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

CARIBBEAN 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 so

on 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.

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SECTION A

Module 1

Answer BOTH questions.

A quadratic equation is given by "* + bx + c : 0, where a, b, c e R. The complex rootsof the equation are cr: 1 - 3i and B.

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

[3 rnarks]

GO ON TO THE NEXT PAGE

1. (a)

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02234020/CAPE 2016

lOx2+2x+l:0

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

[6 marks]


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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

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...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

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A function f is defined by the parametric equations

x:4 cos tandy:3 sin 2tfor0 <t <TE.

Determine the x-coordinates of the two stationary values of f.

-l(c)

022340201CAPE 2016

[7 marks]

Total25 marks


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.1E[.:*.,..'El6-.:r{-

:!r+i-*.:tii:..ft'.-r!ri:

,IiJ,.-hi.:+i+l.,8{-}\j

..=.:

...k$::,.s-------

.14,-E:'.Fi-r'Er'i-J-riii-.-ir},:,--Sr.\:-!(.'-_--s-:--*...,t=..rG.,..t.r:S:...i.-q.

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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


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rI(b) Determine ea sin e'dx.

022340201CAPE 2016

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[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 1

5x2

-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

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022340201CAPE 2016

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[6 marks]


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

.fti.l

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

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(iii) Hence, determine the value of f(r) dxI5

2

[4 marks]

Total25 marks

GO ON TO THE NEXT PAGE02234020/CAPE 2016

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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

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::}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

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(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-

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20

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

[5 marks]


022340201CAPE 2016

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(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]


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

(ii)

[5 marks]

Totat 25 marks


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4. (a)

o2234020tCAPE 2016

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

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

*4x+ I for-l <x< l.

[3 marks]


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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 <x<l

(ii) Determine the series expansion of f up to and including the term in xa

(iii) Hence, approximate f(O.a) correct to 2 decimal places[5 marksl

[3 marks]


(}:.:itt:

022340201CAPE 2016

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..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:.

-=

-=

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(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]


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-lr(ii)

022340201CAPE 2016

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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

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[6 marksl

GO ON TO THE NEXT PAGEo2234020lCAPE 2016

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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

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02234020|CAPE 2016

IJse two iterations of the Newton-Raphson method with initial estimate x, : 1 to

approximate 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


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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 the

terminal.

(i) Determine the number of possible seating arrangements of the passengers who

boarded the bus at the terminal.

[2 marksl

(iD 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 thatcan join the bus.

[4 marks]


022340201CAPE 2016

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5.

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(b) Gavin and his best friend Alexander are two of the five specialist batsmen on his school'scricket 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]


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

:-:H..:-H-.

'::rE:..-.\a_

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

.'i*r

..sr'Ei.-A

r*..

.ts,!!+i

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

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

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

.%..

E:tr=i.

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(c) A matrix A is given as

-1J

0

(i) Find the lAl, determinant of A

022340201CAPE 2016

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[4 marks]


-l

2

0

-1

I46

JL 0223402027

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

[10 marks]

Total25 marks

GO ON TO THE NEXT PAGE022340201CAP8 20t6

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.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

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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

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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]


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

.tl

022340201CAPE 2016

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..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]


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(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]


022340201CAPE 2016

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: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

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.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

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BXTRA SPACE

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

Question No.

02234020/CAPE 2016

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highereducationcentrett.weebly.com...Created Date 5/25/2016 9:17:06 PM