IFORM TP 2016282 ~ ®

TEST CODE 02234020 IMAY/JUNE 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 FOLLOWING INSTRUCTIONS CAREFULLY.

1. This examination paper consists of THREE sections.

2. Each section consists of TWO questions.

3. Answer ALL 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 correct to three significant figures.

7. 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.

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, whererelevant, include the question part beside the answer.

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.

Copyright © 2014 Caribbean Examinations CouncilAll rights reserved.

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SECTION A

Module 1

Answer BOTH questions.

1. (a) A quadratic equation is given by ax' + bx + c = 0, where a, b, C E R. The complex rootsof the equation are a = 1- 3i and 13.

(i) Calculate (a + 13)and (al3).

[3 marks]

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(ii) Hence, show that an equation with roots a ~2 and ~ ~ 2 is given by

lOx2 + 2x + 1 = O.

[6 marks]

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(b) Two complex numbers are given as u = 4 + 2i and v = 1 + 2 {2i .

(i) Complete the Argand diagram below to illustrate u.

6

5

4

3

2

1

-2 -1 1 2 3 4 5 6 7 8-1

-2

[1 mark]

(ii) On the same Argand plane, sketch the circle with equation [z - ul = 3.[2 marks]

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......--':, .. ::,-,-:,',"

,-:::::, ::::::-":::,'::, ::,',-:,-:

·."."0-.-:.' .. ',','::_-~'

:::~:.:::~:::::::St: ::~.::::.~ r,"- -- ,Iii.~~~ :~~:~:

111···&··:··-1:&··:::·"'- ,--,

, " '-,- -,

:::e::::e::::.~ ...~ ...--, ,- --~- - - -~~ " :.:-,:e.....:..:e::.- -- - -' ..- " ..-,~.. : .:~.. ::.:--- - - :, - --:--:-:-:':.:. :-:':.:-:-:-:

-7-

5

(iii) Calculate the modulus and principal argument of z = [ ~ ] .

[6 marks]

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[7 marks]

(c) A function f is defined by the parametric equations

x = 4 cos t and y = 3 sin 2t for 0 :S t:S 1!.

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

Total 25 marks

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---------------~-----------~:------- -,

-::~::::-:---:::

:;:i~I~t---~--~---- --- ---r: ------L-_'_----~:----<""I---

Ii!II

2.

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(a) A function w is defined as w (x, y) = In 1 ~~ ~~ I·

D. Ow

etermme - .ox

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[4 marks]

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I" -10-

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(b) Determine f ell sin e dx.

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[6 marks]

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------ ----------- ------

----- -----------------

--_.--- ----------- ------

---------------------- ------

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ec) Let fex) = X2 + 2x + 3 for 2 ~ x ~ 5.ex - 1) (x2 + 1)

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

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[5 marks]

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(ii) Using partial fractions, show that f(x) = ~1 __ }x .x- x + 1

[6 marks]

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----:------- :::::---------:::: -:------:----- ---:::-- -:::::----

:::--::- :::::::-:::::: :::::::-

---::--:- -:_---::::~-~-=-~-~-~----~-:-:-:-:-:-:

:--::: -::--:::-

-:--::- :------:--:

::::::- -::::::------- -------

~-=-=-=-:-:-= ::-:-:-:-:-:-:

------- -------------- -------

::-:-:--:m:-:i~ll-}~:::::~:::

- l3 -

5

(iii) Hence, determine the value of f f(x) dx.2

[4 marks]

Total 25 marks

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SECTIONB

Module 2

Answer BOTH questions.

3. (a) A sequence is defined by the recurrence relation U = U 1 + x(u )" where u1 = 1, u2 = Xn+ 1 n- n

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

For example, u = u = u1

+ x(u2

)' = 1 + x.3 2 + 1

Given that Us = 13x + 1 and that ulO = 34x + 1, find (u9)'.

[4 marks]

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------1-------.,--------------,---------------1_-_-_----------

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--------------'._---------------------------1_-_-_-_-_-_--_--------------j-------

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(b)n

The nth partial sum of a series, S , is given by S = L r(r - 1).n n

r= 1

(i) n(n2 -1)Show thatS =n 3

[7 marks]

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20

(ii) Hence, or otherwise, evaluate L r(r - 1).10

[5 marks]

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0223402016

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-_--_-_-_-_-_-c--------------

(c)

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, =r np(i) Given that "P: = (n :'r)! show that (2r)! r is equal to the binomial coefficient

"C .r

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[4 marks]

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(ii) Determine the coefficient of the term in x3 in the binomial expansion of(3x + 2)5.

[5 marks]

Total 25 marks

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6,,---;:---4. (a) The function f is defined as f(x) = -\f4x2 + 4x + 1 for -1 < x < 1.

I

(i) Show that f(x) = (1 + 2X)3 .

[3 marks]

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._------

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The series expansion of (1 + xt is given as

l+kx+ k(k-l)x2 + k(k-1)(k-2)x3 + k(k-1)(k-2)(k-3)x4 + ...2! 3! 4!

where k E Rand -1 < x <1.

(ii) Determine the series expansion of f up to and including the term in X4.

[5 marks](iii) Hence, approximate f(O.4) correct to 2 decimal places.

[3 marks]

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I~-=-o:-:-:-:.:.:..-------- -------

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~~:::::

::~~:~~:~~~:~:-.~ .....~:-

:~::~:~::~::~:~:Jt~J.:~:::

I~il~~~S;~~I;SI-.~- ....~---- -- - ---- ---- --- -- - ---- -- ---- - --

I§~~~~~~~EI=-5:···-5-·-=-=~-=--=-)~;;.~=-:-----------------::e:::::::C:::-i"'i<:- .i"'i<:.r~::::::~:::

I!;~I~:;:;:~;;t~;;;;;I~~~~~

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(b) The function hex) = x3 + x-I is defined on the interval [0, 1].

(i) Show that hex) = 0 has a root on the interval [0, 1].

[3 marks]

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(ii)

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Use the iteration x I = 1 with initial estimate XI = 0.7 to estimate the rootn+ x2+1

n

ofh correct to 2 decimal places.

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(c) Use two iterations of the Newton-Raphson method with initial estimate XI = 1 toapproximate the root of the equation g(x) = e4x-3 - 4 in the interval [1,2]. Give youranswer correct to 3 decimal places.

[5 marks]

Total 25 marks

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5.

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SECTIONC

Module 3

Answer BOTH questions.

(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 marks]

(ii) 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]

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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 probability that Gavin andAlexander are the opening pair for a given match?

[5 marks]

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(c) A matrix A is given as

I~! ~lll-l 6 0

(i) Find the IAI, determinant of A.

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[4 marks]

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[10 marks]

Total 25 marks

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

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6.

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(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 marks]

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(iii) Calculate the probability of obtaining at least one head on the coins and an evennumber on the die on a particular attempt.

[4 marks]

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(b) Determine whether y = C1x + C2X2 is a solution to the differential equation

~2y" - xy' + Y = 0, where Cj

and C2

are constants.

[6 marks)

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

3 (~ + x) : = 2y (1 + 2x) is

y=C~(~+X)2, whereCER

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0223402032

[7 marks]

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(ii) Hence, given thaty (1) = 1, solve 3 (x2 + x)~ = 2y (1 + 2x).

END OF TEST

[3 marks]

Total 25 marks

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

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