Cape Pure Mathematics Paper 2 2016

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FORM TP 2016282

02234020

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MAY/JLINE 2016

CARIBBEAN EXAMINATIONS COUNCIL CARIBBEAN ADVANCED PROFICIENCY EXAMINATION@ PURE, MATHEMATICS

UNIT 2 -Paper 02 ANALYSIS, MATRICES AND COMPLEX NUMBERS 2 hours 30 minutes

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 not exact MUST be written corect to three significant flgures.

7

If you need to rewrite

any answer and there is not enough space to do so page, you must use the extra lined page(s) provided at the on the original

back of this booklet. Remember to draw a line through your original answer. 8

If

you use the extra page(s) you MUST write the question number clearly in the box provided at the top of the extra page(s) and, where relevant, include the question part beside the answer.

Examination Materials Permitted Mathematical formulae and tables (provided) - Revised 2012 Mathematical instruments Silent, non-programmable, electronic calculator

DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO

SO

Copyright @ 2014 Caribbean Examinations Council All rights reserved.

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-4SECTION A

Module

1

Answer BOTH questions.

1.

(a)

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

(i)

Calculate (cr + F) and (crB).

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(ii)

Hence, show that an equation with roots

Lo_,

u"O

fr

ir given by

lOx2+2x+l:0

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-6(b)

Two complex numbers are given as u : 4 + 2i and v

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| + 2 \azi

Complete the Argand diagram below to illustrate z.

1

-2

12345678

-1 -1

-2

[1 mark]

(ii)

On the same Argand plane, sketch the circle with equationlz

- ul:3.

[2 marks]

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Calculate the modulus and principal argume nt of

z: [+]

...t+

,ris:

.:s-_---

'--ia':

::{it.

,-F,

r"!i+:

.-=. iE+l

'-'l*.

:-Gtr\l

-.1E.

:-_*-_. :--:l

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

i$,, ,.s.

[6 marks]

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-8(c)

A function f is defined by the parametric equations .1E[.

x:4 cos tandy:3 sin 2tfor0