Pure
Mathematics
CAPE® PAST PAPERS
Macmillan Education
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This revised version published August 2016
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Cover design by Macmillan Publishers Limited
Cover photograph © Caribbean Examinations Council (CXC ®)
Cover image by Mrs Alberta Henry
With thanks to: Krissa Johny
CAPE® Pure Mathematics Past Papers
LIST OF CONTENTS
Unit 1 Paper 02 May 2005
4
Unit 1 Paper 03/B May 2005
10
Unit 2 Paper 02 June 2005
14
Unit 2 Paper 03/B May 2005
19
Unit 1 Paper 01 May 2006
24
Unit 1 Paper 02 May 2006
31
Unit 1 Paper 03/B May 2006
36
Unit 2 Paper 01 May 2006
40
Unit 2 Paper 02 May 2006
46
Unit 2 Paper 03/B May 2006
51
Unit 1 Paper 02 May 2008
55
Unit 1 Paper 03/B May 2008
60
Unit 2 Paper 02 May 2008
63
Unit 2 Paper 03/B May 2008
68
Unit 1 Paper 02 June 2008
72
Unit 1 Paper 03/B June 2008
78
Unit 2 Paper 02 July 2008
82
Unit 2 Paper 03/B June 2008
85
Unit 1 Paper 02 May 2009
89
Unit 1 Paper 03/B June 2009
96
Unit 2 Paper 02 May 2009
99
Unit 2 Paper 03/B June 2009
104
Unit 1 Paper 02 May 2010
107
Unit 1 Paper 03/B June 2010
114
Unit 2 Paper 02 May 2010
118
Unit 2 Paper 03/B June 2010
124
Unit 1 Paper 02 May 2011
128
Unit 1 Paper 03/B June 2011
135
Unit 2 Paper 02 May 2011
138
Unit 2 Paper 03/B June 2011
145
Unit 1 Paper 02 May 2012
148
Unit 1 Paper 032 June 2012
154
Unit 2 Paper 02 May 2012
158
Unit 2 Paper 032 June 2012
165
Unit 1 Paper 02 May 2013
169
Unit 1 Paper 032 June 2013
175
Unit 2 Paper 02 May 2013
180
Unit 2 Paper 032 June 2013
186
Unit 1 Paper 02 May 2014
190
Unit 1 Paper 032 June 2014
197
Unit 2 Paper 02 May 2014
203
Unit 2 Paper 032 June 2014
209
Unit 1 Paper 02 (12 May 2015)
213
Unit 1 Paper 02 (16 June 2015)
219
Unit 1 Paper 032 May/June 2015
225
Unit 2 Paper 02 May/June 2015
229
Unit 2 paper 032 May/June 2015
236
Unit 1 Paper 02 May/June 2016
241
Unit 1 Paper 032 May/June 2016
272
Unit 2 Paper 02 May/June 2016
286
Unit 2 Paper 032 May/June 2016
317
TEST COPE 02134020
FORM T J> 2005253
C i\RIBIIE AN
Mt\ Y/JUi\'E 200S
EXAM I NAT I ONS
CO
NC IL
ADVA NC ED P ROFICIENCY EXAM INAT IO N
I'URE MATHEMATICS
UNIT I - I'AI'ER 02
2 bount
( 25 MAY 2005 (p.m.))
This c:.;.lmill.ltion papct consi51s of THREE sectiou: Module I, Module 2 and Module 3.
Each scc:110n consi.s1S of 2 questions.
The m:a.'1mum m~L f« c:xh section is 40.
The m:l'lmum mlll for lhis ~~:l.mitUI.ion rs 120.
TtuJ c:um•n;acion c;omists of 6 poages.
INSTRUCTIONS TO C;\1\DIOATES
E.1MllnJhOn
I.
1>0 NOT Open this ex:uni•lation paper until i•1Structcd to do liO-
2.
1\nswcr A tL questions from the T tl REt~: $«lions.
3.
Unless othcrwiJ;C suued in the •ucstion, :my numerical answer th:u is not
exact MUST be written correct to three significant fiJul\:s,
M:uc:rials
M.:t.thc.mactC<ll formulx and tables
El«'tron.tc:: cakulauw
(;Qph popcr
Copyright C 2()().1 Caribbean P.x::uninatiom Council
All righu testr\•cd .
02 134020/CAI'E 2005
-2-
S4..'t.tio n A (Module I)
Answer llOTH questions.
I.
(n)
(i)
Complete the table below f<r the function
-2
ll;l II I
8
(ii)
(b)
0
- I
I ft.x) 1. whc,re j{;r) = x (2- x).
2
3
4
I I I0 I I8 I
II
Ske<ch the graph of f(.t)
ior - 2 ,; .<,; 4 .
l 2 mark$)
I
4 marks)
Find the value(s) of the rc~ l number. k. for which the cqu::~t i on k(x2 + S) ~ 6 + 12t -x"l
has equal rools.
[ 6 mar·ksl
I
(c)
(ii)
4 marks)
\Vithou1 using c~ lculatoi'S OEiables. evaluate
Total '20 marks
GOON TO THE NF..XTPAGE
02 134020/CAPE 2005
- 3-
2.
(a)
Prove, by Mathematical Induction, that !On - 1 ts divisible by 9 for all pos1t1ve
[ 9 marks]
integers n.
(b)
A pair of simultaneous equations is given by
px + 2y = 8
- 4x + p 2y = 16
where p
(i)
R.
Find the value of p for which the system has an infinite nu mber of solutions.
[ 3 marks]
Find the solutions for this value of p .
[ 3 marks]
x+ 4
Find the set of real values of x for which x-2 > 5.
[ 5 ma rks]
(ii)
(c)
E
Total 20 marks
Section B (Module 2)
Answer BOTH questions.
3.
The equation of the circle, Q, with centre 0 is x 2 + y 2 - 2x + 2y = 23.
(a)
Express the equation of Q in the form (x - a) 2 + (y- b
(b)
Hence, or otherw ise, state
(i)
(ii)
the coordinates of the centre of Q
the radius of Q.
i = c.
[ 5 marks]
2 ma rks]
[ 1 mark]
(c)
Show that the po int A(4, 3) lies on Q.
3 m a rks]
(d)
F ind the equation of the tangent to Q at the point A .
(e)
The centre of Q is the midpoint of its diameter AB. Find the coordinates of B.
[ 4 marks]
[ 5 marks]
Total 20 ma rks
GO ON TO THE NEXT PAGE
02134020/CAPE 2005
-4-
4.
The diagrams shown below, not drawn to scale, represent
a sector, OABC, of a circle with centre at 0 and a radius of 7 em, whe re angle AOC
n rad.tans.
measures3
a right circ ular cone with vertex 0 and a circu Jar base of radius rem which is formed when
the sector OABC is folded so that OA coincides with OC.
0
A,C
B
A
B
(a)
(i)
Express the arc length ABC in terms of n.
(ii)
Hence, show that
[ 1 mark ]
7
3 marks]
a)
r= -
b)
if hem is the height of the cone, then the exact value of his
6
7 -{35
6
[ 2 marks]
(b)
(i)
(ii)
Show that cos 3
e= 4 cos3 e - 3 cos e.
[ 5 marks]
The position vectors of two points A and B relative to the origin 0 are
a = 4cos2 8i+(6cos8-l)j
b
=
2 cos
ei
- j.
By using the identity in (b) (i) above, find the value of 8, 0::; 8::; ~ , suc h that
a and b are perpendicular.
(c)
Find the modulus of the complex number z
[ 5 marks]
=
25 (2 + 3i)
4 + 3i
[ 4 marks]
Total 20 marks
GO ON TO THE NEXT PAGE
02134020/CAPE 2005
•5•
Sectio n C (Module 3)
;\nswtr BOl'lf (tutstions.
5.
{a)
lim
sin u
I I mark J
{i)
State the vnlue of
(ii)
By means of the substinnion u = 1l'. show that lim
II~
I)
II
.f- )
sin 3x
0
,\
= 3.
4 marksl
(iii)
(b)
lim
X-> 0
sin 3x
~·
4 rnarksl
\
rr y = :_
+ Bx. where A nod Ban: constant~:. sl1ow that
,,~
(c)
l~encc. evaJu~1te
d 1\•
i'it!
X
+
dy
X
'JX ;;
y.
I 4 marks)
'f11edi3gram below. not dr.twn to ~C!IIt•. show~ p:ut of the cuf"\'C y1 = ·b:. Pis the point
on lhc: curve ut which lhe line. y = 2l" cut!( the c ur...,c.
r=lr
Find
(i)
(li)
the coordinate..~; of f'
L 3 nmrk..:;J
the \'Qiume of the solid a;cncrutcd by rot3ting the shaded a~1throog.h 2n radians
::~bou t the .'(·axis,
( ~ mnrks)
Tot:d 20 marks
GO ON 1"0TIIENEXTI'AG1l
02134020/CAPE 2005
- 6-
6.
(>)
Difl'ercnti:uc, with respect to x.
(x! + 7)5 + sin Jx.
(b)
(c)
( 6 m::trks]
Oetenninc the values o f x for which the function y ~ XJ- 9,\'? + 15.l' + 4
(i)
has st~ti011 3ry points
( 3 marks)
(ii)
is increasing
( l marks I
(iii)
is ckcreasing.
I 2 morksl
(i)
(ii)
Usc the substitution 1 e
11 -.t to
show that J
(l" /(.t) dx • fu" /(<1 -x) d.-e.
If } .:j(x)cb: = 12.usethcsubstitutionr=,\' - l
0
r .s nmrk.s)
toevalu~lle/1 3/(X-I)tU.
5
I 3 mnrksJ
TOhll20 marks
END OF TEST
02134020/CAPE 2005
TEST CODE
FORM TP 2005254
CARIBBEAN
02134032
MAY/JUNE 2005
EXAMINATIONS
COUNCIL
ADVANCED PROFICIENCY EXAMINATION
PURE MATHEMATICS
UNIT 1 - PAPER 03/B
If hours
( 20 MAY 2005 (p.m.))
This examination paper consists of THREE sections: Module 1, Module 2, and Module 3.
Each section consists of 1 question.
The maximum mark for each section is 20.
The maximum mark for this examination is 60.
T his examination paper consists of 4 pages.
INSTRUCTIONS TO CANDIDATES
1.
DO NOT open this examination paper until instructed to do so.
2.
Answer ALL questions from the THREE sections.
3.
U nless otherwise stated in the question, any numerical answer that is
not exact MUST be written correct to three significant figures.
Examination materials
Mathematical formulae and tables
Electronic calculator
Graph paper
Copyright © 2004 Caribbean Examinations Council
All rights reserved.
02134032/CAPE 2005
-2 Section A (Module 1)
Answer this question.
1.
(a)
(b)
G iven that 2x2 + 8x + 11
the constants h and k.
the value of EACH of
[ 5 marks]
If p, q, r, s E R , use the fact that (p- q) 2 ;::: 0 to show that p 2 + q 2 ;::: 2 pq.
[ 2 marks]
(i)
Deduce that if p 2 + q 2 = 1, then pq :::; ~-
(ii)
(c)
= 2(x + h) 2 + k for all values of x, find
[ 1 mark]
A club bakes and sellsxcakes, making a profit, in dollars, that is modelled by the function
f(x)
=x 2
(i)
(ii)
- lOx.
Sketch the graph of the function f (x)
=x?- 1Ox .
[ 8 marks]
From your graph, determine
a)
the LEAST number of cakes sold for which a profit is realised
[ 2 marks]
b)
the GREATEST possible loss in dollars
c)
the number of cakes for which the GREATEST possible loss occurs.
[ 1 mark]
[ 1 mark]
Total 20 marks
GO ON TO THE NEXT PAGE
02134032/CAPE 2005
.3•
$c.."Ciicm 8 (Module 2)
A n.swcr this
2.
(a)
<Ju ~tion.
TI1e straigJu lii)C thtough the point P (4, 3) is peq>endicul:;u to J.r + 2y = 5 :md mc<:LS the
given line at N.
JZind
the length or chc 1ine-segn.-:•n fN.
( ii)
(b)
I 6 marks)
the cootdi•Hl.tes of N
(i)
l •w'•rksj
T he to.ble below presents d:uo oolhcted on the movement of the tide at various times
uftcr midnight on a particulor day.
'l'im c Afttr M idnig.hl
(I hours)
(l1 mttres)
High
0
12
Low
6
2
Hish
12
12
Low
18
2
nc:te 1\Jo,·cmcnt
Height
The height. It me tres, con be modtlled by a function of the fom1 h = I' cos (qt)0 + 7
where 1 is the time in hour$ afier midnight. Use dlc dota from the 1able to lind the
valuesof/IMdq.
(12nmrk.<;J
'l'otnl 20 marks
GOON TO T HE NEXT I'AGE
02 134032/CAPil200S
-4 -
S«tion C (Module 3)
;\ nSw('r Lhis QUt'SliOrl.
3.
(a)
(i)
( ii)
lim
x->1
x'- I
x- I ·
I 3 nmrk.sl
[)c,termine the 1"('31 \' 3 hle$ o( X (Qr which the function
1<- 1
JV) =- x1 -x- 2
is cont.inuous:.
(b)
J mnrksl
Differentiate with respect to X . rror. first princip i i.':~. the funccion x: + b.
( 5 marksj
(c)
Initially, the depth of Wtlter in a t:.ltk is 32 m. W~uer drains from the Ulllk chrough a
hole cut in che bouom. At/ minute! afler the water begins droining. the depth o f water
in the tank isx metres. The depth. ofthe water c hanges. with respect to time t. at the rate
equal to (-21- 4).
Find an expression for ,'C i11 tcmls of r.
(i i)
I 5 marks!
l·le.nce. detennine how long it takes for the water to drain completely from the
tonk.
1 4 mnrk.sj
To101120 marks
END OF TEST
02134032/CAPE 2005
TESTCODE02234020
FORM TP 2005256
CA RIB BEAN
MAYIJUNE2005
EXAM I NAT I ONS
COUNC I L
ADVANCED PROFICIENCY EXAMINATION
PURE MATin>MATICS
UNIT 2 - I'APER 02
2 lr!Jurs
01 .JUNE 2005 (p.m.)
·rhis examination paper consistS of THREE SC<;tiuns: Module I. Module 2 and Module 3.
Each scclion consists of 2 questions.
The maximum mnrk tor cnch section is 40.
The maximum mart\ tor this ex:Jmination is 120.
This examinmio'' consisls of 5 pages.
INSTRUCTIONS TO CANDIDAn.S
I.
00 NOT open this cxamino.tion paper until instruc ted to do SQ.
2.
Answer A I.. I.. questions from th-; 'I'I:IJU~£ sections.
3.
Unless Otherwise s:UIIcd in the question. tilly numerical :tnswer thou is not
MUST be written oom:et 10 three significant figures.
e:<~t
Examination Materials
Mathcmmieal fornwloo and tables
Electronic calculator
(iraph paper
Copyright (i) 2004 Ou'ibtc:m Examinations Council
All righ1.s reserved.
02234020/CI\1'6 2005
-2-
Section A (Module 1)
Answer BOTH questions.
1.
(a)
T he diagram below, not drawn to scale, shows two points, P( p , 0.368) and R (3.5, r),
on f (x ) = ex for x E R.
f (x) = ex
y
p
0
X
(i)
Copy the diagram above and on the same axes, sketch the graph of g(x) = ln x .
[ 3 marks]
(ii)
Descri be clearly the relationship betweenf (x)
= ex
and g(x)
= ln x.
[ 3 marks]
(iii)
(b)
U sing a calcul ator, fi nd the value of
a)
r
1 mark]
b)
p.
2 ma rks]
Given that
log0 (be)= x, Iogb (ea) = y, logc(ab) = z and a -:~; b -:~; e,
show that axl.lcZ
(c)
= (abel
Find the values of x
E
R for which eX + 3e-x
3 marks]
= 4.
8 marks]
Total 20 marks
GO ON TO THE NEXT PAGE
02234020/CAPE 2005
- 32.
(a)
A CufVe is g,iven J):lrameuically by x = (3- 21)1, y = ,., - 2t. Find
<ly •
(i)
(b)
1
·~
10
terms o( 1
( 4 murks)
(ii)
the grJdicm of l_he normalt•> the curve at the point 1 c: 2.
(i)
Ell: pres-s 2.\' + 1 in the fc:rm
stnnts. X"' (.\' + 1)
(ii)
l·lence. CY31u•lle
~
.\'
+
...~~
J'.\'2(""
l Z mat•ks)
+ ~ , where i\ , 6 and C arc con·
x+ 1
I 7 marks)
2x+ I
+ I) tl,\'·
'
Tot,al 20 marks
S«tion 8 (ll.·l odule 2)
Ans wer BOTH queslioos.
J.
(a)
I
I
Use the f:K:tthat-- -
(i)
r
r+ 1
=
I
r (r +
1
S. -t
(
)1
--r = l l(r+ l ) 11+1 '
I) to s how 1h111
I
I I mark ·1
(ii)
(b)
5·'
·~'"
. f:-1\'Cn
.
b y ,. =--.,.
F"md A L l_ 11c
I vaI ucs
••tl! c.:ommon rnt1.o. r. o f n g.(.'Qmctn.r :>en.cs 1S
4 + ,._
of x ror which the !\Cries conv.:rges.
'
JIO marks)
(C)
By substituthl_g suit::ablc values of xon both sides o f the expansion o f
( I +xr ==
L" "c,.,' .
r =-0
show th at
"
r.
(i)
.. =z··
~c
I 2 uui.rks)
r::O
L" "C, (-1)' = 0.
(ii)
( 2 murksJ
r=O
Total 20 marks
GO ON TO T ilE NEXT PAGE
02234020/CAPE 2005
· 4·
4.
11lc function./. is given by J(x) • 6 - 4;r - .t'.
(~)
Show that
(i)
f
(ii)
the equ:uionj{x);::;: 0 has a tt:ll root. ex. in the <;loS«! interval I I. 21
4 marksl
is everywhere strictly dececasing
L -'marks-}
(iii)
(b)
If
f .I mnrks"l
a is the only real root of theequ:uionj{.t) = 0.
x. is lhe n'h approximation to a. use dw Newton·Raphson method to show thntlhe
(n + I t itpproxim:uion x• .. 1 is &iven by
2.1;1 + 6
,'(... ,= 3.-;+4.
I 8 rnnrksl
Tot:.liZO mnrks
Section C (Module 3)
Answe.-130Tl:l qutStlons.
5.
(a)
On a particul:u-day, o <::ertain fuel service s-t3.tion offered 100 customers who purchased
premium or regular gusoline, a frre c-hec-k of Lhe t!ngine oil or brnkc Ouid in the ir
\.'o?hi('l('$ Thl_~ S(':l"\'i('('S r~fllli r('.(i by ll\t-~ ('IIS(Otner<> wer1• ~S (olfows·
IS% of the customers purchased premium gasoline. the otherS purchased regular
g{tSoline.
20% of the CU$10illCI'$ WhO pun::h~scd premium g_:.$01i llC reqUCSICd :1 Check for broke fiuid,
the others requested a check for engine o il.
51 oflhecustomei'S who putehased regular gasoline requested a check for engine oil. the
o-the.tS requested a check for brake fluid.
(i)
Copy a•'d comp\e(e the di;tgl't'm below to represent lhe. event srxu:e.
llrllkl.' fluid
/'""\
Engine oil
/'""\
)
(
(
51
'-../
l--.~mhml
g;!$01int:'
l(C'guJa r
gt;tse>lioc
'-../
I 3 marks(
GO ON TO THE NEXT I'AGE
02234020/CAPE 2005
- 5-
( ii)
(b)
Find the probobility that a customer chosen at random
a)
who had purchased premium g:.soline requC$tcd n check for engine oil
b)
who had requested uclted:. of the brake Ouid purchased regular gnsolinc
c)
who had teqoosted :.'I check or the engine oil purcha$t.-d regular gasoline.
I 6 marks]
A bag oontah\S 12 red b.;'llls, 8 blue balls :.mel 4 white balls. Three balls are drawn from
the bng at random wlthoul replaeell:lCnt.
Calculate
r 3 mnrksj
(i)
1.he tota.l number or ways ofchoosing lhc three balls
(ii)
lhe probability that ONE ba!l of EACH colour is drawn
(iii)
the probability that ALL ·r t1JU!I! balls druwn are of lhc SAME colour.
3 mnrks]
[ 5 mark$)
Total 20 mark.~
6.
Col
Pind the vnlues of X for which
X
X
2
(b)
2
2
• 0.
x
[ JOmarks]
1\velve hundted people visited :..n exhibitiOJl or1 hs opening day. 11lerenfcer. the
a ueodance fell each day by 4% of tlte number on the previous day.
(i)
Oblllin an expression for the number of "i!;itors on the n1" day.
L 2 mnrksl
(ii)
Find !he: total m.un ber of visitors for the first n days,
I 3 mnrksl
( iii)
TI'e exhibiti o1~ closed aftet 10 days. Detennine how many people visited
duri.ng tJ1e period for Which it was Opened.
r 3 mnrksl
(iv)
lfthcex hibition h:td been kept opened indefinilely. wh:lt would be !he maximum
number of vis-itors1
I 2 marks I
Totol20 m~lrk$
ENI) Of T EST
02234020/CAPE 2005
•
TEST CODE 02234032
FORM TP 2005257
CA RIIIB EAN
MAY/JUNU 2005
EXAM I NAT I O NS
COUNCIL
ADVANCED PROFICIENCY EXAMINATION
PURE MA TU EMATl CS
UNIT 2 - PA l'E R 03/B
1-j. /:ours
( 23 MAY ZOOS (p.m.))
This examination p41per consists of THREE secriom: Module I. Module 2, and Module 3.
Exh section consists of I question.
The maximum mark foe eoch S«tion is 20.
The. m~imum mark for this examination is 60.
T his examination paper consists of 5 pages.
INSTRUCTIONS TO CANDIDATES
I.
00 NOT open this cx:.minalion paper unlil inslruc(cd to do so.
2.
A 1lswer ALL questions (Yom the THRt;t:: seetioo:s.
3.
UnJcss otherwise sta.te<l in the question. any nuoltric:~J answer tht~t is
r\Ot exa<:t iM UST be wrinen C<ltrect to tJuee si&nific:H*t figures.
Examination matcriols
Mathe•natic:d formuJae and tables
Electronic calcuhuor
Grnph paper
Copyright 0 2004 C~beean Examinations Council
All rights reserved.
02234032/CAPE 2005
- 2St.>ction A (Module 1)
Answer Ui s question.
l.
Table I presents d:tl3 Ql)l:tined from a biobgical investigation that involves two vatiables X
::tJld >'·
Table I
X
20
JO
40
so
y
890
1040
2500
3700
It is believed that x andy are rel:ued by the fonnu la. )' =b.\*.
(a)
(i)
(ii)
(b)
By taking logarithms to OOse 10 o f bolh side$. conver1 y • b.~ to the fonn
Y = nX + d where n nnd dare constonts.
I 4 mark$)
Hence.• express
o)
Y in terms of y
b)
X i1ltenns of x
c)
d in tenus of b.
r 3 mnrksl
Use <he dala f ro m Table I 10 compl«c Thble 2.
Tnble 2
1.30
UiO
).2 1
3.57
( 2 mar-ks)
(c)
Jn the J;r:lph Oil page 3, fog.. X is pJOII~d against JQg 10 )' (Qr ).) $,\' $ 1.7.
(i)
(ii)
Assuming that the 'be&t Slra1ghtline' is drawn to lit the data. determine
a)
the g,rodient of this l!ne
Z mnrksl
b)
the value of/) given thm Ihis line pas.scs thrQUgh (0. I)
4
c)
Lhe \'aluc of each oflhe oonstMts," and d. in P:lfl (tl) (i) above.
l 2 mnr k.o;J
mr~rks l
Using Ihe graph. Qt ozhtrwise. estim::ne the value of x ror which)' is 1800.
I 3 mnrks-1
T()lnl20 mnrk.~
GO ON TO THE NEXT PAGE
02234032/CAPE 2005
.. -~ ..
~~
IJ
\
. . •.
f
I
1
I·
•
.
1-+
I·
tIJ
1
r •
H-
I:E
11~1
1-
1-
1~
.
I:
1
.l
±-
1
1
~lD;~tn,.trr,~nthtiTfi~t§~~~-'E1=j[' Qif:H}:tH,~ ~~~H+H,.i:~~~·h;:l~I~~~~~,Q1:~~~~~f!UlU~
:
:~ r ' !
: ,r
I .
13
~
r
I; 1'1;li ~:
:
:
1\ I
mm 1 I~Hr ,
I~ II IJ
I
1'1
1\
l
It
1
,,
i
I
t
t
i"T
+I.
1·1·1·1·1+
~:.....
-t-i'o.l-1~
N
NT TllB NEXT r AO
022~ 4032/
A P 2005
I·
-4Section B (Module 2)
Answer this question.
2.
Mr John Slick takes out a n investment with a n investment company which requires
making a fixed payment of $A at the beginning of each year. At the end of the investment period, John expects to receive a payout sum of money which is equal to the total
payments made, together with interest added at the end of EACH year at a rate of r % per
annum of the total sum in the fund.
The table below shows information on Mr Slick's investment for the first three years.
Year
Amount at
Beginning of Year ($)
1
A
Interest ($)
A
Payout Sum$
r
X
A+(Ax_r)
100
100
= A(1+-r)
100
=AR
2
A+ AR
(A+ AR)
X _ r_
(A+ AR) + [(A+ AR)
100
X
1~0
J
= (A + AR) (1 + I ~O)
=(A+ AR) R
= AR+ AR2
3
A+AR+AR2
(a)
Write expressions for
(i)
(ii)
2
r
2
(A + AR + AR) x lOO (A+ AR + AR ) R
2
= AR+AR +AR3
the amounts at the beginning of Years 4 and 5
2 marks]
payout sums at the end of Years 4 and 5.
2 marks]
(b)
By using the information in the Table, or otherwise, write an expression for the
amount at the beginning of the nth year.
[ 2 marks]
(c)
Show that the payout sum in (b) above is $ AR (R"- 1) for R > 1.
R-1
(d)
Find the value of A, to the nearest dollar, when n = 20, r = 5 and the payout sum in (c)
above is $500 000.00.
[ 7 marks]
[ 7 marks]
Total 20 marks
GO ON TO THE NEXT PAGE
02234032/CAPE 2005
-5Seclion C . todulf' J)
A~·tr
J.
lhis quesdon.
Theoutput3x I matrix Yin:atcstin;gpror:usmachemtnlpl;..ntiJn:laiC'dtothemputlx l matrix
X by me;u1s of the tqu;IDoo Y : AX. \~rhere
A •
I 2 3 )
(
2 4 S
3
s
•
6
(u)
Show thJI A is non·singu1nr.
5 marks I
(b)
Show lh:Lt X • A-•y ,
3 marks I
(<)
lo;nd A
(d)
Find the mput nutri~ X correspond.i1g to the o.nput rN:tl'\( Y •
1
•
I 9 ruorksl
(
~.9·2 )·
~
( J marks)
Total 20 marks
0 2234032/CAPE 2005
TESTCOOE 02134010
FORM TP 2006257
CA RIBB EAN
MAYIJ UNE 2006
EXA MI NAT IO NS
COUNCIL
ADVANCED PROFICIENCY EXAMINATION
PURE MATHEMATICS
UN IT I - PAI'ER 0 1
2 hours
( 19 MAY 2006 ( p.m.))
This ex::unin:uion paper ~ol'l si.sts of THREE sections: Module I. Module 2. Md Module 3.
Each sectio•' cons-ists of S questions.
The m~i mum m~ for each section is 40.
The maximum mnrk for this examination is 120.
This examination paper consist.<; of7 pages.
INSTRUCTIONS TO CANDIDATES
l.
00 ~o·r open this cxnminarion paper until inst_ructcd 10 do so.
2.
Answer ALl... qucslions from :he TH R£E sec-Lions.
3.
Unless otherwise St:lted in the question. tmy numerical unswer that is
Mt exact ~·l UST be wriuen (:()JTC:Ctto three significant figures.
E.x::amination m3tcrials
M:nh<:-l'l'l<ltiC:LI formuloe and tables
Electronic calculator
Gr.~ph p<lpcr
Copyright C 2005 Caribb!an Ex:unin.ations Coundl®
All rights reserved.
02 134010/CAPE 2006
. 2.
S«tion II (Module I)
Aa~wer
I.
(a)
ALl. qu<'stions.
The funclionj{;c) is given by j{;c) = t' - (p + I ).~ + fJ. p e N.
(i)
S how that(.\'- I) is a foctcx-ofj{x) for all values o f p.
lmur k.S]
(ii)
rr (.\"- 2) is .. U.ctor offl.x). lind the vulue o f p.
Z mnrks)
n
II
(b)
Given th~t
L
r = 4-V• + 1). show t11at
r=l
2
L (3r + I) • .!. n(3n + S).
'c l
4 rnurksl
2
Tott~l
2.
(o)
Le1A={x :2SxS 7}andD=(x : x-4ISI•}.he R.
Find the I..A ttG EST value of It for which 8
(b)
8 marks
c
i\ ,
Lelx. y.kc: R such th m(x+jy)l+kY •x'- +.ty+yl.
I
Find the value of l:.
3 nn).rks )
Totnl? mnr_
ks
3.
(•)
(i)
Find a. b
e
ll such thai
l
X+ 1
- 2 •
a.t' + b
.\'+ 1
• where .t' lf; -
Hence. fi nd the rongc of values of x e rt for which
(1>)
Wi1hou1 chc use of calculatorS or tables. show lh:tt
I.
I l •nnrksl
.2L.
> 2.
x+ I
•
4
..[1. lt 8_,,,
1 .t marks )
= 2" (./2 ).
r :~ mal'ksl
Totnl 10 ma••ks
GO ONTOTHENEXT PAGE
02134010/CAPE 2006
- 34.
The diagram below (not drawn to scale) represents the graph of the function f(x)
- 1 ~ x ~ 1 and p, q E R.
= x 2 + 1,
f(x)
(- l , p)
(q, 2)
-------r--------~---------,,-----~ X
-1
(a)
(b)
0
+1
Find
(i)
the value of p and of q
[ 2 marks]
(ii)
the range of the functi onj(x) for the given domai n.
[ 1 mark]
Determine whether f (x)
(i)
(ii)
(iii)
is surj ective (onto)
1 mark]
is injective (one-to-one)
1 mark]
has an inverse.
[ 1 mark]
Total 6 marks
5.
Find the values of m , n
E
R for which the system of equations
X+ 2y = 1
2x+ my= n
(a)
possesses a unique solution
[ 3 marks]
(b)
is inconsistent
[ 2 marks]
(c)
possesses infi nitely many solutions.
[ 2 marks]
Total 7 marks
02134010/CAPE 2006
GO ON TO THE NEXT PAGE
•4 •
Section 8 (Module 2)
Answc.r ALL q ucslion.s.
6.
In the diagram below (not drawn to scalt). the straight line through the point P(2. 7) and
perpendicular to the line x + 2y = II intersects x + 2y = I I at the poim Q.
)'
Find
<•>
the equation of the line through P a nd Q
2 marks)
(b)
the coordin!ltcs of the point Q
3 marks)
(c)
the EXACT lengLh of the line segment f'Q.
2 marks]
Totn17 m t~ rks
GO ON TO T HE NEXT I' AGE
02134010/CAI'E 200~
·5·
7.
In the dlatr.t.m below (not d.ra"n 10 sc:.;:de), AC • BC. AD • 7 untts. DC= S units.
an&kACB•fndiansW~g)eAOC= ~ ndi~M.
•
7
Fond tho EXACT l<ngth of
"
(a)
AC
I 5 marks)
(b)
AB
(3 marks)
TocaJ 8 trulrk$
3.
<•>
Soh·c the cqu3tion 4 eosl&- 4 sin 0- I • 0
(b)
Show IIMI
(01
0 :s 0 s n.
l -t."OS2(
2t- • tan 2x.
1 +cos
l 5 nuark.sl
I 3 marks I
TotoIS marks
9.
(a)
The roots of the quadr.atie equation J.:! + 6.t + k • 0 w-e -3 + 2i and -3- 2i .
find lhe vnlue or the constant k.
(b)
Fuld the real numbc:rs u and'' such fh:u
( 2 marksJ
l!.:!1! •
3-41
I
+ ,.;.
( 6 marks)
Total 8 mark$
tO.
Gn·cn the \ectof'S p • 2i + 3j :and q = Ji - Zj.
(3)
fiftd 1'. y E R such thalxp + )'(~ • -.3i- llj
I 1 marks)
(b)
show th~n p :..nd (f are pctpendicular
l 2 ma.riu)
Tot:1l 9 marks
GOONTOTIIENEXT PAGE
02134010/CAI'E 2006
s~ct i o n
CCModute 3)
Answer ALL l(Uts tions •
II .
lhl:'l
,_, I
.r+x-2
.x--3x+2'
( .J marks l
(a)
find
(b)
Find the \•a luc.s of x e R sueh that t:tc func tion
/()
9 - x'
·' • ~,,_3)(Jx J-3)
is discontinuous.
I 4 nulrksl
~l'oln l
7 nutrk.<;.
2 -:t
Tile fune tion /(x) i.s defined by j{;t) = -,- (or x E R . x .,_ 0 .
12.
.r
Determine the nature of the crilical·taluc(S) offtx).
(b)
I_
6 mnrk.sl
Differentiate. with respect to x. JV)= s in::(x!).
T oUll9 mur k.."
13.
Titc diagram below ( not dr;_twn to sc:a't ) is a .skt:tch o( the St.-ction o( the (unction
.{Vt) - x (.~- 12) which JX1$:fe.3 tht<>ush the oOsin 0 . A and /J arc the ~lalion;try poinl..$ on d~
curve.
,.
Find
(3)
the eoordintttes of each of the s.tatiQtJ:.ll')' points.'' ~1 nd II
( ~)
the C<Juntion <>f the norm~ I to the c-tuvejV) ;; .'(
S mnrks]
,r- 12) ut the origin.
.a mnrks l
·rotal 9 nu\ rkS
GO ONT011ll! NEXT f'AG€.
02 13401OICAI'E 2006
-7-
14.
16
T he di agram below (not drawn to scale) s hows the shaded area, A, bounded by the curve y = x2
and the lines y
=
1
1,x
-x-
2
= 2 and x = 3.
y
I
y= 2x - l
16
Y = x2
X
r
(a)
Express the shaded area, A , as the difference of two definite integrals.
(b)
Hence, show that A= 16
(c)
F ind the value of A.
f x - dx - ; f
3
2
2
X
d.x +
d.x.
1 mark]
2 marks]
3 marks]
Total 6 marks
15.
Use the resul t
I:
=
f(x)dx
r
f(a- x)dx, a> 0 , to s how th at
(a)
fnx s in x dx = fn(n -x) sin x dx.
(b)
H ence, show that
0
0
(i)
(ii)
r
X
Sin
X
dx
= '!T; fsi n X dx- J:x Sin X cfx
[ 2 marks]
[ 2 marks]
[ 5 marks]
J:xsinxd.x=n.
Total 9 marks
END OF TEST
02 13401 0/CAPE 2006
TEST COOE 02134020
FORM TP 2006258
CARIB il EAN
MAY/JUNE 2006
EXA M I NATIONS
COUNC I L
ADVANCED PROFIC IENCY EXAMINATION
I'URE MA l'IH: MATICS
UNIT 1 - I'AI'ER 02
2 Jrours
( 24 MAY !006 (p.m.))
This exrunin~ti on
P~i>erconsists of THREE
st.etiort;;: Module I. Module 2 :uul Module 3.
&ich seetion cMsislS of 2 queStions.
The maximum mark for each section is 40.
The maximum mark for thb; cxamimllion is 120.
This cxamin:llion consists of 5 pages.
INSTRUCTIONS TO CANDIDATES
I.
1)0 NOT open this cxmn inntion pnpcr until instructed to do so.
2.
Answer At.L questiOilS fr'OJU th: T HREE sections.
3.
Unless otherwise stated in the question. any numeric:.) answer that is not
exact ~·fUST be written COITCC"tlO three significant ligures.
M:uhematical formu lae and tnbles
Electronic <:-alculntor
Crraph p;.lpel'
Copyright <0 2005 Caribbean Examinations Council®
All rights reserved.
02134020/CAI'E 2006
-2S«tion A (Modul~ I )
I.
(o)
Soh c lhc s•mul tanrou.s ~uations
,>+xy=6
I Smarks)
x-3y+ 1 • 0.
(b)
'rhc roots or the equation xl + 4x + I = 0 a.re a and fl. W•thout solving the cquotion.
(i)
(ii)
.sU•lo the vnlues of a+ 13 a•Hl afl
( lmarksl
rind the value of o.Z + f:V-
I 3 mnrks)
find the equation who-se I'OOlS arc I +
(iii)
! and I +-Jr.
r 7 marks)
TolallO marks
•
! , •-21n (It+ 1).
,.,
(a)
~c. by Mathem:uiol lnduction. f'l3l
(b)
E11pru.s.. •n 1mns of" and in the Sll-IPLEST form.
(i)
(10 marl<s(
( Z marks)
2n
1:
(ii)
r
(c)
Pind 11 if
" +I
1 4 marks]
r.
'"
l:
r• " + I
r
>=
100.
I 4 mark$)
Total2.0 marks
CO ON TO "rHE NEXT PAGE
02134020/C"Al'fi 2006
-3-
Sc.'ttioo
n
(Module 2)
Answer BOTH qutSiion.~
3.
(a)
(i)
Find the coordin:ucs of the centre nnd radius oft he circle X!+ 2x +
r -4)' s 4.
[ .a mnrksl
(ii)
By writing X+ l = 3 sin e. sllow th3:t the parametric equations of this circle are
x= - 1+3sin6.y=2+3eos0.
(iii)
(b)
( 5marksJ
Show that the x-<:-oordinates of the poina of intei'SieCtion of this c ircle with
the line .t+)'= I rucx=-1 t~ \'2.
( 4 marks]
Find the gener.tl solutions of the tqltnion cos 0 • 2 sin:O- l.
7 m"rks]
Total 20 marks
4.
(a)
(b)
Given that 4 sin x- cos .t c R sin (x- a). R > 0 and 0° <a< 90°,
( 7 marks]
(i)
lind the values of Rand C'J: co>rrect to one decimal place
(ii)
hence. find ONE value of x between 0° :md 360° for which the curve
y = 4 sin x-cos x has a stationary poinl.
( 2 morks )
L<:IZ 1 = 2- 3i and z, = 3 + 4i.
(i)
(ii)
Find in the form a + bi. "·bE R.
J
o)
z, + lz
I mark
b)
z: 1:.2
J m1lrks)
c)
.!!.
5 markS]
"
Pind the quadratic equation whose roo1s arc l 1 and t~.
[ 2 mMks J
Totnl 20 nulrks
GO ON TO THE NEXT PAGE
02134020/CAPE 2006
-4-
Scclion C (Module J)
Answ~r
5.
<•>
lim
(i)
State the value of
(ii)
Given th at sin 2(x +
a.ndlor &r.
(iii)
801'1-1 questions.
Sx -+ 0
sin 8x
[ I mnr k )
--s:r·
0..-)- sin 2x a 2 cos A sin B. find A nnd Bin terms of x
I 2 marks]
Hence. or otherwise. differentiate with respect to .:r. from first principles.
thefunctiony=sin 2l'.
(b)
I 7marksj
The curvcy = !z:c! + .!_ p:lSSCS through the point p (1 .1 ) ond hus a grodicnt or 5 31 P.
Filld
.<
(i)
the \'OJucs of the constams /rand k
( 5 morksl
(ii)
the equation of the tangent to the curve at the point wherex •
...!...
2
I 5 mar ks)
't otal 20 marks
GOONTOTiiE NEXT PAGE
02134020/CAPE 2006
. 5.
6.
(o)
In the diogram given below (nol drown 10 scule), the :u'e!l Sunder the line y = x. for
0S
xS
I, is divided into n set or n .ectongular strips each of width ! units.
n
J'
----~~o~F===~2---+J---+--~------~---~l_,~-h<"I.'o"J-..x
•
"
(i)
.
"
"
"
Show th:ll tJ1e areaS is tlppl'!).lCimately
I
2
3
tt - l
-::1+-,
+-,+
.... + -or-.
n
n
n
n·
(ii)
n- 1
r
Gi\•en th:ll
L
r= I
(b)
(i)
(ii)
(c)
= ~ rr (n- 1). show thol S • ~(I - ~).
-
Show that forf(x)=
Hence. ev~u:~te
I 6 murksl
-
2<
..-'+ ~ -
J0 1
Find the value of u > 0 if
"
( 4 marks)
( J morks]
flu .t'1 <lx = 1927 •
N
r 5 nlllrks]
1'ot.ul 20 mur-k..;
!::NO OF TEST
02134020/CAPE2006
TEST CODE 02134032
FORM TP2006259
CA R I BBE AN
MAY/JUNE 2006
EXAM I NA TIO NS
COU
C IL
ADVANCED PROFI C IENCY EXAMINATION
I' UltE MA'ni EMATICS
UNIT I - I'AI'ER 0318
( 19M;\ Y !006 (p.m.))
Thls examin:uion paper consists ofTt~REE sectiorti: Module I. Module 2. .and Module 3.
Each section con~iists of 1 question.
The mr.xlmum m~rk for each section is 20.
·rhe mruchnum m:.rk for this eJwminalion is 60.
'rhls ex:unin::nion paper cx.nsists or 4 p:ag(J,
ISS1'RUCTIOSSTO C,\J\"l>ll)AT&'l
1.
00 'OT open Lhis extuninmion paper until in.strucaed 10 do so.
2.
Answer A l.L questions (rom the 1'HRE£ sections.
3.
Unless othcn..,ise su'lttd in the question. any numericoJ :ms""er th:ll is
ll(l( e.xoct MUS'l' be 'olorintn cwrec:t tO three signme:.nc fiaures.
Examination m:neri:ds
Mathematical formulae and tables
Blcctronic calcuiOltOt
Or:.ph paper'
Copynglll 0 2005 C•nbi><Jn Ex:unin31ions Coun<i141
.t\11 n,hu ~nrccl
Ol134032/C-'PE 2006
•
I,
-- -(• I
,,_,
(II I
(h)
1
~·
t\ \OI..,_h ur M lthlt'-lic club hn,.'" five IUhtrltti, N, ~ . W,
I
1
m:ar*sl
lmd ''• l11 hill ltlliJthiJ cmnp. He
IU.tlt'J In lllSlJnrntnt. f of athldt'l M, "• ' oifld )I II) rh)'t.flllf KtiVhtt:J l. 2, 3 :lJld 4
ln:•Jh.l•nt eo lht duagram bdow tft "hk:h A lu ..... . ' 'I ,
2. ), 4) and
I
IC• . IJ
'' U.( ... )).(x.ll.()•A ll
.
(I)
(II)
hU)
-'-
.
1,
n.
Stilet C)Nb rt~n why the ""'"'l"""'tU J fmm A 11• If~~ not" fU"'lJcm..
l lmu rk )
Suuc I WO ch.angeslh!lt thtco••ch would nL>ed In 11111lc ao lhr.t1 the asslgnmem.
I 2 marks]
f. btc:umcs II func:liOfl g!" -+ n.
I \t-t-IIJ
lht function~ A
-+
»••
(II) .._.,«" .. 1 llt't ,., bfdtrt'd ~*"·
I J marbJ
OZ I \4(1\21( 'A I'll 2!KK>
( 10 ON 'I() 'll lll NllXT PAGI!
-3S('ttion B (Module 2)
Answer lhis <iucslion.
2.
(n)
In nn expcrianenl,thc live weigh!, wg:rams.. of a hen w;.s found LObe a line:lr function./.
of the number of days. tl. after the te.•, was placed on a special diet, where 0 5 d s: 50.
At the beginning of the experil'nt.nl, the hen weighed 500 grams and 25 days later it
weighed I SOO gram$.
(i)
Copy olnd c:ompletc the t:tblc. below.
tl (days)
w (gms)
25
soo
( Jtna rkl
(ii)
Octcrminc
I 3 m~u-ks l
n)
the line:tr func1ion.j. such that/(</)= w
b)
the expected weight o f ~ hen 10 days aflcr the diet began.
l 2 mnrk~J
(iii)
Show 1hat (tan o _sec or~ • sinlo - 2 sin 0 tl .
c-os 20
(b)
( ii)
(c)
After how mony days is lhchen expcx:1cd to weigh 2 LSO gmms'? ( 2 marks)
H ence show that I - si_n O • (t:tn
I +sm9
e- sec 0)2
,
[ J marks)
l
4 mu r k.'")
(iivcn the complex 1lumber z. • {3 + l i. find
2
2
( 1 m urk)
(ii)
arg (t)
2 mur-ksl
(iii)
z :c;
2 mnrk.sl
Total 20 rnorks
GOON TO Ti l E NEXT J'AGE
02134032/CAPE 2006
-4-
Section C (Module 3)
Answer this question.
3.
(a)
(i)
(ii)
By expressing X- 4 as
c-r-; + 2) c.Y~- 2), find
lim
x-74
G- 2
x- 4
.y-;-
2
Hence, find lim
2
x -7 4 x - Sx + 4
[ 3 marks]
[ 3 marks]
=10, find J: [f(x) + 4] dx + J: f(x) dx.
[ 7 marks]
(b)
Given that J:J(x) dx
(c)
A bowl is formed by rotating the area between the curves y = x 2 and y = x 2 - 1 for x ~ 0
and 0 ~ y ~ 1 through 2n radians around the y-axis. Calculate
(i)
(ii)
the capacity of the bowl, that is, the amount of liquid it can hold
[ 3 marks]
the volume of material in the bowl.
[ 4 marks]
Total 20 marks
END OF TEST
021 34032/CAPE 2006
•
TESTCOoe02234010
FORM TP 2006260
MAY/JUNE 2006
CAR IBB EAN
EXAM I NA TIO NS
COUNC i l-
ADVANCED PROFICIENCY EXAMINATION
J'URE MA'HIEMATICS
UNIT 2 - I'AI'ER 01
2 ltours
( 22 MAY 2006 (n.n>.J)
This exrunination pnpcr-consisL'> of'rHRUE scctioll>: Module I. Module 2 aod Module 3.
Each section consists or S questions.
1'he mttJCimum mark for each section is 40.
The maximum ma rk for this ex:.~min:uion is l20.
This examh,~tio•' consists of 6 pages.
INSTRUCTIONS TO CANDIDATilS
I.
00 NOT open this examin:nio• paper until instruc ted to do so.
2.
Answer ALL questions from the 'rH ltt::E: sections.
3.
Unless otherwise su•ted in the <;ucstion. My numeric<~.! anS\\'er that is not
e:tact M UST be wrinc.n cotrect to three significjnt ligures.
Examination M:ueritals
Mnthcm:llicul fonnulac ;md tables
Electronic calculatot
Gtoph pnpet
Copyright C 2005 Caribbean Examinations Council®
All righLS reserved.
022340 IO/CAI'Il2006
Section A (l\tlodule 1)
Answer ALt. qut~lions.
1.
Solve. for,"(, the equmio1lS
<•>
I 5 marks)
(b)
l J markll:]
'f'otnl S n_
u1rks
2.
Oirferentiate with respect to X the rollowin,s:
, .
(a)
y • e.!t~i•n'
3 mar-ks)
(b)
y= ton 3x +In V: +4)
4 rmlrksl
Totnl 7 m.arks
3.
(o)
Find Lhc gradient or the c urve xl + .l)' = 2y2 ot l.hc point P (- 2. I),
( 5 n'Ulrk.~)
(b)
Hence. n nd the equation of the nornal to the .:un·e ;11 P.
[ 3 IHilrks)
·rol.al
4.
8 mark~
lfy • sin 2."(+c::os2x.
dy
;zr
(a)
find
(b)
show thm
3
mark~)
a'y
- + 4y;; 0.
th'
4 marks)
Total 7 nmrk.S
5.
Usc 1he substituti01' indicuted in EACH case: to find the following inu:grnls :
(a)
Jsin x cos x tb:;
8
(b)
11 =
sin ."(
6 omrks l
Tolnl tO mark.<;
GOONTOTHENEXT PAGE
022;!4010/CAPE 2006
-3 Scttion 6 (ModuJt Z)
An. ·wcr
.
A t L <JUHiions.
6.
A sequence (u.,) of ~I numbers satisfies u, ... 1 ,.,., = 3(-1)": u 1 = l.
(:1.)
(b)
Show th.at
(i)
( 3 mnrksl
(ii)
( I mnrk
Write the FlRST FOUR terms of th(5 sequence.
( 3 marks)
Total 7
7.
(o)
Verifylhat thesum, s". of the SCric.!t+
1
nulrk.~
~' + ~ + ..., lOIIICffi'IS. isS.. = ~(l -
F>·
I 4 runrksl
(b)
Three consecutive tc.nns. x- d , x otd x + t/, tl > 0. o( :.n nrilhmctic series have sum 2 1
and product JIS. J':Jnd the value of
(i)
(ii)
l 2 marks)
X
the common d iffcnmce d.
4m.arks)
Totnl tO mnrks
(a)
show 1h:11 .t?- 5x - 14 = 0
4 marks)
(b)
find x.
2 nmrksJ
Total 6 marks
9.
(3)
Exp~•nd (I + ux) (2- x)J in powers of x up to the tcml in .il. u e R.
6 rn~rksl
(b)
Given lhat the coeffi cient of the tetm in ,l:!. is zero. find the vaJue of"·
2 marksJ
Totnl 8 mtlrks
GO ONTOTHI!NE.XT PAGEl
022340101CAPE 2006
-' 10.
y
J'•e~
•-;r
The diagf;lm above- ( not drawn 10
sc~'l l e)
$hows the graphs of the two functions y =
e" a.nd
y= - X(tt)
Stole the equation in X that i.s sotisf~ed ot/J (CL Jl). the point or intersection of the two
grophs.
( 2 marksj
(b)
ShOW that (X lieS 1ft the CIOSC<IIIltCMJ I- I, 0 1.
l 7 m;)rks l
1"otul 9
m~1rks
GO ON TO TilE NEXT PAGE
022340 I0/CAPE 2006
-5 -
Section C(Modulc 3)
I I.
A comminee of 4 people is to be selected from a group consisting of 8 m:.l<:s and 4 females.
Determine the number of wnys in which th(' commillt.'C may be formed if it is lo contain
<•>
NO females
2 nuJrksl
(b)
EXt\ CTLY one female
3 marks I
(c)
AT LEAST one fcm~ l e.
4 mnrksj
Total 9 marks
12.
<•>
The leners H, R. 0. SandT :uc co•sonants.
word HARDEST be ruT'J.nged so ttu:t
(i)
(ii)
(b)
I n how m::my W!.l)'S can the l etters of thc
the first1cuer is ::Lconoonan•'?
3marks)
the first ::tnd lastleuerS are consonJJlts?
.lmorksJ
Find the probability that the event in (3) (i} ubove occu rs.
2 mnrks]
Tot:tl 8 marks
13.
The deter"minant ~is g ive n by
a b+c l
b
c
l:. =
C +(l
o+b •
Show th;n 6-= 0 (Ot any ll. b ;Lnd ,. e R.
I 6 mnrks]
Tot:al (, mnrk.o;
14.
(a)
Write lhe following system of equations in the fonn AX ~ D.
.(+y -z =2
2-K - )' +t-= I
3x+2c: a I
(b)
(i)
rind the matrix 8. the m3trix of cof3Ctors of the
(ii)
Calcul;.lt¢ aTA.
(iii)
m~urix
A.
5 morks j
2 marks)
I I·
Deduce the value of A
I mark I
Total tO marks
CO ON TO THE Nf;XT PACE
02234010/CAPE 2006
-~ ·
IS.
A c losed cylinder h:.s a fixed he ight. h cn1. but its rndius. rem. is increasing :u the rnte of
1.5 em per second.
(a)
(b)
\Vritc down a differenti:tl e<.w:nion for r with respect 10 time 1 sees.
( lmn rk l
Find. in tcnns of '1', the rotc of irrreasc with respect to d me 1 of the total surface
area. 1\. of the cylinder when the r:tdius is 4 em and the height is 10 em.
[ 6 marks)
I A = 2T~ + 2Tr!J)
END OF TEST
02234010/CAPE 2~
•
TEST CODE
FORM T P 2006261
02234020
MAYflUJ\&2006
C A R I 6 8 E A N t: X A M I N A T I 0
S
C 0 U N C I I.
ADVANCED PROFIClENCY EXAM INATION
I'UKt: MA.riii;MATICS
UN IT 2 - l'AI'ER 02
2 hrm.n
(
Jl MAY!006 (p.m) )
This e.umint\tion paper consi$C.s ofTIIREB sec1iOM: Module I. Module 2 and Module 3.
EDch scc1ion consists of 2 questions.
The llllt.Ximum mruk for c:.ch se<:tion Is 40.
The. 1r1aximum m"rk. for tJ' is exa.min:uion is 120.
·rhis ex::.mination consists of S pa.gc:J.
U>STRUCI'IOSS TO C\KOI OATE.'>
1.
00 NOT open th11 t),anunaho• paper untJl instruct«! to do so.
2.
Answer ALL tlUC:iiiOt\S ffom lho T liREE sections .
3.
t)nlc.<t$ otherwise $1attd in the <1ucstion. :my numerical UllSWCr that i~ not
cX3CI MU~t"'' be wrillen com:ct to three signilic:anl (i~u n:J.
Ex:.manalion Materials
M.alhemattc31 formulx :md Dbkt
Ekcuonic cakul:nOt
Or>ph paexr
Copyriar,ht C 200,1 C~nbbe:ln Enminalions Councll~
All nghu mervcd.
02234020/CAPE 2006
-2S«-tionA ( Modult I)
Answu BOTil questions.
I.
(0)
If./t<) • x' /,..x. show tlgt
/'(;c)=x'lnX(3/nx+2)
(i)
( S mar ks)
/"(t) • 6.t l1r! x + lOx /11 .1' + 1t.
(H)
[ S nt.'lr ks)
(b)
·n';C
enrolment pnuem of membership
or n country club
follows an exponential
locistic funclion N.
N•
800
. k e R .rE K. .
I • ke"
~here N i.s the number of ~mbc:n enrolled 1 years after the (onl\3tion or the club.
The initi31 membership W3$ .S0pct'Sl(l'lsand :sfcc:r one )·c:~. Ulere are 200 per$0RsenroUed
in the club.
(i)
What i.s the LARGEST nurrbet rt¥hoed by t.he mrmbr.n.h1p of the club?
I 2 tnal"b)
(u)
Calculate lhe EXACT value of .t and o( r
(iii)
How m:any members wi11lh:re be in the club 3 )t'~ a.ftet its (ornQtion?
l l marks)
I 6 marks]
·rota I 20 marks
2.
(o)
1+x
in p!lltial froclion.s.
(;c- I)C;c'+ I)
(I)
P.xpre.s5
(ii)
J lenoe. find
f(x-
I + X..
1)(.>:' +II
dx.
I 6 mnrks)
I J marks)
I
(b)
01\cn lhat 1• •
J•
Jl" r dx. "here n c; N.
I
(u)
(tu)
Showalutl,.=<-n1,._ 1•
~marks)
I 4 marl;,s)
lienee. a.- ochef"" i.sc~ evalu.u /1• wnllnJ )OUr ans"er in terms of~.
( 3 nmrks)
·rot:a l 20 marks
CO ON TO Ti lE NEXT PAGE
02234020/CAI'I!. 2()(](i
- 3Section 8 (Modul<l)
Ans,n-.r BOTII quf':Stions.
3-
(a)
(i)
..L
Show thanhe tetmS or
ln3'
r• I
(li)
ate in arithmetic progressio•.
( 3 mar-ks)
rind 1he sum of the tina 20 :enns of chis sel'ies.
l
4
mnrk.-.-1
2m
(iii)
(b)
L
I 3 marksf
J.-le(I(:C-, show th.at
br 31 =(2m2+ m) In 3.
ral
The kqUC:IlCC: Of positive: lC:nDS.. l;r.J, iS defintd b)' .r. • I •..; + ~·
(i)
<
XI
i•
Show. by nwhe~ in6Ktion.. or ~ftc. ltut x. < 4" (or all positive
( 7 mark$)
1Afe&en n.
(ii)
By constdt:ring x• • 1 -
.¥"•• Of othel"ise. show
thai x. <
I 3mark$)
x•• 1•
TotallO marks
4.
(a)
Sketch 1he furle!tions y =sin x und y= .ron the SAM E axe$.
(b)
Deduce tholthe functionf(;.:) = sin ;c- xl has BXACll~Y two re:.l rootS.
I 5 marks I
( 3 mifrks)
(c)
FirKi lhe interval in which the non- ~!ro root a o(jtx) lu:s.
(d)
Staning with a first approxim:ttion of cx :u .r1 • 0.7. UJC: one tter3hon of the Newton·
RaphJOn method to obtain a bencr ~~i~tion of Cl to 3 decimal places.
I S mark$)
I 4 marks)
Tota120marb
00 ON 1'0 HIE NEXT PAGE
02234020/CAPil2006
-4-
Section C (Module 3)
Answer BOTH questions.
5.
(a)
(i)
(ii)
How many numbers lying between 3 000 and 6 000 can be formed from the
digits, 1, 2, 3, 4 , 5, 6, if no digit is used more than once in forming the number?
[ 5 marks]
Determine the probability that a number in 5 (a) (i) above is even.
[ 5 marks]
(b)
In an experiment, pis the probability of success and q is the probability of failure in a
single trial. For n trials, the probability of x successes and (n- x) fai lures is represented
by nCxr q"-x, n > 0. Apply this model to the following problem.
The probability that John will hit the target at a firing practice is ~. He fires 9 shots.
6
Calcu late the probability that he will hit the target
(i)
AT LEAST 8 times
[ 7 marks]
(ii)
NO MORE than seven times.
[ 3 marks]
Total 20 marks
2
6.
(a)
If A= ( -:
2
-2
-~
) andB = (
~
-1
1
1
~ ).
1
(i)
find AB
3 marks]
(ii)
deduce A- 1•
3 marks]
GO ON TO THE NEXT PAGE
02234020/CAPE 2006
-s(b)
A OUI'l<l)' sells llvee bnnds of gr.w-=<1 mix_ P. Q and R. Eoch bnnd is nude from
thlt<' type< of Jr.>SS. C. Z and B. The numb« oHtlop>ms ofuch type of VOSS in o bag
of each br.and ts sumnwised in the Dbfe belov.
T,-.,. of C ra5.."
Cr.assSH<I
(Kilognlms)
Mi~
C-gnss
Z·&nlS..'~
/J•j!l'll~
Or.u\d I'
2
2
6
Orond Q
•
2
4
Umnd R
0
6
•
Olend
<
:
b
A blend •-' produced by mixingp ba&JofBr.md P,q b;i.gJ ofB,.-.and Qand r~gsofBr.md
R.
(t)
Wnte down.., up«Uion i• ltnns of p. q and'· for the numb« of kilogr.uns
of Z.cr.w on the blend.
I t mark I
(i•)
Lt:t c. : and b represent dk numbtr or l.•lov.ams of C-sr.w. z-pass and
ll-grass respec::thely in the Ncnd. Wntc down a set of THREE equations in p.
q, r.to n::~nl the numbe:roflilogram'l ofEACI I lypc of lnlS5 in the blend.
I J marks)
(iii)
Rewrite the set ofTHREE eq.w.tions in (b) (ti) abQ•d,, in the malrix form MX • D
where M is a 3 by 3 m:urix, X and I) aro oolumn matrices.
I 3 rm1rks)
(iv)
Given that ttr1 cxist.s.. write X ill tcl'm~ of t.r' ••nd 1),
(v)
Given
tll•• ftr': (
3 marks )
-0.2 ~.2 0.3 )
0.35 0.1 -0.15 .
-0.05 0.2 -0.05
calculate bow """'Y bags of EACII bnlnd. P. Q. ond R. ""' roquit<d to
prod""' • blend coot.>ining JO kitovozns of C·Jr.>SS. 30 ~tlograms of Z.cr.w
and SO kilograms of 8-gniSS.
I ~ marks)
TolallO marks
END Of TF.S'l'
022340201CAPE 2006
TESTCOOE 02234032
FORM TP 2006262
CA Rill!I EAN
MAY/JUNE 2006
EXAM I NATIONS
COUNC I L
ADVANCED PROFICIENCY EXAMINATION
PUR£ MATHEMATICS
UNIT 2 - PAI'E R 03/ll
( 22 MAY !O()Ii (p.m.) )
This examination pape-r consists of T HREE st.oc.rioM: Module I. Module 2. and Module 3.
E.:tch section oonsis-L'> o f I question.
The maximum mark for each section is 20 .
The maximum ma rk for this CJCamination is 60.
This examination paper consists of 4 p..'lges.
INSTRUCTIONS TO
CAN DII>ATF.~
I.
00 NOT open this ex~minotion p3per u111il in.stmcted to do so.
2.
Answer ALL questions from !he TI-I RE E st."Ciions.
3.
Unless othe rw ise slated in the question. any numcric.al :m..swcr that is
not ex,tct M US'I' be written cam..-ctto three signific-a•ll figures.
Exumin:uion m:ucri:1ls
Mathematical fonnulae :md lnblcs
Elt•ctronic calculntor
Gr:1ph paper
Copyright <0 2005 Caribbean Exwninations Count-il®
A ll rights reserved.
0 2234032/Ct\ 1'6 2006
-2 -
Section A (Module 1)
Answer this question.
1.
T he rate of inc rease of the number of algae with respect to time, t days, is equal to k timesf(t),
where f (t) is the number of algae at any given time t and k E R.
(a)
Obtain a differe nti al equation involvingfi:t) which may be used to model this situation.
[ 1 mark]
(b)
Given that
6
the number of algae at the beginning is 10
the number of algae doubles every 2 days,
(i)
determine the values ofj(O) andf(2)
(ii)
show that
(iii)
[ 2 marks]
a)
k=
2 zn 2
[10 marks]
b)
fit)
= 106(21'2)
[ 5 marks]
1
determine the approximate number of algae present after 7 days.
[ 2 marks]
Total 20 marks
GO ON TO THE NEXT PAGE
02234032/CAPE 2006
-lSection B (Module l )
Answu tin qu~don.
(a)
ACJr9rMpun:h3SCd:atlbe beginnintofthc yeN. for Pdollars. The va.tu.eof:acarat tbe
tnd or tu(h )~r isestim:ated to be lk value: at the btJtjnnlna:ortht )tar multiplied by
,,
(o-!).tf£N.
(i)
Copy and complete the table be:: low showina the voluc or the ear for lhe first
five years a fte r purchase.
Yf.'nr I
Yt':nr 2
Yror J
p
P( o-.!..)
1'( 1-.!..)2
Vnlue n t the
lh:J:}nnin~
or \'cur
q
Ytl'lr 4
\'~1r
5
q
($)
VAIUt ... the
f.nd of \'tar
($)
1'(1-.!..) 1-+>[P (l-+)J
q
I •
=PI' --r
•
(ii)
(iii)
Describe FULLY the sequc:aee sho\Vn in the t:.blc
I 2 marks)
D<:tc:rminc. in terms of P :md q, the v;~luc of the c:ur n years after purchase.
l l markl
(b)
If the origin:&! value o f t he c.1r w:~s
s:oooo.OO :md the vnlue Ill the end o f the fourth year
wasSS 192.00. find
(i)
the vaJue of q
( 5 marks)
(ai)
the estimated value of the cnr :Jrtcr fhc )'eat'S
I 2 marks]
(m)
the LEAST mtcg.r.J.l value o-· " · lhe number of )'eatS after pureh:ase. foe which
lht: 6tarnakd \alue of lhe ear falls ~low $$00.00.
t 7 marks)
Total :W marks
GOON TOTflENEXTPAGE
022340321CAI'Il2006
-4-
S«lioo C CMO<Iule 3)
A n.o:;wer I his question.
3.
(o)
A box conta:itlS 8 green b.1.lls and 6 red baJis. Five b:.-tlls ~c selected :•t random. Find
the prob<&bility that
(i)
ALL 5 balls 3re green
l 4 murksl
(ii)
EXACTLY 3 o f the five balls are rtd
r 4 marks)
at LEASTON£.of1hc five balls is n:d.
I 3 mn rksJ
(iii)
(b)
Use lhe llletbod o f row reduction ':> echelon form on the augme nted ma trix for the
followi ng system of equations to show that the system is it\Consistent.
( 9 muks]
X+2)'+4!=6
y+2z=3
x+y+2z;; l
Total20 marks
END OF TEST
02234032/CAI'E 2006
TEST CODE 02134020
FORM TP 2008240
CAR I BBEAN
MAY/JUNE 2008
EXAMINATIONS
COUNC IL
ADVANCED PROFICIENCY EXAMINATION
PUR£ MATll£MATICS
UNIT I - PAPER 02
ALCEBRA, CEOM£TRV AND CALCULUS
( 21 MAY 2008 (p.m.))
1bis e:uminatron pcsper consiJu ofTilRf!E a«c.ona.: Module l. Module 2 and Module 3.
Eaeh section consllts of l q~ataon.s.
The maximum m~rk for each Module '' SO.
The: max:amum mule for thts e-xemana110n is 1~.
This exami~tion COn$1Sts o( S pnnted P'l&«·
INSTRUCTIOI'IS TO CANDIIlATES
1.
00 NOT open th11 examination p:1pe:r umal instructed to do so.
'2.
An.swt:r ALL questions fro m the Til REt<; Ketions.
3.
Write your aohii10n1, with full working. in the tnlwcr booklet provided.
4.
Unless othctwisc tte:te<l in the question, ooy numerical answtr lhat is
not cx.acc !\•r UST be written eorrecl to three sianiGcant figures.
Examination Ma iC'rialj PtrmlrtOO
•
•
Gropb paper (Jln>Vld<d)
Malhcnutical fonnube and oabl"" (provtded) Re\'bed 2008
Mathem3.tical inwumc:nu
Silent. non~mmablc:~ elcctrome calculator
~aJ>t 0 2007 CanDbcan EaammabotU Cowlcil e.
All naJ>ts reoaved.
02134020"CAPE 2008
-
2
-
SECTION A (Module 1)
Answer BOTH questions.
1.
(a)
The roots of the cubic equation x3 + 3px1 + qx + r = 0 are 1,-1 and 3. Find the values of
the real constants p, q and r.
(7 marks]
(b)
Without using calculators or tables, show that
(i)
V~6~- < 2
V fTs-- VT
■
.
(ii)
flT+ V T
+
VT- V T
(c)
v t
22 ++ V
T
[5 marks]
V 6~- V 2
(i)
n
Show that E r (r + 1) = ~
r» 1
(ii)
Hence, or otherwise, evaluate
[5 marks]
4.
■VT+ V T
n (n + 1) (n + 2),
n e N.
[5 marks]
50
Z
r(r+
1).
[3 marks]
f = 3t
Total 25 marks
2.
(a)
The roots o f the quadratic equation
2x2 + 4x + 5 = 0 are a and p .
Without solving the equation
(i)
write down the values of a + p and af3
(ii)
calculate
(iii)
[2 marks]
a)
a 2 + p2
[2 marks]
b)
a 3 + p3
[4 marks]
find a quadratic equation whose roots are a 3 and p3.
[4 marks]
GO ON TO THE NEXT PAGE
02134020/CAPE 2008
.).
(b)
(i)
Sol"-e for x the cquatton xVJ - 4£"' = 3.
log, (r- I) •
(5 marks)
(ii)
Find X sueb WI los, (X+ 3) ~
(Hi)
Without the usc of calculators or tables., e\'alu1tc
2
log,. c21 >+ Ioa., <-r>
+
l
log.,<...->
IS marks!
I.
+ ... +
loa., c9a> + log,.
<w·
9
(3 marks)
TotallS marks
SECTION B ~lodul• 1)
Amtwe.r BOTU: questions.
3.
(a)
The lines y • 3x + 4 and 4y • 3¥ + S are inclined at an~leJ a a.nd Jl rto.spectively lO lhe
x-axis.
Jl.
(i)
Sta;e the: values of lin c. and tan
(ii)
Wrtbout ••na t&bl.. or cekula10n. 61'ld the tan"cnl o(lbc an&Jc bct•·caa Ute t"'v
(l marks)
lines.
(b)
)4 marks)
(i)
Prove that 5ln 29
(ii)
Express tan 0 an terms of sin 20 and cos 20.
(iii)
tan 8cos 26 • laD
8.
(J marks)
)2 muks)
Hence show, without usins table:s or C31cuJators. that 1111\ 2-2.s• • .f2 - 1.
14 marks I
(<)
(i)
(ii)
Givm l.bat A. 8 and Care the angles of a tn:male. J't'OVt lh3t
A;8 • cosf
a)
SID
b)
sinB•sUIC
2cos
(3 marks)
T cos B-C
"
(2 marks)
2
Hence, Jbow that
sinA+s:i.nB+sinC • 4cos ~ cos~ cosf .
(5 marks)
Total 2S marks
GO ON TO THE NEXT PAGE
02134020CAPE 2008
-4-
4.
(a)
(b)
In the Canemn pbl>e wi1h ori!lin 0, the coordonaiCI or pooniS P alld Q ate (-2. 0) and
(S. 8) respca>,-ely. The nUdpoino or PQ os M .
(1)
Fond the equatiooorthe lone whoch posscslhroualo Malld os pcrpc,.dieulanoPQ.
(8 marla(
(n)
Hence. Ot" otbt:t'Wise. find the coorchnatu of the cenb'e of lbe c~lc lhrougb P. 0
and Q.
(9 marla(
(i)
Provelhattheliney=:c+ 1 isatanacnttothecu-clcxl+}.t+ 10.'C-l2y+ 11•0.
(6 moria(
(H)
Find the coordinates of the point of contact of thtl tanaent to the circle.
(2
mark.$1
Totall.S marks
SEcno~
c (Module J)
,o\_n,swer BOTH ques-tions.
5.
(a)
Fu~
/ant
X-+)
(b)
x'- 27
T +x 12
(4 marks}
A chcmaeal process is conuoUtd by the Nnc:uon
P • ~ + w:. where u and v are constants.
t
+.
01venth:11 P• - t when 1• 1 BndthemteofchanioofPwuhrespeetto tis - 5 when
1•
(c)
find t11c wlucs of u 3nd v.
(6 marks)
The curve C passes through the point (-1, 0) and 1ta g:rad1cnt at any point (x,y) is g.iven
by
%-J.r'-6.x.
(3 marla]
(i)
Fond lhe equatioo or C.
(io)
Fond the eoordmsoes or the stahO!Wy pooniS or C alld delennone the ll3tu:e or
EACH pomL
[7 marks)
(m)
Skcwh the ppb or C alld label the x-onlm:q>IS
(5 marks)
Total 2.5 marks
00 ON TO THE NEA'T PAGE
02134020/CAPE 2008
- 5(a)
(1>)
Differentiate with respect to x
(i)
d2x
]3 marks]
(ii)
sin' (x' + 4).
14 morks]
(i)
Giva> thnt
I•fix)
dx a 7. evaluate
'
(ii)
I• (2-J(x)J
13 marks)
dx.
'
The nren under the curve y • r +lex- .S. above the x-;u:is and bounded by the
lines x = l and x = 3. is l4
i units
2•
Find the value of the consUUlt k.
(c)
]4 marks]
The diagram below(not drawn to scale) ~entsa.canin theshapeofaclosedcylinder
with a hemi.sphere at one end. The can h.::l.s a volume of 45 n unitY.
r
lo
----·-·;·- 1
(i)
(ii)
Taking r unit$ 8$ the rndius of the cylinder and h units as its heig.ht, show that.
a)
h - !2. - k
b)
A •
i'
(3 marks]
3
s;r
+ 9~:t, wbereA units i.s theextem:al surface area of the can.
13 marks]
Hence, find the value of r for which A i.s a minimum nnd the corresponding
minimum va.Jue of A.
[Volume of a sphere ..
4
3
1t
(5 marks!
rl, .surfnee nrea of o sphere • 4 n r'-. ]
fVo)ume of a cylinder= n: il h, curved surfuce area of a cylinder • 2 n r 1:.]
iotnl 25 n12rks
E!\'D OF TEST
02134020/CAPE 2008
TEST CODE 02134032
FORM T P 2008241
CAR IB BEAN
MAYfJUNE 2008
EXAM I NAT I O N S
CO U NC I L
ADVANCED PROFICIENCY EXAMINATION
PUR£ MATHEMATICS
UNIT I - PAPER 03/8
ALCEBRA, GEOMETRY AND CALCULUS
1 ~ hours
( 16 MAY 2008 (p.m.))
Thu <lWDmaOoo poper eonsuu ofTIJR.EE S«bonJ Module I, Module 2 and Module 3.
Eaeh section cons1w of l quemon.
The maximum mark ror each MOOule is 20.
Th~ maximum mark for this cxll.mination is 60.
This oxamin:nlon consists or 3 printed pages.
IXSTR!ZCTIOI>S TO CA.,' DIDATES
l.
DO NOT opefllhis e:xanu.nabOfl paper untd rm:trueted to do so.
2.
Answer ALL. queniotl$ from the T l-1 REE sections.
3.
Write your 1olutions, with ful l wotlt.ina. in the a.nswcrbooklc:t provided.
4.
UnJeu othc-rw1K stated in the q:uu11on, any numencAl ansY..cr th:a• is
not exact MUST be: wnneo correct 10 three: s1pifiearu fiaures
Enminarion Mater lab l.•t rmlrttd
•
Gmph paper (provided)
Mnlhematieal fonnulae and tables (provided) - Re' bed 2008
Mathmuttical instruments
S1lmt. non-programmable. electronic calculator
Copyriahc 0 2007 CanDbean Euminalions Council®.
All rightl reJCT\'cd.
02134032/CAPE 2008
-2S ECT I O~
A (Module I )
Answer tbls questJon.
I,
(a)
One root of the quadratic equation x:l + 12x + k = 0 is three lim es the o ther and k is 3
constanL
Find
(b)
(i)
the roots of the equ3tion
(3 mar ks!
(ii)
the value of k.
(2 marks(
(i)
The
funerio 1~
Jtx) has the propeny tha t
f(l>r+ 3) = 2fl.x) + 3,
(ii)
(e)
X
G R.
If flO) • 6, find lhe volues of ,1\3) andfl9).
[4 mar ksl
$oJve for X the tquatiOn 5• - 5•- ~ • JS 000.
(4 marksl
A computer nmoufaeturer fi nds that when x m illion dollars are spent o n
profit, P(x). in mfllions of dolla.rs, is given by
/"(.t) - 20 + S
lu~1
resea~h.
the
(A + 3).
(3 m arks)
( i)
What is the profit if 6 million dollars arc spent on research?
(H)
How much should be Spent on reseal'(:h to make a pro{h of 40 mi1lion dollars?
(4 marks)
Total 20 marks
SECTION 8 (Module 2)
Answer [his q uestion.
2.
(a)
Find the values of x in the range 0 ~ x ::_ 2n such that
4cos1 x + lcosx - 5sin2x • 0.
(b)
{I)
( 10 mar ks(
Determine the value of the real n umber t such lh:st the vec-tors p ... 41 + 5j and
q • 31 - tj arc pctpendicular.
12 mark&]
GO ON TO 'THE NEli.'T PAGE
02 134032/CAPE 2008
- 3(ii)
Given thnt vectors u = 21 + 3j and v • I + 5j, fmd the acute angle 6 between
u and v.
14 m~uks)
(iii}
Given that the vector u in (b) (ii) above represents 3 forte F with respect to the
origin 0, and axes Ox and Oy, calculate
a)
the magnitude ofF
(2 marks(
b)
lhe angle tfl of inclination ofF to Ox.
(2 marks(
Total 20 marks
SECTION C (Module 3)
Answer th1s ques-Uon.
3.
(a)
A curve has equation y • x + ~ •
X
~
15 marks(
+ x ::; = y.
(i)
Show !hat x'
(ii)
Pinl.l llu; ~l.l<~lion of lhe nonnnl tO the curve at the polnt wbere x • 4.
(S morks(
(b)
F'nd
J xl +rx-1 d;'f.
I
(c)
The volume of the: liquid in 3. eontainer is Y em'. The liquid leak$ from the container al
lhe m.tc of30t cm:s' per sec, where tis the time in second$.
(4 marksl
(i)
Write down a djfferentinl equation for Vwiih respect to timet 5-ee$.
12 ro:.rks)
(ii)
Find the amount of liquid lost in the 3"' second.
(4 marks(
Tota.l 20 marks
•
Et\'0 OF TEST
02134032/CAPE 2008
FORM TP 2008243
CARIBBEAN
a\
TEST CODE
~
EXAMINATIONS
02234020
MAY/JUNE 2008
COUNCIL
ADVANCED PROFICIENCY EXAMINATION
PURE MATHEMATICS
UNIT 2 - PAPER 02
ANALYSIS, MATRICES AND COMPLEX NUMBERS
2 ¥2 hours
( 28 MAY 2008
(p.m.))
This examination paper consists of THREE sections: Module 1, Module 2 and Module 3.
Each section consists of 2 questions.
The maximum mark for each Module is 50.
The maximum mark for this examination is 150.
This examination consists of 5 printed pages.
INSTRUCTIONS TO CANDIDATES
1.
DO NOT open this examination paper until instructed to do so.
2.
Answer ALL questions from the THREE sections.
3.
Write your solutions, with full working, in the answer booklet provided.
4.
Unless otherwise stated in the question, any numerical answer that is
not exact MUST be written correct to three significant figures.
Examination Materials Permitted
Graph paper (provided)
Mathematical formulae and tables (provided) - Revised 2008
Mathematical instruments
Silent, non-programmable, electronic calculator
Copyright © 2007 Caribbean Examinations Council ®.
All rights reserved.
02234020/CAPE 2008
-
2
-
SECTION A (Module 1)
Answer BOTH questions.
1.
(a)
Differentiate with respect to r
(i)
(ii)
(b)
eAx COS
In
[4 marks]
7LT
+ -.
[4 marks]
r r
Given y = 3 show, by using logarithms, that
= —3~x In 3.
dr
dr
4y
(c)
(i)
[5 marks]
Express in partial fractions
2x2 - 3r + 4
[7 marks]
( r - l ) ( r 2+ l ) ‘
(ii)
Hence, find
2x2 - 3x + 4 dr.
I ( r - l ) ( r 2+ l)
[5 marks]
Total 25 marks
2.
(a)
Solve the differential equation
dy_
dx
(b)
+ y = e1*.
[5 marks]
The gradient at the point (r, y ) on a curve is given by
dy
dr = e4x.
Given that the curve passes through the point (0, 1), find its equation.
(c)
e
Evaluate J, x2 In r dr, writing your answer in terms of e.
[5 marks]
[7 marks]
GO ON TO THE NEXT PAGE
02234020/CAPE 2008
- 3(d)
(i)
Use tbe substinmon ,, • 1 - • to rmc1
J~(u)
13 marks)
Hence.. or otherwise, use the S\lbstuuuon w • M x 10 eval\&ate
.,
J• ~,-1:-+,....,sin-x
15 marks)
dx,
Totall5 marks
SECTI0:-1 B (Modul• 2)
Answer BOTH qut$UOnJ,
3.
(1)
A uquence {u.} isdefuaed by the~ re:Lanon
"·
(1)
(n)
1 •
u. • n. " • • 3.
Sla~ tbe
(c)
13 marks!
lim FOUR terms of tbe l<qUCIICC
Prove by matbematic:al induction. or olberwlse, tMt
u,. (b)
n ._ N
r-n+6
2
18 marks!
•
A 0 1) Wtth firSt tenn a and eommon n1tlo r has l"m 10 infi mty 81 and the sum of the first
four tc:nns is 6S. f'md the values of" a and r.
16 marks I
(i)
(1i)
Wnte down the firs-t FlVE terms 1n the power series expansion o f In (I + x),
J1Ahng the r.mge of values of.~ for wh1ch the serieltl vr.Jid..
13 marks}
a)
UStng the result from (c) ( i) tbove. obtAm a JtmilarexpansJOn for 1n (J -x).
b)
Hence. prove that
12 m.arksl
ID
(l•xJ
l..,-::x = 2 <x + TI "J
+
TI "~ + •.).
13 matksJ
Total 25 marks
•
GO ON TO TilE NEXT PAGE
02234020/CAPE 2008
-
4*
(a)
4
-
(i)
Sh(>w that the function J{x) = x? - 3x + 1 has! a root a in
2],
i the closed
sed interval [1,
[1,2].
[3 marks]
(ii)
Use the Newton-Raphson method to show that
hat if jt, is a first approximation
jproximation to a
imation tc
in the interval [1,2], then a second approximation
to a in the interval [1, 2] is
given by
2xJ- 1
x2
:
•
[5 marks]
3x 1 —3
~
(b)
(i)
(ii)
(iii)
Use the binomial theorem or Maclaurin’s theorem to expand (1 +xY'A
x )'A inascending
in ascending
powers o f x as far as the term in x3, stating the values of* for which
the
expansion
tich
is valid.
[4 marks]
marks]
[4
Obtain a similar expansion for (1 -~x)'A.
Prove that if x is so small that x3 and higher powers o f x can be neglected, then
I 1 —x
.
,
1 ,
•J i +x * ~ x Y x
(iv)
marks]
[4[4marks]
'
[5ESmarks]
m ark s^
iking j:x == -^-,, show, without
vi
Hence, by taking
using calculators or tables,
ables,that
thatVVT
"2~isis
17
approximately
ly equal to i f | § - .
m arksj
[4[4marks]
Total 25 m
marks
arks
SECTION C (Module 3)
Answer BOTH questions.
5.
(a)
A cricket selection committee of 4 members is to be chosen from 5 former batsmen and
3 former bowlers.
In how many ways can this committee be selected so that the committee includes AT
LEAST
(i)
ONE
ONEformer
formerbatsman?
batsman?
(ii)ONE
ONE
batsman
batsman
andand
ONE
ONE
bowler?
bowler?
[8 marks]
[3 marks]
GO ON TO THE NEXT PAGE
02234020/CAPE 2008
-
(b)
5
-
Given the matrices
r 3 1 0^
ii 00 1
,
00 -1 00
v.
J
r 3
A =
1
V.
(i)
(ii)
(iii)
1
r 00i -i00- 1r
-1
-1
^
-1
-1
-1
3 -1
r o0 rn
00 00--3
3
~1 3 -1
V
J
f ii
-1
B -
and
M =
J
-
-1
-1
determine EACH o f the following matrices:
a)
A - B
[2 marks]
b)
AM
[3 marks]
deduce from (i) b) above the inverse A-1 o f the matrix A
[3 marks]
find the matrix X such that AX + B = A.
[6 marks]
Total 25 marks
6.
(a)
(i)
Express the complex number
——
^ ~
5 - zi
(ii)
(iii)
(b)
in the form X (1 —*).
[4 marks]
State the value of X.
[1 mark ]
^ 2 - 33/1 T •
Verify that
:hatf—
is a real number and state its value.
1 55--i' J 1
[5 marks]
The complex number z is represented by the point T in an Argand diagram.
Given that z =
show that
1
3 + it
where t is a variable and z denotes the complex conjugate o f z,
CO
z + z = 6 zz
(ii)
as t varies, 7Ties on a circle, and state the coordinates of the centre o f this circle.
]8 marks]
[7 marks]
Total 25 marks
END OF TEST
02234020/CAPE 2008
TEST CODE 02234032
FORM TP 2008244
C AR I BBEAN
MAY/JUNE 2008
EXAM ! NAT I ON S
COUNC IL
ADVANCED PROFICIENCY EXAMINATION
PURE MATHEMATICS
UNIT2 - PAPER03/B
A..'iALYSIS. MATRICES AND C0~1:PLEX NUMBERS
I
~ hours
( 19 MAY 2008 (p.m.))
Thu exantllUibon ps;per consisu of THREE sections: Module 1, Module 2 :tnd Module 3.
l!ach section consi.scs of l question.
The maximum mArk for c:acb Module: is 20.
The maximum mark for this vcamin.ation is 60.
This exanuaatton consists of 4 pnnt.ed p3g.es.
INSTRUCTIONS TO CANDfDATF.S
DO t'\OT open dus c:xanuaabon papu unbl •nstruc:kd 10 do so.
2.
Answer ALL quesuons from the THREE
3.
Write your solutions. with full working. in the answer booklet provided
4
Unless ot.herwtse stated 111 the qucstton. any numnteal ansv;:cr that
not exact ~t CST be wnnen conec:t 10 thttt S:l.pufieant figures.
Examination Materials
•
sec:tiOnJ.
P~rmltted
Gr>pb paper (p<ovld<d)
Mathemaocal fonnum >nd lablca (P<ovidecl) - R ...·lsecJ 2008
Mathematical 1nstNtnenlS
Silenl. non-proa.ramm..able, elecm'Onic ca1eultnor
Copyn&JII 0 2WI Caribbean Ex.aminauons C'ounal ®.
All ngh" r<$<TVcd
02234032/CAPE 2008
ll
SECTIO:" A ~lodule I)
Am"·er this quC"Sdon.
I.
<•>
(b)
The paramett~~:: equations of a curve: arc ~~~·en by x • 3r ADd y • 61.
(i)
Find the value of :.;: tttlhe point P on the curve where y • 18.
(5 marks)
(1i)
Find the equation of the norm.alto the curve at P.
(3 m>rks(
In "" experiment it was. discovered that the volume, V cmJ, or a cenam iUbstance in n
room after 1 seconds may be determined by the equntion
V • 60e 0,..'.
F1nd dV in 1erms or 1.
dt
(u)
(m)
(3 muks(
De1ennine the r.~te at which the volume
a)
mc:reascs after 10 sccond.s
(I mark I
b)
is lDCJ'CUiDg wben n is ISO W .
(3 m•.rks(
Skeo:h lbegrophofV• 60e ... 2hoWUI& lbepoonl(s) ofm1.....:1i00. where !hey
C:ltlst. wslb tne UC$.
(S markS)
Total lOmarks
GO ON TO THE NEJ\"T PAGE
02234032/CAI'~
2008
-3SECTION B (Module 2)
Answer this question.
2.
(a)
Matthew started a savings account at a local bank by depositing $5 in the first week. In
each succeeding week after the first, he added twice the amount deposited in the previous
week.
(i)
Derive an expression for
a)
the amount deposited in the rth week, in terms of r
b)
the TOTAL amount in the account after n weeks, in terms of 11.
[3 marks]
[3 marks]
(ii)
(b)
Calculate the MINIMUM number, n, of weeks it would take for the amount in the
account to exceed $1000.00 if no withdrawal is made.
[3 marks]
The series S is given by
s =
(i)
(c)
1-
1
2
+ 3 1- +
4
5-1
8
+ 7-1 +
16
Express S as the sum of an AP and a GP.
[3 marks]
(ii)
Find the sum of the first n terms of S.
[3 marks]
(i)
Use the binomial theorem to expand ~ as a power series in y as far as the
term iny".
Y
[2 marks]
(ii)
Given that the Maclaurin series expansion for cos x is
cosx = 1 -
-
;x2
2!
+ -
x4
4!
- -
x6
6!
+
find the first THREE non-zero terms in the power series expansion of sec x.
[3 marks]
Total 20 marks
GO ON TO THE NEXT PAGE
02234032/CA PE 2008
-
4
-
SECTION C (Module 3)
Answer this question.
3.
(a)
(i)
By considering the augmented matrix for the following system of equations,
determine the value o f k for which the system is consistent.
x + 3y + 5z = 2
x+
+ 4y —z = 1
z == kk
y -—66z
(ii)
(b)
[5 marks]
Find ALL the solutions to the system for the value of A: obtained in (i) above.
[4 marks]
The probability that a person selected at random
—
—
—
owns a car is 0.25
is self-employed is 0.40
is self-employed OR owns aacar
carisis 0.6.
0.6.
(i)
personselected
selectedat at
randomowns
owns
a car
ANDis is
Determine the probability that aaperson
random
a car
AND
self-employed.
[4 marks]
(ii)
Stating a reason in EACH case, determine whether the events ‘owns a car’ and
‘is self-employed’ are
a)
independent events
[4 marks]
b)
mutually exclusive events,
[3 marks]
Total 20 marks
END OF TEST
02234032/CAPE 2008
TEST CODE 22134020
F ORM TP 2008240
CARIBBEAN
MAY/JUNE 2008
EXAMINATIONS
COUNCIL
ADVANCED PROFICIENCY EXAMINATION
PIJRE MATEI:EMATICS
IJNIT 1 - PAPER 02
ALGEBRA, GEO~iETRY AND CALCIJLIJS
( 30 JUNE 2008 (a.m.))
This examination paper consists of TRREE sections: Module 1, Module 2 tmd Module 3.
~ch section consists or 2 questions.
The maximwn muk tor esch Module is 50.
The maximum mt1tk for this examinBtioo is 150.
Th.is examination consists of 6 primed pages.
IN$TRUCTIONS TO CANDIDATES
1.
DO No·r open this examination paper w1til inSU"Ueted to do so.
2.
Answer ALL questions from the TDREE se<:tions..
3.
Write yoor solutiOilS, whh full working, in the answer booklet provided,
4.
Unless otherwise stated in the question., any numerical answer that is
not exact MUST be wriuen correct to t.hrc:e sipific:mt figures.
Examination Materials Pcrmjuc;d
Graph paper (provided)
M.athemalical formulae and tables (provided) - Revised 2008
Mathematical instruments
Silent, non-progr.w:unable. electrOnic calculator
Copyright C> 2008 Caribbean Examinations CoW>Cil <8>.
All rights rescrvcd.
22134020/CAPE 2008
SECTION A (Module I)
Answer- HOTH ques-r.:.ions.
I.
(a)
(i)
Detertnine the values of the reaJ number h for which the roots of the qu adratic
equation 4.r- 2/tx + (8- IJ) • 0 are real.
18 m a rksJ
(ii)
The roots of the cubic equa1j on
xl - lST + px - lOS • O
areS-k,SandS+k.
Find We values of the constants p and Jc.
(b)
(i)
(7 marks(
Copy the table below and complete by insenittg the values for the functions
Jt<) •(x+ 2 ( and s(x)=2 ( x-l (.
X
-3
.f(x)
I
g(x)
8
-2
-1
0
I
6
1
2
3
3
2
4
5
6
2
(4 marks(
(ii)
Using a scale of I em to I unit on both axes, draw on the sam t. graph
14 morks(
j(x) and g(x} for - 3 S: :c ;S S.
(iii)
Using the graphs, find the values or X for whiebj{x) . g(x).
(2 marks(
Total 25 m a r ks
GO ON TO THE NEXT PAGE
22134020/CAPE 2008
-
2.
(a)
3
-
Without using calculators or tables, evaluate
2710 + 910
274 + 911
(b)
(i)
(ii)
(c)
[8 marks]
Prove that log m = log,,=10 W , for m, n e N.
l°g10"
[4 marks]
Hence, given that y = (log, 3) (log 4) (log 5) ... (log 32), calculate the exact
value ofy.
[6 marks]
Prove, by the principle of mathematical induction, that
fin) = 7" —1
is divisible by 6, for all n e N.
[7 marks]
Total 25 m arks
SECTION B (Module 2)
Answer BOTH questions.
3.
(a)
Let p = i - j. If q = X,i + 2j, find values of X such that
(i)
q is parallel to p
f1 m ark ]
(ii)
q is perpendicular to p
[2 marks]
(iii)
(b)
(c)
the angle between p and q is
71
[5 marks]
T
ou
—
cos 2A
+ sin 2A — = tan
, A..
Show j-u
that* ™1—
;-----———:—
1 + cos IA + sin 2A
(i)
(ii)
(iii)
[6 marks]
Using the formula for sinyl + sin B, show that if t = 2 cos 0 then
sin (n + 1) 0= t sin n6 —sin (n - 1) 6
[2 marks]
Hence, show that sin 30 = (t2 - 1) sin 9.
[2 marks]
Using (c) (ii) above, or otherwise, find ALL solutions o f sin 3 9 = sin 0, 0 < 0 < n.
[7 marks]
Total 25 m arks
GO ON TO THE NEXT PAGE
22134020/CAPE 2008
.
4.
(a)
(i)
4
-
The line x - 2 y + 4 = 0 cuts the circle, x1+y2- 2 x - 2 0 y + 51 = 0 with centre P, at
the points A and B.
Find the coordinates of P, A and B.
(ii)
[6 marks)
The equation of any circle through A and B is of the form
jc2 + y 2 —2x - 20y + 51 + A (x - 2j/ + 4) = 0
where A, is a parameter.
A new circle C with centre Q passes through P, A and B,
Find
(b)
a)
the value of X
[2 marksl
b)
the equation of circle C
[2 marks]
c)
between the
the centres
the distance, | PQ j,I, between
centres
[3 marks]
d)
ifPQ
PQ cuts
cuts AB
AB at at
M.M.
the distance 1PM || if
[4 marks]
A curve is given by the parametric equations x = 2 + 3 sin t, y = 3 + 4 cos t.
Show that
(i)
the Cartesian equation of the curve is
(* -2 )> + Cv^3)2 = j
9
16
(ii)
[3 marks]
every point on the curve lies within or on the circle
( x - 2 ) 2 + ( y - 3)2 = 25.
(5 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
22134020/CAPE 2008
-5SECTION C (Module 3)
Answer BOTH questions.
5.
(a)
(b)
Use L'Hopital's rule to obtain
(i)
(ii)
(c)
Given thaty =
lim
x
-7
sin 4x
0 sin 5x
[3 marks]
x
l - 4x
a)
fmd dy
dx
b)
show that r
[4 marks]
~ =y2•
[2 marks]
d2
d
Hence, or otherwise, show that r ~ + 2 (x - y) ~ = 0.
[3 marks]
A rectangular box without a lid is made from thin cardboard. The sides of the base are
2x em and 3x em, and its height is h em. The total surface area of the box is 200 cm2 •
(i)
20
3x
Show that h = - - - .
X
5
(ii)
Find the height of the box for which its volume V cm3 is a maximum.
[4 marks]
[9 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
22134020/CAPE 2008
-
6.
(a)
6
-
Use the substitution u = 3x1 + I to find J
xk dx
dr
x2 +
(b)
[6 m arks]
i'
’
A curve C passes through the point (3, -1 ) and has gradient x2 - 4x +■3 at the point (x, 7 )
on C.
[4 m arks]
Find the equation o f C.
(c)
The figure below (not d raw n to scale) shows part o f the line y + 2x = 5 and part of the
curve y = x ( 4 - x ) which meet at A. The line meets Oy at B and the curve cuts Ox at C,
y
A
B s
y v
\y
O
C
X
(i)
Find the coordinates o f A, B and C.
[6 m arks]
(ii)
Hence find the exact value o f the area of the shaded region.
[9 m arks]
Total 25 m arks
END OF TEST
22134020/CAPE 2008
TEST CODE
FORM TP 2008241
CARIBB E AN
22134032
MAY/JUNE 2008
EXAMI N ATIO N S
CO U NCIL
ADVANCED PROFICIENCY EXAMINATION
PURE MATHEMATICS
UNIT I - PAPER 03/B
ALGEBRA, GEOMETRY AND CALCULUS
J
~
lrou n·
( 26 JUNE 2008 (a.m.))
This examitu.tioo paper oonsist.s of TORE£ sections: Module t, Module 2 a nd Module 3.
Each section eons:Uu of J question.
The maximwn nUU'k for each Module is 20.
The maximum mark for this examination is 60.
This examination consist-; of 4 printod pages.
I NSTRUCT I ONS TO C ANDIDATES
I.
DO NOT open this examination paper until instructed to do so.
2.
Ans\Ver ALL questions fro1n lhe TllR££ $c:ctions.
3.
Write your solution$, witb full working, in the answer bookl~ provided.
4.
Untc.s s otherwise stau~d in the question, any numerical answer that is
not exact MUST be wrinen eorroc:t to three significant figures.
E:samjnarion Materials Ptrnajncd
Graph paper (prov;ded)
MatbelU.atical fonnulae and tables (provided)- ltevised 2008
MathematieaJ ins-U'Uinents
Silent, I'IOI'I·progranunable, e lectronic calculator
Copyright C 2008 Caribbean Examinations Counc-f1 ® .
All rights reserved.
22134032/CAPE 2008
-2
-
SECTION A (M odule 1)
Answer this question.
1.
(a)
(b)
(i)
Write log2 2P in terms o fp only.
[2 marks]
(ii)
Solve for x the equation log2 [logj {2x - 2)] = 2.
[3 marks]
The diagram below (not drawn to scale) shows the graph o f the function
fix ) = 3.x- + hx2 + he + m which touches the jc-axis at x = —1.
/M
HA
-r-i,oT
( §3 '°>
’»>
<
(0,-2),
(i)
Determine the values o f the constants h, k and m.
(ii)
[7 marks]
State the range o f values ofx in (-- 00. 0] for \vhich/(x) is a decreasing function.
[2 marks]
100
(c)
Evaluate X (3 r + 2).
r =
1
[6 marks]
Total 20 marks
GO ON TO THE NEXT PAGE
22134032/CAPE 2008
-
3
-
SECTION B (M odule 2)
Answer this question.
2.
The diagram below shows the path o f a com et around the sun S. The path is described by the
parametric equation x = at2 and y = 2at, where a > 0 is a constant.
y/
p
Q
R
s
x
p7
(a)
Show that the Cartesian equation for the path is y'1 = 4ax.
(b)
2a
Given that the gradient m o f the tangent at any point on the path satisfies m■=
-------,,
yy
(i)
show that the equation o f the tangent at (•*,>
(jc,, y j is y y 1l == 2a
*,) in Cartesian
2 a ((x
x + jtj)
form and ty = jc + at2 in parametric form
[5 marks]
(ii)
(iii)
(iv)
[2 marks]
find the equation o f the normal at the point P with parameter ■tl
*1
[3 marks]
f, t2
r2 +
+ 22 =
show that *,2 +
+ tx
= 00 if the normal in (ii) above intersects the path
again at the point P" with param eter f
[6 marks]
find the distance | QR | if the tangent at Pmeets
meets the x-axis at R.
the x-axis at Q and the normal
[4 marks]
Total 20 marks
GO ON TO THE NEXT PAGE
22134032/CAPE 2008
-4SECTIO~
C ( Module J)
~u thi~
J_
(a)
quotion.
(i)
By exp«osinax- 9 0$ (.,r;-• J)(.J";:- 3~ fond lim
9
(11)
II roce. find
......
hm
G-J
x-+9
x'-10.+9
.r;:- 3 .
X
9
13 marbl
14 mubJ
...
(b)
(I)
Find the \'a.lue of" if
(II)
Givc.n that
J'
1
J• ~• dx •
j{x) dx • 7, evaluate
J.' IJix)+ l]cl<+ J,' IJix)-l)cl<.
(c)
(3 marks)
IS marks!
The: fi;prc bdow(not drawn c·oscalt') shov.·s a Mnu~ tJo-,\·1 ~ htchcooraios:bquKl.
11'1e volume
v em3 or liquid is g.ivcn by
v. J... n: h 2 (24 - J,)
3
where h 11 the grcatC$t depth of t11e liquid in em l..1quid 11 poured into the bowl at the r3te ·
or I00 em' per second.
( i)
(u)
dl'
Fmd dJ m tenns of h.
13 markliJ
C&kulate l.be tale at .,·hich h tS mereuut.a when IJ • 2 em. (Lea"-e )'Our ans'\o\'er in
lmn> of lt.)
12 marks)
Total 20 marks
END OF TEST
22134032/CAPE 2008
TEST CODE 22234020
FORM TP 2008243
CARIBBEAN
MAY/JUNE 2008
EXAMINAT I O N S
COUNC I L
ADVANCED PROFICI ENCY E XAMINATION
' ' .•
PURE MATHEMATICS
UNIT 2 - PAPER 02
Al'IALYSIS, MAT RICES AND COMPLEX NUMBERS
1
*
IJOIII'$
(I 5 J ULY 2008 (p.m.))
This examination paper consists of THREE sct.ticms: Module I. Module 2 and Module 3.
Eacb section consists of 2 questioos..
11~e maximwn mark for eaeh Module is 50.
Tbe maximum mark for this examination is l SO.
11Us cxruuiu.Mioo consim of S printed pages.
INSTRUCTI ONS TO C AND1DAT£S
I.
00 NOT open this examjnation pape:t until insU\J:Ctcd to do so.
2.
Answer ALL questions ftom the Tff.R££ sections.
3.
Write yout soh.zliOt)$, with full workins. in the answer booklet provided.
4.
Unless otherwise S-t.3ted in the question, any numerical answer that is
nol exact MUST be written correct to three sig.uificaru figures.
Exan1ination
Material~
Prrmjugd
Graph paper (provided)
Mathematical fonnulae and tables (prOvided) - Revised 2008
M athematical instruments
Silent., non· progrttmmable, elec.tronjc. calculator
Copyriglu Q 2008 Caribbeao Examinations Council ®.
All rights reserved.
22234020/CAPE 2008
-
4.
(a)
-
The sequence {an} o f positive numbers is defined by
__ 3
_ 4 (1 +
+a)
_
an+ \i
(b)
4
4 + an
_
, aa '
’
_
2 '
(i)
Find a2 and ay
[2 marks]
(ii)
Express an+ , - 2 in terms o f an.
[2 marks]
(iii)
that
Given that an < 2for all n, show that
a)
an+l < 2
[3 marks]
b)
an < a n+l.
[6 marks]
Find the term independent o f x in the binomial expansion o f (x2 — ^-)15.
[You may leave your answer in the form of factorials and powers, for example, 151x8*.]
x 85.]
2!
]6 marks]
(c)
Use the binomial theorem to find the difference between 2 10 and (2.002)10 correct to 5
decimal places.
[6 marks]
Total 25 marks
SECTION C (Module 3)
A nswer BOTH questions.
5.
(a)
Four-digit, numbers are formed from the digits 1, 2, 3, 4, 7, 9.
(i)
(ii)
(b)
How many 4-digit numbers can be formed if
a)
the digits, 1, 2, 3, 4, 7, 9, can all be repeated?
[2 marks]
b)
none o f the digits, 1, 2, 3, 4, 7, 9, can be repeated?
[2 marks]
Calculate the probability that a 4-digit number in (a) (i) b) above is even.
[3 marks]
A father and son practise shooting at basketball, and score when the ball hits the basket.
The son scores 75% o f the time and the father scores 4 out o f 7 tries. If EACH takes one
shot at the basket, calculate the probability that only ONE o f them scores.
[6 marks]
GO ON TO THE NEXT PAGE
22234020/CAPE 2008
•5•
(e)
(i)
Find the values of II, k E R suc h lh.at 3 + 4i is a root of the quadrotic equation
~+ll:+k=O.
(ii)
Usc De Moivre's theorem for (CO$ 9
16 marks(
+ i sin 9)l to show that
cos 39 • 4 cos' 9-3 cos&.
16 marks(
Total 25 marks
6.
(a)
Solve for x the equ~tion
I
(b)
X
2
x'
8
- 0.
112 marksl
The Popular Taxi Service in ;'!; certain city provides transportation for tours of the city
using ca~ ooaches and buses. Selection of vehicl~ for tours of dis1aoces (in km) is as
follows:
;c cars.
2y coaches and 3: buses cover 34 km tOurs.
2x cars, 3y coaches and 4: buses cover 49 km tours.
3x cars. 4y coaches and 6: buses cover 7J km tours.
(i)
Expi'C:!l!C: the infnrm:uit~n :.Mvt! :.<: ;~ m~1rix .,.uation
AX = Y
where A is 3 x 3 matrix. X and \' are 3 x 1 mntrices wilh
13 marks!
( ii)
{iji)
Let B •[-4~ ..!0 ~2J.
a)
Calculate All.
(3 mark$)
b)
Deduce the inverse A·• of A.
13 ntftrks)
Hence. or otherwise. detennine the number of cars and buses used in
the 34 lon tours.
(4 marks I
Tol'al 25 marks
llND OF TEST
22234020/CAPE 2008
~
FORM TP 2008244
CA RIBBEA i'>
~
EXAMI NAT IO NS
~TOODE 222J40J2
MAY/JUNE 2008
C O U :SCII..
ADVANCED PROFIC IENCY EXAMINATION
PURE MATBE!\tATICS
UNIT 2 - PAPER 0318
ANALYSIS, i\tATRJCES AND C0~1PLEX NUMBERS
I ~
lumrs
(21 JUNE 2008
(n.m.>)
Th.is cxa.minacjon paper c:onsiSl$ of 'THREE scc.tions: Module I, Module 2 and Module 3.
Elcb sot:bon coni&J.U of I questiOn..
The maxJmWll awtc: for each Module is 20.
The ma.xunum mart for thu ~is 60.
ThJ o.anun~uon consuu of 4 printed pages..
INSTRUCTIOSS TO CA!S"DJDATES
1.
00 NOT open this exami.nation paper until instructed to do so.
2.
Answer A l,.L quc:scions from the Tll REE secdons.
3.
Wri1e your solutions, with fuJI working, in the ll.llJWcr bookJct provided.
4.
Unless ocherwise stated in the question, any nummc:a.l AllSwcr that is
noc exact MUS'l" be "'linen corroc.t to three sianiticant fl$\Jra.
Enmlnadon Maceri•h Pcrmjnf11
Gnph paper (pro''Jdcd)
MalhcnDuooJ fonnulae and llbles (provided) - RniS<d 200i
Malhemon<al INU\Uneoll
Silan. non-J)f'08rlrnmable. elcct.rooic calc:ulator-
Copyriaht 0 2008 Can'bbe.'n Examinations Council®.
All rights reserved.
22234032/CAPE 2008
-
2
-
SECTION A (Module 1)
Answer this question.
1
.
(a)
Given that x = In [y + V (y2 - 1)], y > 1, express y in terms of x.
(b)
Use the substitution u = sin x to find
J
(c)
[5 marks]
[6 marks]
cos3x dx.
Engine oil at temperature T °C cools according to the model
T = 60 e-*' + 10
where t is the time in minutes from the moment the engine is switched off.
(i)
Determine the initial temperature of the oil when the engine is first switched off.
[2 marks]
(ii)
If the oil cools to 32°C after three minutes, determine how long it will take for the
oil to cool to a temperature of 15°C.
[7 marks]
Total 20 marks
SECTION B (Module 2)
Answer this question.
2.
(a)
(i)
Write the general term of the series whose first four terms are
1
1x3
+ — .— + — .— + — -— + ...
3x5
5x7
7x9
[2 marks]
(ii)
Use the method of differences to find the sum of the first n terms.
[5 marks]
(iii)
Show that the series converges and fmd its sum to infinity.
[3 marks]
GO ON TO THE NEXT PAGE
22234032/CAPE 2008
-3
(b)
-
The diagram below (not drawn to scale) shows part o f the suspension o f a bridge
A support cable POQ, is in the shape of a curve with equation
y-
1
10
X3' 2
+ c, where c is a constant.
Starting at P, through O and finishing at Q, 51 vertical cables are bolted 1 metre apart to
the roadway and to the support cable POQ. The shortest vertical cable OAhas a length of
5 metres, where O is the lowest point of the support cable.
The cost, in dollars, o f installing the cable LH at a horizontal distance of r metres from
OA is S i00 plus-S $ h V~r^ w
where h is the height of the point L above O.
P
0
L,
Support
Cable
Vertical
Cables
O
^Roadway
A
H
(i)
ofr,r, the
th cost o f installing the cable LH.
Find, in termsis of
[4 marks]
(ii)
Hence, obtain the total cost of installing the 5 1 vertical cables.
[6 marks)
Total 20 marks
GO ON TO THE NEXT PAGE
22234032/CAPE 2008
-
4
-
SECTION C (Module 3)
Answer this question.
3.
w
(a)
(b)
L c t:L t
( l - 2 i ) a + i)
(1 + i)’
•
(i)
Express z in the form a + b\, where a, b e R.
[5 marks]
(ii)
Calculate the exact value o f j z |.
[3 marks]
Two 3 x 1 matrices X and Y satisfy the equation X = AY, where the matrix
f l
A =
V.
-1
3
4
2
1
n
4
6J
is non-singular.
Find
(i)
A“!
(ii)
Y, when X =
[8 marks]
6
[4 marks]
4
11
Total 20 marks
END OF TEST
22234032/CAPE 2008
TEST CODE
FORM TP 2009234
CARIBBEAN
02134020
MAY/JUNE 2009
EXAMINATIONS
COUNCIL
ADVANCED PROFICIENCY EXAMINATION
PURE MATHEMATICS
UNIT 1 - PAPER 02
ALGEBRA, GEOMETRY AND CALCULUS
c
2% hours
20 MAY 2009 (p.m.) )
This examination paper consists of THREE sections: Module 1, Module 2 and Module 3.
Each section consists of 2 questions.
The maximum mark for each Module is 50.
The maximum mark for this examination is 150.
This examination consists of 7 printed pages.
INSTRUCTIONS TO CANDIDATES
1.
DO NOT open this examination paper until instructed to do so.
2.
Answer ALL questions from the THREE sections.
3.
Write your solutions, with full working, in the answer booklet provided.
4.
Unless otherwise stated in the question, any numerical answer that is not
exact MUST be written correct to three significant figures.
Examination Materials Permitted
Graph paper (provided)
Mathematical formulae and tables (provided) - Revised 2009
Mathematical instruments
Silent, non-programmable, electronic calculator
Copyright © 2009 Caribbean Examinations Council ®.
All rights reserved.
02134020/CAPE 2009
-
2
-
SECTION A (Module 1)
Answer BO TH questions.
1.
(a)
Without the use of tables or a calculator, simplify V~28 ++• V 343 in.
in the form k 'T l',
where k is an integer.
[5 marks]
(b)
Let x andy be positive real numbers such that x ^ y .
(i)
(ii)
Simplify
JC4 - /
[6 marks]
x-y
Hence, or otherwise, show that
4_yt
+ 1)3 + ^ + ^2^ + ^ + i ) y 2 + y i
(y+
+ 1i )) 4/ =
= (( yy+
l ) 3 + ( y + l ) 2y + 0 + l ) j 2 + y .
(y
(iii)
(c)
Deduce that (y + l ) 4 —y 4 < 4 (y + l) 3 .
Solve the equation log4 x = 1 + log2 2x, x > 0.
[4 marks]
[2 marks]
[8 marks]
Total 25 marks
2.
(a)
The roots of the quadratic equation
2X2 + 4x + 5 = 0 are a and (3 .
?
2
Without solving the equation, find a quadratic equation with roots — and — .
[6 marks]
GO ON TO THE NEXT PAGE
02134020/CAPE 2009
-
(b)
3
-
The coach of an athletic club trains six athletes, u, v, w, x, y and z, in his training camp.
He makes an assignment,/, of athletes u, v, x, y and z to physical activities 1, 2, 3 and 4
according to the diagram below in which A = {u, v, w, x, y, zj and B = {1, 2, 3, 4}.
A
/
B
u
V
1
w
2
X
3
y.
►4
2
(c)
(i)
Express / as a set of ordered pairs.
[4 marks]
(ii)
a)
State TWO reasons why / is NOT a function.
[2
b)
Hence, with MINIMUM changes to / construct a function g : A —> B as a
set of ordered pairs.
[4 marks]
c)
Determine how many different functions are possible for g in (ii) b) above.
[2 marks]
marks]
The function /o n R is defined by
Ax) =
x - 3 if jc < 3
x
if x>3.
4
{
Find the value of
/1/(20)]
[3 marks]
(ii) /[/(§ )]
(hi) f [ f (3)].
[2 marks]
(i)
[2 marks]
Total 25 marks
02134020/CAPE 2009
GO ON TO THE NEXT PAGE
-
4
.
SECTION B (Module 2)
Answer BOTH questions.
3,
Answers to this question obtained by accurate drawing wili not be accepted.
(a)
The circle C has equation (jc - 3)2 + (y - 4)2 = 25.
(i)
State the radius and the coordinates o f the centre o f C.
[2 marks]
(ii)
Find the equation of the tangent at the point (6, 8) on C.
[4 marks]
(iii)
(b)
Calculate the coordinates o f the points o f intersection o f C with the straight line
y = 2x + 3.
[7marks]
The points P and Q have position vectors relative to the origin O given respectively
p = - i + 6j and q = 3i + 8j.
(i)
(ii)
by
a)
Calculate, in degrees, the size o f the acute angle G between p and q.
[5 marks]
b)
Hence, calculate the area o f triangle POQ.
[2 marks]
Find, in terms of i and j, the position vector of
a)
M, where M is the m idpoint o f PQ
[2 marks]
b)
R, where R is such that PQRO, labelled clockwise, forms a parallelogram.
[3 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
02134020/CAPE 2009
-
4.
(a)
5
-
The diagram below, which is n o t d ra w n to scale, shows a quadrilateral ABCD in which
AB = 4 cm, BC = 9 cm, AD = x cm and z. BAD = z BCD = 0 and ^ CZM is a nght-angle.
4 cm
A
e
B
9 cm
x cm
e
C
D
(i)
Show that x = 4 cos 0 + 9 sin 0.
(ii)
By expressing x in the form r cos (9 - a), where r is positive and 0 < a < ~ n,
find the MAXIMUM possible value o f x.
(b)
(c)
[4 marks]
[6 marks]
Given that A and B are acute angles such that sin A = — and cos B —---- -, find, w ithout
using tables or calculators, the EXACT values of
5
1^
(i)
sin (,(A + B)
sin
[3 marks]
(ii)
cos
cos ((A - B)
[3 marks]
(lii)
cos 1cos 2A.
[2 marks]
Prove that
tan
X
2
+
71
4
sec x + tan x.
[7 marks]
Total 25 m arks
GO ON TO THE NEXT PAGE
02134020/CAPE 2009
-
6
-
SECTION C (Module 3)
Answer BOTH questions.
5.
lim
x —>2
x3- 8
__________
x2 - 6x + 8
(a)
Find
(b)
The function/on R is defined by
ftx) =
Sketch the graph offlx) for the domain —1 < x < 2.
(ii)
Find
b)
(iii)
(d)
f 3 —x i f x > l
|_1 + x if x < 1.
(i)
a)
(c)
[5 marks]
x
[2 marks]
[2 marks]
hm
Ax)
1+
[2
/i>” i fix),
x —=>I-
marks]
[3 marks]
Deduce that/[x) is continuous at x = 1.
.
J_
'
X 2
Differentiate from first principles, with respect to x, the function v =
.
[6 marks]
'
The function /f,x) is such that / (x) = 3x2 + 6x + k where k is a constant.
Given that/fO) = - 6 and/ ( l ) = -3 , find the function j(x).
[5 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
02134020/CAPE 2009
.. 1·
6.
(a)
Ga\"CD lhll )' • sin 2r • cos 2x. show that
(6 marks]
(b)
•
(c)
•
•
'
e
Oovmt!wJ (x•l)dx=3 J
(x-l)dx,a>O,findlhcvalucortheco.,.tanta.
(6 mark$)
The diagram below (oot drawn to scnle) represents a piece o( thm cardboard 16 em by
10 c:m. Sbadod SC1uares, each of side x em, are removed Crom eac:h comer. The remainder
is folded to fonn a tray.
IOmo
., e:•
(i)
Sbow that the volwne, Vern'. of the b"ly IS a•...·en by
v(ii)
4 (x' - 13.<' .. 40.).
(5 mark.<)
Hence:, find a possible value of x such th"l V 11 3 maxunum.
(8 marksJ
Total 25 marks
£1\'0 OF TEST
02134020/CAPE 2009
TESTCOOE 02134032
FORM TP 2009235
CA RIBB EAN
MAY/.11JKE 2009
EXAMINAT IO NS
COUNC I L
ADVANCED PROFICIENCY EXAMINATION
PURE MATHEMATICS
UNIT I - PAPER 03/B
ALGEBRA, GEOME TRY AND CALCULUS
c
I * h ours
10 J UN& 2009
(p.m.))
This examination paper oonsislS of THREE sections: Module I, Module 2 and Module 3.
Eacb section consists of I qu~stion.
Tbe ma:<.imum. mark for each Module is 20.
The maximum mark for this examination is 60.
nus examinatioo consists of 3 printed pages.
INSTRUCTIONS TO C A NQIDAT£'0
(
I.
00 NO"r open lhis e.'\:amitu'ltion paper until inslt\Jeted to do so.
2.
Answer ALl.. questions from the TH RE£ sections.
3.
Write your solutions. with full woric:ing, in th¢ answer booklet provided.
4.
Unless otherwise stated in the question. 3ny numeric'*l answer thai is not
exact i\1US'r be written conect to three significant figures.
Enmin;\tion 1\ilnteriab: Permitted
Graph paper (provided)
Mathematical fonnuJae and tables (provided) - Revised 2009
Mathematical inst.rwnencs
Silenl, non-progr.unmablc~ electronic ca1cul:nor
Cop)•right 0 2009 Caribbean Ex.a:min::nions Co~.~ncil ®.
All riglns reS«Vcd.
02134032/CAPE 2009
-
2
-
SECTION A (Module 1)
Answer this question.
1.
(a)
Find the set of real values of x for which
[ x - 1 | > | 2x + 1 | .
(b)
[6 marks]
A packaging company makes crates for special purposes. The company finds that the
unit costy(x), in thousands of dollars, of producing crates with a square base of x metres
is
f{x) = (x2 - Ax)2 + 2x2 - 8x.
Using the substitution;-1= x2 - 4x, find the sizes of the crates for which the unit cost is
three thousand dollars.
[7 marks]
(c)
(i)
By taking logarithms, show that for any positive integers p and x,
pte%x - x.
(ii)
[4 marks]
Hence, without using calculators or tables, find the EXACT value of
« (logj 6 + log, 1 5 - 2 logj3)
2
.
[3 marks]
Total 20 marks
SECTION B (Module 2)
Answer this question.
2.
(a)
(i)
Show that the equation of the tangent to the circle
x2 + y2 + 8x + 14 = 0 at the point (p, q) is
(p + 4) (x - p) + q {y - q) = 0.
(ii)
Show that the equation of the tangent can also be written as
px + qy + 4 (jc + p ) + 14 = 0.
(iii)
[3 marks]
[2 marks]
If the tangent at (p, q) on the circle passes through thepoint (-3, 3), find the
values o fp and q.
[7 marks]
GO ON TO THE NEXT PAGE
02134032/CAPE 2009
-
(b)
3
-
A point moves so that at time t its distances from the coordinate axes are given by
x = 2 + 3 cos t and y = 4 + 4 sin t.
(i)
Find the maximum and minimum values o f x and y.
Find the maxii
[4 marks]
(ii)
Find the Cartesian equation o f the curve traced by the point.
[4 marks]
Total 20 marks
SECTION C (Module 3)
Answer this question.
J U + 3t5f - 1
[5 marks]
(a)
Find
(b)
The point P (-1, 5) is a point o f inflexion on the curve y = X s + bx2 + c, where b and c are
constants.
dt.
Find
(i)
the values
b and ocf b and c
theo fvalues
to curve
the curve
(ii)the equation o f the normal tolalthe
at P.at P.
(c)
[5 marks]
[3 marks]
Scientists on an experimental station released a spherical balloon into the atmosphere.
The volume of air in the balloon is increased or decreased as required.
(i)
The radius, r, of the balloon is increasing at the constant rate o f 0.02 cm/s. Find
the rate at which the volume, V cm3, is increasing when r = 3 cm. Express your
answer in terms o f n.
[2 marks]
(ii)
The volume, V cm3, o f the balloon decreases by 6% when the radius decreases by
p%. Find p.
[5 marks]
Total 20 marks
END OF TEST
02134032/CAPE 2009
TEST CODE
FORM TP 2009237
CARIBBEAN
02234020
MAY/JUNE 2009
EXAMINAT IONS
COUNCIL
ADVANCED PROFICIE NCY EXAMINATION
PURE MATHEMATIC S
UNIT 2- PAPER 02
ANALYSIS, MATRICES AND COMPLEX NUMBERS
c
2 Yz hours
27 MAY 2009 (p.m.) )
This examination paper consists of THREE sections: Module 1, Module 2 and Module 3.
Each section consists of 2 questions.
The maximum mark for each Module is 50.
The maximum mark for this examination is 150.
This examination consists of 5 printed pages.
INSTRUCTIONS T O CANDIDATES
1.
DO NOT open this examination paper until instructed to do so.
2.
Answer ALL questions from the THREE sections.
3.
Write your solutions, with full working, in the answer booklet provided.
4.
Unless otherwise stated in the question, any numerical answer that is not
exact MUST be written correct to three significant figures.
Examination Materials Permitted
Graph pape~ (provided)
Mathematical formulae and tables (provided) - Revised 2009
Mathematical instruments
Silent, non-programmable, electronic calculator
Copyright © 2009 Caribbean Examinations Council ® .
All rights reserved.
02234020/CAPE 2009
SJ;;CTION A (Module I)
Answer BOTH q uc$tio.ns.
I.
(a)
(b)
·r
F .md .2!:.
dl' 1
(i)
y • sin: Sx +sin' 3.\" + c<W 3:~
(3 mftrkS)
(ii)
y = .J cos xl
(4 marks)
(iii)
y- )." .
(4 mn.rks)
(i)
Given that y = cos-1 x. where 0 ::S C0$"'1 x $ :t. prove tl1at
t •---:;;:!::
1
""'
~ 1-x'
[Noc.c: cos-'x • arc cos :r]
(ii)
17 mMks )
111e parametric equations of a curve ru-e defmcd in terms of a parameter t by
y •
..rr:=i and x = cos-1 /,
•>
Show that d y ...
d.<
b)
llertee. fi1)d
~
where 0 ~ 1 < I.
..rr+7 .
(4 m!\rksJ
2
in tenns
or t, giving your answer in simplified fonn.
(3 marks)
Total 25 ma rks
2.
marks)
J'~~
..J I - r
that f '~ 1- x' dx "' 0.759.
•
(a)
Sketch lhc region whose area is defined by the imegroJ
(b)
Using FIVE vertical strips, apply the:. trapezium rule to show
dx •
(3
0
(6 m>rks(
(c)
(i)
Use integnnion by psns to show that, if / •
JV I -r
dx .
d.l'. then
19 marksl
00 ON TO TilE NEXT PAGE
022340201CAPE 2009
_
(li)
Deduce that I =
integration.
X
3
-
V 1 —x 2 + sin~' x
- + cc, where c is an arbitrary constant o f
2
[Note: cos-1* = arc cos jc]
[2 m arksj
i
iii)
(iv)
Hence, find
V 1 - x 1 dx
Jo
[3 m arks]
Use the results in Parts (b) and (c) (iii) above to find an approximation to n.
[2 m arks]
Total 25 m arks
SECTIO N B (M odule 2)
A nswer B O T H questions.
3.
(a)
A sequence {tn} is defined by the recurrence relation
tn +
= *tn
. for all n se N.
K
+>
5’5 (i
f\] = 11
11
tn+i
5>
+
Ii =
«n + 5,
(i)
(ii)
Determine
termine tv titi and
and tA
t .,
[3 m arks]
Express
press tn
t in
in terms
terms of
of nn.n.
[5 m arks]
(b)
Find the range o f values o f jt for which the common ratio r o f a convergent geometric
2 jc -3
series is
[8 m arks]
x+4 '
(c)
Let J[r)
j{r) =
(i)
(ii)
} ■, r e N.
r+ 1
Express j{ f) - f i r + 1) in terms o f r.
[3 m arks]
Hence, or otherwise, find
«n
.
s = xZ _____ - ______
S
" ,-i (r+l)(r+2) ■
'
(iii)
[4 m arks]
Deduce the sum to infinity o f the series in (c) (ii) above.
[2 m arks]
Total 25 m arks
GO ON TO THE NEXT PAGE
02234020/CAPE 2009
• 4.
(5 m arks)
(a)
(ia)
The: ooeffieaC:nl of .r '" the expansion of
(I+ 2.')' (I + px)'
is - 26. Find the J)()S..~able vntues of the realoumber p.
(b)
(i)
17 r:n2rks)
Wnte down the firt~l 1:0UR oon-7.ero tenns of the power scnes expanSlon of
ln (1 -+- 2r), SLaUf\llhe ranae of values ofx for whlchlhc .Knet ' ' valid.
12 marksl
(ti)
Usc Mxburm's meo.em to oblaln the M• niREE non-letO tams tn the pow.·cr
sencs cxpam.on 1n x ofsan h-.
(7 m.r·ksl
(iii)
Hence, or other-.1sc, obtam the first THREE non-.rero tcnn~ m the power series
cxpansaon 1ft X of
In (I +sin 2.\)
(4 m:arksl
·rotal lS n1arks
\t;;C'I IO.N' C
~lodul~
l)
An\"'tr 8 0TU q u estions.
s.
(a)
A committee of 4 pct'SOnS •• to be choset'a from 8 persons, mcludang Mr Smllh and h1s
wtfe. Mr Smtlh will ncn jom chc comminoc without hil wife, but has wife will join ilie
comm ittee without him.
Calculate the number of v. ayli ln wh1ch tlte commiuee or 4 persons c.an be ronned.
IS marks)
(b)
Tv,;o balls are dr1wft .,.,\haul rq>lac:etl\ent from a bag ce>ntama~ 12 Nils numbered I to
12
14 marks)
{ i)
the numben: on BOTti balls arc even
(ii)
the number on one ball Is odd and the nwnbet on the other b:all LS even,
14 nl:arks l
GO 0 ' TO THE I\"EXT PAGE
0223402~ CAPE
2009
-
(c)
(i)
5
-
Find complex numbers u = x + iy such that x and y are real numbers and
u2 = -1 5 + 8i.
(ii)
|7 marks]
Hence, or otherwise, solve for z the equation
z2 - (3 + 2i) z + (5 + i) = 0.
[5 marks]
Total 25 m arks
6.
(a)
Solve for x the equation
x
-
3
1
1
-1
(b)
(i)
x
~
-1
5
1
= 0.
1
* -3
[10 m arks]
Given the matrices
r
A =
1
1
1
-1
-2
3
I
4
9
,
30 - 1 2
5 -8
-5
4
B -
2^
3
1
V
(ii)
a)
find AB
]4 marks]
b)
hence deduce the inverse A-1 of the matrix A.
[3 m arks]
A system o f equations is given by
x —y + z — 1
x — 2y + 4z = 5
x + 3y + 9z = 25.
a)
Express the system in the form
Aa' = b , where A is a matrix and x and b are column vectors.
b)
Hence, or otherwise, solve the system o f equations.
[5 marks]
Total 25 m arks
END O F TEST
02234020/CAPE 2009
TEST CODE
FORM TP 2009238
CAR I BBEAN
02234032
MAYfi\JNE 2009
EXAM I NA TI ONS
COUNCIL
ADVANCED PROFICIENCY EXAMINATION
PURE MATHEJ\tATICS
UNIT l - PAPER 03/B
ANALYSIS, J\tATRJCESAND COMPLEX NUMBERS
( 03 JUNt l~
(•.m.))
TbiJ ex~mination piper COilliJcJ of mR££ ICCobonl.: ModWt I, Module 2 and ModWe 3.
Eaeb I«CioD consisu or 1 quesr:ioa.
lbe rnuJmum mart for c.ch Moctule it 20.
The mu.imum mart for chis e~ami.Mtion b 60.
This c:xa:m.inltion consistJ or 3 printed pllet.
JNSTR!ICTIONS TO CANDIDATFS
I.
DO NOT open OUt examination phper until instructed to do so.
2.
AnJwer .A.LL quetdons from lhe T JfJll·; E lcetio.ns.
3.
Write )'OW' to1Ulions, with fuiJ working, in the answer booklet provided.
4.
l)nlca othclwisc stated in che question. any numerieaJ answer that is noc
exaet MUST be written corrcc:t to three Jipi6ettU figures.
E t amip•dop Maltdah PermUted
Graph paper (provided)
~lhcmorical
~lhcmorical
formu!O< oncl toblco (provldcd) - RrviJ<d loot
inRruments
SHcut, non·prognmntoble. <lecuorue calculotor
CopynJllt 0 20Gt Coril>b<an Eumuw,..,. Council ® .
All """' n:a<n...S.
Oll.l4Q)VCAPE 2009
-
2
-
SECTION A (Module 1)
Answer this question.
1.
(a)
Solve the differential equation
* r+ , (/+1) =<* + ») e^^.
(b)
[7 marks]
1 7
m
a
r k
s l
A curve is being cut by an automatic machine. The x and y coordinates of the cu rv e are
connected by the differential equation
- ^ 5— 3 — ---- 4y = 5 sin x +v3 3cos
x. x.
cos
Find the equation of the curve, given that the curve passes th ro u g h the origin an d that
y = e~” —e4* when x~%.
[13 m arks]
Total 20 marks
SECTION B (Module 2)
Answer this question.
2.
(a)
Prove by mathematical induction that
£
r= 1
r3
=
n
n'2i{n
n + l )2
f
4
^
for all integers n > \ .
(b)
[7 m arks]
An A-P. with ten terms has first term 60 and last term -120. F in d the sum o f A L L the
terms.
[4 marks]
GO O N T O THE N E X T PAGE
02234032/CAPE 2009
•
• 3.
(c)
John's starting annual salaty at a company i$ S2S 000. His contract at the company states
that his Mnual salary in subsequent years wiiJ be increased by 2% over the-salary of the
previous year.
Find.. to tbe oet.rett dolbr.
(i)
John's salary for the tenth year with the company
(ii)
the TOTAL amount of money which the company would have paid 10 John at the
end ofbjs first ten yean with the company.
(S marks)
(4 marks)
Total lO marks
SECTION C (Module 3)
ADSWt'r th is quntion.
3.
(a)
There are6 staffmembersand 7 students on thesportseouocil of a college. A committee
of l 0 per$0f\S is to be s.elected to orgat1ize a tou.mamern. Calculate the number of ways in
which the committee. can be selected if the num~r of studentS must be crater 1'han or
equ•l to the number of staff members.
(6 marks)
(b)
A and JJ are ....,ts such \hat
P(A) = 0.6.
P(B) - 0.2
and
P(A
n
B) - 0.1.
Calcuhne-
(c)
(i)
P(A u B)
(2 marks)
(ii)
P(A " B")
(2 marloo)
(iii)
the probability that exactly ONE of A and B will occur.
(4 marks]
(i)
Show thai lhe locus of the complex numbt:r z suc h that
fz + i-1 1• 5
(ii}
is a circle.
14 marks]
find the centre and radius of the circle in (c) (i) above.
12 m arks)
Toa.J lO marks
ENI> OF T EST
022340321CAPE 1009
TEST CODE
FORM TP 2010227
CARIBBEAN
02134020
MAY/JUNE 2010
E XAM I NAT I O N S
COUNCIL
ADVANCED PROFICIENCY EXAMINATION
PURE MATHEMATICS
UNIT 1 – PAPER 02
ALGEBRA, GEOMETRY AND CALCULUS
2 ½ hours
20 MAY 2010 (p.m.)
This examination paper consists of THREE sections: Module 1, Module 2 and Module 3.
Each section consists of 2 questions.
The maximum mark for each Module is 50.
The maximum mark for this examination is 150.
This examination consists of 7 printed pages.
INSTRUCTIONS TO CANDIDATES
1.
DO NOT open this examination paper until instructed to do so.
2.
Answer ALL questions from the THREE sections.
3.
Write your solutions, with full working, in the answer booklet provided.
4.
Unless otherwise stated in the question, any numerical answer that is not
exact MUST be written correct to three significant figures.
Examination Materials Permitted
Graph paper (provided)
Mathematical formulae and tables (provided) – Revised 2009
Mathematical instruments
Silent, non-programmable, electronic calculator
Copyright © 2009 Caribbean Examinations Council ®.
All rights reserved.
02134020/CAPE 2010
-2SECTION A (Module 1)
Answer BOTH questions.
1.
(a)
Find the values of the constant p such that x – p is a factor of
f(x) = 4x3 – (3p + 2) x2 – (p2 – 1) x + 3.
(b)
[5 marks]
Solve, for x and y, the simultaneous equations
log (x – 1) + 2 log y = 2 log 3
log x + log y = log 6.
(c)
Solve, for x ∈ R, the inequality
2x – 3
——— – 5 > 0.
x+1
(d)
[8 marks]
[5 marks]
By using y = 2x, or otherwise, solve
4x – 3 (2x + 1) + 8 = 0.
[7 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
02134020/CAPE 2010
-3-
2.
(a)
(i)
n
1
Use the fact that Sn = Σ r = — n (n + 1) to express
2
r=1
2n
S2n = Σ r in terms of n.
[2 marks]
r=1
(ii)
Find constants p and q such that
S2n – Sn = pn2 + qn.
(iii)
[5 marks]
Hence, or otherwise, find n such that
S2n – Sn = 260.
(b)
[5 marks]
The diagram below (not drawn to scale) shows the graph of y = x2 (3 – x). The coordinates
of points P and Q are (2, 4) and (3, 0) respectively.
y
P
0
1
2
Q
x
3
(i)
Write down the solution set of the inequality x2 (3 – x) < 0.
(ii)
Given that the equation x2 (3 – x) = k has three real solutions for x, write down the
set of possible values for k.
[3 marks]
(iii)
The functions f and g are defined as follows:
[4 marks]
f : x → x2 (3 – x), 0 < x < 2
g : x → x2 (3 – x), 0 < x < 3
By using (b) (ii) above, or otherwise, show that
a)
f has an inverse
b)
g does NOT have an inverse.
[6 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
02134020/CAPE 2010
-4SECTION B (Module 2)
Answer BOTH questions.
3.
(a)
The vectors p and q are given by
p = 6i + 4j
q = –8i – 9j.
(b)
(i)
Calculate, in degrees, the angle between p and q.
(ii)
a)
Find a non-zero vector v such that p.v = 0.
b)
State the relationship between p and v.
[5 marks]
[5 marks]
The circle C1 has (–3, 4) and (1, 2) as endpoints of a diameter.
(i)
Show that the equation of C1 is x2 + y2 + 2x – 6y + 5 = 0.
(ii)
The circle C2 has equation x2 + y2 + x – 5y = 0. Calculate the coordinates of the
points of intersection of C1 and C2.
[9 marks]
[6 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
02134020/CAPE 2010
-
4.
(a)
(i)
(11)
(iii)
(b)
5
-
Solve the equation cos 3A = 0.5 for 0 < A < n.
[4 marks]
Show that cos 3A = 4 cos3A - 3 cos A.
[6 marks]
The THREE roots o f the equation 4p3- 3p - 0.5 = 0 all lie between -1 and 1. Use
the results in (a) (i) and (ii) to find these roots.
[4 marks]
The following diagram, not drawn to scale, represents a painting o f height, h metres, that
is fastened to a vertical wall at a height o f d metres above, and x metres away from, the
level o f an observer, O.
Painting
hm
/\
dm
a
O
x m
v
The viewing angle o f the painting is ( a - P), where a and p are respectively the angles of
inclination, in radians, from the level o f the observer to the top and base o f the painting.
(i)
(ii)
hx
Show that tan ( a - P) = ——
— d~—— — .
x I d (d I h)
[6 marks]
The viewing angle o f the painting, ( a - P), is at a maximum when x = V h (d + h).
Calculate the maximum viewing angle, in radians, when d = 3h.
[5 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
02134020/CAPE 2010
-6SECTION C (Module 3)
Answer BOTH questions.
5.
(a)
(b)
Find
(i)
lim
x→3
x2 – 9
———–
x3 – 27
[4 marks]
(ii)
lim
x→0
tan x – 5x
————— .
sin 2x – 4x
[5 marks]
The function f on R is defined by
f(x) =
(i)
(ii)
(c)
(i)
(ii)
3x – 7, if x > 4
1 + 2x, if x < 4.
Find
a)
lim
f(x)
x → 4+
[2 marks]
b)
lim
f(x).
x → 4–
[2 marks]
Deduce that f(x) is discontinuous at x = 4.
Evaluate
∫
1
1
x–—
x
–1
[2 marks]
2
dx.
[6 marks]
Using the substitution u = x2 + 4, or otherwise, find
∫ x √x
2
+ 4 dx.
[4 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
02134020/CAPE 2010
-
6.
(a)
7
-
Differentiate with respect to x
(i)
(ii)
y = sin (3x + 2) + tan 5x
y =
[3 marks]
x2 + 1
x3 - 1
[4 marks]
,
4
(b)
The function f(x) satisfies f f x ) dx = 7.
4
(i)
Findf
[3 f x ) + 4] dx.
[4 marks]
3
(ii)
Using the substitution u = x + 1, evaluate
f 2f
(x + 1) dx.
[4 marks]
0
(c)
In the diagram below (not drawn to scale), the line x + y = 2 intersects the curve y = x2
at the points P and Q.
2
/y = X
y/
p
XT
o‘
X
x +y = 2
(i)
(ii)
Find the coordinates o f the points P and Q.
[5 marks]
Calculate the area o f the shaded portion o f the diagram bounded by the curve and
the straight line.
[5 marks]
Total 25 marks
END OF TEST
02134020/CAPE 2010
TEST CODE
FORM TP 2010228
CARIBBEAN
02134032
MAY/JUNE 2010
E XAM I NAT I O N S
COUNCIL
ADVANCED PROFICIENCY EXAMINATION
PURE MATHEMATICS
UNIT 1 – PAPER 03/B
ALGEBRA, GEOMETRY AND CALCULUS
1 ½ hours
09 JUNE 2010 (p.m.)
This examination paper consists of THREE sections: Module 1, Module 2 and Module 3.
Each section consists of 1 question.
The maximum mark for each Module is 20.
The maximum mark for this examination is 60.
This examination consists of 4 printed pages.
INSTRUCTIONS TO CANDIDATES
1.
DO NOT open this examination paper until instructed to do so.
2.
Answer ALL questions from the THREE sections.
3.
Write your solutions, with full working, in the answer booklet provided.
4.
Unless otherwise stated in the question, any numerical answer that is not
exact MUST be written correct to three significant figures.
Examination Materials Permitted
Graph paper (provided)
Mathematical formulae and tables (provided) – Revised 2009
Mathematical instruments
Silent, non-programmable, electronic calculator
Copyright © 2009 Caribbean Examinations Council ®.
All rights reserved.
02134032/CAPE 2010
-2SECTION A (Module 1)
Answer this questions.
1.
(a)
(b)
The roots of the cubic equation x3 + px2 + qx + 48 = 0 are α, 2α and 3α. Find
(i)
the value of α
[2 marks]
(ii)
the values of the constants p and q.
[4 marks]
The function f on R is given by
f: x → 3x – 2.
(i)
Show that f is one-to-one.
(ii)
Find the value of x for which
f (f (x)) = f (x + 3).
(c)
[3 marks]
[4 marks]
Prove by mathematical induction that
9n – 1 is divisible by 8
for all n ∈ N.
[7 marks]
Total 20 marks
GO ON TO THE NEXT PAGE
02134032/CAPE 2010
-3SECTION B (Module 2)
Answer this questions.
2.
A surveyor models the boundaries and extent of a triangular plot of land on a Cartesian plane as
shown in the diagram below (not drawn to scale). The line 2x + 3y = 6 meets the y-axis at A and
the x-axis at B.
C is the point on the line 2x + 3y = 6 such that AB = BC.
CD is drawn perpendicular to AC to meet the line through A parallel to 5x + y = 7 at D.
2x
y
+
3y
=
6
A
B
0
x
C
D
(a)
Find
(i)
the coordinates of A, B and C
[6 marks]
(ii)
the equations of the lines CD and AD.
[5 marks]
(b)
Show that the point D has coordinates (2, –8).
[4 marks]
(c)
Calculate the area of triangle ACD.
[5 marks]
Total 20 marks
GO ON TO THE NEXT PAGE
02134032/CAPE 2010
-4SECTION C (Module 3)
Answer this questions.
3.
∫ (cos 5x + tan
2
x) dx.
(a)
Find
(b)
Part of the curve y = x (x – 1) (x – 2) is shown in the figure below (not drawn to scale).
[4 marks]
y
0
(c)
q
p
x
(i)
Find the values of p and q.
(ii)
Hence find the area of the region enclosed by the curve and the x-axis from
[5 marks]
x = 0 to x = q.
[2 marks]
A piece of wire, 60 cm long, is bent to form the shape shown in the figure below, (not
drawn to scale). The shape consists of a semi-circular arc of radius r cm and three sides
of a rectangle of height x cm.
m
rc
x cm
(i)
Express x in terms of r.
(ii)
Show that the enclosed area A cm2 is given by
π .
A = 60r – 2r2 1 + —
4
(iii)
Find the exact value of r for the stationary point of A.
[3 marks]
[3 marks]
[3 marks]
Total 20 marks
END OF TEST
02134032/CAPE 2010
TEST CODE
FORM TP 2010230
CARIBBEAN
02234020
MAY/JUNE 2010
E XAM I NAT I O N S
COUNCIL
ADVANCED PROFICIENCY EXAMINATION
PURE MATHEMATICS
UNIT 2 – PAPER 02
ANALYSIS, MATRICES AND COMPLEX NUMBERS
2 ½ hours
26 MAY 2010 (p.m.)
This examination paper consists of THREE sections: Module 1, Module 2 and Module 3.
Each section consists of 2 questions.
The maximum mark for each Module is 50.
The maximum mark for this examination is 150.
This examination consists of 6 printed pages.
INSTRUCTIONS TO CANDIDATES
1.
DO NOT open this examination paper until instructed to do so.
2.
Answer ALL questions from the THREE sections.
3.
Write your solutions, with full working, in the answer booklet provided.
4.
Unless otherwise stated in the question, any numerical answer that is not
exact MUST be written correct to three significant figures.
Examination Materials Permitted
Graph paper (provided)
Mathematical formulae and tables (provided) – Revised 2009
Mathematical instruments
Silent, non-programmable, electronic calculator
Copyright © 2009 Caribbean Examinations Council ®.
All rights reserved.
02234020/CAPE 2010
-2SECTION A (Module 1)
Answer BOTH questions.
1.
(a)
The temperature of water, x° C, in an insulated tank at time, t hours, may be modelled by
the equation x = 65 + 8e–0.02t. Determine the
(i)
initial temperature of the water in the tank
[2 marks]
(ii)
temperature at which the water in the tank will eventually stabilize
[2 marks]
(iii)
(b)
(i)
time when the temperature of the water in the tank is 70° C.
Given that y = etan
[4 marks]
1
1 π, show that
, where – — π < tan–1 (2x) < —
2
2
–1(2x)
dy
(1 + 4x2) —– = 2y.
dx
(ii)
(c)
[4 marks]
d 2y
Hence, show that (1 + 4x2)2 —–2 = 4y (1 – 4x).
dx
Determine
[4 marks]
4
dx
∫ ———
e +1
x
(i)
by using the substitution u = ex
(ii)
4
by first multiplying both the numerator and denominator of the integrand ———
ex + 1
[6 marks]
by e–x before integrating.
[3 marks]
Total 25 marks
2.
(a)
(i)
d
Given that n is a positive integer, find —– [x (ln x)n].
dx
(ii)
Hence, or otherwise, derive the reduction formula In = x (ln x)n – nIn – 1, where
In =
(iii)
∫ (ln x)
n
[4 marks]
dx.
[4 marks]
∫
Use the reduction formula in (a) (ii) above to determine (ln x)3 dx.
[6 marks]
GO ON TO THE NEXT PAGE
02234020/CAPE 2010
-3(b)
The amount of salt, y kg, that dissolves in a tank of water at time t minutes satisfies the
2y
dy
differential equation —– + ——— = 3.
t + 10
dt
(i)
Using a suitable integrating factor, show that the general solution of this differential
c
equation is y = t + 10 + ————,
where c is an arbitrary constant.
(t + 10)2
(ii)
[7 marks]
Given that the tank initially contains 5 kg of salt in the liquid, calculate the amount
of salt that dissolves in the tank of water at t = 15.
[4 marks]
Total 25 marks
SECTION B (Module 2)
Answer BOTH questions.
3.
(a)
The first four terms of a sequence are
2 x 3,
(b)
(c)
5 x 5,
8 x 7,
11 x 9.
(i)
Express, in terms of r, the r th term of the sequence.
(ii)
If Sn denotes the series formed by summing the first n terms of the sequence, find
Sn in terms of n.
[5 marks]
[2 marks]
The 9th term of an A.P. is three times the 3rd term and the sum of the first 10 terms is 110.
Find the first term a and the common difference d.
[6 marks]
(i)
Use the binominal theorem to expand (1 + 2x)½ as far as the term in x3, stating the
values of x for which the expansion is valid.
[5 marks]
(ii)
x
1
Prove that ——————— = — (1 + x – √ 1 + 2x) for x > 0.
x
1 + x + √ 1 + 2x
(iii)
Hence, or otherwise, show that, if x is small so that the term in x3 and higher powers
of x can be neglected, the expansion in (c) (ii) above is approximately equal to
1
— x (1 – x).
2
[4 marks]
[3 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
02234020/CAPE 2010
-44.
(a)
(i)
By expressing nCr and nCr – 1 in terms of factorials, prove that nCr + nCr – 1 = n + 1Cr.
[6 marks]
(ii)
a)
1
Given that r is a positive integer and f(r) = —, show that
r!
r
f(r) – f(r + 1) = ———
(r + 1)!
b)
Hence, or otherwise, find the sum
n
Sn = Σ
r=1
c)
(b)
[3 marks]
r
——— .
(r + 1)!
[5 marks]
Deduce the sum to infinity of Sn in (ii) b) above.
[2 marks]
(i)
Show that the function f(x) = x3 – 6x + 4 has a root x in the closed interval [0, 1].
[5 marks]
(ii)
By taking 0.6 as a first approximation of x1 in the interval [0, 1], use the NewtonRaphson method to obtain a second approximation x2 in the interval [0, 1].
[4 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
02234020/CAPE 2010
-5SECTION C (Module 3)
Answer BOTH questions.
5.
(a)
(b)
(c)
Calculate
(i)
the number of different permutations of the 8 letters of the word SYLLABUS
[3 marks]
(ii)
the number of different selections of 5 letters which can be made from the letters
of the word SYLLABUS.
[5 marks]
The events A and B are such that P(A) = 0.4, P(B) = 0.45 and P(A
B).
B) = 0.68.
(i)
Find P(A
(ii)
Stating a reason in each case, determine whether or not the events A and B are
[3 marks]
a)
mutually exclusive
[3 marks]
b)
independent.
[3 marks]
(i)
i–1
Express the complex number (2 + 3i) + —— in the form a + ib, where a and b are
i+1
both real numbers.
[4 marks]
(ii)
Given that 1 – i is the root of the equation z3 + z2 – 4z + 6 = 0, find the remaining
roots.
[4 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
02234020/CAPE 2010
-66.
(a)
A system of equations is given by
x+y+z=0
2x + y – z = –1
x + 2y + 4z = k
where k is a real number.
(i)
Write the augmented matrix of the system.
[2 marks]
(ii)
Reduce the augmented matrix to echelon form.
[3 marks]
(iii)
Deduce the value of k for which the system is consistent.
[2 marks]
(iv)
Find ALL solutions corresponding to the value of k obtained in (iii) above.
[4 marks]
(b)
Given A =
(i)
0 –1
–1
0
1
1
1
1
1
.
Find
a)
A2
[4 marks]
b)
B = 3I + A – A2
[4 marks]
(ii)
Calculate AB.
[4 marks]
(iii)
Deduce the inverse, A–1, of the matrix A.
[2 marks]
Total 25 marks
END OF TEST
02234020/CAPE 2010
TEST CODE
FORM TP 2010231
CARIBBEAN
02234032
MAY/JUNE 2010
E XAM I NAT I O N S
COUNCIL
ADVANCED PROFICIENCY EXAMINATION
PURE MATHEMATICS
UNIT 2 – PAPER 03/B
ANALYSIS, MATRICES AND COMPLEX NUMBERS
1 ½ hours
02 JUNE 2010 (a.m.)
This examination paper consists of THREE sections: Module 1, Module 2 and Module 3.
Each section consists of 1 question.
The maximum mark for each Module is 20.
The maximum mark for this examination is 60.
This examination consists of 4 printed pages.
INSTRUCTIONS TO CANDIDATES
1.
DO NOT open this examination paper until instructed to do so.
2.
Answer ALL questions from the THREE sections.
3.
Write your solutions, with full working, in the answer booklet provided.
4.
Unless otherwise stated in the question, any numerical answer that is not
exact MUST be written correct to three significant figures.
Examination Materials Permitted
Graph paper (provided)
Mathematical formulae and tables (provided) – Revised 2009
Mathematical instruments
Silent, non-programmable, electronic calculator
Copyright © 2009 Caribbean Examinations Council ®.
All rights reserved.
02234032/CAPE 2010
-2SECTION A (Module 1)
Answer this questions.
1.
(a)
Express in partial fractions
1 – x2 .
–––––––—
x (x2 + 1)
(b)
[7 marks]
The rate of change of a population of bugs is modelled by the differential equation
dy
—– – ky = 0, where y is the size of the population at time, t, given in days, and k is the
dt
constant. Initially, the population is y0 and it doubles in size in 3 days.
(i)
(ii)
Show that
a)
y = y0 ekt
[7 marks]
b)
1
k = — ln 2.
3
[3 marks]
Find the proportional increase in population at the end of the second day.
[3 marks]
Total 20 marks
GO ON TO THE NEXT PAGE
02234032/CAPE 2010
-3SECTION B (Module 2)
Answer this questions.
2.
(a)
The sum to infinity of a convergent geometric series is equal to six times the first term.
Find the common ratio of the series.
[5 marks]
(b)
Find the sum to infinity of the series Σ ar whose r th term ar is
∞
r=1
2r + 1
——— .
r!
(c)
[8 marks]
1
A truck bought for $15 000 depreciates at the rate of 12 — % each year. Calculate the
2
value of the truck
(i)
after 1 year
[2 marks]
(ii)
after t years
[2 marks]
(iii)
when its value FIRST falls below $5 000.
[3 marks]
Total 20 marks
GO ON TO THE NEXT PAGE
02234032/CAPE 2010
-4SECTION C (Module 3)
Answer this questions.
3.
(a)
Find the number of integers between 300 and 1 000 which can be formed by using the
digits 1, 3, 5, 7 and 9
(i)
if NO digit can be repeated
[3 marks]
(ii)
if ANY digit can be repeated.
[2 marks]
(b)
Find the probability that a number in (a) (ii) above ends with the digit 9.
(c)
A farmer made three separate visits to the chicken farm to purchase chickens. On each
visit he paid $ x for each grade A chicken, $ y for each grade B chicken and $ z for each
grade C. His calculations are summarised in the table below.
Number of Chickens Bought
[3 marks]
Number
of
Visits
Grade A
Grade B
Grade C
Total
Spent
$
1
20
40
60
1 120
2
40
60
80
1 720
3
60
80
120
2 480
(i)
Use the information above to form a system of linear equations in x, y and z.
[3 marks]
(ii)
Express the system of equations in the form Ax = b.
[2 marks]
(iii)
Solve the equations to find x, y and z.
[7 marks]
Total 20 marks
END OF TEST
02234032/CAPE 2010
TEST CODE
FORM TP 2011231
CARIBBEAN
02134020
MAY/JUNE 2011
E XAM I NAT I O N S
COUNCIL
ADVANCED PROFICIENCY EXAMINATION
PURE MATHEMATICS
UNIT 1 – PAPER 02
ALGEBRA, GEOMETRY AND CALCULUS
2 ½ hours
10 MAY 2011 (p.m.)
This examination paper consists of THREE sections: Module 1, Module 2 and Module 3.
Each section consists of 2 questions.
The maximum mark for each Module is 50.
The maximum mark for this examination is 150.
This examination consists of 7 printed pages.
INSTRUCTIONS TO CANDIDATES
1.
DO NOT open this examination paper until instructed to do so.
2.
Answer ALL questions from the THREE sections.
3.
Write your solutions, with full working, in the answer booklet provided.
4.
Unless otherwise stated in the question, any numerical answer that is not
exact MUST be written correct to three significant figures.
Examination Materials Permitted
Graph paper (provided)
Mathematical formulae and tables (provided) – Revised 2010
Mathematical instruments
Silent, non-programmable, electronic calculator
Copyright © 2010 Caribbean Examinations Council
All rights reserved.
02134020/CAPE 2011
-2SECTION A (Module 1)
Answer BOTH questions.
1.
(a)
Without using calculators, find the exact value of
(i)
(ii)
(b)
2
(√75 + √12 ) – (√75 – √12 )
27
–14
x 9
–38
2
[3 marks]
–18
x 81 .
[3 marks]
The diagram below, not drawn to scale, represents a segment of the graph of the function
f(x) = x3 + mx2 + nx + p
where m, n and p are constants.
f(x)
(0,4)
Q
0
1
x
2
Find
(c)
(i)
the value of p
[2 marks]
(ii)
the values of m and n
[4 marks]
(iii)
the x-coordinate of the point Q.
[2 marks]
(i)
By substituting y = log2x, or otherwise, solve, for x, the equation
√ log2x = log2 √ x .
(ii)
[6 marks]
Solve, for real values of x , the inequality
x2 – | x | – 12 < 0.
[5 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
02134020/CAPE 2011
-3-
2.
(a)
The quadratic equation x2 – px + 24 = 0, p ∈ R, has roots a and b.
(i)
(ii)
(b)
Express in terms of p
a)
a + b
[1 mark ]
b)
a2 + b2.
[4 marks]
Given that a2 + b2 = 33, find the possible values of p.
[3 marks]
The function f(x) has the property that
f(2x + 3) = 2f(x) + 3, x ∈ R.
If f(0) = 6, find the value of
(i)
f(3)
[4 marks]
(ii)
f(9)
[2 marks]
(iii)
f(–3).
[3 marks]
(c)
Prove that the product of any two consecutive integers k and k + 1 is an even integer.
[2 marks]
(d)
Prove, by mathematical induction, that n (n2 + 5) is divisible by 6 for all positive
integers n.
[6 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
02134020/CAPE 2011
-4SECTION B (Module 2)
Answer BOTH questions.
3.
(a)
(b)
(i)
Let a = a1i + a2j and b = b1i + b2j with | a | = 13 and | b | = 10. Find the value of
(a + b) . (a – b).
[5 marks]
(ii)
If 2b – a = 11i, determine the possible values of a and b.
[5 marks]
The line L has equation x – y + 1 = 0 and the circle C has equation x2 + y2 – 2y – 15 = 0.
(i)
Show that L passes through the centre of C.
[2 marks]
(ii)
If L intersects C at P and Q, determine the coordinates of P and Q.
[3 marks]
(iii)
Find the constants a, b and c such that x = b + a cos θ and y = c + a sin θ are
parametric equations (in parameter θ) of C.
[3 marks]
(iv)
Another circle C2, with the same radius as C, touches L at the centre of C. Find
the possible equations of C2.
[7 marks]
Total 25 marks
4.
(a)
By using x = cos2θ, or otherwise, find all values of the angle θ such that
8 cos4 θ – 10 cos2 θ + 3 = 0, 0 < θ < p.
(b)
[6 marks]
The diagram below, not drawn to scale, shows a rectangle PQRS with sides 6 cm and
8 cm inscribed in another rectangle ABCD.
A
Q
B
6 cm
P
R
8 cm
θ
D
S
C
GO ON TO THE NEXT PAGE
02134020/CAPE 2011
-5-
(c)
(i)
The angle that SR makes with DC is θ. Find, in terms of θ, the length of the side
[2 marks]
BC.
(ii)
Find the value of θ if | BC | = 7 cm.
[5 marks]
(iii)
Is 15 a possible value for | BC |? Give a reason for your answer.
[2 marks]
1 – cos 2 θ
Show that ————
= tan θ.
[3 marks]
(i)
(ii)
(iii)
sin 2 θ
Hence, show that
a)
1 – cos 4 θ
————
= tan 2 θ.
[3 marks]
b)
1 – cos 6 θ
————
= tan 3 θ.
[2 marks]
sin 4 θ
sin 6 θ
Using the results in (c) (i) and (ii) above, evaluate
n
S (tan r θ sin 2r θ + cos 2r θ)
r=1
where n is a positive integer.
[2 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
02134020/CAPE 2011
-6SECTION C (Module 3)
Answer BOTH questions.
5.
lim
x → –2
x2 + 5x + 6
—————
.
x2 – x – 6
(a)
Find
(b)
The function f on R is defined by
f(x) =
[4 marks]
x2 + 1 if x > 2
bx + 1 if x < 2.
Determine
(c)
(i)
f(2)
[2 marks]
(ii)
lim
f(x)
x → 2+
[2 marks]
(iii)
lim f(x) in terms of the constant b
x → 2–
[2 marks]
(iv)
the value of b such that f is continuous at x = 2.
[4 marks]
The curve y = px3 + qx2 + 3x + 2 passes through the point T (1, 2) and its gradient at T
is 7. The line x = 1 cuts the x-axis at M, and the normal to the curve at T cuts the x-axis
at N.
Find
(i)
the values of the constants p and q
[6 marks]
(ii)
the equation of the normal to the curve at T
[3 marks]
(iii)
the length of MN.
[2 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
02134020/CAPE 2011
-
6
.
(a)
7
-
The diagram below, not drawn to scale, is a sketch o f the section o f the function
f x ) = x (x2 - 12) which passes through the origin O. A and B are stationary points on the
curve.
y
A
f(x) = x(x? - 1 1 )
-
o
X
B
Find
(i)
(b)
the coordinates o f each o f the stationary points A and B
[8 marks]
(ii)
the equation o f the normal to the curve f x ) = x (x2 - 12) at the origin, O
[2 marks]
(iii)
the area between the curve and the positive x-axis.
(i)
Use the result
a
f
I* a
f(x) dx = I" f (a - x) dx, a > 0,
^0
0
rn
fn
to show that I x sin x dx = I (n - x) sin x dx.
0
0
(ii)
[6 marks]
[2 marks]
Hence, show that
a)
b)
•n
I
x sin x dx = n f n sin x dx - f n x sin x dx
n
0
0
[2 marks]
0
x sin x dx = n.
[5 marks]
Total 25 marks
END OF TEST
02134020/CAPE 2011
TEST CODE
FORM TP 2011232
CARIBBEAN
02134032
MAY/JUNE 2011
E XAM I NAT I O N S
COUNCIL
ADVANCED PROFICIENCY EXAMINATION
PURE MATHEMATICS
UNIT 1 – PAPER 03/B
ALGEBRA, GEOMETRY AND CALCULUS
1 ½ hours
08 JUNE 2011 (p.m.)
This examination paper consists of THREE sections: Module 1, Module 2 and Module 3.
Each section consists of 1 question.
The maximum mark for each Module is 20.
The maximum mark for this examination is 60.
This examination consists of 3 printed pages.
INSTRUCTIONS TO CANDIDATES
1.
DO NOT open this examination paper until instructed to do so.
2.
Answer ALL questions from the THREE sections.
3.
Write your solutions, with full working, in the answer booklet provided.
4.
Unless otherwise stated in the question, any numerical answer that is not
exact MUST be written correct to three significant figures.
Examination Materials Permitted
Graph paper (provided)
Mathematical formulae and tables (provided) – Revised 2010
Mathematical instruments
Silent, non-programmable, electronic calculator
Copyright © 2010 Caribbean Examinations Council
All rights reserved.
02134032/CAPE 2011
-2SECTION A (Module 1)
Answer this question.
1.
(a)
Solve, for x, the equation
2x + 22–x = 5.
(b)
[5 marks]
The functions f and g are defined on R by
f: x → 3x + 5 and g: x → x – 7.
(c)
(i)
Show that f is one-to-one.
[3 marks]
(ii)
Solve, for x, the equation f (g (2x + 1)) = f (3x – 2).
[4 marks]
A car manufacturer finds that when x million dollars are spent on research, the profit, P(x),
in millions of dollars, is given by
P(x) = 15 + 10 log4 (x + 4).
(i)
What is the expected profit if 12 million dollars are spent on research?
[3 marks]
(ii)
How much money should be spent on research to make a profit of 30 million
dollars?
[5 marks]
Total 20 marks
SECTION B (Module 2)
Answer this question.
2.
(a)
L1 and L2 are lines with equations 2x – y = 5 and x – 2y = 1, respectively.
C is a circle with equation x2 + y2 – 12x + 6y + 20 = 0.
(i)
Show that L1 and L2 intersect at a point P on C.
[3 marks]
(ii)
Find the point Q, other than P, at which the line L1 intersects C.
[4 marks]
(iii)
Find the equation of the tangent to C at P.
[4 marks]
GO ON TO THE NEXT PAGE
02134032/CAPE 2011
-3(b)
(i)
Show that sin 3A = 3 sin A – 4 sin3A.
(ii)
Given the vectors u = 2 sin θi + cos 2θj and υ = cos2 θi + sin θj, 0 < θ < p, find
the values of θ for which u and υ are perpendicular.
[4 marks]
[5 marks]
Total 20 marks
SECTION C (Module 3)
Answer this question.
3.
(a)
Find
(b)
(i)
lim
x3 – 4x
——— .
x→2
x–2
[4 marks]
Differentiate, with respect to x,
x
——— .
3x + 4
(ii)
[4 marks]
Hence, or otherwise, find
16
dx.
∫ ————
(3x + 4)
[3 marks]
2
(c)
A packaging company wishes to make a closed cylindrical container of thin material to
hold a volume, V, of 10 cm3. The outside surface of the container is S cm2, the radius is
r cm and the height is h cm.
(i)
20
Show that S = 2p r2 + —– .
r
(ii)
Hence, find the exact value of r for which S has a MINIMUM value.
[V = p r2 h, S = 2p r2 + 2p rh]
[3 marks]
[6 marks]
Total 20 marks
END OF TEST
02134032/CAPE 2011
TEST CODE
FORM TP 2011234
CARIBBEAN
02234020
MAY/JUNE 2011
E XAM I NAT I O N S
COUNCIL
ADVANCED PROFICIENCY EXAMINATION
PURE MATHEMATICS
UNIT 2 – PAPER 02
ANALYSIS, MATRICES AND COMPLEX NUMBERS
2 ½ hours
25 MAY 2011 (p.m.)
This examination paper consists of THREE sections: Module 1, Module 2 and Module 3.
Each section consists of 2 questions.
The maximum mark for each Module is 50.
The maximum mark for this examination is 150.
This examination consists of 7 printed pages.
INSTRUCTIONS TO CANDIDATES
1.
DO NOT open this examination paper until instructed to do so.
2.
Answer ALL questions from the THREE sections.
3.
Write your solutions, with full working, in the answer booklet provided.
4.
Unless otherwise stated in the question, any numerical answer that is not
exact MUST be written correct to three significant figures.
Examination Materials Permitted
Graph paper (provided)
Mathematical formulae and tables (provided) – Revised 2010
Mathematical instruments
Silent, non-programmable, electronic calculator
Copyright © 2010 Caribbean Examinations Council
All rights reserved.
02234020/CAPE 2011
-2SECTION A (Module 1)
Answer BOTH questions.
1.
(a)
(b)
dy
Find —– if
dx
(i)
x2 + y2 – 2x + 2y – 14 = 0
[3 marks]
(ii)
y = ecos x
[3 marks]
(iii)
y = cos2 6x + sin2 8x.
[3 marks]
1
Let y = x sin — , x ≠ 0.
x
Show that
(c)
(i)
1
dy
x —– = y – cos (—)
x
dx
[3 marks]
(ii)
d 2y
x4 —–2 + y = 0.
dx
[3 marks]
1 .
A curve is given by the parametric equations x = √ t , y – t = ——
√t
(i)
Find the gradient of the tangent to the curve at the point where t = 4.
[7 marks]
(ii)
Find the equation of the tangent to the curve at the point where t = 4.
[3 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
02234020/CAPE 2011
-3-
2.
(a)
1
Let Fn(x) = —
n!
∫
x
tn e–t dt .
0
(i)
Find F0(x) and Fn(0), given that 0! = 1.
[3 marks]
(ii)
1
Show that Fn (x) = Fn – 1 (x) – — xn e–x.
n!
[7 marks]
(iii)
Hence, show that if M is an integer greater than 1, then
x2 x3
xM ) + (ex – 1).
ex FM (x) = – ( x + — + — + ... + —–
2! 3!
M!
(b)
(i)
2x2 + 3
Express ———
in partial fractions.
(x2 + 1)2
(ii)
Hence, find
∫
2x2 + 3
———
dx.
(x2 + 1)2
[4 marks]
[5 marks]
[6 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
02234020/CAPE 2011
-4SECTION B (Module 2)
Answer BOTH questions.
3.
(a)
(b)
1
1
The sequence of positive terms, {xn}, is defined by xn + 1 = x2n + — , x1 < — , n > 1.
4
2
(i)
1 for all positive integers n.
Show, by mathematical induction, that xn < —
2
[5 marks]
(ii)
By considering xn + 1 – xn , show that xn < xn + 1.
(i)
Find the constants A and B such that
[3 marks]
A
2 – 3x
B
—————— ≡ —— + ——– .
1
–
x
(1 – x) (1 – 2x)
1 – 2x
[3 marks]
(ii)
Obtain the first FOUR non-zero terms of the expansion of each of (1 – x)–1 and
(1 – 2x)–1 as power series of x in ascending order.
[4 marks]
(iii)
Find
a)
the range of values of x for which the series expansion of
2 – 3x
——————
(1 – x) (1 – 2x)
b)
(iv)
is valid
[2 marks]
the coefficient of xn in (iii) a) above.
[2 marks]
The sum, Sn, of the first n terms of a series is given by
Sn = n (3n – 4).
Show that the series is an Arithmetic Progression (A.P.) with common
difference 6.
[6 marks]
Total 25 marks
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02234020/CAPE 2011
-54.
(a)
A Geometric Progression (G.P.) with first term a and common ratio r, 0 < r < 1, is such
that the sum of the first three terms is 26
— and their product is 8.
3
(b)
(i)
1
13
Show that r + 1 + — = —– .
r
3
(ii)
Hence, find
[4 marks]
a)
the value of r
[4 marks]
b)
the value of a
[1 mark ]
c)
the sum to infinity of the G.P.
[2 marks]
Expand
2
———
, |x|<1
ex + e–x
in ascending powers of x as far as the term in x4.
(c)
[5 marks]
1
Let f(r) = ———– , r ∈ N.
r (r + 1)
(i)
Express f(r) – f (r + 1) in terms of r.
(ii)
Hence, or otherwise, find
n
Sn =
(iii)
S
r=1
[3 marks]
3
——————
.
r (r + 1) (r + 2)
[4 marks]
Deduce the sum to infinity of the series in (c) (ii) above.
[2 marks]
Total 25 marks
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02234020/CAPE 2011
-6SECTION C (Module 3)
Answer BOTH questions.
5.
(a)
n
r
is defined as the number of ways of selecting r distinct objects from a given set of
n distinct objects. From the definition, show that
(i)
n
r
(ii)
n+1
r
(iii)
=
n
n–r
=
n
r
[2 marks]
n
r–1
+
[4 marks]
Hence, prove that
8
6
+
8
5
x
8
3
+
8
2
is a perfect square.
(b)
(c)
[3 marks]
(i)
Find the number of 5-digit numbers greater than 30 000 which can be formed with
the digits, 1, 3, 5, 6 and 8, if no digit is repeated.
[3 marks]
(ii)
What is the probability of one of the numbers chosen in (b) (i) being even?
[5 marks]
(i)
a)
Show that (1 – i) is one of the square roots of –2i.
[2 marks]
b)
Find the other square root.
[1 mark ]
(ii)
Hence, find the roots of the quadratic equation
z2 – (3 + 5i) z + (8i – 4) = 0.
[5 marks]
Total 25 marks
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02234020/CAPE 2011
-76.
(a)
The matrix A =
1
2
–1
1
–3
–3
1
2
–2
.
(i)
Show that | A | = 5.
[3 marks]
(ii)
Matrix A is changed to form new matrices B, C and D. Write down the determinant
of EACH of the new matrices, giving a reason for your answer in EACH case.
a)
Matrix B is formed by interchanging row 1 and row 2 of matrix A and then
interchanging column 1 and column 2 of the resulting matrix. [2 marks]
b)
Row 1 of matrix C is formed by adding row 2 to row 1 of matrix A. The
other rows remain unchanged.
[2 marks]
c)
Matrix D is formed by multiplying each element of matrix A by 5.
[2 marks]
(b)
Given the matrix M =
12
2
–9
–1
–1
2
5
0
–5
,
Find
(c)
(i)
AM
[3 marks]
(ii)
the inverse, A–1 , of A.
[2 marks]
(i)
Write the system of equations
x + y + z = 5
2x – 3y + 2z = –10
–x – 3y – 2z = –11
in the form Ax = b.
[1 mark ]
(ii)
Show that x = A–1b.
[2 marks]
(iii)
Hence, solve the system of equations.
[2 marks]
(iv)
a)
Show that (x, y, z) = (1, 1, 1) is a solution of the following system of equations:
x + y + z = 3
2x + 2y + 2z = 6
3x + 3y + 3z = 9
b)
Hence, find the general solution of the system.
[1 mark ]
[5 marks]
Total 25 marks
END OF TEST
02234020/CAPE 2011
TEST CODE
FORM TP 2011235
CARIBBEAN
02234032
MAY/JUNE 2011
E XAM I NAT I O N S
COUNCIL
ADVANCED PROFICIENCY EXAMINATION
PURE MATHEMATICS
UNIT 2 – PAPER 03/B
ANALYSIS, MATRICES AND COMPLEX NUMBERS
1 ½ hours
01 JUNE 2011 (a.m.)
This examination paper consists of THREE sections: Module 1, Module 2 and Module 3.
Each section consists of 1 question.
The maximum mark for each Module is 20.
The maximum mark for this examination is 60.
This examination consists of 3 printed pages.
INSTRUCTIONS TO CANDIDATES
1.
DO NOT open this examination paper until instructed to do so.
2.
Answer ALL questions from the THREE sections.
3.
Write your solutions, with full working, in the answer booklet provided.
4.
Unless otherwise stated in the question, any numerical answer that is not
exact MUST be written correct to three significant figures.
Examination Materials Permitted
Graph paper (provided)
Mathematical formulae and tables (provided) – Revised 2010
Mathematical instruments
Silent, non-programmable, electronic calculator
Copyright © 2010 Caribbean Examinations Council
All rights reserved.
02234032/CAPE 2011
-2SECTION A (Module 1)
Answer this question.
1.
(a)
A target is moving along a curve whose parametric equations are
x = 4 – 3 cos t, y = 5 + 2 sin t,
where t is the time. The distances are measured in metres.
Let θ be the angle which the tangent to the curve makes with the positive x-axis.
(b)
(i)
2π
Find the rate at which θ is increasing or decreasing when t = —– seconds.
3
[7 marks]
(ii)
What are the units of the rate of increase?
[1 mark ]
(iii)
Find the Cartesian equation of the curve.
[2 marks]
Find the general solution of the differential equation
dy
d2y
—–2 – 3 —– – 4y = 8x2.
dx
dx
[10 marks]
Total 20 marks
SECTION B (Module 2)
Answer this question.
2.
(a)
(i)
Show that the equation x2 + 8x – 8 = 0 has a root, α, in the interval [0, 1].
[3 marks]
(ii)
By taking x 0 = 0 as the first approximation for α and using the formula
8–x
8
2
n
three times, find a better approximation for α.
xn + 1 = ———
(b)
(i)
(ii)
[3 marks]
Write down the first FOUR non-zero terms of the expansions of ln (1 – x) and e–x
in ascending powers of x, stating for EACH expansion the range of values of x for
which it is valid.
[3 marks]
x2
x3
If –1 < x < 1 and y = x + — + — + ..., prove that
2
3
x = 1 – e–y.
[2 marks]
GO ON TO THE NEXT PAGE
02234032/CAPE 2011
-3(c)
In a model for the growth of a population, pn is the number of individuals in the population
at the end of n years. Initially, the population consists of 1000 individuals. In EACH
calendar year (January to December), the population increases by 20% and on 31 December,
100 individuals leave the population.
(i)
Calculate the values of p1 and p2.
[2 marks]
(ii)
Obtain an equation connecting pn + 1 and pn.
[1 mark ]
(iii)
Show that pn = 500(1.2)n + 500.
[6 marks]
Total 20 marks
SECTION C (Module 3)
Answer this question.
3.
(a)
Let A =
5 –6 –6
–1
4
2
3 –6 –4
(i)
Show that A2 – 3A + 2I = 0.
[6 marks]
(ii)
1
Deduce that A–1 = — (3I – A).
2
[4 marks]
(iii)
Hence, find the solution of the system of equations
5x – 6y – 6z = 10
–x + 4y + 2z = – 4
3x – 6y – 4z = 8.
[3 marks]
(b)
1
2+i
If z = ——, find the real and imaginary parts of z + —.
z
1–i
(c)
1
If z + — is written in the form r (cos θ + i sin θ) where r is the real and positive, find r
z
and tan θ.
[4 marks]
[3 marks]
Total 20 marks
END OF TEST
02234032/CAPE 2011
TEST CODE
FORM TP 2012231
CARIBBEAN
02134020
MAY/JUNE 2012
E XAM I NAT I O N S
COUNCIL
ADVANCED PROFICIENCY EXAMINATION
PURE MATHEMATICS
UNIT 1 – Paper 02
ALGEBRA, GEOMETRY AND CALCULUS
2 hours 30 minutes
10 MAY 2012 (p.m.)
This examination paper consists of THREE sections: Module 1, Module 2 and Module 3.
Each section consists of 2 questions.
The maximum mark for each Module is 50.
The maximum mark for this examination is 150.
This examination consists of 6 printed pages.
READ THE FOLLOWING INSTRUCTIONS CAREFULLY.
1.
DO NOT open this examination paper until instructed to do so.
2.
Answer ALL questions from the THREE sections.
3.
Write your solutions, with full working, in the answer booklet provided.
4.
Unless otherwise stated in the question, any numerical answer that is not
exact MUST be written correct to three significant figures.
Examination Materials Permitted
Graph paper (provided)
Mathematical formulae and tables (provided) – Revised 2012
Mathematical instruments
Silent, non-programmable, electronic calculator
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO.
Copyright © 2011 Caribbean Examinations Council
All rights reserved.
02134020/CAPE 2012
-2SECTION A (Module 1)
Answer BOTH questions.
1.
(a)
The expression f (x) = 2x3 – px2 + qx – 10 is divisible by x – 1 and has a remainder –6 when
divided by x + 1.
Find
(b)
(i)
the values of the constants p and q
[7 marks]
(ii)
the factors of f (x).
[3 marks]
Find positive integers x and y such that
2
( √ x + √ y ) = 16 + √ 240 .
(c)
(i)
[8 marks]
Solve, for real values of x, the inequality
| 3x – 7 | < 5.
(ii)
[5 marks]
Show that no real solution, x, exists for the inequality | 3x – 7 | + 5 < 0.
[2 marks]
Total 25 marks
2.
(a)
The function f on R is defined by
f: x → x2 – 3.
(i)
Find, in terms of x, f (f (x)).
(ii)
Determine the values of x for which
[3 marks]
f (f (x)) = f (x + 3).
(b)
[6 marks]
The roots of the equation 4x2 – 3x + 1 = 0 are α and β.
Without solving the equation
(i)
write down the values of α + β and αβ
[2 marks]
(ii)
find the value of α2 + β2
[2 marks]
(iii)
2 and —
2 .
obtain a quadratic equation whose roots are —
2
α
β2
[5 marks]
GO ON TO THE NEXT PAGE
02134020/CAPE 2012
-3(c)
Without the use of calculators or tables, evaluate
(i)
(ii)
9
10
1
3
5
7
log10 ( —
) + log10 ( —
) + log10 ( —
) + log10 ( —
) + log10 ( — )
3
5
7
9
r
∑ log10 ( —–– ).
r+1
r=1
[3 marks]
99
[4 marks]
Total 25 marks
SECTION B (Module 2)
Answer BOTH questions.
3.
(a)
(i)
Given that cos (A + B) = cos A cos B – sin A sin B and cos 2θ = 2 cos2 θ – 1, prove
that
1
cos 3θ ≡ 2 cos θ [cos2 θ – sin2 θ – —].
2
(ii)
[7 marks]
Using the appropriate formula, show that
1
— [sin 6θ – sin 2θ] ≡ (2 cos2 2θ – 1) sin 2θ.
2
(iii)
(b)
�
Hence, or otherwise, solve sin 6θ – sin 2θ = 0 for 0 < θ < —.
2
Find ALL possible values of cos θ such that 2 cot2 θ + cos θ = 0.
[5 marks]
[5 marks]
[8 marks]
Total 25 marks
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02134020/CAPE 2012
-44.
(a)
(b)
(i)
Determine the Cartesian equation of the curve, C, defined by the parametric
equations y = 3 sec θ and x = 3 tan θ.
[5 marks]
(ii)
Find the points of intersection of the curve y = √ 10x with C.
[9 marks]
Let p and q be two position vectors with endpoints (–3, 4) and (–1, 6) respectively.
(i)
Express p and q in the form xi + yj.
[2 marks]
(ii)
Obtain the vector p – q.
[2 marks]
(iii)
Calculate p●q.
[2 marks]
(iv)
Let the angle between p and q be θ. Use the result of (iii) above to calculate θ in
degrees.
[5 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
02134020/CAPE 2012
-5SECTION C (Module 3)
Answer BOTH questions.
5.
(a)
(i)
x3 + 8
Find the values of x for which ———
is discontinuous.
x2 – 4
(ii)
Hence, or otherwise, find
lim
x → –2
(iii)
(b)
[3 marks]
lim
sin x
—— = 1, or otherwise, find,
x→0
x
2x3 + 4x
———— .
sin 2x
[5 marks]
The function f on R is defined by
f (x) =
(i)
(ii)
(c)
x3 + 8 .
———
x2 – 4
By using the fact that
lim
x→0
[2 marks]
x2 + 1, x > 1,
4 + px, x < 1.
Find
a)
lim
x → 1+
b)
the value of the constant p such that
f (x)
[2 marks]
lim
f (x) exists.
x→1
[4 marks]
Hence, determine the value of f (1) for f to be continuous at the point x = 1.
[1 mark ]
A chemical process in a manufacturing plant is controlled by the function
v
M = ut2 + —
t2
where u and v are constants.
35
Given that M = –1 when t = 1 and that the rate of change of M with respect to t is –— when
4
t = 2, find the values of u and v.
[8 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
02134020/CAPE 2012
-6-
6.
(a)
dy
dx
(i)
Given that y = √ 4x2 – 7, show that y —– = 4x.
(ii)
Hence, or otherwise, show that
d2y
dx
dy
dx
y —–2 + —–
(b)
[3 marks]
2
= 4.
[3 marks]
The curve, C, passes through the point (–1, 0) and its gradient at the point (x, y) is given
by
dy
dx
—– = 3x2 – 6x.
(i)
Find the equation of C.
[4 marks]
(ii)
Find the coordinates of the stationary points of C.
[3 marks]
(iii)
Determine the nature of EACH stationary point.
[3 marks]
(iv)
Find the coordinates of the points P and Q at which the curve C meets the x-axis.
[5 marks]
(v)
Hence, sketch the curve C, showing
a)
the stationary points
b)
the points P and Q.
[4 marks]
Total 25 marks
END OF TEST
IF YOU FINISH BEFORE TIME IS CALLED, CHECK YOUR WORK ON THIS TEST.
02134020/CAPE 2012
TEST CODE
FORM TP 2012232
CARIBBEAN
02134032
MAY/JUNE 2012
E XAM I NAT I O N S
COUNCIL
ADVANCED PROFICIENCY EXAMINATION
PURE MATHEMATICS
UNIT 1 – Paper 032
ALGEBRA, GEOMETRY AND CALCULUS
1 hour 30 minutes
08 JUNE 2012 (p.m.)
This examination paper consists of THREE sections: Module 1, Module 2 and Module 3.
Each section consists of 1 question.
The maximum mark for each Module is 20.
The maximum mark for this examination is 60.
This examination consists of 4 printed pages.
READ THE FOLLOWING INSTRUCTIONS CAREFULLY.
1.
DO NOT open this examination paper until instructed to do so.
2.
Answer ALL questions from the THREE sections.
3.
Write your solutions, with full working, in the answer booklet provided.
4.
Unless otherwise stated in the question, any numerical answer that is not
exact MUST be written correct to three significant figures.
Examination Materials Permitted
Graph paper (provided)
Mathematical formulae and tables (provided) – Revised 2012
Mathematical instruments
Silent, non-programmable, electronic calculator
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO.
Copyright © 2011 Caribbean Examinations Council
All rights reserved.
02134032/CAPE 2012
-2SECTION A (Module 1)
Answer this question.
1.
(a)
The roots of the cubic equation x3 – px – 48 = 0 are α, 2α and –3α.
Find
(b)
(i)
the value of α
[3 marks]
(ii)
the value of the constant p.
[4 marks]
Prove by mathematical induction that
9n – 1 is divisible by 8 for all integers n > 1.
(c)
[6 marks]
Let m and n be positive integers.
(i)
1
Prove that lognm = ——– .
logmn
(ii)
Hence, solve for x, the equation
log2 x + 2 logx2 = 3.
[3 marks]
[4 marks]
Total 20 marks
GO ON TO THE NEXT PAGE
02134032/CAPE 2012
-
3
-
SECTION B (Module 2)
Answer this question.
2.
(a)
The diagram below (not drawn to scale) shows the graph o f the circle,C, whose equation
is x2 + y2 - 6x + 2y - 15 = 0.
y
.P(7,2)
o
>x
Q'
(i)
(b)
Determine the radius and the coordinates o f the centre o f C.
[3 marks]
(ii)
Find the equation o f the tangent to the circle at the point P (7, 2).
[5 marks]
(iii)
Find the coordinates o f the point Q (Q / P) at which the diameter through P cuts
the circle.
[2 marks]
(i)
Express f(9) = 3 V3~ cos 0 - 3 sin 0 in the form R cos (0 + a) where R > 0 and 00
is acute.
[4 marks]
(ii)
(c)
Hence, obtain the maximum value off(d).
[2 marks]
The vector PQ = i - 3j is parallel to the vector OR with |OR| = V 5. Find scalars a and b
such that OR = a i + bj.
[4 marks]
Total 20 marks
GO ON TO THE NEXT PAGE
02134032/CAPE 2012
-4-
SECTION C (Module 3)
Answer this question.
3.
(a)
(i)
By expressing x – 4 as (√ x + 2) (√ x – 2), find
lim
x→4
(ii)
√x–2
——— .
x–4
[3 marks]
Hence, or otherwise, find
lim
x→4
√x–2
—————
.
x2 – 5x + 4
[5 marks]
(b)
Find the gradient of the curve y = 2x3 at the point P on the curve at which y = 16.
[3 marks]
(c)
The diagram below (not drawn to scale) represents an empty vessel in the shape of a
right circular cone of semi-vertical angle 45°. Water is poured into the vessel at the rate
of 10 cm3 per second. At time, t, seconds after the start of the pouring of water, the height
of the water in the vessel is x cm and its volume is V cm3.
(i)
Express V in terms of t only.
[1 mark ]
(ii)
Express V in terms of x only.
[2 marks]
(iii)
Find, correct to 2 decimal places, the rate at which the water level is rising after
5 seconds.
[6 marks]
Total 20 marks
1
3
[The volume of a right circular cone of height h and radius of base r is V = — p r2 h.]
END OF TEST
IF YOU FINISH BEFORE TIME IS CALLED, CHECK YOUR WORK ON THIS TEST.
02134032/CAPE 2012
TEST CODE
FORM TP 2012234
CARIBBEAN
02234020
MAY/JUNE 2012
E XAM I NAT I O N S
COUNCIL
ADVANCED PROFICIENCY EXAMINATION
PURE MATHEMATICS
UNIT 2 – Paper 02
ANALYSIS, MATRICES AND COMPLEX NUMBERS
2 hours 30 minutes
25 MAY 2012 (p.m.)
This examination paper consists of THREE sections: Module 1, Module 2 and Module 3.
Each section consists of 2 questions.
The maximum mark for each Module is 50.
The maximum mark for this examination is 150.
This examination consists of 7 printed pages.
READ THE FOLLOWING INSTRUCTIONS CAREFULLY.
1.
DO NOT open this examination paper until instructed to do so.
2.
Answer ALL questions from the THREE sections.
3.
Write your solutions, with full working, in the answer booklet provided.
4.
Unless otherwise stated in the question, any numerical answer that is not
exact MUST be written correct to three significant figures.
Examination Materials Permitted
Graph paper (provided)
Mathematical formulae and tables (provided) – Revised 2012
Mathematical instruments
Silent, non-programmable, electronic calculator
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO.
Copyright © 2011 Caribbean Examinations Council
All rights reserved.
02234020/CAPE 2012
-2SECTION A (Module 1)
Answer BOTH questions.
1.
(a)
(i)
(ii)
(b)
Given the curve y = x2 ex,
a)
dy
d2y
find —– and ——
dx
dx2
[5 marks]
b)
dy
find the x-coordinates of the points at which —– = 0
dx
[2 marks]
c)
d2y
find the x-coordinates of the points at which ——
=0
dx2
[2 marks]
Hence , determine if the coordinates identified in (i) b) and c) above are at the
[7 marks]
maxima, minima or points of inflection of y = x2 ex.
A curve is defined by the parametric equations x = sin–1 √ t , y = t2 – 2t.
Find
(i)
the gradient of a tangent to the curve at the point with parameter t
[6 marks]
(ii)
1
the equation of the tangent at the point where t = —.
2
[3 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
02234020/CAPE 2012
-32.
(a)
(i)
Express
x2 – 3x
——————
in partial fractions.
(x – 1) (x2 + 1)
(ii)
Hence, find
∫
(b)
(i)
[7 marks]
x2 – 3x
——————
dx.
3
x – x2 + x – 1
[5 marks]
Given that sin A cos B – cos A sin B = sin (A – B) show that
cos 3x sin x = sin 3x cos x – sin 2x.
(ii)
If Im =
∫ cos
Jm =
∫ cos
[2 marks]
x sin 3x dx and
m
m
x sin 2x dx,
prove that (m + 3) Im = mJm–1 – cosm x cos 3x.
(iii)
Hence, by putting m = 1, prove that
4
(iv)
[7 marks]
Evaluate
∫
∫
–�4
cos x sin 3x dx =
0
–�4
sin 2x dx.
∫
–�4
0
3
sin 2x dx + — .
2
[2 marks]
[2 marks]
0
Total 25 marks
GO ON TO THE NEXT PAGE
02234020/CAPE 2012
-4SECTION B (Module 2)
Answer BOTH questions.
(a)
(b)
For a particular G.P., u6 = 486 and u11 = 118 098, where un is the nth term.
(i)
Calculate the first term, a, and the common ratio, r.
[5 marks]
(ii)
Hence, calculate n if Sn = 177 146.
[4 marks]
The first four terms of a sequence are 1 × 3, 2 × 4, 3 × 5, 4 × 6.
(i)
Express, in terms of r, the rth term, ur, of the sequence.
(ii)
Prove, by mathematical induction, that
n
1 n (n + 1) (2n + 7),
∑ ur = —
6
r=1
(c)
n ∈ N.
A
3.
[2 marks]
[7 marks]
(i)
Use Maclaurin’s Theorem to find the first three non-zero terms in the power series
expansion of cos 2x.
[5 marks]
(ii)
Hence, or otherwise, obtain the first two non-zero terms in the power series
expansion of sin2 x.
[2 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
02234020/CAPE 2012
-5-
4.
(a)
n
in terms of factorials.
r
(i)
Express
[1 mark ]
(ii)
Hence, show that
(iii)
Find the coefficient of x in
(iv)
Using the identity (1 + x)2n = (1 + x)n (1 + x)n, show that
n
n
= n–r .
r
[3 marks]
8
4
2n
n
2
2
3
x –—
x
2
2
.
[5 marks]
2
2
= c 0 + c 1 + c 3 + . . . + c n – 1 + c n , where cr =
n
r .
[8 marks]
(b)
Let f (x) = 2x3 + 3x2 – 4x – 1 = 0.
(i)
Use the intermediate value theorem to determine whether the equation f (x) has
any roots in the interval [0.2, 2].
[2 marks]
(ii)
Using x1 = 0.6 as a first approximation of a root T of f (x), execute FOUR iterations
of the Newton–Raphson method to obtain a second approximation, x2, of T.
[6 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
02234020/CAPE 2012
-6SECTION C (Module 3)
Answer BOTH questions.
5.
(a)
(b)
(c)
How many 4-digit even numbers can be formed from the digits 1, 2, 3, 4, 6, 7, 8
(i)
if each digit appears at most once?
[4 marks]
(ii)
if there is no restriction on the number of times a digit may appear?
[3 marks]
A committee of five is to be formed from among six Jamaicans, two Tobagonians and
three Guyanese.
(i)
Find the probability that the committee consists entirely of Jamaicans.
[3 marks]
(ii)
Find the number of ways in which the committee can be formed, given the
following restriction: There are as many Tobagonians on the committee as there
are Guyanese.
[6 marks]
Let A be the matrix
1
2
1
0
1
–1
3
–1
1
.
(i)
Find the matrix B, where B = A2 – 3A – I.
[3 marks]
(ii)
Show that AB = –9I.
[1 mark ]
(iii)
Hence, find the inverse, A–1, of A.
[2 marks]
(iv)
Solve the system of linear equations
B
x
y
z
=
3
–1
2
.
[3 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
02234020/CAPE 2012
-76.
(a)
(i)
Draw the points A and B on an Argand diagram,
√2
1+i
where A = —— and B = —— .
1–i
1–i
(ii)
[6 marks]
(1 + √ 2 + i)
3� .
Hence, or otherwise, show that the argument of ———–—— is EXACTLY —–
1–i
8
[5 marks]
(b)
(i)
Find ALL complex numbers, z, such that z2 = i.
(ii)
Hence, find ALL complex roots of the equation
[3 marks]
z2 – (3 + 5i) z – (4 – 7i) = 0.
(c)
[5 marks]
Use de Moivre’s theorem to show that
cos 6 θ = cos6 θ – 15 cos4 θ sin2 θ + 15 cos2 θ sin4 θ – sin6 θ .
[6 marks]
Total 25 marks
END OF TEST
IF YOU FINISH BEFORE TIME IS CALLED, CHECK YOUR WORK ON THIS TEST.
02234020/CAPE 2012
TEST CODE
FORM TP 2012235
CARIBBEAN
02234032
MAY/JUNE 2012
E XAM I NAT I O N S
COUNCIL
ADVANCED PROFICIENCY EXAMINATION
PURE MATHEMATICS
UNIT 2 – Paper 032
ANALYSIS, MATRICES AND COMPLEX NUMBERS
1 hour 30 minutes
01 JUNE 2012 (a.m.)
This examination paper consists of THREE sections: Module 1, Module 2 and Module 3.
Each section consists of 1 question.
The maximum mark for each Module is 20.
The maximum mark for this examination is 60.
This examination consists of 4 printed pages.
READ THE FOLLOWING INSTRUCTIONS CAREFULLY.
1.
DO NOT open this examination paper until instructed to do so.
2.
Answer ALL questions from the THREE sections.
3.
Write your solutions, with full working, in the answer booklet provided.
4.
Unless otherwise stated in the question, any numerical answer that is not
exact MUST be written correct to three significant figures.
Examination Materials Permitted
Graph paper (provided)
Mathematical formulae and tables (provided) – Revised 2012
Mathematical instruments
Silent, non-programmable, electronic calculator
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO.
Copyright © 2011 Caribbean Examinations Council
All rights reserved.
02234032/CAPE 2012
-2SECTION A (Module 1)
Answer this question.
–1
1.
(a)
x (x – 1) 3
Given that y = ————
,
1 + sin3 x
dy
by taking logarithms of both sides, or otherwise, find —– in terms of x.
dx
(b)
(i)
1
Sketch the curve y = √ 1 + x3 , for values of x between – — and 1.
2
(ii)
Using the trapezium rule, with 5 intervals, find an approximation to
∫
(c)
(i)
[3 marks]
1
√ 1 + x3 dx.
[5 marks]
0
Use integration by parts to find
∫x
(ii)
[4 marks]
2
cos x dx.
[6 marks]
�
Hence, find the area under the curve y = x2 cos x, between x = 0 and x = — .
2
[2 marks]
Total 20 marks
GO ON TO THE NEXT PAGE
02234032/CAPE 2012
-3SECTION B (Module 2)
Answer this question.
2.
(a)
(b)
5
(i)
1
Write down the binomial expansion of 1 + — x
4
(ii)
Hence, calculate (1.025)5 correct to three decimal places.
.
[4 marks]
[4 marks]
Let f (x) = x2 – 5x + 3 and g(x) = ex be two functions.
(i)
Sketch the graphs for f (x) and g(x) on the same coordinate axes for the domain
–1 < x < 2.
[4 marks]
(ii)
Using x1 = 0.3 as an initial approximation to the root x of f (x) – g(x) = 0, execute
TWO iterations of the Newton-Raphson method to obtain a better approximation,
x3, of x correct to four decimal places.
[6 marks]
(iii)
Assuming that x3 is the true root of f (x) – g(x) = 0, calculate the relative error of
x 1.
[2 marks]
Total 20 marks
GO ON TO THE NEXT PAGE
02234032/CAPE 2012
-4SECTION C (Module 3)
Answer this question.
3.
(a)
A computer programmer is trying to break into a company’s code. His program generates
a list of all permutations of any set of letters that it is given, without regard for duplicates.
For example, given the letters TTA, it will generate a list of six 3-letter permutations
(words).
If the program generates a list of all 8-letter permutations of TELESTEL, without regard
for duplicates,
(b)
(i)
how many times will any given word be repeated in the list?
[5 marks]
(ii)
in how many words will the first four letters be all different?
[5 marks]
(i)
Find the inverse of the matrix
1
1
1
A =
(ii)
1
3
6
.
[5 marks]
Find a 3 × 1 matrix, Y, such that
A
(iii)
1
2
3
3
–1
2
= Y.
[2 marks]
Hence, find a 3 × 3 matrix B such that
BY =
6
–2
4
.
[3 marks]
Total 20 marks
END OF TEST
IF YOU FINISH BEFORE TIME IS CALLED, CHECK YOUR WORK ON THIS TEST.
02234032/CAPE 2012
TEST CODE
FORM TP 2013233
CARIBBEAN
02134020
MAY/JUNE 2013
E XAM I NAT I O N S
COUNCIL
CARIBBEAN ADVANCED PROFICIENCY EXAMINATION®
PURE MATHEMATICS
UNIT 1 – Paper 02
ALGEBRA, GEOMETRY AND CALCULUS
2 hours 30 minutes
14 MAY 2013 (p.m.)
This examination paper consists of THREE sections: Module 1, Module 2 and Module 3.
Each section consists of 2 questions.
The maximum mark for each Module is 50.
The maximum mark for this examination is 150.
This examination consists of 6 printed pages.
READ THE FOLLOWING INSTRUCTIONS CAREFULLY.
1.
DO NOT open this examination paper until instructed to do so.
2.
Answer ALL questions from the THREE sections.
3.
Write your solutions, with full working, in the answer booklet provided.
4.
Unless otherwise stated in the question, any numerical answer that is not
exact MUST be written correct to three significant figures.
Examination Materials Permitted
Graph paper (provided)
Mathematical formulae and tables (provided) – Revised 2012
Mathematical instruments
Silent, non-programmable, electronic calculator
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO.
Copyright © 2012 Caribbean Examinations Council
All rights reserved.
02134020/CAPE 2013
-2SECTION A (Module 1)
Answer BOTH questions.
1.
(a)
(b)
Let p and q be two propositions. Construct a truth table for the statements
(i)
p→q
(ii)
~ (p ˄ q).
[1 mark]
[2 marks]
A binary operator ⊕ is defined on a set of positive real numbers by
y ⊕ x = y2 + x2 + 2y + x – 5xy.
Solve the equation 2 ⊕ x = 0.
[5 marks]
(c)
Use mathematical induction to prove that 5n + 3 is divisible by 2 for all values of n ∈ N.
[8 marks]
(d)
Let f(x) = x3 – 9x2 + px + 16.
(i)
Given that (x + 1) is a factor of f(x), show that p = 6.
[2 marks]
(ii)
Factorise f(x) completely.
[4 marks]
(iii)
Hence, or otherwise, solve f(x) = 0.
[3 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
02134020/CAPE 2013
-32.
(a)
Let A = {x : x ∈ R, x > 1}.
A function f : A → R is defined as f(x) = x2 – x. Show that f is one to one.
(b)
Let f(x) = 3x + 2 and g(x) = e2x.
(i)
(ii)
(c)
[7 marks]
Find
a)
f –1(x) and g–1(x)
b)
f֯֯ [g(x)] (or f֯֯ ◦ g(x)).
[4 marks]
[1 mark]
Show that (f֯֯ ◦ g)–1 (x) = g–1 (x) ◦ f –1 (x).
[5 marks]
Solve the following:
(i)
3x2 + 4x +1 < 5
[4 marks]
(ii)
| x + 2 | = 3x + 5
[4 marks]
Total 25 marks
SECTION B (Module 2)
Answer BOTH questions.
3.
(a)
(b)
(i)
2 tan θ
Show that sin 2θ = ————.
1 + tan2 θ
[4 marks]
(ii)
Hence, or otherwise, solve sin 2θ – tan θ = 0 for 0 < θ < 2π.
[8 marks]
(i)
Express f (θ) = 3 cos θ – 4 sin θ in the form r cos (θ + α) where
π
r > 0 and 0° < α < —.
2
(ii)
(iii)
[4 marks]
Hence, find
a)
the maximum value of f (θ)
[2 marks]
b)
1
the minimum value of ——— .
8 + f (θ)
[2 marks]
Given that the sum of the angles A, B and C of a triangle is π radians, show that
a)
sin A = sin (B + C)
]3 marks]
b)
sin A + sin B + sin C = sin (A + B) + sin (B + C) + sin (A + C).
[2 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
02134020/CAPE 2013
-44.
(a)
(b)
A circle C is defined by the equation x2 + y2 – 6x – 4y + 4 = 0.
(i)
Show that the centre and the radius of the circle, C, are (3, 2) and 3, respectively.
[3 marks]
(ii)
a)
Find the equation of the normal to the circle C at the point (6, 2).
[3 marks]
b)
Show that the tangent to the circle at the point (6, 2) is parallel to the
y-axis.
[3 marks]
Show that the Cartesian equation of the curve that has the parametric equations
x = t2 + t, y = 2t – 4
is 4x = y2 + 10y + 24.
(c)
[4 marks]
The points A (3, –1, 2), B (1, 2, –4) and C (–1, 1, –2) are three vertices of a parallelogram
ABCD.
→
→
(i)
Express the vectors AB and BC in the form xi + yj + zk.
[3 marks]
(ii)
Show that the vector r = – 16j – 8k is perpendicular to the plane through A, B
[5 marks]
and C.
(iii)
Hence, find the Cartesian equation of the plane through A, B and C.
[4 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
02134020/CAPE 2013
-
5
-
SECTION C (Module 3)
Answer BOTH questions.
5.
(a)
X+ 2 X< 2
A functionf(x) is defined as f(x) = -j x2 ’ x > 2
[ x2,
x>2
(1)
(ii)
'
Find lim f x ) .
x^ 2
[4 marks]
Determine whether f(x) is continuous at x = 2. Give a reason for your answer.
[2 marks]
T .
x2 + 2x + 3
,
dy -4x3 - 10x2 - 14x + 4
(b)Let y = — —5— —=3—. Show that —— = --------------------------—
-—
------v7
*
(x2 + 2)3
dx
(x2 + 2)4
dx
(x + 2)
(c)
[5 marks]
The equation of an ellipse is given by
x = 1 - 3 cos 0, y = 2 sin 0, 0 < 0 < 2n.
dy
Find —jxr in terms of 0.
(d)
[5 marks]
The diagram below (not drawn to scale) shows the curve y = x2 + 3 and the line y = 4x.
y
Q
p
o
(i)
(ii)
X
Determine the coordinates of the points P and Q at which the curve and the line
intersect.
[4 marks]
Calculate the area of the shaded region.
[5 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
02134020/CAPE 2013
-
6.
(a)
(i)
(ii)
-
By using the substitution u = 1 - x, find
J x (1 - x)2 dx.
[5 marks]
Given that f t ) = 2 cos t, g(t) = 4 sin 5t + 3 cos t,
show that
(b)
6
J f(t) + g(t)] dt = J f(t) dt + J g(t) dt.
[4 marks]
A sports association is planning to construct a running track in the shape o f a rectangle
surmounted by a semicircle, as shown in the diagram below. The letter x represents the
length o f the rectangular section and r represents the radius o f the semicircle.
-
X
-
r
The perimeter o f the track must be 600 metres.
(i)
(ii)
(c)
(i)
Show that r =
600 - 2x
2 +n
[2 marks]
Hence, determine the length, x, that maximises the area enclosed by the track.
[6 marks]
Let y = -x sin x - 2 cos x + A x + B, where A and B are constants.
Show that y " = x sin x.
(ii)
[4 marks]
Hence, determine the specific solution o f the differential equation
y ” = x sin x,
given that when x = 0, y = 1 and when x = n, y = 6.
[4 marks]
Total 25 marks
END OF TEST
FINISH BEFORE TIME IS CALLED, CHECK YOUR WORK ON THIS TEST.
02134020/CAPE 2013
TEST CODE
FORM TP 2013234
CARIBBEAN
02134032
MAY/JUNE 2013
E XAM I NAT I O N S
COUNCIL
CARIBBEAN ADVANCED PROFICIENCY EXAMINATION®
PURE MATHEMATICS
UNIT 1 – Paper 032
ALGEBRA, GEOMETRY AND CALCULUS
1 hour 30 minutes
12 JUNE 2013 (p.m.)
This examination paper consists of THREE sections: Module 1, Module 2 and Module 3.
Each section consists of 1 question.
The maximum mark for each Module is 20.
The maximum mark for this examination is 60.
This examination consists of 5 printed pages.
READ THE FOLLOWING INSTRUCTIONS CAREFULLY.
1.
DO NOT open this examination paper until instructed to do so.
2.
Answer ALL questions from the THREE sections.
3.
Write your solutions, with full working, in the answer booklet provided.
4.
Unless otherwise stated in the question, any numerical answer that is not
exact MUST be written correct to three significant figures.
Examination Materials Permitted
Graph paper (provided)
Mathematical formulae and tables (provided) – Revised 2012
Mathematical instruments
Silent, non-programmable, electronic calculator
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO.
Copyright © 2012 Caribbean Examinations Council
All rights reserved.
02134032/CAPE 2013
-2SECTION A (Module 1)
Answer this question.
1.
(a)
Let p and q be two propositions.
(i)
State the converse of (p ˄ q) → (q ˅ ~ p).
(ii)
Show that the contrapositive of the inverse of (p ˄ q) → (q ˅ ~ p) is the converse
of (p ˄ q) → (q ˅ ~ p).
[3 marks]
[1 mark]
(b)
Solve the equation log2 (x + 3) = 3 – log2 (x + 2).
(c)
The amount of impurity, A, present in a chemical depends on the time it takes to purify. It
is known that A = 3e4t – 7e2t – 6 at any time t minutes. Find the time at which the chemical
is free of impurity (that is when A = 0).
[6 marks]
(d)
On the same axes, sketch the graphs of f(x) = 2x + 3 and g(x) = |2x + 3|.
Show clearly ALL intercepts that may be present.
[5 marks]
[5 marks]
Total 20 marks
GO ON TO THE NEXT PAGE
02134032/CAPE 2013
-
3
-
SECTION B (Module 2)
Answer this question.
2.
(a)
4
3
A is an acute angle and B is an obtuse angle, where sin (A) =and cos (B) =
- =- y
TO
Without finding the values of angles A and B, calculate cos (3A).
.
[5 marks]
(b)
Solve the equation 4 cos 20 - 14 sin 0 = 7 for values of 0 between 0 and 2n radians.
[8 marks]
(c)
An engineer is asked to build a table in the shape of two circles C and C2 which intersect
each other, as shown in the diagram below (not drawn to scale).
Cl
R
Q
Cl
The equations of Cx and C2 arex2+ y2 + 4x + 6y - 3 = 0 and x2+ y2 + 4x + 2y - 7 = 0
respectively.
A leg of the table is attached at EACH of the points Q and R where the circles intersect.
Determine the coordinates of the positions of the legs of the table.
[7 marks]
Total 20 marks
GO ON TO THE NEXT PAGE
02134032/CAPE 2013
-
4
-
SECTION C (Module 3)
Answer this question.
3.
(a)
The diagram below shows the graph of a function, fx ).
f(x)
-3
-2
,1
0
(i)
(ii)
1
2\
3
x
Determine for the function
li"
a)
xx ^^ o0 f x )
b)
b)
xxx ^ 22 f(x).
li"
[1 mark]
[2 marks]
State whether f is continuous
State whether
at x =f 2.
is continuous at x = 2. Justify your answer.[2marks]
GO ON TO THE NEXT PAGE
02134032/CAPE 2013
-5-
(b)
1
Differentiate f(x) = —— from first principles.
√ 2x
(c)
Find the x-coordinates of the maximum and minimum points of the curve
f(x) = 4x3 + 7x2 – 6x.
(d)
[5 marks]
[7 marks]
y2
x2
A water tank is made by rotating the curve with equation — + —– = 1 about the x-axis
25
4
between x = 0 and x = 2.
Find the volume of water that the tank can hold.
[3 marks]
Total 20 marks
END OF TEST
IF YOU FINISH BEFORE TIME IS CALLED, CHECK YOUR WORK ON THIS TEST.
02134032/CAPE 2013
TEST CODE
FORM TP 2013236
CARIBBEAN
02234020
MAY/JUNE 2013
E XAM I NAT I O N S
COUNCIL
CARIBBEAN ADVANCED PROFICIENCY EXAMINATION®
PURE MATHEMATICS
UNIT 2 – Paper 02
ANALYSIS, MATRICES AND COMPLEX NUMBERS
2 hours 30 minutes
29 MAY 2013 (p.m.)
This examination paper consists of THREE sections: Module 1, Module 2 and Module 3.
Each section consists of 2 questions.
The maximum mark for each Module is 50.
The maximum mark for this examination is 150.
This examination consists of 6 printed pages.
READ THE FOLLOWING INSTRUCTIONS CAREFULLY.
1.
DO NOT open this examination paper until instructed to do so.
2.
Answer ALL questions from the THREE sections.
3.
Write your solutions, with full working, in the answer booklet provided.
4.
Unless otherwise stated in the question, any numerical answer that is not
exact MUST be written correct to three significant figures.
Examination Materials Permitted
Graph paper (provided)
Mathematical formulae and tables (provided) – Revised 2012
Mathematical instruments
Silent, non-programmable, electronic calculator
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO.
Copyright © 2012 Caribbean Examinations Council
All rights reserved.
02234020/CAPE 2013
-2SECTION A (Module 1)
Answer BOTH questions.
1.
(a)
Determine the derivative, with respect to x, of the function ln (x2y) – sin y = 3x – 2y.
[5 marks]
(b)
Let f (x, y, z) = 3 yz2 – e4x cos 4z –3y2 – 4.
∂f / ∂y
∂z
∂z
Given that —– = – ———, determine —– in terms of x, y and z.
∂f / ∂z
∂y
∂y
(c)
Use de Moivre’s theorem to prove that
cos 5θ = 16 cos5 θ – 20 cos3 θ + 5 cos θ.
(d)
[5 marks]
[6 marks]
(i)
Write the complex number z = (–1 + i)7 in the form reiθ, where r = | z | and
θ = arg z.
[3 marks]
(ii)
Hence, prove that (–1 + i)7 = –8(1 + i).
[6 marks]
Total 25 marks
2.
(a)
(b)
∫ sin x cos 2x dx.
(i)
Determine
(ii)
Hence, calculate
Let f(x) = x | x | =
∫
–�2
[5 marks]
sin x cos 2x dx.
[2 marks]
0
x2 ; x > 0
.
–x2 ; x < 0
Use the trapezium rule with four intervals to calculate the area between f(x) and the x-axis
for the domain –0.75 < x < 2.25.
[5 marks]
(c)
(i)
2
4
2x2 + 4 = ———
Show that ———–
– ———–
.
x2 + 4
(x2 + 4)2
(x2 + 4)2
(ii)
2x2 + 4 dx. Use the substitution x = 2 tan θ.
Hence, find ———–
(x2 + 4)2
∫
[6 marks]
[7 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
02234020/CAPE 2013
-3SECTION B (Module 2)
Answer BOTH questions.
3
3.
(a)
The sequence{an} is defined by a1 = 1, an+1 = 4 + 2 √ an .
Use mathematical induction to prove that 1 < an < 8 for all n in the set of positive
integers.
[6 marks]
(b)
1
Let k > 0 and let f(k) = —.
k2
(i)
Show that
a)
2k + 1 .
f(k) – f(k + 1) = ————
2
k (k + 1)2
b)
∑
n
k=1
(iii)
1 – ———
1
—
2
k
(k + 1)2
1
= 1 – ———
.
(n + 1)2
(i)
(ii)
[5 marks]
Hence, or otherwise, prove that
∞
2k + 1 = 1.
∑ ————
2
k
(k + 1)2
k=1
(c)
[3 marks]
[3 marks]
Obtain the first four non-zero terms of the Taylor Series expansion of cos x in
π
ascending powers of (x – —).
4
[5 marks]
π
Hence, calculate an approximation to cos (—–).
16
[3 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
02234020/CAPE 2013
-44.
(a)
(i)
Obtain the binomial expansion of
4
4
√ (1 + x) + √ (1 – x)
up to the term containing x2.
(ii)
(b)
(c)
[4 marks]
1 compute an approximation of √4 17 + √4 15 to four
Hence, by letting x = —–,
16
decimal places.
[4 marks]
Show that the coefficient of the x5 term of the product (x + 2)5 (x – 2)4 is 96.
[7 marks]
(i)
Use the Intermediate Value Theorem to prove that x3 = 25 has at least one root in
the interval [2, 3].
[3 marks]
(ii)
The table below shows the results of the first four iterations in the estimation of
the root of f(x) = x3 – 25 = 0 using interval bisection.
The procedure used a = 2 and b = 3 as the starting points and pn is the estimate of
the root for the nth iteration.
n
an
bn
pn
f(pn)
1
2
3
2.5
–9.375
2
2.5
3
2.75
–4.2031
3
2.75
3
2.875
–1.2363
4
2.875
3
2.9375
0.3474
5
6
......
......
Complete the table to obtain an approximation of the root of the equation x3 = 25
correct to 2 decimal places.
[7 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
02234020/CAPE 2013
-5SECTION C (Module 3)
Answer BOTH questions.
5.
(a)
Three letters from the word BRIDGE are selected one after the other without replacement.
When a letter is selected, it is classified as either a vowel (V) or a consonant (C).
Use a tree diagram to show the possible outcomes (vowel or consonant) of the THREE
selections. Show all probabilities on the diagram.
[7 marks]
(b)
(i)
The augmented matrix for a system of three linear equations with variables x, y
and z respectively is
A=
1
–5
1
1
1
–5
–1
1
3
1
2
3
By reducing the augmented matrix to echelon form, determine whether or not the
system of linear equations is consistent.
[5 marks]
(ii)
The augmented matrix for another system is formed by replacing the THIRD row
of A in (i) above with (1 –5 5 | 3).
Determine whether the solution of the new system is unique. Give a reason for
your answer.
[5 marks]
(c)
A country, X, has three airports (A, B, C). The percentage of travellers that use each of the
airports is 45%, 30% and 25% respectively. Given that a traveller has a weapon in his/
her possession, the probability of being caught is, 0.7, 0.9 and 0.85 for airports A, B, and
C respectively.
Let the event that:
•
•
the traveller is caught be denoted by D, and
the event that airport A, B, or C is used be denoted by A, B, and C respectively.
(i)
What is the probability that a traveller using an airport in Country X is caught with
a weapon?
[5 marks]
(ii)
On a particular day, a traveller was caught carrying a weapon at an airport in
Country X. What is the probability that the traveller used airport C? [3 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
02234020/CAPE 2013
-66.
(a)
(i)
Obtain the general solution of the differential equation
dy
dx
cos x —– + y sin x = 2x cos2 x.
(ii)
(b)
[7 marks]
15 √ 2 π2
π
Hence, given that y = ———— , when x = —, determine the constant of the
32
4
integration.
[5 marks]
The general solution of the differential equation
y" + 2y' + 5y = 4 sin 2t
is y = CF + PI, where CF is the complementary function and PI is a particular integral.
(i)
a)
Calculate the roots of
λ2 + 2λ + 5 = 0, the auxiliary equation.
b)
Hence, obtain the complementary function (CF), the general solution of
y" + 2y' + 5y = 0.
(ii)
[2 marks]
[3 marks]
Given that the form of the particular integral (PI) is
up(t) = A cos 2t + B sin 2t,
16
4
Show that A = – —– and B = —–.
17
17
(iii)
[3 marks]
Given that y(0) = 0.04 and y'(0) = 0, obtain the general solution of the differential
equation.
[5 marks]
Total 25 marks
END OF TEST
IF YOU FINISH BEFORE TIME IS CALLED, CHECK YOUR WORK ON THIS TEST.
02234020/CAPE 2013
TEST CODE
FORM TP 2013237
CARIBBEAN
02234032
MAY/JUNE 2013
E XAM I NAT I O N S
COUNCIL
CARIBBEAN ADVANCED PROFICIENCY EXAMINATION®
PURE MATHEMATICS
UNIT 2 – Paper 032
ANALYSIS, MATRICES AND COMPLEX NUMBERS
1 hour 30 minutes
05 JUNE 2013 (a.m.)
This examination paper consists of THREE sections: Module 1, Module 2 and Module 3.
Each section consists of 1 question.
The maximum mark for each Module is 20.
The maximum mark for this examination is 60.
This examination consists of 4 printed pages.
READ THE FOLLOWING INSTRUCTIONS CAREFULLY.
1.
DO NOT open this examination paper until instructed to do so.
2.
Answer ALL questions from the THREE sections.
3.
Write your solutions, with full working, in the answer booklet provided.
4.
Unless otherwise stated in the question, any numerical answer that is not
exact MUST be written correct to three significant figures.
Examination Materials Permitted
Graph paper (provided)
Mathematical formulae and tables (provided) – Revised 2012
Mathematical instruments
Silent, non-programmable, electronic calculator
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO.
Copyright © 2012 Caribbean Examinations Council
All rights reserved.
02234032/CAPE 2013
-2SECTION A (Module 1)
Answer this question.
1.
(a)
A firm measures production by the Cobb-Douglas production function
–1
P(k(t), l(t)) = 20k 4 l
–34
where k is the capital (in millions of dollars) and l is the labour force (in thousands of
workers).
Let l = 3 and k = 4.
Assume that the capital is DECREASING at a rate of $200 000 per year and that the labour
force is INCREASING at a rate of 60 workers per year.
dP
∂P
dk
∂P
dl
dP
Given that —– = —– • —– + —– • —– , calculate —– .
dt
∂k
dt
∂l
dt
dt
(b)
[6 marks]
∫
Let Fn(x) = cosn x dx.
By rewriting cosn x as cos x cosn–1 x or otherwise, prove that
(c)
1
n–1
Fn(x) = — cosn–1 x sin x + ——— Fn – 2 (x).
n
n
[6 marks]
Find the square root of the complex number z = 2 + i.
[8 marks]
Total 20 marks
GO ON TO THE NEXT PAGE
02234032/CAPE 2013
-3SECTION B (Module 2)
Answer this question.
4
2.
(a)
(i)
1
Show that the binomial expansion of 1 + — x
2
is
3 x2 + —
1 x3 + —–
1 x4 .
1 + 2x + —
2
2
16
(b)
(c)
[4 marks]
(ii)
Hence, compute 1.3774 correct to two decimal places.
(i)
Derive the first three non-zero terms in the Maclaurin expansion of ln (1 + x).
[4 marks]
(ii)
Hence, express the Maclaurin expansion of ln (1 + x) in sigma notation.
[2 marks]
[4 marks]
A geometric series is given by
x2 + —
x3 + —
x4 + . . .
x + —
2
4
8
(i)
Determine the values of x for which the series is convergent.
[3 marks]
(ii)
Hence, or otherwise, if the series is convergent, show that S2 < 4.
[3 marks]
Total 20 marks
GO ON TO THE NEXT PAGE
02234032/CAPE 2013
-4SECTION C (Module 3)
Answer this question.
3.
(a)
A system of equations Ax = b is given by
1 1 –1
2 –1 3
1 –2 –2
(b)
(i)
Calculate | A |.
(ii)
Let the matrix C =
x
y
z
=
6
–9
3
[3 marks]
8 7 –3
4 –1 3
2 –5 –3
a)
Show that CTA – 18I = 0.
[4 marks]
b)
Hence or otherwise, obtain A–1.
[2 marks]
c)
Solve the given system of equations for x, y and z.
[4 marks]
To make new words, three letters are selected without replacement from the word TRAVEL
and are written down in the order in which they are selected.
(i)
How many three-letter words may be formed?
[2 marks]
(ii)
For a three-letter word to be legal, it must have at least one vowel (that is a, e, i,
o or u). What is the probability that a legal word is formed on a single attempt?
[5 marks]
Total 20 marks
END OF TEST
IF YOU FINISH BEFORE TIME IS CALLED, CHECK YOUR WORK ON THIS TEST.
02234032/CAPE 2013
TEST CODE
FORM TP 2014240
CARIBBEAN
02134020
MAY/JUNE 2014
E XAM I NAT I O N S
COUNCIL
CARIBBEAN ADVANCED PROFICIENCY EXAMINATION®
PURE MATHEMATICS
UNIT 1 – Paper 02
ALGEBRA, GEOMETRY AND CALCULUS
2 hours 30 minutes
13 MAY 2014 (p.m.)
READ THE FOLLOWING INSTRUCTIONS CAREFULLY.
1.
This examination paper consists of THREE sections.
2.
Answer ALL questions from the THREE sections.
3.
Each section consists of TWO questions.
4.
Write your solutions, with full working, in the answer booklet provided.
5.
Unless otherwise stated in the question, any numerical answer that is not
exact MUST be written correct to three significant figures.
Examination Materials Permitted
Graph paper (provided)
Mathematical formulae and tables (provided) – Revised 2012
Mathematical instruments
Silent, non-programmable, electronic calculator
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO.
Copyright © 2013 Caribbean Examinations Council
All rights reserved.
02134020/CAPE 2014
-2SECTION A
Module 1
Answer BOTH questions.
1.
(a)
Let p, q and r be three propositions. Construct a truth table for the statement
(p → q) ˄ (r → q).
[5 marks]
(b)
A binary operator ⊕ is defined on a set of positive real numbers by
y ⊕ x = y3 + x3 + ay2 + ax2 – 5y – 5x + 16 where a is a real number.
(i)
State, giving a reason for your answer, if ⊕ is commutative in R.
(ii)
Given that f(x) = 2 ⊕ x and (x – 1) is a factor of f(x),
[3 marks]
a)
find the value of a [4 marks]
b)
factorize f(x) completely.
(c)
[3 marks]
Use mathematical induction to prove that
n
12 + 32 + 52 + ..... + (2n – 1)2 = — (4n2 – 1) for n ∈ N.
3
[10 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
02134020/CAPE 2014
-32.
(a)
The functions f and g are defined as follows:
f(x) = 2x2 + 1
x–1
g(x) = ——
√ 2
(i)
where 1 < x < ∞, x ∈ R.
Determine, in terms of x,
a)
f 2(x) [3 marks]
b)
f֯֯ [g(x)].
[3 marks]
(ii)
Hence, or otherwise, state the relationship between f֯֯ and g.
[1 mark]
(b)
a+b
Given that a3 + b3 + 3a2b = 5ab2, show that 3 log ——– = log a + 2 log b. [5 marks]
2
(c)
Solve EACH of the following equations:
(i)
1
ex + —
– 2 = 0
ex
[4 marks]
(ii)
log2 (x + 1) – log2 (3x + 1) = 2
[4 marks]
(d)
Without the use of a calculator, show that
√3+1
√2–1
√2+1
√3–1
——— + ——— + ——— + ——— = 10.
√3–1
√2+1
√2–1
√3+1
[5 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
02134020/CAPE 2014
-4SECTION B
Module 2
Answer BOTH questions.
3.
(a)
(i)
cot y – cot x
sin (x – y)
Prove that –————— = ————– .
cot x + cot y
sin (x + y)
(ii)
Hence, or otherwise, find the possible values for y in the trigonometric equation
(b)
(i)
cot y – cot x
–————— = 1, 0 < y < 2π,
cot x + cot y
1 ,0<x<—
π .
when sin x = —
2
2
(ii)
[8 marks]
Express f(θ) = 3 sin 2θ + 4 cos 2θ in the form r sin (2θ + α) where
π
r > 0 and 0 < α < —.
2
[4 marks]
[4 marks]
Hence, or otherwise, determine
a)
the value of θ, between 0 and 2π radians, at which f(θ) is a minimum
[4 marks]
b)
1
the minimum and maximum values of ———– .
7 – f(θ)
[5 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
02134020/CAPE 2014
-5Let L1 and L2 be two diameters of a circle C. The equations of L1 and L2 are x – y + 1 = 0
and x + y – 5 = 0, respectively.
(i)
Show that the coordinates of the centre of the circle, C, where L1 and L2 intersect
are (2, 3).
[3 marks]
4.
(a)
(ii)
A and B are endpoints of the diameter L1. Given that the coordinates of A are
(1, 2) and that the diameters of a circle bisect each other, determine the coordinates
[3 marks]
of B. A point, p, moves in the x – y plane such that its distance from C (2, 3) is always
(iii)
√ 2 units. Determine the locus of p.
[3 marks]
(b)
The parametric equations of a curve, S, are given by
1
t
x = –—— and y = –——
.
1+t
1 – t2
(c)
Determine the Cartesian equation of the curve, S.
[6 marks]
The points P (3, –2, 1), Q (–1, λ, 5) and R (2, 1, –4) are three vertices of a triangle PQR.
→ →
→
(i)
Express EACH of the vectors PQ, QR and RP in the form xi + yj + zk.
[4 marks]
(ii)
Hence, find the value of λ, given that PQR is right-angled with the side PQ as
hypotenuse.
[6 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
02134020/CAPE 2014
-6SECTION C
Module 3
Answer BOTH questions.
5.
(a)
Let f(x) be a function defined as
f(x) =
ax + 2, x < 3 .
ax2,x > 3
(i)
Find the value of a if f(x) is continuous at x = 3.
(ii)
x2 + 2
Let g(x) = —————
.
2
bx + x + 4
(b)
Given that
(i)
dy
1 . Using first principles, find —–
Let y = ——
.
dx
√x
(ii)
dy
x
If y = ———
, determine an expression for —– .
dx
√ 1 +x
(c)
lim
lim
2g (x) =
g(x), find the value of b.
x→1
x→0
Simplify the answer FULLY.
[4 marks]
[5 marks]
[8 marks]
[4 marks]
The parametric equations of a curve are given by
x = cos θ, y = sin θ, 0 < θ < 2π.
dy
dx
Find —– in terms of θ.
Simplify the answer as far as possible.
[4 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
02134020/CAPE 2014
-76.
(a)
The gradient of a curve which passes through the point (–1, –4) is given by
dy
dx
—– = 3x2 – 4x + 1.
(i)
Find
a)
the equation of the curve
[4 marks]
b)
the coordinates of the stationary points and determine their nature.
[8 marks]
(b)
(ii)
Sketch the curve in (a) (i) a) above, clearly marking ALL stationary points and
intercepts.
[4 marks]
The equation of a curve is given by f(x) = 2x √ 1 + x2 .
3
(i)
∫
Evaluate f(x) dx.
[5 marks]
0
(ii)
Find the volume generated by rotating the area bounded by the curve in (b) (i)
above, the x-axis, and the lines x = 0 and x = 2 about the x-axis.
[4 marks]
Total 25 marks
END OF TEST
IF YOU FINISH BEFORE TIME IS CALLED, CHECK YOUR WORK ON THIS TEST.
02134020/CAPE 2014
TEST CODE
FORM TP 2014241
CARIBBEAN
02134032
MAY/JUNE 2014
E XAM I NAT I O N S
COUNCIL
CARIBBEAN ADVANCED PROFICIENCY EXAMINATION®
PURE MATHEMATICS
UNIT 1 – Paper 032
ALGEBRA, GEOMETRY AND CALCULUS
1 hour 30 minutes
11 JUNE 2014 (p.m.)
READ THE FOLLOWING INSTRUCTIONS CAREFULLY.
1.
This examination paper consists of THREE sections.
2.
Answer ALL questions from the THREE sections.
3.
Each section consists of ONE question.
4.
Write your solutions, with full working, in the answer booklet provided.
5.
Unless otherwise stated in the question, any numerical answer that is not
exact MUST be written correct to three significant figures.
Examination Materials Permitted
Graph paper (provided)
Mathematical formulae and tables (provided) – Revised 2012
Mathematical instruments
Silent, non-programmable, electronic calculator
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO.
Copyright © 2013 Caribbean Examinations Council
All rights reserved.
02134032/CAPE 2014
-2SECTION A
Module 1
Answer this question.
1.
(a)
The binary operation * is defined on a set {e, a, b, c, d, f} as shown in the table below.
For example, a * b = d.
*
e
a
b
c
d
f
e
e
a
b
c
d
f
a
a
e
d
f
b
c
b
b
f
e
d
c
a
c
c
d
f
e
a
b
d
d
c
a
b
f
e
f
f
b
c
a
e
d
(i)
State, giving a reason, if * is commutative.
[2 marks]
(ii)
Name the identity element for the operation *.
[1 mark]
(iii)
Determine the inverse of
a)
d
[1 mark]
b)
c.
[1 mark]
(b)
Let α, β and γ be the roots of the equation 2x3 + 4x2 + 3x – 1 = 0.
(i)
Calculate EACH of the following:
a)
1
1
1
—–
+ —–
+ —–
α2
β2
γ2
[5 marks]
b)
1 + ——
1 + ——
1
——
α2 β2 β2 γ2 γ2 α2
[2 marks]
1
1
1
(ii)
Hence, or otherwise, find the equation whose roots are —–
, —–, —– .
α2 β2 γ2
[2 marks]
GO ON TO THE NEXT PAGE
02134032/CAPE 2014
-3(c)
An answer sheet is provided for this question.
1
The diagram below shows the graph of the function g(x) = —— .
x–1
4
3
2
1
_4
_3
_2
_1
1
_
_
_
_
2
3
4
x
1
2
3
4
(i)
On the answer sheet provided, sketch the graphs of |g(x)| and f(x) = x – 1, showing
clearly the intercepts and the asymptotes.
[5 marks]
(ii)
Hence, or otherwise, obtain the value of x such that f(x) = g(x).
[1 mark]
Total 20 marks
GO ON TO THE NEXT PAGE
02134032/CAPE 2014
-4SECTION B
Module 2
Answer this question.
2.
(a)
P (1, 3, 2), Q (–1, 2, 3) and R (1, 3, 5) are the vertices of a triangle.
→
→
(i)
Find the displacement vectors PQ and PR.
(ii)
Hence, determine
→
→
[4 marks]
a)
| PQ | and | PR |
b)
the cosine of the acute angle between PQ and PR
c)
the area of triangle PQR.
[4 marks]
(b)
→
→
[3 marks]
[4 marks]
π
π
π – —,
Given that — = —
show without the use of a calculator, that the EXACT value
12 3
4
π is 2 – √ 3 .
of tan —
[5 marks]
12
Total 20 marks
GO ON TO THE NEXT PAGE
02134032/CAPE 2014
-5SECTION C
Module 3
Answer this question.
3.
sin 8x
lim ———
.
x → 0 2x
(a)
Evaluate
(b)
The equation of a curve is given by
[4 marks]
y = x3 + x2 + 2.
(i)
Determine the coordinates of the points on the curve where the gradient is 1.
[6 marks]
(ii)
Determine the equation of the normal which intersects the curve at (–1, 2).
[4 marks]
(c)
The diagram below (not drawn to scale) shows the design of a petal drawn on a square
tile of length 1 metre.
y
1
y = √x
Petal
y = x2
0
1
x
The design may be modelled by the finite region enclosed by the curves y = √ x and
y = x2 where x and y are lengths measured in metres.
Calculate the area of the petal.
[6 marks]
Total 20 marks
END OF TEST
IF YOU FINISH BEFORE TIME IS CALLED, CHECK YOUR WORK ON THIS TEST.
02134032/CAPE 2014
TEST CODE
FORM TP 2014241
CARIBBEAN
02134032
MAY/JUNE 2014
E XAM I NAT I O N S
COUNCIL
CARIBBEAN ADVANCED PROFICIENCY EXAMINATION®
PURE MATHEMATICS
UNIT 1 – Paper 032
Answer Sheet for Question 1 (c)
Candidate Number .............................................
y
4
3
2
1
-4
-3
-2
-1
0
1
2
3
4
x
-1
-2
-3
-4
ATTACH THIS ANSWER SHEET TO YOUR ANSWER BOOKLET
02134032/CAPE 2014
TEST CODE
FORM TP 2014243
CARIBBEAN
02234020
MAY/JUNE 2014
E XAM I NAT I O N S
COUNCIL
CARIBBEAN ADVANCED PROFICIENCY EXAMINATION®
PURE MATHEMATICS
UNIT 2 – Paper 02
ANALYSIS, MATRICES AND COMPLEX NUMBERS
2 hours 30 minutes
28 MAY 2014 (p.m.)
READ THE FOLLOWING INSTRUCTIONS CAREFULLY.
1.
This examination paper consists of THREE sections.
2.
Answer ALL questions from the THREE sections.
3.
Each section consists of TWO questions.
4.
Write your solutions, with full working, in the answer booklet provided.
5.
Unless otherwise stated in the question, any numerical answer that is not
exact MUST be written correct to three significant figures.
Examination Materials Permitted
Graph paper (provided)
Mathematical formulae and tables (provided) – Revised 2012
Mathematical instruments
Silent, non-programmable, electronic calculator
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO.
Copyright © 2013 Caribbean Examinations Council
All rights reserved.
02234020/CAPE 2014
-2SECTION A
Module 1
Answer BOTH questions.
1.
(a)
(i)
Differentiate, with respect to x,
x
y = ln (x2 + 4) – x tan–1 ( — ).
2
(ii)
A curve is defined parametrically as
x = a cos3 t, y = a sin3 t.
Show that the tangent at the point P (x, y) is the line
(b)
[5 marks]
y cos t + x sin t = a sin t cos t.
[7 marks]
Let the roots of the quadratic equation x2 + 3x + 9 = 0 be α and β.
(i)
Determine the nature of the roots of the equation.
[2 marks]
(ii)
Express α and β in the form reiθ, where r is the modulus and θ is the argument,
where –π < θ < π.
[4 marks]
(iii)
Using de Moivre’s theorem, or otherwise, compute α3 + β3.
[4 marks]
(iv)
Hence, or otherwise, obtain the quadratic equation whose roots are α3 and β3.
[3 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
02234020/CAPE 2014
-3-
2.
(a)
∫
Let Fn (x) = (ln x)n dx.
(i)
Show that Fn (x) = x (ln x)n – n Fn – 1(x).[3 marks]
(ii)
Hence, or otherwise, show that
(b)
(i)
F3 (2) – F3 (1) = 2 (ln 2)3 – 6 (ln 2)2 + 12 ln 2 – 6.
y2 + 2y + 1
By decomposing —————
into partial fractions, show that
y4 + 2y2 + 1
1
2y
y2 + 2y + 1
—————
= ———
+ ———–
.
y2 + 1
(y2 + 1)2
y4 + 2y2 + 1
1
(ii)
Hence, find
∫
0
[7 marks]
y2 + 2y + 1 dy.
—————
y4 + 2y2 + 1
[7 marks]
[8 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
02234020/CAPE 2014
-4SECTION B
Module 2
Answer BOTH questions.
3.
(a)
(i)
Prove, by mathematical induction, that for n ∈ N
(b)
(ii)
1
1
1
1
1
Sn = 1 + — + —2 + —3 + ...... + ——
n–1 .
n – 1 = 2 – ——
2
2
2
2
2
Hence, or otherwise, find
[8 marks]
lim
S .
[3 marks]
n→∞ n
Find the Maclaurin expansion for
f(x) = (1 + x)2 sin x
up to and including the term in x3.
[14 marks]
Total 25 marks
4.
(a)
(i)
For the binomial expansion of (2x + 3)20, show that the ratio of the term in x6 to
3
the term in x7 is — .
4x
(ii)
a)
Determine the FIRST THREE terms of the binomial expansion of (1 + 2x)10.
b)
Hence, obtain an estimate for (1.01)10.
(b)
(n + 1) !
n!
n!
Show that ————– + ————————– = ——————.
(n – r + 1) !r!
(n – r) !r!
(n – r + 1) ! (r – 1) !
[5 marks]
[7 marks]
[6 marks]
(c)
(i)
Show that the function f(x) = –x3 + 3x + 4 has a root in the interval [1, 3].
[3 marks]
(ii)
By taking x1 = 2.1 as a first approximation of the root in the interval [1, 3], use the
Newton–Raphson method to obtain a second approximation, x2, in the interval
[1, 3].
[4 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
02234020/CAPE 2014
-5SECTION C
Module 3
Answer BOTH questions.
5.
(a)
(i)
Five teams are to meet at a round table. Each team consists of two members AND
one leader. How many seating arrangements are possible if each team sits together
with the leader of the team in the middle?
[7 marks]
(ii)
In an experiment, individuals were asked to colour a shape by selecting from two
available colours, red and blue. The individuals chose one colour, two colours or
no colour.
In total, 80% of the individuals used colours and 600 individuals used no colour.
a)
b)
Given that 40% of the individuals used red and 50% used blue, calculate the
probability that an individual used BOTH colours.
[4 marks]
(b)
Determine the TOTAL number of individuals that participated in the
experiment.
[2 marks]
A and B are the two matrices given below.
A=
1
3
2
x
0
1
–1
1
2
B=
2
01
2
3
1
5
4
2
(i)
Determine the range of values of x for which A–1 exists.
[4 marks]
(ii)
Given that det (AB) = –21, show that x = 3.
[4 marks]
(iii)
Hence, obtain A–1.
[4 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
02234020/CAPE 2014
-66.
(a)
(i)
Show that the general solution of the differential equation
y' + y tan x = sec x
(b)
is
(ii)
y = sin x + C cos x.
[10 marks]
π
2 and x = —.
Hence, obtain the particular solution where y = ——
4
√2
[4 marks]
A differential equation is given as y" – 5y' = xe5x. Given that a particular solution is
[11 marks]
yp(x) = Ax2 e5x + Bxe5x, solve the differential equation.
Total 25 marks
END OF TEST
IF YOU FINISH BEFORE TIME IS CALLED, CHECK YOUR WORK ON THIS TEST.
02234020/CAPE 2014
TEST CODE
FORM TP 2014244
CARIBBEAN
02234032
MAY/JUNE 2014
E XAM I NAT I O N S
COUNCIL
CARIBBEAN ADVANCED PROFICIENCY EXAMINATION®
PURE MATHEMATICS
UNIT 2 – Paper 032
ANALYSIS, MATRICES AND COMPLEX NUMBERS
1 hour 30 minutes
04 JUNE 2014 (a.m.)
READ THE FOLLOWING INSTRUCTIONS CAREFULLY.
1.
This examination paper consists of THREE sections.
2.
Answer ALL questions from the THREE sections.
3.
Each section consists of ONE question.
4.
Write your solutions, with full working, in the answer booklet provided.
5.
Unless otherwise stated in the question, any numerical answer that is not
exact MUST be written correct to three significant figures.
Examination Materials Permitted
Graph paper (provided)
Mathematical formulae and tables (provided) – Revised 2012
Mathematical instruments
Silent, non-programmable, electronic calculator
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO.
Copyright © 2013 Caribbean Examinations Council
All rights reserved.
02234032/CAPE 2014
-2SECTION A
Module 1
Answer this question.
1
1.
(a)
Let l =
∫
–1
1
———
dx.
1 + e–x
(i)
Use the trapezium rule with two trapezia of equal width to obtain an estimate of
[3 marks]
l.
(ii)
Evaluate the integral l by means of the substitution u = ex.
(b)
The diagram below (not drawn to scale) shows an open rectangular box with a partition
in the middle.
The dimensions of the box, measured in centimetres, are x, y, and z. The volume of the
box is 384 cm3.
(i)
The pieces from which the box is assembled are cut from a flat plank of wood.
Show that the TOTAL area of the pieces cut from the plank, A cm2, is given by
[7 marks]
(ii)
768 1152
A = xy + —— + —— .
y
x
[5 marks]
∂A
∂A
The minimum value of A occurs where —– = 0 and —– = 0 simultaneously.
∂x
∂y
a)
∂A
∂A
Determine —– and —– .
∂x
∂y
b)
∂A = 0 and —–
∂A = 0 are both satisfied by
Hence, show that the equations —–
∂x
∂y
x = 12, y = 8. [2 marks]
[3 marks]
Total 20 marks
GO ON TO THE NEXT PAGE
02234032/CAPE 2014
-3SECTION B
Module 2
Answer this question.
2.
(a)
(i)
1
1
2 .
Show that ——— – ——— = ———
2r + 1
2r – 1
4r2 – 1
(ii)
2
2n .
Hence, or otherwise, show that Σ ———
= ———
2
–
1
4r
2n
+1
r=1
[3 marks]
n
(b)
[5 marks]
An arithmetic progression is such that the fifth and tenth partial sums are S5 = 60 and
S10 = 202 respectively.
(i)
Calculate the first term, a, and the common difference, d.
[5 marks]
(ii)
Hence, or otherwise, calculate the 15th term, u15.
[2 marks]
(i)
Show that the function f(x) = e–x – 2x + 3 has a root, α, in the closed interval
[1, 2].
[2 marks]
(ii)
Apply linear interpolation ONCE in the interval [1, 2] to find an approximation to
the root, α.
[3 marks]
(c)
Total 20 marks
GO ON TO THE NEXT PAGE
02234032/CAPE 2014
-4SECTION C
Module 3
Answer this question.
3.
(a)
A bag contains 2 red balls, 3 blue balls and 1 white ball. In an experiment, 2 balls are
drawn at random from the bag without replacement.
(i)
Use a tree diagram to show the possible events and their corresponding probabilities.
[5 marks]
(b)
(ii)
Calculate the probability that the second ball drawn is blue.
[4 marks]
The current flow in a particular circuit is defined by the differential equation
di
L —– + Ri = V,
dt
where i is the current at time t, and V, R and L are constants representing the voltage,
resistance and self-inductance respectively.
The switch in the circuit is closed at time t = 0 and i(0) = 0.
(i)
By solving the differential equation using an appropriate integrating factor, verify
R t
V
L ).
that i = — (1 – e– —
[8 marks]
R
(ii)
The steady-state current in the circuit is lim i. Use the result of (b) (i) above to
t→∞
lim
evaluate
i.
[3 marks]
t→∞
Total 20 marks
END OF TEST
IF YOU FINISH BEFORE TIME IS CALLED, CHECK YOUR WORK ON THIS TEST.
02234032/CAPE 2014
TEST CODE
FORM TP 2015265
CARIBBEAN
02134020
MAY/JUNE 2015
E XAM I NAT I O N S
COUNCIL
CARIBBEAN ADVANCED PROFICIENCY EXAMINATION®
PURE MATHEMATICS
UNIT 1 – Paper 02
ALGEBRA, GEOMETRY AND CALCULUS
2 hours 30 minutes
12 MAY 2015 (p.m.)
READ THE FOLLOWING INSTRUCTIONS CAREFULLY.
1.
This examination paper consists of THREE sections.
2.
Answer ALL questions from the THREE sections.
3.
Each section consists of TWO questions.
4.
Write your solutions, with full working, in the answer booklet provided.
5.
Unless otherwise stated in the question, any numerical answer that is not
exact MUST be written correct to three significant figures.
5265
Examination Materials Permitted
Graph paper (provided)
Mathematical formulae and tables (provided) – Revised 2012
Mathematical instruments
Silent, non-programmable, electronic calculator
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO.
Copyright © 2014 Caribbean Examinations Council
All rights reserved.
02134020/CAPE 2015
-2SECTION A
Module 1
Answer BOTH questions.
1.
(a)
Let p and q be any two propositions.
(i)
State the inverse and the contrapositive of the statement p → q.
(ii)
Copy and complete the table below to show the truth table for
[2 marks]
p → q and ~q → ~p.
p
q
T
T
T
F
F
T
F
F
~p
p→q
~q
(b)
(iii)
~q → ~p
[4 marks]
Hence, state whether the compound statements p → q and ~q → ~p are logically
equivalent. Justify your response. [2 marks]
The polynomial f(x) = x3 + px2 – x + q has a factor (x – 5) and a remainder of 24 when
divided by (x – 1).
(i)
Find the values of p and q. [4 marks]
(ii)
Hence, factorize f(x) = x3 + px2 – x + q completely.
(c)
[5 marks]
Given that S(n) = 5 + 52 + 53 + 54 + ... + 5n, use mathematical induction to prove that
4S(n) = 5n+1 – 5 for n ∈ N.
[8 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
02134020/CAPE 2015
-32.
(a)
The relations f : A → B and g : B → C are functions which are both one-to-one and onto.
Show that (g ° f) is
(i)
one-to-one
[4 marks]
(ii)
onto.
[4 marks]
(b)
Solve EACH of the following equations:
(i)
4 – ——
4 = 0
3 – —–
(9)x
(81)x
[7 marks]
(ii)
| 5x – 6 | = x + 5
[5 marks]
(c)
The population growth of bacteria present in a river after time, t hours, is given by
N = 300 + 5t.
Determine
(i)
the number of bacteria present at t = 0
[1 mark]
(ii)
the time required to triple the number of bacteria.
[4 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
02134020/CAPE 2015
-4SECTION B
Module 2
Answer BOTH questions.
3.
(a)
(i)
Show that cos 3x = 4 cos3 x – 3 cos x.
[6 marks]
(ii)
Hence, or otherwise, solve
cos 6x – cos 2x = 0 for 0 < x < 2π.
(b)
(i)
Express f(2θ) = 3 sin 2θ + 4 cos 2θ in the form r sin (2θ + α) where
[9 marks]
(ii)
π
r > 0 and 0 < α < —.
2
[6 marks]
1
Hence, or otherwise, find the maximum and minimum values of ———–
.
7 – f(2θ)
[4 marks]
Total 25 marks
4.
(a)
The circles C1 and C2 are defined by the parametric equations as follows:
C1:
x = √10 cos θ – 3;
y = √10 sin θ + 2
C2: x = 4 cos θ + 3;
y = 4 sin θ + 2.
(i)
Determine the Cartesian equations of C1 and C2 in the form (x – a)2 + (y – b)2 = r2.
[4 marks]
(b)
(ii)
Hence or otherwise, find the points of intersection of C1 and C2.
[9 marks]
A point P (x, y) moves so that its distance from the fixed point (0, 3) is two times the
distance from the fixed point (5, 2). Show that the equation of the locus of the point
P (x, y) is a circle.
[12 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
02134020/CAPE 2015
-5SECTION C
Module 3
Answer BOTH questions.
5.
(a)
Let f be a function defined as
sin (ax)
——— if x ≠ 0,
x
f(x) =
4
if x = 0
a≠0
If f is continuous at x = 0, determine the value of a.
(b)
Using first principles, determine the derivative of f(x) = sin (2x).
(c)
2x
If y = ——— show that
√1 + x2
[4 marks]
[6 marks]
y
1+x
[7 marks]
——
+ ————
= 0.
2
2 2
3y
(1 + x )
[8 marks]
Total 25 marks
dy
dx
(i)
x —– = ———
2
(ii)
d2y
dx
GO ON TO THE NEXT PAGE
02134020/CAPE 2015
-66.
The diagram below (not drawn to scale) shows the region bounded by the lines
(a)
y = 3x – 7, y + x = 9 and 3y = x + 3.
(a)
(i)
Show that the coordinates of A, B and C are (4, 5), (3, 2), and (6, 3) respectively.
[5 marks]
(ii)
Hence, use integration to determine the area bounded by the lines.
[6 marks]
(b)
The gradient function of a curve y = f(x) which passes through the point (0, –6) is given
by 3x2 + 8x – 3.
(i)
Determine the equation of the curve.
[3 marks]
(ii)
Find the coordinates and nature of the stationary point of the curve in (b) (i) above.
[8 marks]
(iii)
Sketch the curve in (b) (i) by clearly labelling the stationary points.
[3 marks]
Total 25 marks
END OF TEST
IF YOU FINISH BEFORE TIME IS CALLED, CHECK YOUR WORK ON THIS TEST.
02134020/CAPE 2015
TEST CODE
FORM TP 2015317
CARIBBEAN
02134022
MAY/JUNE 2015
E XAM I NAT I O N S
COUNCIL
CARIBBEAN ADVANCED PROFICIENCY EXAMINATION®
PURE MATHEMATICS
UNIT 1 – Paper 02
ALGEBRA, GEOMETRY AND CALCULUS
2 hours 30 minutes
16 JUNE 2015 (p.m.)
READ THE FOLLOWING INSTRUCTIONS CAREFULLY.
1.
This examination paper consists of THREE sections.
2.
Answer ALL questions from the THREE sections.
3.
Each section consists of TWO questions.
4.
Write your solutions, with full working, in the answer booklet provided.
5.
Unless otherwise stated in the question, any numerical answer that is not
exact MUST be written correct to three significant figures.
Examination Materials Permitted
Graph paper (provided)
Mathematical formulae and tables (provided) – Revised 2012
Mathematical instruments
Silent, non-programmable, electronic calculator
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO.
Copyright © 2015 Caribbean Examinations Council
All rights reserved.
02134022/CAPE 2015
-2SECTION A
Module 1
Answer BOTH questions.
1.
(a)
Let p, q and r be propositions.
Copy and complete the truth table below for the propositions (p ˄ q) → r and
(p → r) ˄ (q → r).
(i)
p
q
r
T
T
T
T
T
F
T
F
T
T
F
F
F
T
T
F
T
F
F
F
T
F
F
F
p˄q
p→r
q→r
(p ˄ q) → r
(p → r) ˄ (q → r)
(ii)
[5 marks]
Hence, determine whether or not (p ˄ q) → r and (p → r) ˄ (q → r) are
logically equivalent. Justify your response.
[2 marks]
(b)
Use mathematical induction to prove that 10n+1 + 3(10n) + 5 is divisible by 9 for all natural
numbers. [8 marks]
(c)
Solve the equation x3 − 6x2 − 69x + 154 = 0.
[10 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
02134022/CAPE 2015
-3-
2.
(a)
(i)
(ii)
Show that
1
logax
may be rewritten as
ln a
ln x
, where a > 0, a ≠ 1.
[5 marks]
Hence, or otherwise, solve the equation:
1
1
1
1
+
+
+
=1
log5 x
log2 x
log4 x
log3 x
[4 marks]
(b)
The diagrams below (not drawn to scale) show the graphs of f(x) = x2 − 4x + 3 and
g(x) = 2 − 3x.
(i)
On the same axes, sketch the graphs | f(x) | and | g(x) |.
[4 marks]
(ii)
Hence, or otherwise, solve the equation | x2 – 4x + 3 | = | 2 − 3x |.
[6 marks]
(c)
A function f is such that f -1(x) = √ x + 1 + 1.
Determine the value of f (f -1[ f (1)]).
[6 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
02134022/CAPE 2015
-4SECTION B
Module 2
Answer BOTH questions.
3.
(a)
Given that tan2 x =
sin2 x
, show that sin x = ±
1 − sin2 x
tan x
1 + tan 2 x
.
[6 marks]
Note: You are not required to simplify the expression sin2x.
(b)
(i)
(ii)
Show that f (θ) = √3 sin θ + cos θ may be expressed as
0 ≤ θ ≤ π– .
2
where
[6 marks]
Hence, or otherwise,
a)
b)
solve the equation f (θ) = √2 for 0 ≤ θ ≤ 2π
[5 marks]
determine the maximum value of f and the smallest positive value of θ for
which it occurs.
[3 marks]
π
(c)
Without the use of a calculator or tables, find the exact value of cos —.
[5 marks]
12
Total 25 marks
4.
(a)
Calculate the distance between the points of intersection of the line 3x − 2y + 6 = 0 and the
circle x2 + y2 = 9.
[8 marks]
(b)
A circle, C, is defined by the equation 3x2 + 3y2 − 2y − 4x = 0.
(i)
Find the centre and radius of C.
[5 marks]
(ii)
Find the equation of the tangent to C at the point (0,0) .
[4 marks]
(c)
Let a = ci + j + 2k and b = 2i − 2j + k.
–.
Find c given that the angle between the vectors a and b is π
3
[8 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
02134022/CAPE 2015
-5SECTION C
Module 3
Answer BOTH questions.
5.
d2y
dx
(a)
Given that y = tan–1 (1 − x2), find —2 .
(b)
The equation of a circle is given by the parametric equations
[5 marks]
x = a + b sin t and y = c + d cos t where b ≠ 0.
Determine
(i)
dy
[4 marks]
— in terms of t
dx
(ii)
the value of t for which the circle has vertical tangents.
(c)
A function f is defined as f (x) = 3 +
[3 marks]
6
x – 2.
(i)
Determine
a)
whether the function f has turning points
[4 marks]
b)
the vertical and horizontal asymptotes of f.
[4 marks]
Hence, sketch the graph of f.
(ii)
[5 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
02134022/CAPE 2015
-66.
(a)
1
The diagram below (not drawn to scale) shows the curve f(x) = —.
x2
(i)
Determine the area of the region, A.
Calculate the volume of the solid that results from revolving the region, A, about
the x-axis.
[5 marks]
(b)
(c)
(ii)
[3 marks]
y
dy = − −.
The gradient function for a curve y = f(x) is given as —
dx
x
Given that the curve passes through the point (1,1), determine an expression for f(x).
[7 marks]
By using the substitution, y = x2, show that
∫
1
y+
y3
dy = ∫
2
dx .
1 + x2
[5 marks]
(d)
Given that sin 2θ = 2 sinθ cosθ and cos 2θ = 1 − 2 sin2 θ,
find ∫sin2x cos2x dx.
[5 marks]
Total 25 marks
END OF TEST
IF YOU FINISH BEFORE TIME IS CALLED, CHECK YOUR WORK ON THIS TEST.
02134022/CAPE 2015
TEST CODE
FORM TP 2015266
CARIBBEAN
02134032
MAY/JUNE 2015
E XAM I NAT I O N S
COUNCIL
CARIBBEAN ADVANCED PROFICIENCY EXAMINATION®
PURE MATHEMATICS
UNIT 1 – Paper 032
ALGEBRA, GEOMETRY AND CALCULUS
1 hour 30 minutes
5266
10 JUNE 2015 (p.m.)
READ THE FOLLOWING INSTRUCTIONS CAREFULLY.
1.
This examination paper consists of THREE sections.
2.
Answer ALL questions from the THREE sections.
3.
Each section consists of ONE question.
4.
Write your solutions, with full working, in the answer booklet provided.
5.
Unless otherwise stated in the question, any numerical answer that is not
exact MUST be written correct to three significant figures.
Examination Materials Permitted
Graph paper (provided)
Mathematical formulae and tables (provided) – Revised 2012
Mathematical instruments
Silent, non-programmable, electronic calculator
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO.
Copyright © 2014 Caribbean Examinations Council
All rights reserved.
02134032/CAPE 2015
-2SECTION A
Module 1
Answer this question.
1.
(a)
Let p and q be two propositions.
(i)
Copy and complete the truth table below.
p
q
T
T
T
F
F
T
F
F
~p
~q
p∨q
~p ∧ ~q
(p ∨ q) ∨ (~p ∧ ~q)
[5 marks]
(ii)
Hence, or otherwise, state if (p ∨ q) ∨ (~p ∧ ~q) is a tautology. Justify your
response.
[1 mark]
(b)
Solve the equation log4 (x2 + 1) – log2 (2x – 1) = 0.
(c)
xy
Let x, y, k ∈ R. An operation * is defined as x * y = ———— where x + y ≠ k.
x+y–k
Show that * is associative.
[8 marks]
[6 marks]
Total 20 marks
GO ON TO THE NEXT PAGE
02134032/CAPE 2015
-3SECTION B
Module 2
Answer this question.
2.
(a)
ABCD is a parallelogram such that A has coordinates (0, 0). The position vectors B and
D are a and b respectively. Using vector algebra show that
(AC)2 + (BD)2 = (AB)2 + (BC)2 + (DC)2 + (AD)2.
(b)
(c)
[7 marks]
The equation of a circle, C1, is given by x2 + y2 – 4x – 6y + 3 = 0.
Find the equation of the tangent to C1 at the point (5, 2).
[6 marks]
3
4
Angles 2A and 2B are obtuse angles with sin (2A) = — and cos (B) = —.
5
5
Without calculating the value of A and B find the EXACT value of
(i)
cos (2A)
[2 marks]
(ii)
cos (A + B).
[5 marks]
Total 20 marks
GO ON TO THE NEXT PAGE
02134032/CAPE 2015
-4SECTION C
Module 3
Answer this question.
3.
1 – cos2 (2x) .
lim —————
x2
x→0
(a)
Evaluate
[4 marks]
(b)
A spherical shaped ice cream is melting in such a way that its volume is decreasing at a
rate of 2 cm3 per minute. Find the rate at which the radius decreases when the diameter
is 8 cm.
[6 marks]
(c)
The diagram below (not drawn to scale) is a sketch of the curve y = x2 + 3 and the line
y = 4x.
(i)
Find the coordinates of P and Q, the points of intersection of the curve and the
line.
[5 marks]
(ii)
Calculate the volume of the solid generated when the shaded region is rotated
completely about the x-axis.
[5 marks]
Total 20 marks
END OF TEST
IF YOU FINISH BEFORE TIME IS CALLED, CHECK YOUR WORK ON THIS TEST.
02134032/CAPE 2015
TEST CODE
FORM TP 2015268
CARIBBEAN
02234020
MAY/JUNE 2015
E XAM I NAT I O N S
COUNCIL
CARIBBEAN ADVANCED PROFICIENCY EXAMINATION®
PURE MATHEMATICS
UNIT 2 – Paper 02
ANALYSIS, MATRICES AND COMPLEX NUMBERS
2 hours 30 minutes
27 MAY 2015 (p.m.)
5268
READ THE FOLLOWING INSTRUCTIONS CAREFULLY.
1.
This examination paper consists of THREE sections.
2.
Answer ALL questions from the THREE sections.
3.
Each section consists of TWO questions.
4.
Write your solutions, with full working, in the answer booklet provided.
5.
Unless otherwise stated in the question, any numerical answer that is not
exact MUST be written correct to three significant figures.
Examination Materials Permitted
Graph paper (provided)
Mathematical formulae and tables (provided) – Revised 2012
Mathematical instruments
Silent, non-programmable, electronic calculator
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO.
Copyright © 2014 Caribbean Examinations Council
All rights reserved.
02234020/CAPE 2015
-2SECTION A
Module 1
Answer BOTH questions.
1.
(a)
Three complex numbers are given as
z2 = √ 3 + 3i and z3 = –2 + 2i.
z1 = 1 + (7 – 4 √ 3 )i,
(i)
z3
Express the quotient —
z in the form x + iy where x, y ∈ R.
[3 marks]
2
(ii)
π
Given that arg w = arg z3 – [arg z1 + arg z2], | z1 | = 1 and arg z1 = — rewrite
12
z3
iθ
w = —–
z z in the form re where r = | w | and θ = arg w.
[6 marks]
1 2
(b)
A complex number v = x + iy is such that v2 = 2 + i. Show that
2+√5
x2 = ———–.
2
(c)
[7 marks]
The function f is defined by the parametric equations
e–t
x = ————
√ 1 – t2
and
y = sin–1 t
for –1 < t < 0.5.
(i)
dy
et (1 – t2)
Show that —– = ————.
dx
t2 + t – 1
[6 marks]
(ii)
Hence, show that f has no stationary value.
[3 marks]
Total 25 marks
GO ON TO THE NEXT PAGE
02234020/CAPE 2015
-32.
Let 4x2 + 3xy2 + 7x + 3y = 0.
(a)
(i)
Use implicit differentiation to show that
(ii)
dy
8x + 3y2 + 7
—– = —————.
[5 marks]
dx
3(1 + 2xy)
Show that for f(x, y) = 4x2 + 3xy2 + 7x + 3y
∂2f (x, y)
∂f (x, y)
∂2f (x, y)
∂2f (x, y)
6 ———– – 10 = ————
————
+
————.
∂x2
∂y
∂y2
∂y ∂x
(b)
[5 marks]
The rational function
18x2 + 13
f(x) = ————
9x2 + 4
is defined on the domain –2 < x < 2.
b
Express f(x) in the form a + ———
where a, b ∈ R.
9x2 + 4
(i)
[2 marks]
2
(ii)
Given that f(x) is symmetric about the y-axis, evaluate
∫
f(x) dx.
[6 marks]
–2
(c)
Let h be a function of x.
(i)
Show that
∫
hn+1
hn ln h dh = ———–
[–1 + (n + 1) ln h] + C,
(n + 1)2
where –1 ≠ n ∈ Z and C ∈ R.
(ii)
Hence, find
sin2 x cos x ln (sin x) dx.
∫
[5 marks]
[2 marks]
Total 25 marks
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02234020/CAPE 2015
-4SECTION B
Module 2
Answer BOTH questions.
3.
(a)
The nth term of a sequence is given by
2n + 1
Tn = ———–.
√n2 + 1
lim
T .
[5 marks]
n→∞ n
(i)
Determine
(ii)
9
1
Show that T4 = — 1 + —
4
16
(iii)
Hence, use the binomial expansion with x = — to approximate the value of T4 for
—
1
—
2
.
[3 marks]
1
16
terms up to and including x3. Give your answer correct to two decimal places.
[4 marks]
(b)
A series is given as
4
5
3
2 + — + — + —– + . . .
9
16
4
(i)
Express the nth partial sum Sn of the series in sigma notation.
[2 marks]
∞
(ii)
(c)
2
1 converges to π
Hence, given that ∑ —
—, show that Sn diverges as n →∞.
2
6
n=1 n
[4 marks]
Use the method of induction to prove that
n
n(n2 – 1)
∑ r(r – 1) = ————.
3
r=1
[7 marks]
Total 25 marks
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02234020/CAPE 2015
-54.
(a)
A function is defined as g(x) = e3x+1.
(i)
Obtain the Maclaurin series expansion for g(x) up to and including the term in x4.
[6 marks]
(ii)
(b)
(i)
Hence, estimate g(0.2) correct to three decimal places.
[3 marks]
Let f(x) = x – 3 sin x – 1.
Use the intermediate value theorem to show that f has at least one root in the
interval [–2, 0].
[3 marks]
Use at least three iterations of the method of interval bisection to show that
(ii)
f(–0.538) ≈ 0 in the interval [–0.7, –0.3].
(c)
(≈ 0 means approximately equal to 0) [8 marks]
Use the Newton–Raphson method with initial estimate x1 = 5.5 to approximate the root
of g(x) = sin 3x in the interval [5, 6], correct to two decimal places.
[5 marks]
Total 25 marks
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02234020/CAPE 2015
-6SECTION C
Module 3
Answer BOTH questions.
5.
(a)
Ten students from across CARICOM applied for mathematics scholarships. Three of the
applicants are females and the remaining seven are males. The scholarships are awarded
to four successful students. Determine the number of possible ways in which a group of
FOUR applicants may be selected if
(i)
no restrictions are applied
[1 mark]
(ii)
at least one of the successful applicants must be female.
[3 marks]
(b)
Numbers are formed using the digits 1, 2, 3, 4 and 5 without repeating any digit. Determine
(i)
the greatest possible amount of numbers that may be formed
[4 marks]
(ii)
the probability that a number formed is greater than 100.
[3 marks]
(c)
A system of equations is given as
2x + 3y – z = –3.5
x – y + 2z = 7
1.5x + 3z = 9
(i)
Rewrite the system of equations as an augmented matrix.
[2 marks]
(ii)
Use elementary row operations to reduce the system to echelon form.
[5 marks]
(iii)
Hence, solve the system of equations.
(iv)
Show that the system has no solution if the third equation is changed to
1.5x – 1.5y + 3z = 9.
[3 marks]
[4 marks]
Total 25 marks
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02234020/CAPE 2015
-76.
(a)
Alicia’s chance of getting to school depends on the weather. The weather can be either rainy
or sunny. If it is a rainy day, the probability that she gets to school is 0.7. In addition, she
goes to school on 99% of the sunny school days. It is also known that 32% of all school
days are rainy.
(i)
Construct a tree diagram to show the probabilities that Alicia arrives at school.
[3 marks]
(ii)
What is the probability that Alicia is at school on any given school day?
[3 marks]
(iii)
Given that Alicia is at school today, determine the probability that it is a rainy day.
[4 marks]
(b)
(i)
Show that the equation y + xy + x2 = 0 is a solution of the differential equation
(ii)
dy
y – x2
—– = ———–.
dx
x(1 + x)
[5 marks]
A differential equation is given as y′′ – 2y = 0.
a)
Find the general solution of the differential equation.
b)
Hence, show that the solution which satisfies the boundary conditions
√2
y (0) = 1 and y′ —–
2
[3 marks]
= 0 is
1
y = ——–
e√2 x + e2–√2 x .
2
e +1
[7 marks]
Total 25 marks
END OF TEST
IF YOU FINISH BEFORE TIME IS CALLED, CHECK YOUR WORK ON THIS TEST.
02234020/CAPE 2015
TEST CODE
FORM TP 2015269
CARIBBEAN
02234032
MAY/JUNE 2015
E XAM I NAT I O N S
COUNCIL
CARIBBEAN ADVANCED PROFICIENCY EXAMINATION®
PURE MATHEMATICS
UNIT 2 – Paper 032
ANALYSIS, MATRICES AND COMPLEX NUMBERS
1 hour 30 minutes
03 JUNE 2015 (a.m.)
READ THE FOLLOWING INSTRUCTIONS CAREFULLY.
1.
This examination paper consists of THREE sections.
2.
Answer ALL questions from the THREE sections.
3.
Each section consists of ONE question.
4.
Write your solutions, with full working, in the answer booklet provided.
5.
Unless otherwise stated in the question, any numerical answer that is not
exact MUST be written correct to three significant figures.
Examination Materials Permitted
5269
Graph paper (provided)
Mathematical formulae and tables (provided) – Revised 2012
Mathematical instruments
Silent, non-programmable, electronic calculator
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO.
Copyright © 2014 Caribbean Examinations Council
All rights reserved.
02234032/CAPE 2015
-2SECTION A
Module 1
Answer this question.
1.
(a)
3π
A complex number z1 is such that | z1 | = 2 and arg z1 = —–.
4
(i)
Identify the coordinates of z1 on an Argand diagram.
(ii)
On the same axes, connect z1 to the origin with a line segment and label the angle
that represents arg z1.
[2 marks]
(iii)
On the same axes, sketch the locus of the point z2 which moves in the complex
plane such that | z1 – z2 | = 1.
[2 marks]
(b)
[3 marks]
Use the trapezium rule with five ordinates to find an approximate value of
(c)
(i)
∫
2
√ 4 + x3 dx.
[6 marks]
0
Determine
(ii)
∫
x
sin–1 (—)
2
———–— dx.
[5 marks]
√4 – x2
Hence, calculate
∫
x
sin–1 (—)
2 dx.
———–—
[2 marks]
0
√4 – x2
1
Total 20 marks
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02234032/CAPE 2015
-3SECTION B
Module 2
Answer this question.
2.
(a)
(i)
Determine the Taylor series expansion of
f(x) = ex cos x
π
centred at — up to and including the first two non-zero terms.
2
π
Hence, estimate f(—).
[2 marks]
6
(ii)
[4 marks]
(b)
The twentieth term of an arithmetic progression is 35 and the sum of the first 19 terms is
285. Calculate the sum of the first five terms.
[7 marks]
(c)
The numbers n – 4, n + 2, 3n + 1 are consecutive terms of a geometric sequence. Given
that the corresponding series converges, determine the common ratio.
[7 marks]
Total 20 marks
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02234032/CAPE 2015
-4SECTION C
Module 3
Answer this question.
3.
(a)
A board game involves tossing TWO fair dice and ONE fair coin. The face shown on the
coin determines the action of the next player.
If a HEAD is observed on the coin, the total on the dice is as observed. If a TAIL is
observed on the coin, the number on each die must be 3 or less. If any of the numbers is
more than 3, the die is thrown again until a 1, 2 or 3 is shown.
(i)
Copy and complete the table below to show the possible totals of the throws.
DIE 1
2
3
4
1
2
3
4
5
2
3
4
5
6
3
4
5
6
7
4
5
6
7
8
5
6
DIE 2
HEAD
1
5
6
TAIL
1
2
3
(ii)
[2 marks]
What is the probability that the sum of the numbers on the dice is EVEN on any
turn in the game?
[2 marks]
(iii)
Determine the probability of obtaining a HEAD and an EVEN total on the dice.
[4 marks]
(iv)
State, giving a reason for your answer, whether the events of obtaining a HEAD
and an EVEN total on the dice are independent.
[2 marks]
GO ON TO THE NEXT PAGE
02234032/CAPE 2015
-5(b)
A differential equation is given as y′ + y = 2 sin x.
(i)
Determine the general solution of the differential equation.
[8 marks]
(ii)
Hence, or otherwise, obtain the particular solution given that when x = 0, y = 1.
[2 marks]
Total 20 marks
END OF TEST
IF YOU FINISH BEFORE TIME IS CALLED, CHECK YOUR WORK ON THIS TEST.
02234032/CAPE 2015
TEST CODE
FORM TP 2016279
CARIBBEAN
02134020
MAY/JUNE 2016
E XAM I NAT I O N S
COUNCIL
CARIBBEAN ADVANCED PROFICIENCY EXAMINATION®
PURE MATHEMATICS
UNIT 1 – Paper 02
ALGEBRA, GEOMETRY AND CALCULUS
2 hours 30 minutes
READ THE FOLLOWING INSTRUCTIONS CAREFULLY.
1.
This examination paper consists of THREE sections.
2.
Each section consists of TWO questions.
3.
Answer ALL questions from the THREE sections.
4.
Write your answers in the spaces provided in this booklet.
5.
Do NOT write in the margins.
6.
Unless otherwise stated in the question, any numerical answer that is not
exact MUST be written correct to three significant figures.
7.
If you need to rewrite any answer and there is not enough space to do so
on the original page, you must use the extra lined page(s) provided at the
back of this booklet. Remember to draw a line through your original
answer.
8.
If you use the extra page(s) you MUST write the question number
clearly in the box provided at the top of the extra page(s) and, where
relevant, include the question part beside the answer.
Examination Materials Permitted
Mathematical formulae and tables (provided) – Revised 2012
Mathematical instruments
Silent, non-programmable, electronic calculator
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO.
Copyright © 2014 Caribbean Examinations Council
All rights reserved.
02134020/CAPE 2016
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SECTION A
Module 1
Answer BOTH questions.
1.
(a)
Let f(x) = 2x3 – x2 + px + q.
(i)
Given that x + 3 is a factor of f(x) and that there is a remainder of 10, when f(x) is
divided by x + 1 show that p = –25 and q = –12.
[7 marks]
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(ii)
Hence, solve the equation f(x) = 0.
[6 marks]
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(b)
Use mathematical induction to prove that 6 n – 1 is divisible by 5 for all natural
numbers n.
[6 marks]
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(i)
Given that p and q are two propositions, complete the truth table below:
p
q
T
T
T
F
F
T
F
F
p→q
p˅q
p˄q
(p ˅ q) → (p ˄ q)
[4 marks]
(ii)
State, giving a reason for your response, whether the following statements are
logically equivalent:
∙ p→q
∙ (p ˅ q) → (p ˄ q)
................................................................................................................................
.................................................................................................................................
[2 marks]
Total 25 marks
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2.
(a)
Solve the following equation for x:
log2 (10 – x) + log2 x = 4
[6 marks]
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(b)
x+3
A function f is defined by f(x) = ——– , x ≠ 1.
x–1
Determine whether f is bijective, that is, both one-to-one and onto.
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[8 marks]
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DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA - 11 (c)
Let the roots of the equation 2x3 – 5x2 + 4x + 6 = 0 be α, β and γ.
(i)
State the values of α + β + γ, αβ + αγ + βγ and αβγ.
[3 marks]
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(ii)
Hence, or otherwise, determine an equation with integer coefficients which has
1 and —
1 .
1 , —
roots —
α2
β2
γ2
Note: (αβ)2 + (αγ)2 + (βγ)2 = (αβ + αγ + βγ)2 – 2αβγ (α + β + γ)
α2 + β2 + γ2 = (α + β + γ)2 – 2 (αβ + αγ + βγ)
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[8 marks]
Total 25 marks
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SECTION B
Module 2
Answer BOTH questions.
3.
(a)
(i)
Show that sec2 θ =
cosec θ
.
cosec θ – sin θ
[4 marks]
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(ii)
Hence, or otherwise, solve the equation
for 0 < θ < 2π.
4
cosec θ
= —
3
cosec θ – sin θ
[5 marks]
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(b)
(i)
Express the function f(θ) = sin θ + cos θ in the form r sin (θ + α), where
π .
r > 0 and 0 < θ < —
2
[5 marks]
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DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA - 17 (ii)
Hence, find the maximum value of f and the smallest non-negative value of θ at
which it occurs.
[5 marks]
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(c)
Prove that
tan A + tan B + tan C – tan A tan B tan C
tan (A + B + C) = ————————————————— .
1 – tan A tan B – tan A tan C – tan B tan C
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[6 marks]
Total 25 marks
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4.
(a)
(i)
x
Given that sin θ = x, show that tan θ = ————
, where 0 < θ < π . .
2
2
√1 – x
[3 marks]
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Hence, or otherwise, determine the Cartesian equation of the curve defined
π
parametrically by y = tan 2t and x = sin t for 0 < t < 2 .
[5 marks]
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(b)
2
1
Let u = –3 and v = 1
5
2
be two position vectors in R3.
(i)
Calculate the lengths of u and v respectively.
(ii)
Find cos θ where θ is the angle between u and v in R3.
[3 marks]
[4 marks]
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A point, P (x, y), moves such that its distance from the x-axis is half its distance from the
origin.
Determine the Cartesian equation of the locus of P.
[5 marks]
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(d)
The line L has the equation 2x + y + 3 = 0 and the circle C has the equation x2 + y2 = 9.
Determine the points of intersection of the circle C and the line L.
[5 marks]
Total 25 marks
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DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA - 25 SECTION C
Module 3
Answer BOTH questions.
5.
(a)
Use an appropriate substitution to find
∫
–1
(x + 1)3 dx.
[4 marks]
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(b)
The diagram below represents the finite region R which is enclosed by the curve y = x3 – 1
and the lines x = 0 and y = 0.
Calculate the volume of the solid that results from rotating R about the y-axis.
[5 marks]
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Given that
∫
a
0
∫
1
0
f(x) dx =
∫
a
f(a – x) dx
a > 0, show that
0
1
ex
————
dx = — .
x
2
e + e1 – x
[6 marks]
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(d)
An initial population of 10 000 bacteria grow exponentially at a rate of 2% per hour,
where y = f (t) is the number of bacteria present t hours later.
(i)
Solve an appropriate differential equation to show that the number of bacteria
present at any time can be modelled by the equation y = 10 000 e 0.02t .
[7 marks]
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Determine the time required for the bacteria population to double in size.
[3 marks]
Total 25 marks
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6.
(a)
Find the equation of the tangent to the curve f(x) = 2x3 + 5x2 – x + 12 at the point where
x = 3.
[4 marks]
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A function f is defined on R as
x2 + 2x + 3
f(x) =
ax + b
(i)
Calculate the
x<0
x>0
lim
f(x)
x → 0–
and
lim
f(x) .
x → 0+
[4 marks]
(ii)
Hence, determine the values of a and b such that f(x) is continuous at x = 0.
[5 marks]
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(iii)
lim
If the value of b = 3, determine a such that f ʹ (0) = t → 0
f(0 + t) – f(0)
t
[6 marks]
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Use first principles to differentiate f(x) = √ x with respect to x.
[6 marks]
Total 25 marks
END OF TEST
IF YOU FINISH BEFORE TIME IS CALLED, CHECK YOUR WORK ON THIS TEST.
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02134020/CAPE 2016
TEST CODE
FORM TP 2016280
CARIBBEAN
02134032
MAY/JUNE 2016
E XAM I NAT I O N S
COUNCIL
CARIBBEAN ADVANCED PROFICIENCY EXAMINATION®
PURE MATHEMATICS
UNIT 1 – Paper 032
ALGEBRA, GEOMETRY AND CALCULUS
1 hour 30 minutes
READ THE FOLLOWING INSTRUCTIONS CAREFULLY.
1.
This examination paper consists of THREE sections.
2.
Each section consists of ONE question.
3.
Answer ALL questions.
4.
Write your answers in the spaces provided in this booklet.
5.
Do NOT write in the margins.
6.
Unless otherwise stated in the question, any numerical answer that is not
exact MUST be written correct to three significant figures.
7.
If you need to rewrite any answer and there is not enough space to do so
on the original page, you must use the extra lined page(s) provided at the
back of this booklet. Remember to draw a line through your original
answer.
8.
If you use the extra page(s) you MUST write the question number
clearly in the box provided at the top of the extra page(s) and, where
relevant, include the question part beside the answer.
Examination Materials Permitted
Mathematical formulae and tables (provided) – Revised 2012
Mathematical instruments
Silent, non-programmable, electronic calculator
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO.
Copyright © 2015 Caribbean Examinations Council
All rights reserved.
02134032/CAPE 2016
‘‘*’’Barcode Area”*”
Sequential Bar Code
SECTION A
Module 1
Answer this question.
1.
(a)
(i)
Using the substitution y = ex, or otherwise, solve the equation
e2x − 21ex − 100 = 0.
[6 marks]
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(ii)
Solve the inequality | x + 5 | ≥ | 3x + 2 |.
[4 marks]
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(b)
Let p and q be two propositions. Prove that (p ˅ q) ˄ ~ p ≡ ~ p ˄ q.
[6 marks]
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Use a direct proof to show that the sum of two odd numbers is even.
[4 marks]
Total 20 marks
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SECTION B
Module 2
Answer this question.
2.
(a)
(i)
π
Express cos θ – √ 3 sin θ in the form r cos (θ + α), where r > 0 and 0 < α < 2 .
[4 marks]
(ii)
Hence, or otherwise, find the general solution of the equation
cos θ – √ 3 sin θ = 0.
[4 marks]
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DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA -9(b)
Two lines L1 and L2 are defined parametrically by
L1: x = 1 + t, y = 2 + 4t, z = –t
and
L2: x = 2u, y = 3 + u, z = –2 + 3u
where t and u are scalars.
(i)
Find the vector equations of L1 and L2.
[3 marks]
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(ii)
Hence, or otherwise, determine whether the lines intersect, are parallel or are
skewed.
Show ALL working to support your answer.
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[9 marks]
Total 20 marks
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SECTION C
Module 3
Answer this question.
3.
(a)
Diagram A below, not drawn to scale, shows the design of a trough. The cross-section
of the trough, which has the shape of a trapezium, is shown in Diagram B. All lengths are
in metres.
The trough must be made using the dimensions shown, but the angle θ may vary.
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DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA - 13 (i)
Given that the volume, V, is the product of the cross-sectional area and the length
of the trough, show that
V = 10 cosθ (1 + sinθ)
[5 marks]
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(ii)
Hence, or otherwise, determine the MAXIMUM possible volume of the trough.
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[10 marks]
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(b)
Calculate the area enclosed between the curve y = x2 and the line y = 4.
[5 marks]
Total 20 marks
END OF TEST
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TEST CODE
FORM TP 2016282
CARIBBEAN
02234020
MAY/JUNE 2016
E XAM I NAT I O N S
COUNCIL
CARIBBEAN ADVANCED PROFICIENCY EXAMINATION®
PURE MATHEMATICS
UNIT 2 – Paper 02
ANALYSIS, MATRICES AND COMPLEX NUMBERS
2 hours 30 minutes
READ THE FOLLOWING INSTRUCTIONS CAREFULLY.
1.
This examination paper consists of THREE sections.
2.
Each section consists of TWO questions.
3.
Answer ALL questions from the THREE sections.
4.
Write your answers in the spaces provided in this booklet.
5.
Do NOT write in the margins.
6.
Unless otherwise stated in the question, any numerical answer that is not
exact MUST be written correct to three significant figures.
7.
If you need to rewrite any answer and there is not enough space to do so
on the original page, you must use the extra lined page(s) provided at the
back of this booklet. Remember to draw a line through your original
answer.
8.
If you use the extra page(s) you MUST write the question number
clearly in the box provided at the top of the extra page(s) and, where
relevant, include the question part beside the answer.
Examination Materials Permitted
Mathematical formulae and tables (provided) – Revised 2012
Mathematical instruments
Silent, non-programmable, electronic calculator
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO.
Copyright © 2014 Caribbean Examinations Council
All rights reserved.
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SECTION A
Module 1
Answer BOTH questions.
1.
(a)
A quadratic equation is given by ax2 + bx + c = 0, where a, b, c ∈ R. The complex roots
of the equation are α = 1 – 3i and β.
(i)
Calculate (α + β) and (αβ).
[3 marks]
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(ii)
1
1
Hence, show that an equation with roots —— and —— is given by
α–2
β–2
10x2 + 2x + 1 = 0.
[6 marks]
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(b)
Two complex numbers are given as u = 4 + 2i and v = 1 + 2 √ 2i .
(i)
Complete the Argand diagram below to illustrate u.
[1 mark]
(ii)
On the same Argand plane, sketch the circle with equation |z – u| = 3.
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[2 marks]
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DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA -75
(iii)
u
Calculate the modulus and principal argument of z = — .
v
[6 marks]
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(c)
A function f is defined by the parametric equations
x = 4 cos t and y = 3 sin 2t for 0 < t < π.
Determine the x-coordinates of the two stationary values of f.
[7 marks]
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DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA -92x + y
A function w is defined as w (x, y) = ln ——— .
x – 10
∂w .
Determine ——
∂x
2.
(a)
[4 marks]
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(b)
Determine
∫e
2x
sin ex dx.
[6 marks]
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DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA - 11 (c)
x2 + 2x + 3
Let f(x) = ——————
for 2 < x < 5.
(x – 1) (x2 + 1)
(i)
Use the trapezium rule with three equal intervals to estimate the area bounded by
f and the lines y = 0, x = 2 and x = 5.
[5 marks]
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(ii)
3
2x .
Using partial fractions, show that f(x) = —— – ——–
x–1
x2 + 1
[6 marks]
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DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA - 13 (iii)
Hence, determine the value of
∫
5
f(x) dx.
2
[4 marks]
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SECTION B
Module 2
Answer BOTH questions.
3.
(a)
A sequence is defined by the recurrence relation un + 1 = un – 1 + x(un)′, where u1 = 1, u2 = x
and (un)′ is the derivative of un.
For example, u3 = u2 + 1 = u1 + x(u2)′ = 1 + x.
Given that u8 = 13x + 1 and that u10 = 34x + 1, find (u9)′.
[4 marks]
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DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA - 15 (b)
n
The nth partial sum of a series, Sn, is given by Sn = ∑ r(r – 1).
r=1
(i)
n(n2 – 1) .
Show that Sn = ————
3
[7 marks]
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(ii)
20
Hence, or otherwise, evaluate ∑ r(r – 1).
10
[5 marks]
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(c)
(i)
2r
Pr nPr
n!
Given that Pr =
show that ——— is equal to the binomial coefficient
(n – r)!
(2r)!
n
Cr .
n
[4 marks]
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(ii)
Determine the coefficient of the term in x3 in the binomial expansion of
(3x + 2)5.
[5 marks]
Total 25 marks
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4.
(a)
6
The function f is defined as f(x) = √ 4x2 + 4x + 1 for –1 < x < 1.
(i)
1
Show that f(x) = (1 + 2x) –3
.
[3 marks]
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The series expansion of (1 + x)k is given as
2
3
4
1 + kx + k (k – 1) x + k (k – 1) (k – 2) x + k (k – 1) (k – 2) (k – 3) x + ...
2!
3!
4!
where k ∈ R and –1 < x <1.
(ii)
Determine the series expansion of f up to and including the term in x4.
(iii)
Hence, approximate f(0.4) correct to 2 decimal places.
[5 marks]
[3 marks]
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DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA - 21 (b)
The function h(x) = x3 + x – 1 is defined on the interval [0, 1].
(i)
Show that h(x) = 0 has a root on the interval [0, 1].
[3 marks]
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1
with initial estimate x1 = 0.7 to estimate the root
xn + 1
(ii)
Use the iteration xn+ 1 =
of h correct to 2 decimal places.
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[6 marks]
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(c)
Use two iterations of the Newton–Raphson method with initial estimate x1 = 1 to
approximate the root of the equation g(x) = e4x – 3 – 4 in the interval [1, 2]. Give your
answer correct to 3 decimal places.
[5 marks]
Total 25 marks
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DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA - 24 -
DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA - 25 SECTION C
Module 3
Answer BOTH questions.
5.
(a)
A bus has 13 seats for passengers. Eight passengers boarded the bus before it left the
terminal.
(i)
Determine the number of possible seating arrangements of the passengers who
boarded the bus at the terminal.
[2 marks]
(ii)
At the first stop, no passengers will get off the bus but there are eight other persons
waiting to board the same bus. Among those waiting are three friends who must
sit together.
Determine the number of possible groups of five of the waiting passengers that
can join the bus.
[4 marks]
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(b)
Gavin and his best friend Alexander are two of the five specialist batsmen on his school’s
cricket team.
Given that the specialist batsmen must bat before the non-specialist batsmen and that
all five specialist batsmen may bat in any order, what is the probability that Gavin and
Alexander are the opening pair for a given match?
[5 marks]
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DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA - 27 (c)
A matrix A is given as
2
0
–1
(i)
1
4
6
–1
3
0
Find the |A|, determinant of A.
[4 marks]
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(ii)
Hence, or otherwise, find A–1, the inverse of A.
[10 marks]
Total 25 marks
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(a)
Two fair coins and one fair die are tossed at the same time.
(i)
Calculate the number of outcomes in the sample space.
[3 marks]
(ii)
Find the probability of obtaining exactly one head.
[2 marks]
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(iii)
Calculate the probability of obtaining at least one head on the coins and an even
number on the die on a particular attempt.
[4 marks]
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DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA - 31 (b)
Determine whether y = C1x + C2x2 is a solution to the differential equation
x2
— y'' – xy' + y = 0, where C1 and C2 are constants.
2
[6 marks]
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(c)
(i)
Show that the general solution to the differential equation
dy
3 (x2 + x) —– = 2y (1 + 2x) is
dx
3
y = C √ (x2 + x)2, where C ∈ R
[7 marks]
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(ii)
dy
Hence, given that y (1) = 1, solve 3 (x2 + x) —– = 2y (1 + 2x).
dx
[3 marks]
END OF TEST
Total 25 marks
IF YOU FINISH BEFORE TIME IS CALLED, CHECK YOUR WORK ON THIS TEST.
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02234020/CAPE 2016
TEST CODE
FORM TP 2016283
CARIBBEAN
02234032
MAY/JUNE 2016
E XAM I NAT I O N S
COUNCIL
CARIBBEAN ADVANCED PROFICIENCY EXAMINATION®
PURE MATHEMATICS
UNIT 2 – Paper 032
ANALYSIS, MATRICES AND COMPLEX NUMBERS
1 hour 30 minutes
READ THE FOLLOWING INSTRUCTIONS CAREFULLY.
1.
This examination paper consists of THREE sections.
2.
Each section consists of ONE question.
3.
Answer ALL questions.
4.
Write your answers in the spaces provided in this booklet.
5.
Do NOT write in the margins.
6.
Unless otherwise stated in the question, any numerical answer that is not
exact MUST be written correct to three significant figures.
7.
If you need to rewrite any answer and there is not enough space to do so
on the original page, you must use the extra lined page(s) provided at the
back of this booklet. Remember to draw a line through your original
answer.
8.
If you use the extra page(s) you MUST write the question number
clearly in the box provided at the top of the extra page(s) and, where
relevant, include the question part beside the answer.
Examination Materials Permitted
Mathematical formulae and tables (provided) – Revised 2012
Mathematical instruments
Silent, non-programmable, electronic calculator
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO.
Copyright © 2015 Caribbean Examinations Council
All rights reserved.
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SECTION A
Module 1
Answer this question.
1.
(a)
Find the equations of the tangents to the curve y = (x – 1) that are parallel to the
(x + 1)
line x – 2y = 1.
[8 marks]
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DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA -5(b)
(i)
Show that
∫x
n
sin x dx = –x n cos x + n
∫x
n–1
[3 marks]
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cos x dx.
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(ii)
Hence, or otherwise, given that
∫x
calculate
n
∫
0
cos x dx = x n sin x – n
–π2
∫x
n–1
sin x dx,
x 3 sin x dx.
[5 marks]
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DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA -6-
DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA -7(c)
Find the derivative of
x 2 tan –1 e 2x + ln x 2
with respect to x for the domain x > 0.
[4 marks]
Total 20 marks
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SECTION B
Module 2
Answer this question.
2.
(a)
Find
∞

1
 1 
∑  sin  n  − sin  n + 1  
n =1
.
[6 marks]
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(i)
Express the number 0.15 = 0.1515151515... as a geometric series with the first
15
term a = 100 .
[4 marks]
(ii)
Hence, express 0.15 as a ratio of integers.
[2 marks]
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(c)
(i)
Show that the equation
f (x) = e x sin x – 2x
has a root in the interval 0.5 < x < 1.
[3 marks]
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DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA - 11 (ii)
Use linear interpolation to approximate the root of f in the interval 0.5 < x < 1,
correct to 1 decimal place.
[5 marks]
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SECTION C
Module 3
Answer this question.
3.
(a)
A farmer has 300 acres of land on which THREE crops, x, y and z are to be cultivated. The
costs of cultivating x, y and z are, respectively, $30, $40 and $50 per acre and the farmer
has a total of $11000 to spend on cultivation.
For each acre of crop x, y and z, 10, 15 and 40 labour hours, respectively, are required.
A maximum of 6000 labour hours are available.
(i)
Represent the information given with a system of linear equations.
[3 marks]
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DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA - 13 (ii)
By converting the equations into matrix form, determine how many acres per crop
the farmer may cultivate.
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[8 marks]
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DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA - 15 (b)
The vibration of a spring with a mass of 4 kg attached is described by the differential
equation
4
d 2x
+ 400 x = 0,
dt
where x metres is the displacement of the mass at time t seconds.
The initial displacement of the mass is 0.25 m and the initial velocity is 0 (i.e. x′ (0) = 0).
Determine an expression for the displacement, x metres, of the mass at time t seconds.
[9 marks]
Total 20 marks
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