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FORM TP 2016279
TEST CODE O2I34O2O
_l
MAY/JTINE 2016
CARIBBEAN EXAMINATIONS COUNCIL
CARIBBEAN ADVANICED PROFICIENCY EXAMINATION@
PURE MATHEMATICS
UNITl-Paper02
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.
J
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 signiflcant 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
rf you use the extra page(s) you MUsr 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.
C opyri ght
L
., X, ff
Examinati on s c ounc i I
o
02134020|CAPE 2016
I
!,fPJfrt
tiltil illl ilt] ilil tilt tilt
lil ilil !1il tilil ilil til
0213402003
J
:-1
r
4
-|
SECTION A
Module 1
Answer BOTH questions.
(a)
l.
Let f(x) :2x3 - f + Px + q.
(i)
Given that x + 3 is a factor of f(x) and that there is a remainder of 10, when (x) is
divided by x + I show that p: )5 and q: -12.
+*.tl:.
it.
-a-l
.:r.-.
:f,i.
,:=.$+.
a,s:
',-_'
-r-\-.'
.iA'.
-I+j
E
-t!+.
.ts-li+.
!$:
:!ti+,
.\'l
fi
--E-t]E+:
.-e1
.-4.
to
"s,
:.r
.-::
f-i+
-:.1(
.=
,...=,
.jEi
jti
JxJ
-g
_L-
'fi
--r.f
[7 marks]
e
'€
GO ON TO THE NEXT PAGE
02t34020lC.APE 2016
L
r
ilil !il til ilil lilll lill lllll lllll llll llll lilllill
0213402004
I
ac
...aa--a
..a',:
.l-.-
r
5
(ii)
-l
Hence, solve the equation (r):0.
[6 marks]
GO ON TO THE NEXT PAGE
02t34020/cAPE 2016
L
I
|ilil ffi ililt ilil tilil tilt t!ilt ilil ilil ilt ill lllt
02'13402005
J
t-
-6(b)
-l
Use mathematical induction to prove that 6'- 1 is divisible by 5 for all natural
numbers n.
[6 marks]
GO ON TO THE NEXT PAGE
02t34020lcAPE 2016
L
I
ll]ililfl ilil fiil llill ]lll ffi lllll llll llll lil lill
0213402006
I
t-
-l
-7 -
(c)
(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
pvq
p^q
(p v q)--- (p n q)
[4 marks]
(ii)
State, giving a reason for your response, whether the following statements are
logically equivalent:
p--+q
(pvq)---(pnq)
[2 marks]
Total25 marks
GO ON TO THE NEXT PAGE
02134020/0APE 2016
L
I
iltil lilt ililt ilil tilil tilt ililt ilil tilil ilfl til ll]
0213402007
I
r
2.
-8(a)
-l
Solve the following equation for x
log, (10 --r) + logrx: 4
[6 marks]
GO ON TO THE NEXT PAGE
02r34020lcAPE 2016
L
r
tilil til ilil ililr ililr lllll lil llll llll lllll llil llll
0213402008
_t
.- .,ftr
r
-l
-9-
(b)
A tunction f is defined by f(x) :
++ , x + |
Determine whether f is bijective, that is, both one-to-one and onto
GO ON TO THE NEXT PAGE
02134020/CAPE 20t6
L
I
il]il illl ilil ililt ililt ilil ffi rilr r]ll ffi flil ilt
0213402009
J
r
-l
- l0-
[8 marks]
GO ON TO THE NEXT PAGE
02134020/0APE 2016
L
I
lililt ilil till ilil tilil tilt illl llll illll llll llll
0213402010
lll
J
r
-l
- ll (c)
Let the roots ofthe equation 2x3 - 5l + 4x + 6 : 0 be o, B and y
(D
State the values of a + I + y, o0 * oy + By and opy.
[3 marks]
GO ON TO THE NEXT PAGE
o2t34020tcAPE 2016
L
I
llil t]ll tllll ffi ilil tilt ffi il]t il] til ilt tilt
0213402011
J
r
-12-
(ii)
-l
Hence, or otherwise, determine an equation with integer coefficients which has
roots
1
)
o-
I
e
and
I
)
Y-
Note: (og)' + (oy)' + (Fy)' : (ap + ay + Ey)' - 2o-Ft(cr+ B + y;
a, +9, + l, :(o + p + y), -2 (ap + oy + 0y)
GO ON TO THE NEXT PAGE
02r34020/cAPE 2016
L
I
tililtililt ilil| illl lilll lllll llll lill lllll lllll lll llll
0213402012
J
r
-13-
.l
[8 marks]
Total25 marks
GO ON TO THE NEXT PAGE
02t34020lcAPE 2016
L
I
|ilil ffi ililr lill ililr rilr ilfl il]t il] ilil til H
0213402013
I
r
-t4-
-l
SECTION B
Module 2
Answer BOTH questions.
3.
(a)
(i)
Show that sec2 0 :
cosec 0
cosecd-sin0
[4 marks]
GO ON TO THE NEXT PAGE
02t34020lcAPE 2016
L
r
ilil til ilil ilil| ililr ilil llfl llil illl lllil lil lil
0213402014
J
r
Itr_-.:
(ii)
ftf.
Et
.--t+-:
-l
-15tot-t" d,
cosecA-sind,
Hence, or otherwise, solve the equatio,
4
J
for 0 < 0<2n.
..t4.,
f,*.
j*i--:
.'-l{-:,
El
\:
.Ir;t
---\r--.
€{.
i-I*
-:
-_-E_-_.
:--ts+:
..€:l
.E.
]A--l
::'Q',
:,-,=*- -
[5 marks]
.G
.-.fi:
GO ON TO THE NEXT PAGE
02t34020/cAPE 2016
L
Iilil liltil!] lill lillllilllllllllllllllll lllll lil llll
0213402015
J
t-
_l
-t6-
(b)
(D Express the function f@) : sin 0 + cos d in the form r sin (0 * a), where
r>Oand}<e<TE
-2
,-t+-.
..G.
-Q..
::$:.
-ao
[5 marks]
GO ON TO THE NEXT PAGE
02t34020/cAPE 2016
L
I
llllil lilr ilil ilil tilil tilt ilfl tilt t]] tilt lil
0213402016
ill
J
t-
-l
-t7 -
(ii)
Hence, find the maximum value of f and the smallest non-negative value of 0 at
which it occurs.
[5 marks]
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02t34020/cAPE 20t6
L
I
tiltil illl ilt] ilil tilil ilil illl illll ult tilt ll]
0213402017
il
_l
t-
-l
-18-
(c)
Prove that
tan(A+B+C):
* tan C - tan A tan B tan C
1 - tan A tan B - tan A tan C - tala B tan C
tan A -r tan B
r-H
i.trt
GO ON TO THE NEXT PAGE
02r34020/cAPE 2016
L
r
iltil ffi ililt ilfl tilil tilt illl tilt tll] ilil flil ll]
0213402018
J
r
-t9-
_l
[6 marksl
Total25 marks
GO ON TO THE NEXT PAGE
02t34020lcAPE 2016
L
I
ffill llll lllll lllll lllll lllll llll lllll lllll llil llil lil
0213402019
I
-jjjjil
:_-:-_-1
t4.
-l
-20-
(a)
(i)
Given that sin d: x, show that tan e: ___L, where 0 < e < a
^tt
-7
2
[3 marks]
GO ON TO THE NEXT PAGE
02r34020tcAPE 2016
L
I
ilil ffi il] tffi ililt ilfl l]ll ililt !]ll ililt ilt
0213402020
il
-J
r
-l
-2t -
(ii)
Hence, or otherwise, determine the Cartesian equation of the curve deflned
parametrically by y : tan 2t and x : sin r for 0 ., .
IE
T.
:-.
!l
[5 marks]
GO ON TO THE NEXT PAGE
021340201CAPE 2016
L
I
liltil lllll lllll lllll lilll lill lill llll lllll lllll lil llll
0213402021
I
r
-l
11
(b) Let u :
ill
lrj
2
and v:
1
be two position vectors in RP
5
(i)
Calculate the lengths of u and v respectively.
(ii)
Find cos d where d is the angle between u and v in R3
[3 marks]
[4 marks]
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02t34020tcAPE 2016
L
r
rilil lil ilil ril tilil ilfl ffi ilil ffi ll]t til flt
0213402022
J
r
-23-
(c)
-l
A point, P (x, y), moves such that its distance from the x-axis is half its distance from the
oflgln.
Determine the Cartesian equation of the locus of P
.'.IE+:
jH-.
:j-a.
-E*.
i-r:i.
[5 marksl
GO ON TO THE NEXT PAGE
02r34020lCAPE 2016
L
I
ilil lllll lllll lllll lill lill lllll llll lllll lllll llllllll
0213402023
_t
t-
-24(d)
-l
The line Z has the equation 2x + y + 3 : 0 and the circle Chas the equation x2 + 1? : 9
Determine the points of intersection of the circle C and the line L.
[5 marksl
Total25 marks
GO ON TO THE NEXT PAGE
02134020/CAPF. 2016
L
r
rilil lilt flil ililt ililt ililt ffi ililt ilfl illil ilil til
0213402024
J
r
-l
a<
SECTION C
.lii
-IEi
.*
-r*
Module 3
t5
-lrg
,*
.k
.ha.
Answer BOTH questions.
iHr
.II+:r
.l*i--l
-!{+il
I
5.
(a)
Use an appropriate substitution to find
J
f, * f f
dx
E+-
.1;:
:s-:i
rtl
:-A:j
.-!r.---
: :
:-..:.-:
:r={
-.{*}-
:s'.
1-r
.4-'
'tt+.,
f;1
.'F+-
.-=.
.Liiii.
.!ii.
{iii.
'-li'il..
.'H.
..*.-.
.*.r
..QI:
!+
[4 marksl
r$..
::.q.
--:
=
i.E
..1*t.
i.!(.
.--Ea:
;--t'.-rk-,
;8.
tiir
E
iEi-:
rEI
rt{Jrr-.
.
.-t+
l-lilr
.t.i
_]*
:.e
S.
..ft
GO ON TO THE NEXT PAGE
02t34020lcAPE 2016
L
I
iltil llll lllll lllll lilll lill lllll llll lllll lllll llllllll
0213402025
I
t-
-l
-26(b)
The diagram below represents the finite region R which is enclosed by the curve/ : x3 - |
and the lines x:0 andy:6.
v
0
y:# -r
x
I
Calculate the volume of the solid that results from rotating R about the y-axis
,.E
-!i+i
[5 marks]
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02134020/CAPE 2016
L
I
illilt tilfi ililt ilt! ilfl ]il tilil il]t tilfi ilil ilfi
0213402026
til
J
'.-.
'.:
: '...1
r-- l
l:: _-l]
::... I
:'-. ]
r
.\1
(c)
:-::xr
Given that
rd
fr
'ij:i+
I
-l
:
)
Io (r) dx J, n" - x) dx a O,show that
d
A +7-
d-x
I
2
.,fi
.{i
t=
.'.o
....Q
lli
a-:g
1\r
.:IA
n.i*
-
-tfai
.-E*
-tR
:':t-t*
f;
*
:.l\
.=
e:
a.
.O:
'.4-l
:=
_.at
..-E
.:::E
;ta
'.I*i
E
E
..\
.-t=
i.+.i,
..s
e
....€'
[6 marksl
,]:.ft
GO ON TO THE NEXT PAGE
02t34020lcAPE 2016
L
il|]il ililililI ilil tilil til[llfl rill l]ll lill lill llll
0213402027
I
r
-28(d)
-l
An initial population of 10 000 bacteria grow exponentially at a rate of 2%o per hour,
where y : f (t) is the number of bacteria present / hours later.
(i)
iNr-r'
,'lEt
:.ft..
.!rir.-
Solve an appropriate differential equation to show that the number of bacteria
present at any time can be modelled by the equation y : l0 000 e0.o2t
..t5..
:-j{---:
',tsi
:-Hir:
::kl
-:\--'liil'
..F{j
jJi.;---
'Si.
_-E_:
r-Fr.-.-
..€:.,
r*.l
ii$:ij
:+:i
;-t*:.'
rOl
[7 marks]
GO ON TO THE NEXT PAGE
02t34020/cAPE 2016
L
I
ll]il illl ilt] ilil tilfl il]t ull ililt til ilil flil ilt
0213402028
_t
:.'R.:
rG:i
,:Ei
=
=
t-
-l
-29 -
(iD Determine the time required for the bacteria population to double in size
=
l:S,
..1*I.
..*.
i-iit
.Jxt:
.iliil
.8,
,.ki
rl-
,...Ir}
::lt:
.:F{
-E--
i-tsi
rE
li.Q
=t.Q
=
r..E!..-.
iIi;I:
irs:
...:S.
..-.8.
.*
----E-.
rtti:
;--tr+
.ts
nI+.
!+
r-:t!ii.
-J\
,:ld.
'-\-]
---N
i-:l+r -S*.
''ti
uo
---A.l
..--r.-t
[3 marksl
Total25 marks
.S'
..fi.,
GO ON TO THE NEXT PAGE
02t34020lcAPE 2016
L
I
lflil !illt ililt ililt tilil tilt ffi lill l]ll lill llil llll
0213402029
I
r
6.
-l
-30-
(a)
Find the equation of the tangent to the curve f(x) :2x3 + 5f - x + 12 at the point where
x:3
Iil.
I\1- j
'{iii:
Ef'.
.H.-
TI!I
*r---
\rE-.:
N-j
t--
:.......:
.;8.:j
.Irtir
:f;{:.,
[4 marks]
'ta:',::
Et
Eii:
EEI
tsr-.,
J*r---.
F4:
iX-il
rt.-
S+,,j
$...
,E::;
sr,
GO ON TO THE NEXT PAGE
02t34020lcAPE 2016
L
I
ilIil ilfl ililt lil tilil tilt illl tilt
til iltil ilt til
0213402030
J
Ei.:.
f-----.
tr-_-_-_._.
:i=
'1=
;=
ljjji-:
r
(b)
A function f is defined on R as
iltt
ir*
i--St
f(r) :
tiri
':l€
-
-l
-31 -
l.+i
;-!.5i
.-hir
i+2x+3
-x<0
ax+b
x>0
,'z
IIT+
(i)
li.lt
Calculate the
i-Ed
lim
x --- 0-
f(x)
and
lim
f(x)
r --- 0*
:-_s--'l
J-,
r.E
i:Ei
i.rts
i-A-- \a,
..4
[4 marksl
(iD Hence, determine the values of a and D such that (r) is continuous at x : 0.
[5 marks]
GO ON TO THE NEXT PAGE
o2t34020lcAPE 2016
L
r
iltil ffi ilil ilil rilil lflr lllll llll lll! lllll lil llll
0213402031
I
r
-l
32-
(iii)
If the value of b :3, determine a such that f '(0) :
lim
(0+r-f(0)
I ---+ 0
t
'.N..
..ITT.:.
jEi:
.'S(r.
.ta-:.
--JE1--'
..tri:
-
l:a-:.
:U.i
.-.ha-.
rI$::
:-1\J-j
--l*ii:
.'E!'-.
-s_---l
_E_-_l
-H-:,
rI}-:-
E.r
iAl
---*J-:
tf,ir
.=-.
:fn.
:tliij
itlit-:r
:i(1
.'*;
'f=:l
..Ql.:
..fn
..Gti
,.Q:l
--i#i:
H.:
ftij
.E-:.
p-;
'lri.-
:E;.
Ei-
.tr-:
-1ri-:
itti:
.ti.::
-Jr+---
:t*.:
:I-i-
*--,
.t=t"
','A_'-'
-\J_---
[6 marks]
GO ON TO THE NEXT PAGE
02t34020/cAPE 2016
L
I
[]il Ull ilU lffi illlt lut !]il illlt ffi ilfl ll] fft
0213402032
J
:ei.'
r
-l
-33-
(c)
Use first principles to differentiate f(x)
: {Twith respect to x
[6 marks]
Total25 marks
END OF TEST
IF YOU FINISH BEFORE TIME IS CALLED, CHECK YOUR WORI( ON THIS TEST.
02134020|CAPE 2016
L
r
tilil ffi ffir ilil rilil ilil ilil illr lllll lffi llfl lill
0213402033
J
r
-l
-34-
:,_-_,_,'__.
i...,
::r
..::t
EXTRA SPACE
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If you use this extra page, you MUST write the question number clearly in the box provided.
:liti.
La-:
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iti[.1
ki
lri'.'
Question No.
.irir-:
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02134020/CAPE 2016
r
rilril ilil rffi ilil rilil tilr ilfl il]t
ilil ilil til ll]
0213402034
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l_ijijjjr
:--.=
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::=
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Question No.
-----)hi.
i-:E+:
:it{j
--!{i l
:i.S{.,
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[:E+:
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a:=
=
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02r34020/cAPE 2016
L
I
tilil ull ilu tffi ilil ll]t !il llll
!il lllll llll lil
0213402035
_l
r
-36-
-l
EXTRA SPACE
If you use this extra page, you MUST write the question number clearly in the box provided.
Question No.
L
02t34020lcAPE 2016
r
tilil ltil ilil tilt tilil ilil ffi il]t ilfl ililt til til
0213402036
J
I-jjjir
[i=
-----.
|f -'''''
--t
I:i:--:
trr-l
t.---t-----
l-.
t------f-----'f-------
f--ji:
r
-37 -
-l
t-------
t-.:St':
EXTRA SPACE
f.-rr.i.
F.f,(:'
f-::Et-:
i.!a,.
If you use this extra page, you MUST write the question number clearly in the box provided.
l15+l
i- rtt
trH.
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Question No.
F+;F+-,
l.
{iaj
l..si
rE--lr.Li,
f:-€i.
IE
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|..Q.
| ''''-'
-=
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.i\f
...E15.
-Ei--'
':t{i'
L.-ts*:-
:.E:
.-!i+.
.'-.+rf-.
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---.lril
:f(i
'.-+-,
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...G,:
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.rS:
...- f
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02t34020lC.APE 2016
L
I
fiil t]ll ilu tffi ililt tilt illl tilt ffi ilil til ilt
0213402037
J
r
-38-
-l
EXTRA SPACE
:.1$.:
-lET-.,
If you use this extra page, you MUST write the question number clearly in the box provided.
.E:i
.lil::
.'lA-:.
JrAr
l-i:r
-lSiir
ti{--:
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Question No.
--rhi--:
.{$,i
.s{:
-ir--.1
FEI.:]
:N-:i
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