Surname Forename(s) Centre Number Candidate Number

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For Performance Measurement

ZIMBABWE SCHOOL EXAMINATIONS COUNCIL

General Certificate of Education Ordinary Level

MATHEMATICS 4030/2
PAPER 2
JUNE 2017 SESSION 2 hours 30 minutes
Candidates answer on the question paper.
Additional materials: Geometrical instruments
Mathematical tables/ Noil-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 T O 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 comer 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.

I N F O R M A T I O N 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.

This question paper consists of 30 printed pages.

Copyright: Zimbabwe School Examinations Council, J2017.

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Section A [64 marks]

Answer all questions in this section.

I 4
Simplify 2 - — x —, giving the answer as a mixed number.

Answer: (a) [2]

Find the Highest Common Factor (H.C.F) of

23 x 32 x 5 x 74,

2 3 x 33 x 5- x 7 2 ,

2 4 x 3 x 5 x 73,

leaving the answer in index form.

Answer (b) [2]

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1 (c) Find the Lowest Common Multiple (L.C.M) of 3x2y, Sx^y1 and 8.vy\

Answer: (c) [2]

(d) (i) Express 248 as a product of its prime factors.

(ii) Find the number by which 248 must be multiplied to make it a

perfect square.

Answer: (d) (i) [2]

00 [1]

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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) [2]

Anesu changed 500 South African rands into United States dollars when the
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$ [3]

(ii) Calculate the amount in United States dollars that Anesu received.

Answer: (b) (ii) US$ [2]

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2 (c)
S

The Venn diagram shows the universal set, and subsets P and Q. The
number of elements in each region is as shown.

Find

(i) n(P),

Answer: (c) (i) n(P)= [1]

(ii) n(Q'), where Q 1 is the compliment of set Q.

Answer: (ii) n(Q') = [1]

(iii) the value of w if the number of elements in the universal set, is twice
the number of elements in Q.

Answer: (in) w= [3]

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It is given that 244„ + 32„ = 331,,.

Find the value of n.

Answer: (a) n- [2]

m~ - m - 1 2
Simplify
m3 -9m

Answer: (b) [3

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The length of each side of an equilateral triangle is 8 cm.

(i) Calculate the area of the triangle.

Answer: (c) (i) cm 2 [2]

(ii) Express the area of the triangle in square metres.

2
Answer: (ii) m [2]

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4 (a) H varies directly as -^Q and H = 51 when Q = 289.

Find the

(i) formula connecting H, Q and a constant k,

Answer: (a) (i) H= [1]

(ii) value of Q when H = 81.

Answer: (a) (ii) Q= [2]

em
(b) Make m the subject of the formula T - 2 j t — ,
8

Answer: (b) m [3]

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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. RPT = y" and R X T = 2 y ° .

Find

(i) RT P in terms of_y,

Answer: (c) (i) RTP = [1]

(ii) the value of v.

Answer: (ii) v= [3]

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10

(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 [1]

22
(ii) T a k e n to b e — .

Calculate the least possible value of the circumference of the circle.

Answer: (a) (ii) cm [2]

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11

(I) State the order of the matrix (3 l).

Answer: (b) (i) [1]

(ii) Evaluate (3 i)^

Answer: (ii) [1]

'-2 S\
Matrix G =
0 6

Find G " , the inverse of matrix G.

Answer: (c) [2]

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12

Solve the equation 3x2 — 4JC — 11 - 0 , giving the answers correct to 2

significant figures.

Answer: (d) x= or [5]

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6 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.

(a) Construct

(i) triangle ABC in which AB = 8 cm, AC = 6,5 cm and BAC = 60°,

[Line AB has been drawn on page 14J [3]

(ii) the locus of points equidistant from AB and BC, [2]

(iii) the perpendicular bisector of BC. [2]

(b) (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. [2]

(ii) Measure and write down the length of BC. [1]

(c) Describe the locus represented by the perpendicular bisector of

BC in (a)(iii). [2]

DO NOT W R I T E IN T H I S SPACE

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14

Answer the whole of question 6 on this page.

Answer: (a)
(<*) (i)
(0 on diagram [3]

(ii) on diagram [2]

(Hi)1 on diagram [2]

(b) (0 on diagram [2]

(W BC = cm [1]

(c)

[2]

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Section B [36 marks]

Answer any three questions in this section.

Each question carries 12 marks.

7 A luxury bus has 100 units of seating area. There are two types of seats. Ordinary and
First Class.

Let the number of Ordinary seats be * and First Class seats be y.

(a) Ordinary seats take up 1 unit of seating area and First Class seats take up 1,5
units of seating area.

Form an inequality which satisfies this condition and show that it reduces to
2at + 3>'s200.

Answer: (a) [2]

(b) There must be at least 10 First Class seats.

Write down an inequality which satisfies this condition.

Answer: (b) [1]

(c) There must also be at least twice as many Ordinary seats as First Class seats.
Write down an inequality which satisfies this condition.

Answer: (c) [1]

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Answer d, e and f of question 7 on the grid on page 17.

(d) The point (x; y) represents x Ordinary seats and y First Class seats.

Draw the graphs of the inequalities in

(0 (a), [1]

(H) (b), [1]

(Hi) (c). [1]

(e) Show, by shading the unwanted regions, the region in which (x; y) must
lie. [2]

(f) A luxury bus company which uses this type of luxury bus charges $ 15 for
each Ordinary seat and $25 for each First Class seat for a certain trip.

Use the graph to find the greatest possible amount of money that the
company would receive from this trip. [3]

DO NOT W R I T E IN T H I S SPACE

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Answer: (d) (i) on graph [1]

(ii) on graph [1]

(Hi) on graph [1]

(e) on graph [2]

(f) $ [3]

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18

8 (a) Evaluate log7l'2 -

Answer: (a) [2]

In the diagram, EH = 11 cm, HG = 8 cm, EG = 15 cm, EGF =• 44° and

EFG - 1 1 0 ° .

(i) Calculate

EF,

Answer: (i) EF = cm [3]

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19

8 (b) Calculate

(ii) EHG, giving the answer to the nearest degree,

Answer: (ii) EHG = [3]

(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: (iv) [2]

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Answer the whole of this question on the grid provided on page 21.

Triangle ABC has vertices at A(l; 1), B(3; 1) and C(2; 3).

(a) (i) Draw and label triangle ABC. [1]

(ii) Triangle ABC is mapped onto triangle A1B1C1 by a transformation

(I 0\
represented by the matrix I I.

Draw and label triangle AiBjCi. [3]

(iii) An enlargement of factor -1—, centre (0; 0) maps triangle ABC onto
triangle A2B2C2.

Draw and label triangle A2B2C2. [3]

(b) (i) Describe completely the transformation represented by

mat
" X (2 in
PI

(ii) Write down the matrix that represents the enlargement in (a)(iii). [1]

(c) A translation | maps point B onto point B3.

Write down the coordinates of point B3. [1]

DO NOT W R I T E IN T H I S SPACE

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21

1 r
\
£
1
r

(>

TT
-J—
1 .

1
«-

-4- i
> "i 4 ; / 4 A

-2

-4

_1_

-6
-L | I

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22

Answer: (a) 0) on graph [1]

00 on graph [31

(ih) on graph [3]

(b) (i)

— P]

00 ( ) [1]

O'O B3 ( ; ) [l]

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23

The following is an incomplete table of values for the function y = j(3 - 2x - A*2).

X -4 -3 -2 -1 0 1 2 3
y -l 0 0,6 0,8 0,6 0 -1 P

(a) Calculate the value of p.

Answer: (a) p = [1]

Answer the following questions on the grid on page 24.

(b) Draw the graph of y - --(3 - 2x - x 2 ). [4]

(c) By drawing a suitable tangent, estimate the gradient of the curve at x = 0. [2]

(d) Use the graph to

(i) solve the equation j ( 3 - 2x - x2) = -0,5, [3]

(ii) find an estimate of the area bounded by the x-axis and the curve. [2]

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24
10

V
1

4-
- 1,5 -r-
;

. 11 1

4-

-L

i
(\
u 4 Xi
J. ,
Cm
U.J
A

1 ^ r

I I-M-
I i i T

)-

Answer: (b) o n graph [4]

(c) [2]

(d) (i) x = or [3]

(W [2]

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25

In the diagram OA = 10a and OB = 106. M is tlie mid-point of OA. T is a point on

AT T
AB such that — = - .
AB 5

MTP and OBP are straight lines.

(a) Express, in terms of a and/or b,

(i) AB,

Answer: (a) (i) AB = [1]

(ii) AT,

Answer: (ii) AT = []]

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26

11 (a) (iii) MT

Answer: (a) (iii) MT - [2]

(b) It is given that OP = fcOB.

Express OP in terms of b and k.

Answer: (b) OP = [I]

(c) It is also given that OP - OM + hMT.

Show that OP - (5 - h)a + 6hb

Answer: (c) [2]

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27
Use the results from (b) and (c) to find the value of h and the value of k.

Answer: (d) h =

*- [4]

Hence express OP in terms of b.

Answer: (e) OP = [1]

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28

12 The heights of 60 children were recorded. Below is an incomplete frequency and

frequency density table of the results.

Height 110<hsl20 120<hsl25 125<h*130 130<h*145 145<hsl50

(h cm)
Frequency 12 18 S 12 10

Frequency 1,2 3,6 m 0,8 2

density

(a) State the modal class,

Answer; (a) [1]

(b) Find the value of m.

Answer: (b) [1]

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29

12 (c) Calculate an estimate of the mean height.

Answer: (c) cm [3]

(d) If two children were chosen at random, calculate the probability that one had a
height of not more than 120 cm and the other had a height greater than 145 cm.

Answer: (d) [3]

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30

12 (e) Answer this p a r t of question 12 on the grid below.

Draw a histogram which represents this information.

±
—

3,5

2,5'

|
Oh
1,5
III
rT*

|I1I

+p

0,5'

i-prm-mrr a
120 125 130 135 140 145 150
Height (cm)

Answer: (e) on the graph [4]

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