stfgcmathminds.weebly.com · 2019. 9. 17. · created date: 5/25/2016 9:17:06 pm

r %. TEST CODE 02234020 -lMAY/JLINE 2016FORM TP 2016282CARIBBEAN EXAMINATIONS COUNCIL

CARIBBEAN ADVANCED PROFICIENCY EXAMINATION@

PURE, MATHEMATICS

UNIT 2 -Paper 02

ANALYSIS, MATRICES AND COMPLEX NUMBERS

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

READ THE F'OLLOWING INSTRUCTIONS CAREFULLY.

1. This examination paper consists of THREE sections.

2. Each section consists of TWO questions.

3. AnswerALL 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 notexact MUST be written corect to three significant flgures.

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.

If you use the extra page(s) you MUST 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.

7

8

Copyright @ 2014 Caribbean Examinations CouncilAll rights reserved.

I tilil lffi rilr ilil rilil rilr rilll lill lllll lllr rfl lil0223402003L 022340201CAPE 2016 _l

DadRectangle

-lI- -4-SECTION A

Module 1

Answer BOTH questions.

A quadratic equation is given by "* + bx + c : 0, where a, b, c e R. The complex roots

of the equation are cr: 1 - 3i and B.

(i) Calculate (cr + F) and (crB).

[3 rnarks]

GO ON TO THE NEXT PAGE

1. (a)

02234020tC4P8 20t6

ililtil ffi till ilil tilil tlililil iltil ilil ilil ilililt IL 0223402004

r 5 -l(ii)

02234020/CAPE 2016

lOx2+2x+l:0

Hence, show that an equation with roots Lo _, u"O fr ir given by

[6 marks]


ililil tilIilffi ilil tilflil[]ilililil tll] til til JL 0223402005

r -6-Two complex numbers are given as u : 4 + 2i and v : | + 2 \azi

(D Complete the Argand diagram below to illustrate z.

-2 -1-1

-2

(ii) On the same Argand plane, sketch the circle with equationlz - ul:3.


-l(b)

1

12345678

[1 mark]

[2 marks]

02234020/CAPE 2016

ill]il ffi tililIil tilil ilil ilililt ililililil il JL 0223402006

...t+,ris:.:s--_---'--ia':::{it.,-F,r"!i+:.-=.iE+l'-'l*.:-Gt-r\l

-.1E.:-_*-_.:--:l

..t=l'r-'Q}l,:.:E:,

i$,,,.s.

-lr -7 -(iiil Calculate the modulus and principal argume nt of z: [+]

[6 marks]

GO ON TO THE NEXT PAGE02234020|CAPE 20t6

_lL I llllil ffi tilt ilil tilil tilt !ilfl tilt t]il ilil til til0223402007

t- -8-A function f is defined by the parametric equations

x:4 cos tandy:3 sin 2tfor0

r -9- -l

.t..

.qi.

.R.

.-€i:,.ftr

2. (a)

022340201CAP8 20t6

A tunction w is defined as w (x. v): In I Z:llvg go l} \^rJ,,, ,,,I x- to I

Determine 0w0x

[4 marksl


I iltil illl tilt ililt tilil ilil ilfl tilt t]il til llil llll JL 0223402009

rI(b) Determine ea sin e'dx.

022340201CAPE 2016

- 10-

illril ffi rililril tilil tilt ililil[Il] lilt !ilI til

[6 marks]


-l

JL 0223402010

-.-

=;:=

=

=i.i;4.'l. J*,t

:.f,d

..t5.'Jr1-liii:.Hk..tt+.I$:.h,:-t-.jFS:-E_.:;in.:-Ft':_A_-'--\f,'tiA-.:-\9-':-{':::=

rla-!Ia-'E-F+-.'-2ii...+.+.

',*1,-.lii.-.\-.:{HiE'i',t-+'

.os,:::t

=

::.*:r+.€,.t*-:gd-l

:]rri--E+..1*(-

_,___-t_1

ts..r!i-.:Ell.8,.-Jr+:

F+,-t-:'rr\.-:

r 15x2

-l(c) Let f(x) : -212i-3 1o,

(i) Use the trapezium rule with three equal intervals to estimate the area bounded byf and the linesy : 0, x:2 and x: 5.

[5 marksl

GO ON TO THE NEXT PAGE02234020tCAPB 2016

l lillil llll illt ilil1 tilil !ilt llll tilt il] ilililil flL 0223402011 J

-lr(iD

022340201CAPE 2016

-12-

I ll]ililll tillllilllilll illlll]ll lill ill lllll lll llll

[6 marks]


Using partial fractions, show that f(r) : * - ;i

.fti.l

.ei.*j-A:l.\r.-.-..Q..

IL 0223402012

-lr -13-(iii) Hence, determine the value of f(r) dxI

5

2

[4 marks]

Total25 marks

GO ON TO THE NEXT PAGE02234020/CAPE 2016

ililil ilililil[! til[il[]ilill llll llil lil lllL 0223402013 I

-lr -t4-SECTION B

Module 2


A sequence is defined by the recurrence relation u n + 1: u, _.r I x(u,)' , where u r: l, ur.: x

and (u,)' is the derivative of u,.

For example,u3: u2*r: ur* x(ur)': | + x.

Given that ur: l3x + I and that u,o: 34x + l, flnd (ar)'.

[4 marksl


3. (a)

022340201CAP8 20t6

_lL ililtil ililtil ilil tilil tilt ilililflll] tilt ililll0223402014

::}i'----!rl

-\.{*t.*..t{,.%-_.lri..,E.h;

.Ql,.Q.

iisf l

rJ...f(.----Ea-j!altrij*.,--.

.-.trr',

=.,.1\::EJI.t*.:-\:

::-^i(,iix.j1l:

.:e.:_rar:'\Jr..k:

_1r(i) Show that S :

-15-

n(nz - l)J

ililil til tililtil tilil til[]ilil[]!ililflil til

(b) The n'h partial sum of a series, Sn, is given UV S,: I r(r - l).n

ts:+

[7 marks]


-JL 0223402015

n

'l*t..:..ift-

-lr -t6-20

(ii) Hence, or otherwise, evaluate I r(r - l).

[5 marks]


022340201CAPE 2016

r ll]il illl till ilil lilll lill lllll ll]l lllll lill lil lll _tL 0223402016

l0

-lr -17 -(c) (i)

02234020/0APE 2016

Given that"P :r,C

, nl ,. show that ':P 'P(n - r)! tr is equal to the binomial coefficient

[4 marks]


1 ||ilil uiltil ilil tilil il]flililut il] il]ilil ffiL 0223402017 _l

_lr - 18 -Determine the coefficient of the term in x3 in the binomial expansion of(3x + 2)s.

(ii)

[5 marks]

Totat 25 marks


o2234020lcAPE 2016

ililililllllllllllllllllllllllllllllllllllllllllllllllllll _tL 0223402018

.lr -19-

ih....G.:*-

4. (a)

o2234020tCAPE 2016

The function f is defined as f(-r):

(i) Show that (x) : (t + Z4!

*4x+ I for-l

-lr -20 -The series expansion of (1 +,r)k is given as

t+tGa k(k-l)x2 1 k(k-l)(k-2)x3 a k(k-l)(k-2)(k-3)xa "..2t 3! 4l

whereteRand-1

..r:E:.i:-I$-:.:fr..

:-_--_-'

=6.. --trr+--'

..-Eh.

,.l'k"!+i.r-r}:.'-'Ff:rlsa:.:-^ :

i+i-a{!i-j

i.=.:l.^:'--\J-

:a:

,.n=

i'E..:lr},.-ts,..1$..,:lic':l!\a-i-t:'''-i+rrts.i-!i+1.{*.n:ti+i'-'-\--.$:.:--.lir--l

l:'I.;'l.e}L:ErIo,I.s.+-------I..l-_r --1-------t-- ___;j......-l

J..-:..t---r'r

fN1.:€'lI.;i,1-----\--'l--Ezr--:

t$;:$].i+j.:=f,;-:i:l=iia.S--ts,-.

':it.r$Q

rlsr:4l:.

-=

-=

-lr -2t -(b)

02234020/CAPE 2016

The function h(x) : x3 + x - 1 is deflned on the interval [0, l].

(i) Show that h(x) :0 has a root on the interval [0, 1].

[3 marks]


JL I tilil tffi ilil flil tilt !ilt t]ll tilt ull ilil il] il0223402021

-lr(ii)

022340201CAPE 2016

"la

I ilil illl ilil| ilil ililt lilt !il il]t !il lffi lil lil


Use the iteration x with initial estimate x : 0.7 to estimate the rootn+1 *n'*l

I

of h correct to 2 decimal places.

IL 0223402022

2..l-i:.

.lr -23-

I tiltil ffi tilt fiil tilil ]lil ull tilt ffi llil lil tilt

[6 marksl

GO ON TO THE NEXT PAGEo2234020lCAPE 2016

JL 0223402023

s..nIIT..,ft'.'---i*--'vl-..:_Ei:_-'j\i-

:.*i'.h{{:rl+t lj,+.

.--\r--.si+:-:t;::-:F1..eil:-:t...€,.:e[.

..E.,rr+..S...:{..#..!ri-:.E-t+.._r>1l=

i-!El:,..trr-..1Er'!q:r+'-

:r+,::c}.: .

t.-i

iE,t\{:.

Fr.tii-'!t"'1*,t

it\:f:rA--

.E::::

.-s.

.ri,,

1-:l

r -24- -l(c)

02234020|CAPE 2016

IJse two iterations of the Newton-Raphson method with initial estimate x, : 1 toapproximate the root of the equation g(x) :

"4x-3 - 4 in the interval ll,2). Give your

answer correct to 3 decimal places.

[5 marks]

Total2S marks


JL I ffiil Ull il]t tilt tilil ilil tffi tilt tffi tilt til il0223402024

.rt?.-IIf--

ElS+-

ts..{ii-:l+.,tia:..\--1

s{.:E--j

.t=,.--s*.'.E.,.9-.1:S.,

=

==.

.:.-:.

.Itt-,fr.--'

.U+.-.lFi-j

I{ij--Et-,

>...:riii-.

--'"rd.s.!l

-lI- -25 -SECTION C

Module 3


(a) A bus has 13 seats for passengers. Eight passengers boarded the bus before it left theterminal.

(i) Determine the number of possible seating arrangements of the passengers whoboarded the bus at the terminal.

[2 marksl

(iD At the first stop, no passengers will get off the bus but there are eight other personswaiting to board the same bus. Among those waiting are three friends who mustsit together.

Determine the number of possible groups of five of the waiting passengers thatcan join the bus.

[4 marks]

GO ON TO THE NEXT PAGE022340201CAPE 2016

ililtilffi tililtilrililrilrillllillffi llllllilllll

5.

IL 0223402025

r -26-(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 thatall five specialist batsmen may bat in any order, what is the probabiliry that Gavin andAlexander are the opening pair for a given match?

[5 marks]


ililil 1il tililtil rilil tilr ililil[]ilililililt

-l

JL 0223402026

,N..ft.i::*-'.'I4-

:-:H..:-H-.

'::rE:..-.\a_

"1I+.-..h{i,lsij-.-ng:

.'i*r

..sr'Ei.-Ar*..

.ts,!!+i

-,I+-tr'.\-:*:'*.-.-E

.-tr+..G..-4..,Q..S:

'.i]ri-.r.r*---rlrii._ttiltri-.'.=;'\i-'

.%..

E:tr=i.

r -27 -(c) A matrix A is given as

-1J

0

(i) Find the lAl, determinant of A

022340201CAPE 2016

I ilil ffi tilt ilil tilil iltil ilfl tilt ffi tilt illt il]

[4 marks]


-l

2

0

-1

I46

JL 0223402027

-lt- -28 -(ii) Hence, or otherwise, find A-r, the inverse of A.

[10 marks]

Total25 marks

GO ON TO THE NEXT PAGE022340201CAP8 20t6

JL I lllril lil rffi ilil tilil tilt til il!il ull ilil ril til0223402028

.I\[.rE'..tzf--.-JEil

H:,r-F*-:'

_-a-lrELlrt.

''A-il.-_\d r,.,f,:

'- -'':l=!l:rI}-.g...X.,::#'.._-AXir

'I+'.b*.E.

i.l-ii.-I*1.Ei.

..1Et-.ft.:ii-'---E-_.t=G'itr.o.Q

-lr -29-6. (a) Two fair coins and one fair die are tossed at the same time

(i) Calculate the number of outcomes in the sample space.

(ii) Find the probability of obtaining exactly one head.

[3 marks]

[2 marksl

GO ON TO THE NEXT PAGE02234020|CAPE 2016

I lilll llll ililt ilil tilil illlt ililt ilil ffi ilil llil il JL 0223402029

-ll- -30-(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]


--F[',l!\-i.

.tl

022340201CAPE 2016

I ilil Ull till lllll lilll lill !]ll lill lil lllil lll llll JL 0223402030

..S,

:ci:.A\---.t+-

L'i:

:-:!1:i*iiEi

ji!$j

:..t Iii5+.:H.--.t-.-

::Iq'

.'!$.--E-:r_j _ _'-F1'.

:.€:..:k.ii.s.----r\---r!!rl--

.-ICj

.S.".:$-.:ir1::.{i+-ll'.E--S+--.*jj!i+lifrl-,!-.t+.i-\t-l,.9---N,.

-=::::ii..rs,r.t(-r._- -!:+ -:_-_---Y -rr-th-:.rit.-lt -.t\a.-.

;=i.E+-l:.I+J:,

-t-i:.:-\-:''I(nj+:j

..t:.R

t:-s:..E:

r - 31 -(b) Determine whethery : Cr* * C{'is a solution to the differential equation

* r" - xy' + y: o, where C, and Crarcconstants'2"

-l

[6 marks]


L I ilil illl tilt ilil tilil till]ll iltil tilil ilfl til til0223402031 J

r -32-(c) (i) Show that the general solution to the differential equation

-l

Y:C

3(f+i*:2y(r+2x) is

'.'l 1* * ry, where c e R

[7 marks]

GO ON TO THE NEXT PAGE022340201CAPE 2016

IL I rilril illl tffi illil tilil tilt t]ll tilr ilil llill ril lfl

:X[.+j.n4:t:.r^-:}rA--5i-:'.'J{+:.E'.-.k".-.r\i...

.Fr..--FLl

s;.t--..Q.-ieil1A.i.--\J-:. -!*l-:

+Et::s.-

:.{ta.--tri .'-lra -E:l{..i..kj

-:t!ia-:_

.-ft).,

.ty.--1\--.-l(-.rl:

-i\:

iHi...G,:.E-..(!rrr+t--.

.r\1,

.--8.

..:=S.,-t t--,:Ei-- -f-i:.t=.'=irEi-j-.*):..t-i{-,:-It 'rd,

:-t-.

0223402032

..itl

.r*t,f;(-r.8.

-lr -33-(ii) Hence, given that y(1) : 1, solve 3 (f + x)Y : 2y (l + 2x).

[3 marks]

Total25 marksEND OF TEST

IF YOU FINTSH BEFORE TIME IS CALLED, CHECK YOUR WORI( ON THIS TEST.

02234020/CAPE 2016

l lffi llilililllll tilil tilflllililflil tffi til il

'_---A-\J

=

L 0223402033 J

t- -34-BXTRA SPACE

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

Question No.

02234020/CAPE 2016

I tilil illl tilt ilil tilil tilt lil lill illl lllll llll llll

-l

,k.:r=i --

L 0223402034 _l

stfgcmathminds.weebly.com · 2019. 9. 17. · created date: 5/25/2016 9:17:06 pm

Documents