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r %. TEST CODE O2I34O2OMAY/JTINE 2016

_lFORM TP 2016279

CARIBBEAN EXAMINATIONS COUNCIL

CARIBBEAN ADVANICED PROFICIENCY EXAMINATION@

PURE MATHEMATICS

UNITl-Paper02

ALGEBRA, GEOMETRY AND CALCULUS

2 hours 30 minutes

Examination Materials Permitted

Mathematical formulae and tables (provided) - Revised 2012Mathematical instrumentsSilent, non-programmable, electronic calculator

DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO.

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READ THE FOLLOWING INSTRUCTIONS CAREFULLY.

This examination paper consists of THREE sections.

Each section consists of TWO questions.

Answer ALL questions from the THREE sections.

Write your answers in the spaces provided in this booklet.

Do NOT write in the margins.

Unless otherwise stated in the question, any numerical answer that is notexact MUST be written correct to three signiflcant figures.

If you need to rewrite any answer and there is not enough space to do soon the original page, you must use the extra lined page(s) provided at theback of this booklet. Remember to draw a line through your originalanswer.

rf you use the extra page(s) you MUsr write the question numberclearly in the box provided at the top of the extra page(s) and, whererelevant, include the question part beside the answer.

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Examinati on s c ounc i I

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Rectangle

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SECTION A

Module 1

Answer BOTH questions.

l. (a) 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) isdivided by x + I show that p: )5 and q: -12.

[7 marks]

GO ON TO THE NEXT PAGE

02t34020lC.APE 2016

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:-1

+*-.tl:.it.-a-l.:r.-.

:f,i.

,:=-.$+.a,s:',-_'-r-\-.'.iA'.-I+jE-t!+.

.ts--li+.!$::!ti+,.\'lfi--E--t-

]E+:.-e1.-4.

to"s,

-:.r.-::f-i+

-:.1(.=

,...=,

.jEijtiJxJ-g_L-

'fi--r.f

e'€ac

...-aa--a

..a',:

.l-.-

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(ii) Hence, solve the equation (r):0.

[6 marks]

GO ON TO THE NEXT PAGE02t34020/cAPE 2016

I |ilil ffi ililt ilil tilil tilt t!ilt ilil ilil ilt ill llltL 02'13402005 J

t- -6-

(b) Use mathematical induction to prove that 6'- 1 is divisible by 5 for all naturalnumbers n.

[6 marks]


02t34020lcAPE 2016

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IL 0213402006

t- -7 - -l(c) (i) Given that p and q are two propositions, complete the truth table below:

p q p ---+ q pvq p^q (p v q)--- (p n q)

T T

T F

F T

F F

[4 marks]

(ii) State, giving a reason for your response, whether the following statements arelogically equivalent:

p--+q

(pvq)---(pnq)

[2 marks]

Total25 marks

GO ON TO THE NEXT PAGE02134020/0APE 2016

I iltil lilt ililt ilil tilil tilt ililt ilil tilil ilfl til ll] IL 0213402007

-lr2. (a)

02r34020lcAPE 2016

Solve the following equation for x

log, (10 --r) + logrx: 4

-8-

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[6 marks]


.- .-

,ftr

_tL 0213402008

-lrA tunction f is defined by f(x) : ++ , x + |

-9-

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(b)

02134020/CAPE 20t6

Determine whether f is bijective, that is, both one-to-one and onto


JL 0213402009

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[8 marks]

GO ON TO THE NEXT PAGE02134020/0APE 2016

JL 0213402010

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(c)

o2t34020tcAPE 2016

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]


I llil t]ll tllll ffi ilil tilt ffi il]t il] til ilt tiltL 0213402011 J

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(ii) Hence, or otherwise, determine an equation with integer coefficients which has

I)

Y-andI

e1)

o-roots

Note: (og)' + (oy)' + (Fy)' : (ap + ay + Ey)' - 2o-Ft(cr+ B + y;

a, +9, + l, :(o + p + y), -2 (ap + oy + 0y)

02r34020/cAPE 2016

I tililtililt ilil| illl lilll lllll llll lill lllll lllll lll llll


JL 0213402012

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I |ilil ffi ililr lill ililr rilr ilfl il]t il] ilil til H

[8 marks]

Total25 marks


.l

02t34020lcAPE 2016

L 0213402013 I

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3. (a) (i) Show that sec2 0 :

-t4-

SECTION B

Module 2


cosec 0cosecd-sin0

[4 marks]


02t34020lcAPE 2016

r ilil til ilil ilil| ililr ilil llfl llil illl lllil lil lil JL 0213402014

Itr_-.:ftf.Et.--t+-:

..t4.,f,*.j*i--:.'-l{-:,

El\:.Ir;t---\r--.€{.i-I* -:

-_-E_-_.

:--ts+:..€:l.E.]A--l

::'Q',

:,-,=*- -

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Iilil liltil!] lill lillllilllllllllllllllll lllll lil llll

[5 marks]


-l(ii) Hence, or otherwise, solve the equatio, tot-t" d, ,cosecA-sind

for 0 < 0<2n.

4J

02t34020/cAPE 2016

JL 0213402015

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.-.fi:

t- -t6-

(b) (D Express the function f@) : sin 0 + cos d in the form r sin (0 * a), where

r>Oand}<e<TE-2

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[5 marks]


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,-t+-...G.-Q..

::$:.-ao

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(ii) Hence, find the maximum value of f and the smallest non-negative value of 0 atwhich it occurs.

[5 marks]

GO ON TO THE NEXT PAGE02t34020/cAPE 20t6

I tiltil illl ilt] ilil tilil ilil illl illll ult tilt ll] il _lL 0213402017

t-(c) Prove that

tan(A+B+C):

-18-

tan A -r tan B * tan C - tan A tan B tan C

-l

1 - tan A tan B - tan A tan C - tala B tan C

GO ON TO THE NEXT PAGE02r34020/cAPE 2016

JL r iltil ffi ililt ilfl tilil tilt illl tilt tll] ilil flil ll]0213402018

r-Hi.trt

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I ffill llll lllll lllll lllll lllll llll lllll lllll llil llil lil

[6 marksl

Total25 marks


02t34020lcAPE 2016

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4. (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 PAGE02r34020tcAPE 2016

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(ii) Hence, or otherwise, determine the Cartesian equation of the curve deflnedIE

parametrically by y : tan 2t and x : sin r for 0 ., . T.

[5 marks]


:-.

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021340201CAPE 2016

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(b) Let u : and v: be two position vectors in RP

(i)

(ii) Find cos d where d is the angle between u and v in R3

-l2

1

5

Calculate the lengths of u and v respectively.

[3 marks]

[4 marks]

GO ON TO THE NEXT PAGE02t34020tcAPE 2016

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.'.IE+:jH-.:j-a.-E*.i-r:i.

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(c) 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

[5 marksl


02r34020lCAPE 2016

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t- -24- -l(d)

02134020/CAPF. 2016

The line Z has the equation 2x + y + 3 : 0 and the circle Chas the equation x2 + 1? : 9Determine the points of intersection of the circle C and the line L.

[5 marksl

Total25 marks


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.lii-IEi.*-r*

t5-lrg

,*iHr.k.ha..II+:r.l*i--l-!{+ilE+-

.1;::s-:irtl:-A:j.-!r.---

: ::-

-..:.-:

:r={-.{*}-:s'.1-r.4-''tt+.,f;1

.'F+-

.-=.

.Liiii.

.!ii.{iii.'-li'il..

.'H...*.-.

.*.r

..QI:!+r$..::.q.

--:

=i.E..1*t.i.!(..--Ea:

;--t'-.-rk-,;8.tiirE

iEi-:rEIrt{-. Jrr-..-t+l-lilr

.t.i_]*

:.eS...ft

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SECTION C

Module 3


I

[4 marksl


5. (a) Use an appropriate substitution to find J f,

* f f dx

I iltil llll lllll lllll lilll lill lllll llll lllll lllll llllllll

02t34020lcAPE 2016 IL 0213402025

t- -26-

The diagram below represents the finite region R which is enclosed by the curve/ : x3 - |and the lines x:0 andy:6.

y:# -r

-l(b)

02134020/CAPE 2016

v

x0 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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L 0213402026

:-::xrrdfr'ij:i+

llia-:g1\r.:IAn.i*- -tfai

.-E*

-tR:':t-t*

f;*:.l\

.=e:a..O:'.4-l

:=_.at

..-E

.:::E;ta'.I*iE.-t=E..\

i.+.i,..se....€'

,]:.ft

r .\1 -l(c) Given that Io (r) dx : J, n" - x) dx a ) O,show that

dI Id-x

A +7- 2

[6 marksl

GO ON TO THE NEXT PAGE02t34020lcAPE 2016

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.,fi.{it=.'.o....Q

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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) Solve an appropriate differential equation to show that the number of bacteriapresent at any time can be modelled by the equation y : l0 000 e0.o2t

[7 marks]


-l(d)

02t34020/cAPE 2016

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iNr-r','lEt

:.ft...!rir.-..t5..:-j{---:',tsi:-Hir:

::kl-:\---'liil'..F{jjJi.;---

'Si._-E_:

r-Fr.-.-..€:.,r*.lii$:ij:+:i

;-t*:.'rOl:.'R.:rG:i,:Ei

==

=l:S,..1*I...*.i-iit.Jxt:.iliil.8,,.kirl-,...Ir}

::lt:.:F{-E--

i-tsi

rEli.Q

=t.Q

=

r..E!..-.iIi;I:irs:...:S..*..-.8.----E-.rtti:;--tr+.tsnI+.!+

r-:t!ii.-J\

,:ld.'-\-]---N

i-:l+-r -S*.

''tiuo---A.l..--r.-t

t- -29 -

(iD Determine the time required for the bacteria population to double in size

-l

[3 marksl

Total25 marks


02t34020lcAPE 2016

IL I lflil !illt ililt ililt tilil tilt ffi lill l]ll lill llil llll0213402029

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..fi.,

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6. (a) Find the equation of the tangent to the curve f(x) :2x3 + 5f - x + 12 at the point wherex:3

[4 marks]

GO ON TO THE NEXT PAGE02t34020lcAPE 2016

I ilIil ilfl ililt lil tilil tilt illl tilt til iltil ilt til JL 0213402030

Iil.I\1- j'{iii:Ef'.

.H.-

TI!I*r---\r-E-.:N-jt--

:.......:.;8.:j.Irtir:f;{:.,' ::ta:',

EtEii:

EEItsr-.,J*r---.

F4:iX-ilrt.-S+,,j$...,E::;

sr,Ei.:.

f-----.tr-_-_-_._.

:i=

'1=

;=ljjji-:

ilttir*i--Sttiri':l€- l.+i;-!.5i.-hir

,'zIIT+

li.lti-Ed:-_s--'lJ-,

r.Ei:Eii.rtsi-A--- \a,..4

r -31 - -l(b) A function f is defined on R as

i+2x+3 -x<0f(r) :

ax+b x>0

lim f(x)x --- 0-

o2t34020lcAPE 2016

r iltil ffi ilil ilil rilil lflr lllll llll lll! lllll lil llll

(i) Calculate the and

[4 marksl

(iD Hence, determine the values of a and D such that (r) is continuous at x : 0.

[5 marks]


lim f(x)r --- 0*

IL 0213402031

r 32-

(iii) If the value of b :3, determine a such that f '(0) :

-llim

I ---+ 0

(0+r-f(0)t

[6 marks]


JL I []il Ull ilU lffi illlt lut !]il illlt ffi ilfl ll] fft0213402032

'.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.riAl---*J-:

tf,ir

.=-.

:fn.:tliijitlit-: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_---:ei.'

-lr -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

r tilil ffi ffir ilil rilil ilil ilil illr lllll lffi llfl lill JL 0213402033

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EXTRA SPACE

If you use this extra page, you MUST write the question number clearly in the box provided.

Question No.

02134020/CAPE 2016

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i...,

::r..::t-ir\'.-.J*Irft..-:liti.La-:JrS-:5ii:!|rr--.iti[.1

kilri'.'.irir-:tirt-.HN+ri:.'ES.

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Fi-:rili-..

*iAjjA'-:

=r{*}.S+r'.'_\'.-.

I.li',{ii-i.E:L;rE{-+:1

!ErrLii:.1!!t-l!(-J--a*rS":''S!:.E.,$:-S:r

tlri-:i]=['-f,qtxr.'_,t-_-l

ts:,Ef-liiu.rfii.-t+.*;T.:t+.reti:8..

F,.l_ijijjjr

:--.=

l r-1-ji

::=

ri:\ft--ji'+rt"r.-8,.jjSf-:

..t-.r--J€-::i*.,.-Fi..-k-.-----)hi.

i-:E+::it{j--!{i l:i.S{.,! -r-.---.

[:E+:l..si.t.Ei,.$,[.a:ai:.r.'

:rn-

ri.=...*-.:.:Id:.:rs.'----:_-.

:4.n-'H+-l'F'.:-si-_:

:Irt.. tii--_-lrt j.-!{..-- ir-:__El

..fi.-

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E-.ltsa-

.:-iEf.

..ti.---lrr-_

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:.t+:..s...k16...e.a:=

=

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EXTRA SPACE


Question No.

02r34020/cAPE 2016

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EXTRA SPACE


Question No.

02t34020lcAPE 2016

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EXTRA SPACE


Question No.

02t34020lC.APE 2016

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[i=t -----.| '''''f - ---I:i:--:trr-lt.----t-----l-.t-------f-----'-f-------f--ji:t-------t-.:St':f.-rr.i.F.f,(:'f-::Et-:

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EXTRA SPACE


Question No.

021340?arcAPE 2016

L J0213402038

highereducationcentrett.weebly.com...Created Date 5/11/2016 4:52:42 PM

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