JUNE 2017 SESSION
2 hours 30 minutes
Candidates answer on the question paper.
Additional materials: Geometrical instruments
Mathematical tables/ Non-programmable electronic calculator
Graph/plain paper
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.
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 three 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. Answers in degrees should be given correct 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 Non-programmable electronic calculators may be used to evaluate explicit numerical expressions.
Question 1
(a)
Two similar square-based pyramids have base areas of \(9 cm^2\) and \(25 cm^2\).
Find the ratio of their volumes, in the form \(a:b\), where \(a\) and \(b\) are integers such that \(a<b\).
**Answer: (a)**
(b)
Anesu changed 500 South African rands into United States dollars when bank exchange rate was US\$1 = R12,50. The bank charged \(3\%\) of the amount that had been changed as commission.
#### (i)
Calculate the bank's commission in US\$.
**Answer: (b)(i)** US\$ \_\_\_\_
#### (ii)
Calculate the amount in United States dollars that Anesu received.
**Answer: (b)(ii)** US\$ \_\_\_\_
Question 2
(a)
It is given that \(244_n + 32_n = 331_n\).
Find the value of \(n\).
**Answer: (a)** \(n=\) \_\_\_\_
(b)
Simplify \(\frac{m^2 - m - 12}{m^3 - 9m}\).
**Answer: (b)**
[3]
(c)
The Venn diagram shows the universal set, \(\xi\), and subsets \(P\) and \(Q\). The number of elements in each region is as shown.
Find
#### (i)
\(n(P)\).
**Answer: (c)(i)** \(n(P)=\)
#### (ii)
\(n(Q')\), where \(Q'\) is the complement of set \(Q\).
**Answer: (c)(ii)** \(n(Q')=\)
#### (iii)
the value of \(w\) if the number of elements in the universal set, \(\xi\), is twice the number of elements in \(Q\).
**Answer: (c)(iii)** \(w=\) \_\_\_\_
Question 3
(a)
The length of each side of an equilateral triangle is 8 cm.
#### (i)
Calculate the area of the triangle.
**Answer: (c)(i)** \_\_\_\_ \(cm^2\)
#### (ii)
Express the area of the triangle in square metres.
**Answer: (c)(ii)** \_\_\_\_ \(m^2\)
(b)
H varies directly as \(\sqrt{Q}\) and \(H=51\) when \(Q=289\).
Find the
#### (i)
formula connecting \(H\), \(Q\) and a constant \(k\),
**Answer: (a)(i)** \(H=\)
#### (ii)
value of \(Q\) when \(H=81\).
**Answer: (a)(ii)** \(Q=\) \_\_\_\_
(c)
Make \(m\) the subject of the formula \(T=2\pi\sqrt{\frac{em}{g}}\).
**Answer: (b)** \(m=\) \_\_\_\_
Question 4
(a)
The radius of a circle is 32 cm measured to the nearest centimetre.
#### (i)
Write down the least possible value of the radius.
**Answer: (a)(i)** \_\_\_\_ cm
#### (ii)
Take \(\pi\) to be \(\frac{22}{7}\).
Calculate the least possible value of the circumference of the circle.
**Answer: (a)(ii)** \_\_\_\_ cm
(b)
#### (i)
State the order of the matrix \(\begin{pmatrix}3 & 1\end{pmatrix}\).
**Answer: (b)(i)**
#### (ii)
Evaluate \(\begin{pmatrix}3 & 1\end{pmatrix}\binom{4}{3}\).
**Answer: (b)(ii)**
(c)
Matrix \(\mathbf{G}=\begin{pmatrix}-2 & 5 \\ 0 & 6\end{pmatrix}\).
Find \(\mathbf{G}^{-1}\), the inverse of matrix \(\mathbf{G}\).
**Answer: (c)**
(d)
Solve the equation \(3x^2 - 4x - 11 = 0\), giving the answers correct to 2 significant figures.
\_\_\_\_ or \_\_\_\_
Question 5
(a)
In the diagram, the points \(R, T\) and \(X\) are on the circumference of a circle centre \(O\). The diameter \(XR\) is produced to \(P\) and \(PT\) is a tangent to the circle at \(T\). \(\angle RPT = y^\circ\) and \(\angle RTX = 2y^\circ\).
Find
#### (i)
\(\angle RTP\) in terms of \(y\).
**Answer: (c)(i)** \(\angle RTP =\)
#### (ii)
the value of \(y\).
**Answer: (c)(ii)**
(b)
Answer the whole of this question on a sheet of plain paper on page 14.
Use ruler and compasses only for all constructions and show clearly all construction lines and arcs.
All constructions should be on a single diagram.
#### (i)
Construct
##### (i)
triangle ABC in which \(AB=8 cm\), \(AC=6.5 cm\) and \(\angle BAC=60^\circ\), [Line \(AB\) has been drawn on page 14]
##### (ii)
the locus of points equidistant from AB and BC,
##### (iii)
the perpendicular bisector of BC.
#### (ii)
##### (i)
Shade the region, inside the triangle, containing the set of points which are nearer to BC than AB and also nearer to C than B.
##### (ii)
Measure and write down the length of BC.
**Answer: (b)(ii)** \(BC=\) cm
(c)
Describe the locus represented by the perpendicular bisector of BC in (a)(iii).
Question 6
Answer the whole of question 6 on this page.
**Answer:**
(a) (i) on diagram
(ii) on diagram
(iii) on diagram
(b) (i) on diagram
(ii) \(BC=\) cm
(c)
Question 7
(a)
Evaluate \(\log_7 7^{-2} - \log_5 \frac{1}{5}\).
**Answer: (a)**
(b)
In the diagram, \(EH=11 cm\), \(HG=8 cm\), \(EG=15 cm\), \(\angle EGF=44^\circ\) and \(\angle EFG=110^\circ\).
#### (i)
Calculate EF.
**Answer: (b)(i)** \(EF=\) \_\_\_\_ cm
#### (ii)
Calculate \(\angle EHG\), giving the answer to the nearest degree.
**Answer: (b)(ii)** \(\angle EHG=\)
#### (iii)
the shortest distance from E to GF produced.
**Answer: (b)(iii)** \_\_\_\_ cm [2]
#### (iv)
the bearing of F from G, given that E is due west of G and E, F, G and H are on level ground.
**Answer: (b)(iv)** [2]
Question 8
Answer the whole of this question on the grid provided on page 21. Triangle ABC has vertices at \(A(1;1), B(3;1)\) and \(C(2;3)\).
(a)
#### (i)
Draw and label triangle ABC.
#### (ii)
Triangle ABC is mapped onto triangle \(A_1B_1C_1\) by a transformation represented by the matrix \(\begin{pmatrix}1 & 0 \\ 2 & 1\end{pmatrix}\).
Draw and label triangle \(A_1B_1C_1\).
#### (iii)
An enlargement of factor \(-1\frac{1}{2}\), centre \((0;0)\) maps triangle ABC onto triangle \(A_2B_2C_2\).
Draw and label triangle \(A_2B_2C_2\).
(b)
#### (i)
Describe completely the transformation represented by matrix \(\begin{pmatrix}1 & 0 \\ 2 & 1\end{pmatrix}\) in (a)(ii).
#### (ii)
Write down the matrix that represents the enlargement in (a)(iii).
(c)
A translation \(\binom{3}{-4}\) maps point B onto point \(B_3\).
Write down the coordinates of point \(B_3\).
**Answer:**
(a) (i) on graph