}}\).
\(\_\_\_\_\)
Answer
Question 19
(a)
In the diagram, two small triangles are shaded.
Shade one more small triangle, so that the diagram will then have one line of symmetry.
(b)
In the diagram, two small squares are shaded.
Shade two more small squares, so that the diagram will then have rotational symmetry of order 2.
Question 20
The pie chart shows the distribution of an athlete's daily exercise programme.
(a)
Calculate the value of \(x\).
(b)
If the athlete spent 18 minutes jogging, calculate the
#### (i)
time she spent on weight lifting.
#### (ii)
total time spent exercising.
Answer
(a) \(x=\) \(\_\_\_\_\)
(b)
(i) \(\_\_\_\_\)
(ii) \(\_\_\_\_\)
Question 21
A machine puts beads of different colours and sizes into packets. The beads are selected at random from a large container and the selection of each bead for a packet is independent of all others. The table shows information on the contents of six packets.
| Packet | 1 | 2 | 3 | 4 | 5 | 6 | Total | | :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | Total number of beads | 15 | 14 | 19 | 18 | 16 | 18 | 100 | | Number of blue beads | 6 | 5 | 8 | 6 | 8 | 7 | |
(a)
Calculate the relative frequency of the machine selecting a blue bead.
Answer
(b)
Calculate how many blue beads you would expect in a packet of 30 beads.
Answer
(c)
The probability that the machine selects a red bead is 0.17.
Calculate the probability that the machine does not select a red bead.
Answer
Question 22
The diagram at the bottom of the page shows the lines \(A B\) and \(B C\).
(a)
By measuring an angle, find reflex angle \(A B C\).
Answer \(A B C=\)
(b)
The point \(D\) is on the opposite side of \(A C\) to \(B\). \(C D=C B\) and \(A D=10 \mathrm{~cm}\). On the diagram, construct quadrilateral \(A B C D\).
(c)
On the diagram, construct the locus of points, inside the quadrilateral \(A B C D\), that are