MINISTRY OF EDUCATION, BOTSWANA
in collaboration with
UNIVERSITY OF CAMBRIDGE LOCAL EXAMINATIONS SYNDICATE
Botswana General Certificate of Secondary Education
0563/03
MATHEMATIGS
October/November 2006
Paper 3
Additional
Materials: Answer paper
Electroniccalculator
2 hours 30 minutes
,
Graph paper (3 sheets)
MathematicalTables(optional)
Geometrical instruments
Head the lollowing carefully before you start.
Write your answers on the separate answer paper provided.
Start each question on a fresh side of the page.
Write your centre number, candidate number and name on each sheet of answer paper you use.
Answer all questions.
Allworking must be clearly shown. The working shor.rld be done on the same sheet as the rest of
the answer. Marks will be given for working which shows that you know how to solve the problem
even if you get the answer wrong.
At the end of the examination, fasten all of your work securely together using the string provided.
Do not use staples, paper clips; highlighters, glue or correction fluid.
The number of marks is given in brackets [ ] at the end of each question or part question.
The total of the marks for this paper is 125.
Electronic calculators may be used.
lf the degree of accuracy is not specified in the question and if the answer is not exact, the answer
should de given to three significant figures. Answers in degrees should be given to one decimal
place.
L
ln any question where the value of n is required, use the value from your calculator or take
3.142.
This document consists of 11 printed pages and 1 blank page.
zr
as
2
Mathematical formulae for paper 3
Surface area and volume of solids
Name of solid
Volume
Total surface area
cone
nrZ +
nrl
\nrztt
{ base area x height
pyramid
sphere
tor'
4nr2
sinA_sinB-sinC
abc
a-b="
sinA sin B
Area of a triangle
Cosine Rule
sin C
'2- Lab sinC
az
=bz + c2 -Zbc cosA
.
cosA
=
b2+c2-a2
2b,
Statistics
Variance
=
E(x-
Ef (x - i\2
7\2
Z,f
n
-,
Standard deviation (SD) =
friliIice
=
o*
n
-(i)'
w.rr,
I/
.\l
3
(uj' e shop charge s lX6oValue
(i)
(ii)
(b)
Added Tax (VAT) on all the goods it sells.
A photocopier costs P4370 before VAL
Calculate its selling price.
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A computer is sold at P5998.30.
How much does it cost without VAT included?
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A salesperson receives 37o of the sales before VAI as commission.
Calculate the sales if the person received a commission of Pl323.
The population of a town was 7.8
(a)
x
104 and
12)
it was expected to double in twenty years.
Calculate the population that was expected at the end
standar{ form.
of twenty years, giving the answer
m
t2]
(b)
Twenty years later, the actual poputation of the town was 1.49 x l0s.
What is the difference between the expected value and the actual value of the population? Give the
answer in standard form.
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(c)
Calculate the actual percentage growth in the population during the twenty years.
(a)
Make r the subject of the formula
a(l + r\
'^-- t-r
(b) lvrite
(c)
;IJ - *,
Factorise 6a2 * ab
as a single fraction.
- b2 completely.
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4
.. A triangular farm AEC is such that AB = 20km, AC
=
(a)
and
OAC
_
SA,.
using a scare of I cm to represent
2km, construct a scare diagram
of the farm.
(b) (i)
(ii)
(c)
14.2ym
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Measure and wrire down rhe
Iengrh af BC.
II]
What is the acual length of BC?
IU
Calculate the area of the farm.
121
(d)
ffi1:H?J:lx'#iTi;*i,Tfl,i's
the same disrance from
A
and.c,zrnd
is arso 3 km from.AB.
t41
5
Moso newspaper costs px per copy
and Lesedi newspaper costs
(a)
The National Library buys 3
copies of Moso and 4 copies
of Lesedi every week.
(i)
(ii)
(b)
pJ, per
copy.
write down
If
an expression in terms of
-r and y for the total weekly
cost in
the total rveekly cost is Pt 7,
fom
an equation in x and
)
pula.
to represent this
I1l
information.
information,
(iiy
write down the price of
a copy
of Lesedi
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newspaper.
It takes the National.Library weens
to spend a ceftain ;
a
It takes the schoor Lituw;;#;to
-- newspapers'
spend the same r*fr:::'on
uullt'
Find the Ieast possibr" uuiu" of the
amount.
l
p)
(c)(i)Solvetheequationsinparts(a)and(b)simuItaneously'
(d)
tr
The school Library spends a total
of P12qo. byr 2 copies of Moso
and 3 copies of Lesedi every
Form an equation in x and to
week.
;l represent this
Ill
,
[z]
Tiie diagram shows part of a circular measuring instrument with cenre O.
fhe instrument has a needle OA such that its tip rotates about O and OA= 8cm.
The tip of the needle moves from A to B such that AOB = 41 .8".
(a)
Calculate
(0
the area of the minor sector A0B,
12]
. (ii)
the shortest distance from A to B.
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{i)
Show that the length of the minor uc AB = 5.84cm, correct to 3 significant figures.
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(lr)
The tip of the needle takes 0.43 seconds to move from A to B.
Calculate the speed, in m/s, of the tip of the needle,
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(b)
a^
,ldnswer
the whole of this question on a sheet of graph paper.
Using a scale of
(a) (i)
(ii)
(iii)
(b)
cm to represent 1 unit on each axis, drawx- andy-axes each from
Draw and label triangle A with vertices at (-2,0), (-2, 4) and (-4, 3),
Draw and label triangle B with vertices at (-3, -1), (-7,
-l)
and
(-6,
I ).
Describe fully the transformation which maps triangle A onto triangle B.
Rotate triangle B through 90o clockwise about (-6, -4).
Label the image C.
(c) (i)
(ii)
(iii)
(d)
1
Draw and label triangle D with vertices ar (5, 0), (7,
*2
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tll
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121
l)
and (7 , -3).
Describe fully the transformation which maps triangle.4 onto triangle D.
Enlarge triangle D using scale factor
Label the image E.
-8 to 8.
and centre (5,
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[2]
-2).
Describe fully the single transformation that maps triangle E onto triangle A.
w1
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7
I
The frequency table below shows the heights, in centimetres, of a group of
Id standard two pupils.
!
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(a)
(b)
Number of pupils
8l
2
83
I
85
6
86
4
87
3
Find the probability that a pupil chosen at random is at least g6 cm tall.
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Two pupils are chosen at random, one after the other from the group.
What is the probability that
(i)
(ii)
(c)
Height (cm)
they are both 85cm rall,
one is
8l cm tall and the orher is 87cm tall.
showing your working clearly, calculate the variance of the heights,
t21
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I
"rThe diagram below shows a conical shelter with base radius OC = Rcm,and slant height AC
= le)syy1.
The shelter is made of a conical plastic top and the rest is fibreglass.
The base radius pB of the plastic top is
The
(a)
(b)
ratio r: R=2:5.
Show thatAB =79.6cm.
The outside surface area of the plastic top is 1 .6 m2.
Calculate the outside Surface area of the fibreglass in m2.
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9
10 A machine prints pages in either black and white or in colour.
"t
(a)
(b)
In one minute, the machine prints x pages in black and white.
Express, in terms of x, the time, in seconds, the machine takes to print one page
in black and white.
In one minute, the machine can print 2 more pages in black and white than in
colour.
Express, in terms of x,
(i)
(iD
(c)
[l]
the number of pages the machine can print in colour in onJminute,
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the time, in seconds, it takes the machine to print one page in colour.
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It takes 2'5 seconds more to print a page in colour than to print it in black and
(i)
white.
Form an equation to represent this information and show that it reduces
to
x2-zx-4g=o.
(ii)
(iii)
,
Solve the equarion
* -X*
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48 = 0.
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How long, in seconds, does the machine take to print 1 page in colour?
tl
l
11 Answer the whole of this question on a sheet of graph paper.
The length of 47 steel nails were measured to the nearest millimetre and the results
are shown in the table
below.
Length (mm)
Number
nails
\q,,
of
39
10
44-
45-50
51--53
t2
I
Thc boundaries fcr the firsi iwo classes arrc 38.5
-s1:56
7
-6s
9
- 44.5 airti 44.5 - 50.5 respeciiveiy.
Write down the boundaries for the next three classes.
(b) Draw a histogram such that the lengths of the nails in the range 3g
rectangle of width 2cm and height 5 cm.
(c)
57
The Iongest 38Vo of thenails are rejected.
Estimate the length of the lougest nail thar is acceptable.
ttl
-
44cm arc represerited by
a
t4)
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IO
12
T$.diurytT shows a concrete slab ABCDEF in the form of a right-angled triangular prism.
DEA = CFB = 90o, DE= 50cm, AE= l50cm and AB= 210cm as shown.
All measurements iue correct to the nearest 10cm.
D
50
E
(a)
Write down the upper bounds for the lengths of
(i) DE,
(ii) AE,
(iii) AB.
(b)
(c)
(d)
(e)
Calculate. in m3, the upper bound for the volume of eonerete needed to
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tU
nna-k-e
tlre sla,b,
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The concrete is a mixture of cement, sand and crushed stones in the ratio of 2 : 6: 7 by volume.
Calculate, in m3, the upper bound for the volume of cement needed to make the slab.
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The density of cement is l400kg/m3.
Calculate the upper bound for the mass of cement needed for the slab,
t21
Cernent is sold in 50kg bags.
How many bags of cement should be bought for the slab?
t2)
11
13 Answer the whole of this question on a sheet of graph paper.
A city council plans to build
(a)
(b)
(c)
shops and offices for rental.
The number of shops to be built is.r and each will have a floor area
Express, in terms of x, the total floor area required for the shops.
of lZ5m2.
The number of offices to be built is y and each will have a floor area of 50 m2.
Express, in terms of y, the total floor area required for the offices.
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The council has only 75Am2 of land to build offiq:s and shops.
Write down an inequality in.r and y and show that it reduces to
5x+2y<30.
(d)
The council wishes to rent out each shop for P10000 per month and each office for
P7000 per month.
The council expects to make at least P70000 per month from the rentals.
write down an inequality in x and y and show that it reduces to
10x+7y>70.
(e)
(f)
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The council decides that there will be ai least one shop.
Write down another inequality to represent this information.
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using a.scale of I cm to represent I unit on each axis, show, by shading the unwanted regions, the set
of points satisfying 5x + 2y € 30, lOx + 7y > 70 and the inequality in part (e).
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G) How much money will the council make if they rent twice
as many offices as shops?
121