r -l - MATH MINDS · 2019. 9. 17. · r % TEST CODE 02234020-l FORM TP 2018301 MAY/JUNE 2018...

$: r -l - MATH MINDS · 2019. 9. 17. · r % TEST CODE 02234020-l FORM TP 2018301 MAY/JUNE 2018 CARIBBEAN EXAMINATIONS COUNCIL C ARIB B EAI\ AD VA NC E D PROF'IC IE N C Y XA1VIINATIO$
r % TEST CODE 02234020-lMAY/JUNE 2018FORM TP 2018301

CARIBBEAN EXAMINATIONS COUNCIL

C ARIB B EAI\ AD VA NC E D PROF'IC IE N C Y E XA1VIINATIO N@

PTJRE MATHEMATICS

UNIT 2 -Paper 02

ANALYSTS, MATRTCES AI\D COMPLEX NUMBERS

2 hours 30 minutes

Examination Materials Permitted

Mathematical formulae and tables (provided) - Revised 2012

Mathematical instrumentsSilent, non-programmable, electronic calculator

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

READ THE FOLLOWING INSTRUCTIONS CAREFULLY.

1. This examination paper consists of THREE sections

2.

3.

4.

5.

6.

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 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

onthe original page, you must use the extra page(s) provided at the back ofthis booklet. Remember to draw a line through your original answer.

8. 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, where

relevant, include the question part beside the answer.

Copyright @ 2Ol7 Caribbean Examinations CouncilAll rights reserved.

r ililllililllillflillilllffillllllllllllllllllllllllll|_02234020/CAPE 2018 0223402003 I

-lr -4-

SECTION A

Module I

Answer BOTH questions.

A curve P is defined parametrically as x = #r, y: *,l. (a) (i)

Determine the gradient of the curve at the point ( + , +)

[5 marksl

GO ON TO THE NEXT PAGE

022340201CAPE 2018

IIilill]ilffi ililililt]llfifl]illlllllil]fllllilfl JL 0223402004

-lr 5

(iD Hence, or otherwise, determine the x and y intercepts of the tangent that touches

the curve at the Point

[4 marksl


(+,+)

022340201CAP8 2018

L rlllilffi ilillllllilililllllllllllllllllllffi llllllll IIt'f,':&

0223402005

r -6-

(b) Let the function/(x, y) = sin (Ix) sin (alcy). Determine

02234020/CAPE 2018

ililililfi |ff ilil ilil ilil[ffi lil[fl ilflilfl lllt

-ld f@,v\

0x 0y

[3 marksl


_tL 0223402006

r -7 -

(c) Use DeMoivre'stheoremto showthatsin 50: sin5 0-lO sin3 0 cos2 0 * 5 cosa 0sin0

[5 marks]


022340201CAP8 2018

L tlllilffi illlfflllllllllllllillllilllllllllllllilllll

-l

Ir.Y.'r$

0223402007

r -8-

(i) Write the complexnumber z=(l -r) in theformreia where r:lz I and0: arg(z).

[3 marksl


-l(d)

02234020/CAPE 2018

1|illlflil]r|IilillrflrlllrL 0223402008 J

-lr 9-

(ii) Hence, show that (1 - i)': 16(1 - i)

[5 marksl

Total25 marks


022340201CAP8 2018

ililil lllll lill lllll lllll llll lllll lfll lllll lllll lil llll IL 0223402009

-lr -10-

2. (a) Determine

I(i) xt cos (x3)dx

GO ON TO THE NEXT PAGE02234020/CAPE 2018

I ffiil lllilffi lffil tflilIillll illlr lill illil llilil JL 0223402010

-lr - 11-

rlllilffi ililffililillllllllllllllllfllllllllllllllll

[8 marksl


022340201CAP8 2018

L Il(r.jis

0223402011

-lr(ii) I

02234020/CAPE 2018

-t2-

ililllllill |ililflt illt ]il[Ilfll]Iilfltil il]ffit

[5 marksl


2re

Jr - "o'

dx.

JLY.!F

L 0223402012

r

L

(b) (i)

022340201CAPE 2018

-13-

ililil[ffi ililllllilillfillllllllllllllllllllllllllil

GO ON TO TIIE NEXT PAGE

-lUse partial fractions to showthat #+ = + -#,

lQY.,"4

0223402013 I

-lr -14-

[8 marksl


ilililr ffi ililffi ililil]t ffi illlt flIl fl]illilil JL 0223402014

r

L

-15- -l(ii) Hence, determine

f x4+ Il;641y ax

rlllil ffi flilffililllllllilffi illllllllllllllfllll]

[4 marksl

Total25 marks


02234020|CAPE 2018 IlQ'f,T/t

0223402015

-lr -16-

SECTION B

Module 2


Asequence {a,} is suchthat or= 0 dfrddn*r: Jr+q

(a) (i) State the third term, ar, of the sequence.

02234020/CAPE 2018

ilililillll lill tffi fl il ]]il]ililfl]ilrff ]il Iil

[2 marksl


3

JL 0223402016

-lr -t7 -

(ii) Use mathematical induction to prove that x, is increasing and that it is bounded

above by 3, that is,4, ( a*, and o, ' 3 for all z € N'


02234020|CAP8 2018

L ilililtillillllillllilllllllllilllllllllllllllllllllil IIrJ,ii+

0223402017

-lr -18-

I ffifi l]ll lffi illlt ilil lil flfl lr]r illl il] ilt til

[8 marksl

GO ON TO THE NEXT PAGE022340201CAPE 2018

L 0223/i02018 J

r(-1)o/o

kl

-ts' -l(b) (i) Letf (x): "-". By calculating the first three non-zero terms and assuming that

the pattem continues, show that the Maclaurin series expansion of f (x) may be

@

expressed as/(x) :Ik=O


02234020/CAPE 2018

r tflfl ilfl ililt ilil ilil il|il llill ffilllll lrlll ill lil IL 0223402019

-lr -20 -

r ilil| lill llll lril ilil ]flr ilfi ilIll ilfl lilr ffi lllt

[8 marksl

GO ON TO THE NEXT PAGE02234020(CAP8 2018

IL 0223402020

r

L

-21 -

(iD Hence, or otherwise, determine the values of x for which the expansion is valid.

[3 marksl

-l

(c) Determine the sum of the ,".i", ,;, (.,"( ; )

.i,, ( * )

[4 marks]

Total25 marks


02234020|CAPE 2018

ililil |ililililffilililItffi fl]l llll lill lll lil IItr,'1lF

0223402021

-lr -22 -

4. (a) Determine the coefficient of the term in x7 in the expression

022340201CAPE, 2018

ilililililffi ffiililillilllillllllllllllllll lllllll

(. - *)'

[4 marksl


L 0223402022 I

r

L

-23 -

(b) By expressing nC, and 'C,-, in terms of factorials, show that"C, * 'C,-r- '*'C,

-l

[6 marksl


02234020/CAPE 2018

r lllil ffi lllfllil ilil lfll lllilut ffil rllll fiil lllr Il(l.t'i.ts

0223402023

r(c) (i)

-24 -

Use the intermediate value theorem to show that the equation 4 cos r - x3 + 2:0has a root in the interval (1, 1.5).

[3 marksl


I ilil| lllll lllfl rllll ilil ]lll lllll lllll lffi lllll llll llll

-l

02234020/CAPE 2018 IL 0223402024

-lr -25 -

(ii) Use linear interpolation to approximate the value of the root of the equation4 cos x - f + 2 = 0 in the interval (1, 1.5), correctto two decimal places.

GO ON TO THE NEXT PAGEo2234020lCAPE 2018

L ililil]ilililil1 ilfi il[ffi illil fliilffi ]il tilr IEtv*

0223402025

r -26 -

illllillll flffi ffiilill til[]illll llil fiil ]il llll

[8 marksl


-l

022340201CAP8 2018

JL 0223402026

r

L

-27 - -l(d)

022340201CAPE 2018

The equation 3d = | - 2 ln.r is known to have a root in the interval (0, 1).

Taking xr= 0.2 as the first approximation of the root, use the NeMon-Raphson methodto find a second approximation, rr, of the root in the interval (0, 1).

[4 marksl

Total25 marks


ilililililililffi ililil 1ffi flil fl]flililr]ilililil JETjI+

0223402027

r -28 -

SECTION C

Module 3


5. (a) Events A and.B are such that P(A): 0.4, P(B) :0.45 and P(A w B) = 0'68

(i) Calculate P(A a B).

-l

[3 marksl

(ii) Determine whether the events A and B are independent. Justiff your response.

[3 marksl


022340201CAPE 2018

ililfi]ilffi lllllilillilrillllllllllllllllllllllll IL 0223402028

r

L

-29 -

A committee of 4 persons is to be chosen from 8 persons, including Mr Smith and his

wife. Mr Smith will not join the committee without his wife, but his wife will join the

committee without him.

Calcutate the number of possible ways the committee of 4 persons can be formed

[5 marksl

GO ON TO TFTE NEXT PAGE

-l(b)

02234020/CAPE 2018

ffi il ilil fl ll tffi ilil[lllilffi ilI ililil1 ilil||r II,Y.l{E

0223402029

r -30-

How rnany odd numbers greater than 500 000 can be made from the digits 2, 3, 4, 5 , 6, 7 ,

without repetitions?

[5 marksl


-l(c)

02234020/CAPE 20r8

l lllil ffi ffi 1 |ril ililil1fi|il |flflfl ilil[]ilfl IL 0223402030

r - 31 - -l(d) Two matrices, are given as

(D

-2 3-)3 4 | n:0 6)

5

02

A

1

By finding AB, deduce that A-t : gg B.

[5 marksl


IL 0223402031

(rs t2 -i)| -s 24 2ol

[.0 4 s)

ffi

-lr -32-

(ii) Hence, or otherwise, solve the system of equations given as

7

l14l=z)

-2 3l341o 6)

G

lj

[4 marksl

Total25 marks


02234020/CAPE 2018

I llllll ffi lill ltil ilIil lilt ilfll ilIlr ffi ilil ilI tilr JL 0223402032

r -33-

A differential equation is given as y' cos x = y sin x + sin 2x.

(i) Show that the general solution of the differential equation is

!r..r-cosrt.Csecx.2

-l6. (a)

022340201CAP8 2018GO ON TO THE NEXT PAGE

ililil 1ililililffi ilfi illl llil illlffiff lilll ilil llll IL 0223402033

-lr -34-

(ii) Hence, or otherwise, solve the initial value problemy'cos x:y sin x + sin 2x, Y (0):0.

[9 marks]

[2 marks]


022340201CAP8 2018

tflilllllllilltllI!ilfllllilillllllllffi llllllllllll IL 0223402034

r

L

-35-

A differential equation is given as yo + 2y' + y: xe-'.

(D Determine the solution of the complementary equation yn + 2y' 't y: 0


-l(b)

02234020/CAPE 2018

[4 marksl

ililil |]flril rffi ilil ]il ilflililffiilIl flil!1r IftT,#

0223402035

-lr -36-

(ii) Given that the particular solution of a differential equation has the form

ye= (Af + Bx2)e-', or otherwise, determine the general solution of the differential

equation.


illlil rilll lril illll tililil[]ilur ffi ilfl il lllt JL 0223402036

-lr -37 -

[10 marksl

Total25 marks

END OF TEST

IF YOU FINISII BEFORE TIME IS CALLED, CHECK YOUR WORK ON THIS TEST.

022340201CAPE 2018

il|1ilffi il illiltil ilil fl il1il| il|ffiffil llil ll] llllL 0223402037 J

r -l - MATH MINDS · 2019. 9. 17. · r % TEST CODE 02234020-l FORM TP 2018301 MAY/JUNE 2018...

Documents

Transcript of r -l - MATH MINDS · 2019. 9. 17. · r % TEST CODE 02234020-l FORM TP 2018301 MAY/JUNE 2018...