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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.

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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

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

the curve at the Point

[4 marksl

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(+,+)

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

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[3 marksl

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

[5 marks]

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

[3 marksl

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-l(d)

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

[5 marksl

Total25 marks

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2. (a) Determine

I(i) xt cos (x3)dx

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[8 marksl

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[5 marksl

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2re

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dx.

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

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[8 marksl

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f x4+ Il;641y ax

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[4 marksl

Total25 marks

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SECTION B

Module 2

Answer BOTH questions.

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

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

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[2 marksl

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(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'

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[8 marksl

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-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

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expressed as/(x) :Ik=O

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02234020/CAPE 2018

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[8 marksl

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

[3 marksl

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

.i,, ( * )

[4 marks]

Total25 marks

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

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[4 marksl

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

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[6 marksl

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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

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(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.

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[8 marksl

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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

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SECTION C

Module 3

Answer BOTH questions.

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).

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[3 marksl

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

[3 marksl

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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

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-l(b)

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

without repetitions?

[5 marksl

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-l(c)

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(D

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

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02

A

1

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

[5 marksl

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

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-2 3l341o 6)

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[4 marksl

Total25 marks

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02234020/CAPE 2018

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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)

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

[9 marks]

[2 marks]

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

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

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-l(b)

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[4 marksl

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(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.

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[10 marksl

Total25 marks

END OF TEST

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

022340201CAPE 2018

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