ZIMBABWE SCHOOL EXAMINATIONS COUNCIL <br> General Certificate of Education Ordinary Level
**MATHEMATICS**
PAPER 2 JUNE 2020 SESSION
Candidates answer on the question paper Additional materials: Mathematical Instruments Mathematical Tables Non programmable Electronic Calculator
4004/2
2 hours 30 minutes
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. Write your Centre and Candidate number in the box on the top right corner of every page of this paper. Check that all the pages are in the booklet and ask the invigilator for a replacement if there are duplicate or missing pages. Answer all questions in Section A and any four questions from Section B. 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. Decimal answers in degrees should be given correct to one decimal place. Mathematical tables and 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. This question paper consists of 26 printed pages and 2 blank page(s). Copyright: Zimbabwe School Examinations Council. J2020.
**SECTION A (52 Marks)**
Answer all questions in this section
Question 1
(a)
The difference between two fractions is \(3 \frac{2}{3}\). The smaller fraction is \(2 \frac{1}{4}\). Find the other fraction.
Answer (a) \(\_\_\_\_\)
(b)
The population of a certain country is 24,9 million. Express this population in standard form.
Answer (b) \(\_\_\_\_\)
(c)
Increase \(\$ 40,00\) in the ratio \(8: 5\).
Answer (c) \(\_\_\_\_\)
(d)
Evaluate \((7 \sqrt{5})^{2}\).
Answer (d) \(\_\_\_\_\) [1]
| Candidate Name | Centre Number | Candidate Number | | :---: | :--- | :--- | | 3 | | | | | | |
(e)
Simplify \(11011_{2}+243_{5}\), giving the answer in base five.
Answer (e) \(\_\_\_\_\)
Question 2
(a)
Two similar cups have diameters of 6 cm and 10 cm.
#### (i)
Write down the ratio of their volumes.
Answer (a)(i) \(\_\_\_\_\)
#### (ii)
Given that the volume of the smaller cup is \(100 \mathrm{~cm}^{3}\), calculate the volume of the larger cup.
Answer (a)(ii) \(\_\_\_\_\)
(b)
A wooden block is in the form of a prism whose cross-section is a parallelogram with base 35 cm, perpendicular height 20 cm and length \(1,2 \mathrm{~m}\). Calculate the
#### (i)
surface area of the cross-section of the block.
Answer (b)(i) \(\_\_\_\_\) [2]
#### (ii)
volume of the block,
Answer (b)(ii) \(\_\_\_\_\)
#### (iii)
mass of the block if \(3 \mathrm{~cm}^{3}\) of the block weigh \(2,5 \mathrm{~g}\).
Answer (b)(iii) \(\_\_\_\_\)
Question 3
(a)
Find the value of \(y\) for which
\[ \left(\begin{array}{ll} y & 4 \end{array}\right)\binom{-2}{3}=\left(\begin{array}{l} -2 \end{array}\right) . \]
Answer (a) \(\_\_\_\_\) [2]
(b)
Find the matrix \(\mathbf{P}\) such that \(P\left(\begin{array}{cc}3 & -4 \\ 2 & 0\end{array}\right)=\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)\).
Answer (b) \(\_\_\_\_\)
(c)
The equation of a straight line is \(y+3 x=-4\)
Find the
#### (i)
coordinates of the point where the line crosses the \(y\)-axis,
Answer (c)(i) \(\_\_\_\_\)
#### (ii)
gradient of the line,
Answer (c)(ii) \(\_\_\_\_\)
#### (iii)
equation of a line parallel to the line \(y+3 x=-4\) and passing through the point \((3 ; 5)\).
Answer (c)(iii) \(\_\_\_\_\)
Question 4
Answer the whole of this question on the blank space below. 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
#### (i)
triangle ABC in which \(\mathrm{AB}=7 \mathrm{~cm}, B . \hat{A} C=45^{\circ}, \mathrm{BC}=8 \mathrm{~cm}\),
Answer (a)(i) on the diagram \(\_\_\_\_\)
#### (ii)
the locus of points equidistant from B and C,
Answer (a)(ii) on the diagram \(\_\_\_\_\) \(\_\_\_\_\)
#### (iii)
the locus of points 5 cm from C.
Answer (a)(iii) on the diagram \(\_\_\_\_\) \(\_\_\_\_\)
(b)
A point R in triangle ABC is such that it is nearer B than C and is less than 5 cm from C. Show by shading, the region in which R must lie.
Answer (b) on the diagram \(\_\_\_\_\) \(\_\_\_\_\)
(c)
Measure and write down the size of \(A \hat{B} C\).
Answer (c) \(\_\_\_\_\) \(\_\_\_\_\)
Question 5
(a)
Show that the equation \(x-3=\frac{5}{3 x}\) reduces to \(3 x^{2}-9 x-5=0\).
Answer (a). \(\_\_\_\_\)
(b)
Solve the equation \(3 x^{2}-9 x-5=0\).
Give the answers correct to 2 decimal places.
Answer (b) \(\_\_\_\_\)
(c)
In the diagram, ABCD is a cyclic quadrilateral with centre \(\mathrm{O} . \mathrm{AB}\) is produced to M. \(M \hat{B} C^{=} 70^{\circ}\) and \(B \hat{D} C^{=} 40^{\circ}\). Find
#### (i)
\(B \hat{C} D\),
Answer (c)(i)
#### (ii)
\(A \hat{B} D\).
Answer (c)(ii)
#### (iii)
\(A \hat{D} O\).
Answer (c)(iii) \(\_\_\_\_\) [2]
**SECTION B (48 Marks)**
Answer any four questions from this section.
Question 6
Each question carries 12 marks
In the diagram, ABO is a triangle in which M is the mid-point of OB and N lies on AB such that \(A N=\frac{1}{5} A B\). ON and AM intersect at X. \(\overrightarrow{O A}=a\) and \(\overrightarrow{O B}=b\).
(a)
Express, in terms of \(a\) and/or \(b\)
#### (i)
\(O \vec{X}\),
Answer (a)(i) \(\_\_\_\_\)
#### (ii)
\(A \vec{N}\),
Answer (a)(ii) \(\_\_\_\_\) [1]
#### (iii)
\(\overrightarrow{O N}\),
Answer (a)(iii)
#### (iv)
\(A \vec{M}\).
Answer (a)(iv)
(b)
Given that \(\overrightarrow{A X}=h A \vec{M}\), show that \(O \vec{X}=(1-h)^{a}+\frac{1}{2} b b\)
Answer (b) \(\_\_\_\_\) \(\_\_\_\_\)
(c)
If \(\overrightarrow{O X}=k \overrightarrow{O N}\), express \(\overrightarrow{O X}\) in terms of \(a, b\) and \(k\).
Answer (c) \(\_\_\_\_\)
| Candidate Name | Centre Number | Candidate Number | | :--- | :--- | :--- | | 12 | | |
(d)
Use the results of (b) and (c) to find the numerical values of \(h\) and \(k\).
Answer (d) \(\_\_\_\_\) \(\_\_\_\_\) [3]
(e)
Hence, or otherwise find the ratio of area of triangle OAX to area of triangle OAM.
Answer (e) \(\_\_\_\_\) [1]
| Candidate Name | Centre Number | Candidate Number | | :--- | :--- | :--- | | 12 | | |
(d)
Use the results of (b) and (c) to find the numerical values of \(h\) and \(k\).
Answer (d) \(\_\_\_\_\) \(\_\_\_\_\) [3]
(e)
Hence, or otherwise find the ratio of area of triangle \(O A X\) to area of triangle OAM.
Answer (e) \(\_\_\_\_\) [1]
Question 7
Answer the whole of this question on the grid below. Use a scale of 2 cm to 1 unit on both axes.
|  | □ | □ | |  | | | □ | | | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | | | | | | | □ | | |  | | □ □ | □ | □ | | | CI | | : : | | \(\_\_\_\_\) | | □ | | □ | □ | | | | | | こ: : | |  | | □ | | | | | | | | |  | □ | | | \(\_\_\_\_\) | | □ | |  | | | □ — : | | | | | | | | | | |  | □ | □ | | | | \(\_\_\_\_\) |  | | □ | | | | | | | | | | | | □ | □ | □ | | | | | |  | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |
(a)
Triangle PQR has vertices at \(\mathrm{P}(1 ; 3), \mathrm{Q}(2 ; 1)\) and \(\mathrm{R}(4 ; 3)\).
Draw and label triangle PQR. \(\_\_\_\_\) \(\_\_\_\_\)
(b)
Triangle \(P_{1} Q_{1} R_{1}\) is the image of triangle \(P Q R\) under a reflection in the line \(y=-x\).
#### (i)
Draw the line \(y=-x\).
Answer (b)(i) on diagram \(\_\_\_\_\)
#### (ii)
Draw and label triangle \(P_{1} Q_{1} R_{1}\). \(\_\_\_\_\)
Answer (b)(ii) on diagram \(\_\_\_\_\)
#### (iii)
A transformation, G, maps triangle \(P Q R\) onto triangle \(P_{2} Q_{2} R_{2}\) with vertices at \(P_{2}(1 ;-6), Q_{2}(2 ;-2)\) and \(R_{2}(4 ;-6)\).
Draw and label triangle \(P_{2} Q_{2} R_{2}\).
Answer (b)(iii) on diagram \(\_\_\_\_\) \(\_\_\_\_\)
#### (iv)
Describe fully the single transformation, G, which maps triangle PQR onto triangle \(P_{2} Q_{2} R_{2}\).
Answer (b)(iv) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
(c)
The point \(R_{3}(-1 ; 2)\) is the image of \(R\) under a translation.
#### (i)
Find the translation vector.
Answer (c)(i) \(\_\_\_\_\)
#### (ii)
Write down the coordinates of \(P_{3}\) and \(Q_{3}\) the images of \(P\) and \(Q\) respectively, under the same translation.
Answer (c)(ii) \(\_\_\_\_\) \(\_\_\_\_\)