CAPE® Pure Mathematics Past Papers 2005-2014

Caribbean Examinations Council

Pure Mathematics

CAPE® PAST PAPERS

Macmillan Education 4 Crinan Street, London, N1 9XW A division of Macmillan Publishers Limited Companies and representatives throughout the world www.macmillan-caribbean.com ISBN 978-0-230-48274-6 AER © Caribbean Examinations Council (CXC ®) 2015 www.cxc.org www.cxc-store.com The author has asserted their right to be identified as the author of this work in accordance with the Copyright, Design and Patents Act 1988. First published 2014 This revised version published 2015 All rights reserved; no part of this publication may be reproduced, stored in a retrieval system, transmitted in any form, or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publishers. Designed by Macmillan Publishers Limited and Red Giraffe Cover design by Macmillan Publishers Limited Cover photograph © Caribbean Examinations Council (CXC ®) Cover image by Mrs Alberta Henry With thanks to: Krissa Johny

CAPE® Pure Mathematics Past Papers LIST OF CONTENTS Unit 1 Paper 02 May 2005

4

Unit 1 Paper 03/B May 2005

10

Unit 2 Paper 02 June 2005

14

Unit 2 Paper 03/B May 2005

19

Unit 1 Paper 01 May 2006

24

Unit 1 Paper 02 May 2006

31

Unit 1 Paper 03/B May 2006

36

Unit 2 Paper 01 May 2006

40

Unit 2 Paper 02 May 2006

46

Unit 2 Paper 03/B May 2006

51

Unit 1 Paper 02 May 2008

55

Unit 1 Paper 03/B May 2008

60

Unit 2 Paper 02 May 2008

63

Unit 2 Paper 03/B May 2008

68

Unit 1 Paper 02 June 2008

72

Unit 1 Paper 03/B June 2008

78

Unit 2 Paper 02 July 2008

82

Unit 2 Paper 03/B June 2008

85

Unit 1 Paper 02 May 2009

89

Unit 1 Paper 03/B June 2009

96

Unit 2 Paper 02 May 2009

99

Unit 2 Paper 03/B June 2009

104

Unit 1 Paper 02 May 2010

107

Unit 1 Paper 03/B June 2010

114

Unit 2 Paper 02 May 2010

118

Unit 2 Paper 03/B June 2010

124

Unit 1 Paper 02 May 2011

128

Unit 1 Paper 03/B June 2011

135

Unit 2 Paper 02 May 2011

138

Unit 2 Paper 03/B June 2011

145

Unit 1 Paper 02 May 2012

148

Unit 1 Paper 032 June 2012

154

Unit 2 Paper 02 May 2012

158

Unit 2 Paper 032 June 2012

165

Unit 1 Paper 02 May 2013

169

Unit 1 Paper 032 June 2013

175

Unit 2 Paper 02 May 2013

180

Unit 2 Paper 032 June 2013

186

Unit 1 Paper 02 May 2014

190

Unit 1 Paper 032 June 2014

197

Unit 2 Paper 02 May 2014

203

Unit 2 Paper 032 June 2014

209

TEST CODE

FORM TP 2005253 CARIBBEAN

02134020

M AY /JUNE 2005

EXAMINATIONS

COUNCIL

ADVANCED PROFICIENCY EXAMINATION PURE MATHEMATICS UNIT 1 - PAPER 02 2 hours

( 25 MAY 2005 (p.m.))

This examination pape r consists of THREE sections: Module l , Module 2 and Module 3. Each section consists of 2 questions. The max imum mark for each section is 40 . T he maximum mark for thi s examination is 120 . Th is exa mination consists of 6 pages .

INSTRUCTIONS TO CANDIDATES

1.

DO NOT open this examination paper until instructed to do so.

2.

Answer ALL questions from the THREE sections.

3.

U nless otherw ise stated in the question, any numerical answer that is not exact MUST be writte n correct to three significant figures.

Examination Materials Mathematical formulae and tables Electronic calcul ator G raph paper

Copy right © 2004 Caribbean Examinations Counci l All rights reserved. 02 134020/CAPE 2005

- 2-

Section A (Module 1) Answer BOTH questions.

1.

(a)

(i)

Complete the table below for the function

ll;lll (ii) (b)

(c)

-1

-2 8

I

0

I

I

Sketch the graph of f(x)

1 I

I f(x) I, wheref(x) = x (2- x).

2 I

0

3 I

4

I

8

I

[ 2 marks]

I for -2 ::=:; x ::=:; 4.

[ 4 marks]

Find the value(s) of the real number, k, for which the equation k(x 2 + 5) has equal roots.

x2 [ 6 marks]

[ 4 marks]

( i)

(ii)

= 6 + 12x -

Without using calculators or tables, evaluate

[ 4 marks] Total 20 marks

GO ON TO THE NEXT PAGE 021 34020/CAPE 2005

- 3-

2.

(a)

Prove, by Mathematical Induction, that !On - 1 ts divisible by 9 for all pos1t1ve [ 9 marks] integers n.

(b)

A pair of simultaneous equations is given by px + 2y = 8 - 4x + p 2y = 16

where p (i)

R.

Find the value of p for which the system has an infinite nu mber of solutions. [ 3 marks] Find the solutions for this value of p .

[ 3 marks]

x+ 4 Find the set of real values of x for which x-2 > 5.

[ 5 ma rks]

(ii)

(c)

E

Total 20 marks

Section B (Module 2) Answer BOTH questions. 3.

The equation of the circle, Q, with centre 0 is x 2 + y 2 - 2x + 2y = 23. (a)

Express the equation of Q in the form (x - a) 2 + (y- b

(b)

Hence, or otherw ise, state (i)

(ii)

the coordinates of the centre of Q the radius of Q.

i = c.

[ 5 marks]

2 ma rks]

[ 1 mark]

(c)

Show that the po int A(4, 3) lies on Q.

3 m a rks]

(d)

F ind the equation of the tangent to Q at the point A .

(e)

The centre of Q is the midpoint of its diameter AB. Find the coordinates of B. [ 4 marks]

[ 5 marks]

Total 20 ma rks

GO ON TO THE NEXT PAGE

02134020/CAPE 2005

-4-

4.

The diagrams shown below, not drawn to scale, represent a sector, OABC, of a circle with centre at 0 and a radius of 7 em, whe re angle AOC

n rad.tans. measures3

a right circ ular cone with vertex 0 and a circu Jar base of radius rem which is formed when the sector OABC is folded so that OA coincides with OC. 0

A,C

B

A

B

(a)

(i)

Express the arc length ABC in terms of n.

(ii)

Hence, show that

[ 1 mark ]

7

3 marks]

a)

r= -

b)

if hem is the height of the cone, then the exact value of his

6

7 -{35

6

[ 2 marks] (b)

(i) (ii)

Show that cos 3

e= 4 cos3 e - 3 cos e.

[ 5 marks]

The position vectors of two points A and B relative to the origin 0 are a = 4cos2 8i+(6cos8-l)j

b

=

2 cos

ei

- j.

By using the identity in (b) (i) above, find the value of 8, 0::; 8::; ~ , suc h that

a and b are perpendicular. (c)

Find the modulus of the complex number z

[ 5 marks]

=

25 (2 + 3i) 4 + 3i

[ 4 marks] Total 20 marks

GO ON TO THE NEXT PAGE

02134020/CAPE 2005

-5-

Section C (Module 3) Answer BOTH questions.

5.

(a)

S tate the value of

(i)

lim u~

0

sin u u

B y means of the substitution u

(ii)

[ 1 mark]

= 3x, show that

lim

x~O

sin 3x x

= 3. 4 marks]

(iii)

(b)

If y 2 X

(c)

Hence, evaluate

= -AX

d2y dx2 +

lim x

~

0

sin 3x sin Sx

[ 4 marks]

+ Bx, whe re A and B are constants, show that X

ely dx

= y.

[ 4 marks]

The diagram be low, not drawn to scale, s hows part of the curve y2 = 4x. Pi s the point o n the curve at whi ch the line y = 2x c uts the c urve.

X

Find ( i)

(ii)

the coordina tes of P

[ 3 marks]

the volu me of the solid generated by rotating the s haded area through 2rc radians abo ut the x-axis. [ 4 marks]

Total 20 marks

GO ON TO THE NEXT PAGE 02 134020/CAPE 2005

- 6-

6.

(a)

Differentiate, with respect to x, (x2 + 7i + sin 3x.

(b)

(c)

[ 6 marks]

Determine the values of x for w hich the functio n y (i)

has stationary points

(ii)

IS mcreasmg

(iii )

is decreasing. USe the SUbstitution t

(i i)

If

f(x) dx

9x 2 + 15x + 4 [ 3 marks] 2 marks]

( i)

r

= x3 -

= a- X tO ShOW that

r

[ 2 marks] j (x) cJx = fOa j (a- X) dx. [ 4 marks]

= 12, use the substitution t =X -

1 to evaluate

Jl

5

3f(x- 1) dx. [ 3 marks]

Total 20 marks

ENDOFTEST

021 34020/CAPE 2005

TEST CODE

FORM TP 2005254 CARIBBEAN

02134032

MAY/JUNE 2005

EXAMINATIONS

COUNCIL

ADVANCED PROFICIENCY EXAMINATION PURE MATHEMATICS UNIT 1 - PAPER 03/B If hours

( 20 MAY 2005 (p.m.))

This examination paper consists of THREE sections: Module 1, Module 2, and Module 3. Each section consists of 1 question. The maximum mark for each section is 20. The maximum mark for this examination is 60. T his examination paper consists of 4 pages.

INSTRUCTIONS TO CANDIDATES 1.

DO NOT open this examination paper until instructed to do so.

2.

Answer ALL questions from the THREE sections.

3.

U nless otherwise stated in the question, any numerical answer that is not exact MUST be written correct to three significant figures.

Examination materials Mathematical formulae and tables Electronic calculator Graph paper

Copyright © 2004 Caribbean Examinations Council All rights reserved. 02134032/CAPE 2005

-2 Section A (Module 1) Answer this question.

1.

(a)

(b)

G iven that 2x2 + 8x + 11 the constants h and k.

the value of EACH of [ 5 marks]

If p, q, r, s E R , use the fact that (p- q) 2 ;::: 0 to show that p 2 + q 2 ;::: 2 pq. [ 2 marks]

(i)

Deduce that if p 2 + q 2 = 1, then pq :::; ~-

(ii)

(c)

= 2(x + h) 2 + k for all values of x, find

[ 1 mark]

A club bakes and sellsxcakes, making a profit, in dollars, that is modelled by the function f(x)

=x 2

(i) (ii)

- lOx.

Sketch the graph of the function f (x)

=x?- 1Ox .

[ 8 marks]

From your graph, determine a)

the LEAST number of cakes sold for which a profit is realised [ 2 marks]

b)

the GREATEST possible loss in dollars

c)

the number of cakes for which the GREATEST possible loss occurs. [ 1 mark]

[ 1 mark]

Total 20 marks

GO ON TO THE NEXT PAGE

02134032/CAPE 2005

-3Section B (Module 2) Answer this question.

2.

(a)

The straight line through the point P (4, 3) is perpendi cu lar to 3x + 2y = 5 and meets the given line at N. Find (i)

(ii) (b)

the coordinates of N

6 marks]

the length of the line-segment PN.

2 marks]

The table below presents data collected on the movement of the tide at various times after midnight on a particular day. Time After Midnight (t hou rs)

(h metres)

High

0

12

Low

6

2

High

12

12

Low

18

2

Tide Movement

Height

The height, h metres, can be modelled by a function of the form h = p cos (qt/ + 7 where t is the time in hours after midnight. Use the data from the table to find the [12 marks] values of p and q. Total 20 mar ks

GO ON TO THE NEXT PAGE 02134032/CAPE 2005

-4Section C (Module 3) Answer this question.

3.

(a)

lim

[ 3 marks]

(i)

Find

(ii)

Determine the real values of x for which the f unction

x--71

- 3x - l f() X x2 - x - 2 is continuo us.

3 marks]

(b)

Differentiate with respect to x, from first principles, the function x 2 + 2x. [ 5 marks]

(c)

Initially, the depth of water in a tank is 32 m. Water drains f rom the tank through a hole c ut in the bottom. At t minutes after the water begins draini ng, the depth of water in the tank is x metres. The depth of the water changes, w ith respect to time t, at the rate equal to (-2t- 4). (i) (ii)

Find an expression for x in terms oft.

[ 5 marks]

Hence, determine how long it takes fo r the water to drain completely from the tank. [ 4 marks]

Total 20 marks

END OF TEST

02134032/CAPE 2005

TEST CODE

FORM TP 2005256

02234020

MAY/JUNE 2005

CARIBBEAN EXAMINATIONS

COUNCIL

ADVANCED PROFICIENCY EXAMINATION PURE MATHEMATICS UNIT 2 - PAPER 02 2 hours

(ot JUNE 2005 (p.m.0 Thi s examination paper consists of THREE sections: Module 1, Module 2 and Module 3. Each section consists of 2 questions. The maximum mark for each section is 40. The maximum mark for this examination is 120. This examination consists of 5 pages.

INSTRUCTIONS TO CANDIDATES 1.

DO NOT open this examination paper until instructed to do so.

2.

Answer ALL questions from the THREE sections.

3.

U nless otherwise stated in the question, any numerical answer that is not exact MUST be writte n correct to three significant figures.

Examination Materials M athematical formu lae and tables Electronic calculator Graph paper

Copyright © 2004 Caribbean Examinations Council All rights reserved. 02234020/CAPE 2005

-2-

Section A (Module 1) Answer BOTH questions. 1.

(a)

T he diagram below, not drawn to scale, shows two points, P( p , 0.368) and R (3.5, r), on f (x ) = ex for x E R. f (x) = ex y

p

0

X

(i)

Copy the diagram above and on the same axes, sketch the graph of g(x) = ln x . [ 3 marks]

(ii)

Descri be clearly the relationship betweenf (x)

= ex

and g(x)

= ln x. [ 3 marks]

(iii)

(b)

U sing a calcul ator, fi nd the value of a)

r

1 mark]

b)

p.

2 ma rks]

Given that log0 (be)= x, Iogb (ea) = y, logc(ab) = z and a -:~; b -:~; e, show that axl.lcZ

(c)

= (abel

Find the values of x

E

R for which eX + 3e-x

3 marks]

= 4.

8 marks] Total 20 marks

GO ON TO THE NEXT PAGE 02234020/CAPE 2005

-3-

2.

(a)

A curve is given parametrically by x = (3- 2t)2 , y = t3 - 2t. Find

1x

(i)

the gradient of the norma l to the curve at the point t = 2.

(ii)

(b)

4 marks]

in terms oft

A B Express 22x + 1 in the form - + 2 x (x + 1) x x stants.

(i)

J

2

(ii)

He nce, evalu ate

X

I

2 marks]

+ ____f_ , whe re A , B and C are conx + 1 [ 7 marks]

2x + 1

2(

X +

[ 7 marks]

. ) dx· 1

Total 20 marks

Section B (Module 2) Answer BOTH questions.

3.

(a)

1

n

s n =L r=I

(c)

~ oo,

I

= r (r +

1 (r(r 1 )=1 -+ I) n+I .

Deduce, that as n

(ii)

(b)

1

Use the fact that- - - r r+ 1

(i)

l) to show that [ 5 marks]

[ 1 mark]

S 11 ~ I.

. r, o f a geometn. c sen.es IS . g1. ven b y r Th e common ratiO, of x for which the series converges.

Sx- . =2 4

+x

p·10d A LL the va I ues [10 marks]

By substituting suitable values of x on both sides of the expansion of n

(l + x)"

=L

"C,.x,.,

r= 0

show that n

L

(i)

"c,. = 2"

[ 2 marks]

r=O 11

L

(ii )

"c,. (- I /

=

o.

[ 2 marks]

r=O Total 20 marks

GO ON TO THE NEXT PAGE 02234020/CAPE 2005

-4-

4.

The fu nction,f, is given by f(x) (a)

- 4x - x3.

Show that 4 marks]

(i)

f is everywhere strictly decreasing

(i i)

the equationf(x) = 0 has a real root, a, in the closed interval [1, 2]

a is the only real root of the equation fix)= 0.

(iii) (b)

=6

[ 4 marks] [ 4 marks]

If xn is the nth approximation to a, use the Newton-Raphson method to show that the (n + l)st approximation x 11 + 1 is given by

2x;': + 6

XII + I

= _3_ 2_4_ .

[ 8 marks]

x, +

Total 20 marks

Section C (Module 3) Answer BOTH questions.

5.

(a)

On a particular day, a certain fuel service station offered 100 c ustomers who purchased prentium or regular gasoline, a free check of the engine oil or brake fluid in their vehicles. The serv ices required by these customers were as follows: 15% of the customers purchased pre mium gasoline, the others purchased regular gasoline. 20% of the customers who purchased prentium gasoline requested a check for brake fluid, the others requested a check for engine oil. 51 of the custome rs who purchased regular gasoline requested a check for engine oil, the others requested a check for brake fluid. (i)

Copy and complete the diagram below to represent the event space. Brake fluid

Engine oil Premium gasoline

51

Regular gasoline

[ 3 marks] GO ON TO THE NEXT PAGE 02234020/CAPE 2005

-5(ii)

(b)

Find the probability that a customer chosen at random a)

who had purchased premium gasoline requested a check for engine oi l

b)

who had requested a check of the brake fluid purchased regular gasoline

c)

who had requested a check of the engine oil purchased regular gasoline. [ 6 marks]

A bag contains 12 red balls, 8 blue balls and 4 white balls. Three balls are drawn from the bag at random without replacement. Calculate the total number of ways of choosing the three balls

[ 3 marks]

(ii)

the probability that ONE ball of EACH colour is drawn

[ 3 marks]

(iii)

the probability that ALL THREE balls drawn are of the SAME colour. [ 5 marks]

(i)

Total 20 marks

6.

(a)

(b)

Find the values of x for which X

1

1

X

2

1

2

2 x

=

0.

[10 marks]

Twelve hundred people visited an exhibition on its opening day. attendance fell each day by 4% of the number on the previous day. (i) (ii)

Thereafter, the

Obtain an expression for the number of visitors on the n1h day.

2 marks]

Find the total number of visitors for the first n days.

3 marks]

(iii)

The exhibition closed after 10 days. Determine how many people visited [ 3 marks] during the period for which it was opened.

(iv)

If the exhibition had been kept opened indefinitely, what would be the maximum [ 2 marks] number of visitors?

Total 20 marks

END OF TEST

02234020/CAPE 2005

TEST CODE

FORM TP 2005257

02234032

MAY /JUNE 2005

CARIBBEAN

EXAMINATIONS

COUNCIL

ADVANCED PROFICIENCY EXAMINATION PURE MATHEMATICS UNIT 2 - PAPER 03/B

(

23 MAY 2005 (p.m.))

This examination paper consists of THREE sections: Module 1, Module 2, and Module 3. Each section consists of 1 question. The maximum mark for each section is 20. The maximum mark for this examination is 60. This examination paper consists of 5 pages.

INSTRUCTIONS TO CANDIDATES

1.

DO NOT open this examination paper until instructed to do so.

2.

Answer ALL questions from the THREE sections.

3.

U nless otherwise stated in the question, any numerical answer that is not exact MUST be written correct to three sig nificant figures.

Examination materials Mathematical formu lae and tables Electronic calculator Graph paper

Copyright © 2004 Caribbean Examinations Council All rights reserved. 02234032/CAPE 2005

-2Section A (Module 1) Answer this question.

1.

Table 1 presents data obtained from a biological investigation that involves two variables x andy. Table 1 X

20

30

40

50

y

890

1640

2500

3700

It is believed that x and y are related by the formul a, y (a)

(i)

(ii)

(b)

= bxn.

By taking logarithms to base 10 of both sides, convert y Y = nX + d where n and d are constants.

= bxn

to the form [ 4 marks]

Hence, express a)

Y in terms of y

b)

X in terms of x

c)

din terms of b.

[ 3 marks]

Use the data from Table 1 to complete Table 2. Table 2 log x 10

log y 10

1.30

1.60 3.21

3.57 [ 2 marks]

(c)

In the graph on page 3, log 10 x is plotted against log 10 y for 1.3 :S x :S 1.7. (i)

(ii)

Assuming that the ' best straight line ' is drawn to fit the data, determine a)

the gradient of this line

[ 2 marks]

b)

the value of b given that this line passes through (0, l)

[ 4 ma rks]

c)

the value of each of the constants, nand d, in Part (a) (i) above. [ 2 marks] Using the graph, or otherw ise, estimate the value of x for which y is 1800. [ 3 marks] Total 20 marks

GO ON TO THE NEXT PAGE 02234032/CAPE 2005

VI

N 0 0

~

?5

w

~

w

N

N

0

~

'"tl

~

~

0

z ....,

0 0

a

L"I

X

01

~OJ

9.1

s·1 17"1

w

-4Section B (Module 2) Answer this question. 2.

Mr John Slick takes out a n investment with a n investment company which requires making a fixed payment of $A at the beginning of each year. At the end of the investment period, John expects to receive a payout sum of money which is equal to the total payments made, together with interest added at the end of EACH year at a rate of r % per annum of the total sum in the fund. The table below shows information on Mr Slick's investment for the first three years.

Year

Amount at Beginning of Year ($)

1

A

Interest ($) A

Payout Sum$

r

X

A+(Ax_r) 100

100

= A(1+-r) 100 =AR 2

A+ AR

(A+ AR)

X _ r_

(A+ AR) + [(A+ AR)

100

X

1~0

J

= (A + AR) (1 + I ~O) =(A+ AR) R = AR+ AR2 3

A+AR+AR2

(a)

Write expressions for (i)

(ii)

2

r

2 (A + AR + AR) x lOO (A+ AR + AR ) R 2 = AR+AR +AR3

the amounts at the beginning of Years 4 and 5

2 marks]

payout sums at the end of Years 4 and 5.

2 marks]

(b)

By using the information in the Table, or otherwise, write an expression for the amount at the beginning of the nth year. [ 2 marks]

(c)

Show that the payout sum in (b) above is $ AR (R"- 1) for R > 1. R-1

(d)

Find the value of A, to the nearest dollar, when n = 20, r = 5 and the payout sum in (c) above is $500 000.00. [ 7 marks]

[ 7 marks]

Total 20 marks

GO ON TO THE NEXT PAGE 02234032/CAPE 2005

-5Section C (Module 3) Answer this question. 3.

The output 3 x 1 matrix Yin a testing process in a chemical plant is related to the input 3 x 1 matrix X by means of the equation Y = AX, w here

A=

1 2 2 4 ( 3 5

(a)

Show that A is non-singular.

(b)

Show that X= A Y.

(c)

Find A-

(d)

Find the input matrix X corresponding to the output matrix Y

5 marks]

- I

3 marks]

1 •

[

~ ( !~

).

9 marks]

3 marks] Total 20 marks

END OF TEST

02234032/CAPE 2005

TEST CODE

FORM TP 2006257 CARIBBEAN

02134010

MAY/JUNE 2006

EXAMINATIONS

COUNCIL

ADVANCED PROFICIENCY EXAMINATION PURE MATHEMATICS UNIT 1 - PAPER 01

c

2 hours

19 MAY 2006 (p.m.))

T his examination paper consists of THREE sections: Module 1, Module 2, and Module 3. Each section consists of 5 questions. The maximum mark for each section is 40. The maximum mark for this examination is 120. This examination paper consists of 7 pages.

INSTRUCTIONS TO CANDIDATES 1.

DO NOT open this examinati on paper until instructed to do so.

2.

Answer ALL questions from the THREE sections.

3.

Unless otherwise stated in the question, any numerical answer that is not exact MUST be written correct to three significant figures.

Examination materials Mathematical formulae and tables Electronic calculator Graph paper

Copyright © 2005 Caribbean Examinations Council® All rights reserved. 02 13401 0/CAPE 2006

-2Section A (Module I ) Answer ALL questions.

1.

(a)

The functionf(x) is give n by f (x) = .0

-

(i)

Show that (x - 1) is a factor ofj(x) for all values of p.

[ 2 marks]

(ii)

If (x- 2) is a factor of j(x), find the value of p .

[ 2 marks]

n

(b)

(p + 1)x2 + p, p E N.

Given that

L

r= l

n

r = ..!!_(n + 1), show that 2

L

(3r + 1) =.!. n(3n + 5).

[ 4 marks]

2

r=l

Total 8 marks

2.

(a)

LetA= {x : 2:::; x:::; 7} and B = {x : I x- 4 1:::; h}, hE R. Find the L ARGEST value of h for which B c A.

[ 6 marks]

Find the value of k.

[ 3 marks] Total 9 ma rks

3.

(a)

(i)

(ii)

(b)

3x

ax + b - , where x X+ 1

Find a, b E R such that - - - 2 = X+ 1

Hence, find the range of values of x

E

':1;

-1.

[ 2 marks]

R for which ~ > 2.

Without the use of calcul ators or tables, show that

X+ 1

12

4

2

4

[ 4 marks]

-

= 2 (..J 2 ).

X 8 -1/J

[ 4 marks] Total tO ma rks

GO ON TO THE NEXT PAGE 0213401 0/CAPE 2006

- 34.

The diagram below (not drawn to scale) represents the graph of the function f(x) - 1 ~ x ~ 1 and p, q E R.

= x 2 + 1,

f(x)

(- l , p)

(q, 2)

-------r--------~---------,,-----~ X

-1

(a)

(b)

0

+1

Find (i)

the value of p and of q

[ 2 marks]

(ii)

the range of the functi onj(x) for the given domai n.

[ 1 mark]

Determine whether f (x) (i)

(ii) (iii)

is surj ective (onto)

1 mark]

is injective (one-to-one)

1 mark]

has an inverse.

[ 1 mark] Total 6 marks

5.

Find the values of m , n

E

R for which the system of equations

X+ 2y = 1 2x+ my= n

(a)

possesses a unique solution

[ 3 marks]

(b)

is inconsistent

[ 2 marks]

(c)

possesses infi nitely many solutions.

[ 2 marks] Total 7 marks

02134010/CAPE 2006

GO ON TO THE NEXT PAGE

-4-

Section B (Module 2) Answer ALL questions. 6.

In the diagram below (not drawn to scale), the straight line through the point P(2, 7) and perpendicular to the line x + 2y = 11 intersects x + 2y = 11 at the point Q. y

Find (a)

the equation of the line through P and Q

(b)

the coordinates of the point Q

3 marks]

(c)

the EXACT le ngth of the line segment PQ.

2 marks]

[ 2 marks]

Total 7 marks

GO ON TO THE NEXT PAGE 0213401 0/CAPE 2006

- 5-

7.

In the diagram below (not drawn to scale), AC = BC, AD 2 angle A CB = ~ radians and angle ADC = ; radians.

=7

uni ts, DC

=8

units,

B

D

Find the EXACT length of (a)

AC

[ 5 marks]

(b)

AB.

[ 3 marks] Total 8 marks

8.

(a)

Solve the equati on 4 cos 2 8- 4 sin 8- 1 = 0 for 0 ~ 8 ~ n.

(b)

Show that

2 1 - cos 2.x 2.x = tan x. 1 + cos

[ 5 marks] [ 3 marks]

Total 8 marks

9.

(a)

The roots of the quadratic equatio n x2 + 6x + k = 0 are -3 + 2i and -3- 2i . [ 2 marks] Find the value of the constant k.

(b)

Find the real numbers u and v such that u + 2 i 3 - 4i

= 1 +vi.

6 marks] Total 8 marks

10.

Given the vectors p = 2i + 3j and q

= 3i -

2j ,

= - 3i -

(a)

find x, y

(b)

show that p and q are perpendicular.

E

R such that xp + yq

11j

[ 7 marks]

[ 2 marks] Total 9 marks

02 134010/CAPE 2006

GO ON TO THE NEXT PAGE

-6Section C (Module 3) Answer ALL questions.

11.

lim

(a)

Find

(b)

Find the values of x j(x)

[ 3 marks]

X---7

=

E

R such that the function

9 -xz

(x2 -3)(JxJ-3)

is d iscontinuous.

[ 4 marks] Total 7 marks

12.

(a)

The functionj(x) is defined by j(x) =

2-x

-----;r-

for x

E

R ,x

-::f:.

0.

Determine the nature of the critical value(s) ofj(x). (b)

Differentiate, with respect to x , j(x)

6 marks]

= sin2 (x2 ) .

3 marks] Total 9 marks

13.

The diag ram below (not drawn to scale) is a sketch of the section of the f unction =x (x2 - 12) whi ch passes through the orig in 0. A and B are the stationary poi nts on the curve.

j(x)

y

(A)

f (x) = x(x 2 - 12)

(B)

Find (a)

the coordinates of each of the stationary points, A a nd B

(b)

the equation of the normal to the curvej(x)

=x (x2 -

5 marks]

12) at the orig in .

4 marks]

Total 9 marks

02 1340 10/CAPE 2006

GO ON TO THE NEXT PAGE

-7-

14.

16 T he di agram below (not drawn to scale) s hows the shaded area, A, bounded by the curve y = x2 and the lines y

=

1 1,x -x-

2

= 2 and x = 3.

y

I

y= 2x - l 16

Y = x2 X

r

(a)

Express the shaded area, A , as the difference of two definite integrals.

(b)

Hence, show that A= 16

(c)

F ind the value of A.

f x - dx - ; f 3

2

2

X

d.x +

d.x.

1 mark] 2 marks] 3 marks]

Total 6 marks

15.

Use the resul t

I:

=

f(x)dx

r

f(a- x)dx, a> 0 , to s how th at

(a)

fnx s in x dx = fn(n -x) sin x dx.

(b)

H ence, show that

0

0

(i)

(ii)

r

X

Sin

X

dx

= '!T; fsi n X dx- J:x Sin X cfx

[ 2 marks]

[ 2 marks]

[ 5 marks]

J:xsinxd.x=n.

Total 9 marks

END OF TEST

02 13401 0/CAPE 2006

TEST CODE

FORM TP 2006258 CARIBBEAN

02134020

MAY/JUNE 2006

EXAMINATIONS

COUNCIL

ADVANCED PROFICIENCY EXAMINATION PURE MATHEMATICS UNIT 1 - PAPER 02 2 hours ( 24 MAY 2006 (p.m.))

This examination paper consists of THREE sections: Module 1, Module 2 and Module 3. Each section consists of 2 questions. The maximum mark for each section is 40 . The maximum mark for this examination is 120. This examination consists of 5 pages.

INSTRUCTIONS TO CANDIDATES

1.

DO NOT open this examination paper until instructed to do so.

2.

Answer ALL questions from the THREE sections.

3.

U nless otherwise stated in the question, any numerical answer that is not exact MUST be written correct to three significant figures.

Examination Materials Mathematical formul ae and tables Electronic calculator Graph paper

Copyri ght © 2005 Caribbean Examinations Council® All rights reserved. 02134020/CAPE 2006

- 2-

Section A (Module 1) Answer BOTH questions.

1.

(a)

Solve the simultaneous equations

x 2 +xy = 6 X- 3y + 1 = 0. (b)

[ 8 marks]

The roots of the equation x 2 + 4x + 1 = 0 are a and state the values of a+

(i)

13.

Without solving the equation,

13 and al3

[ 2 marks]

find the value of a 2 + 132

(ii) (iii)

[ 3 marks]

find the equation whose roots are 1 +

~

and 1 +

i-·

[ 7 marks] Total 20 marks

II

2.

=~n (n + 1).

(a)

Prove, by Mathematical Induction, that r~ r

(b)

Express, in terms of nand in the SIMPLEST form,

1

[10 marks]

2n

(i)

Lr r= 1

[ 2 marks]

2n

L

(ii)

[ 4 marks]

r.

r= n + 1

2n

(c)

Find n if

L

r=n+J

r

= 100.

[ 4 marks] Total 20 marks

GO ON TO THE NEXT PAGE

02134020/CAPE 2006

- 3-

Section B (Module 2) Answer BOTH questions.

3.

(a)

(i)

(ii)

Find the coordinates of the centre and radius of the circle x 2 + 2x + y 2 - 4y = 4. [ 4 marks] By writing x + 1 = 3 sin 8, show that the parametric equations of this circle are -1 + 3 Sin 8, y = 2 + 3 COS 8. [ 5 marks)

X=

(iii)

Show that the x-coordinates of the points of intersection of this circle with the line x + y = 1 are x = - 1 ± ~ 4 marks]

f2.

(b)

[

F ind the general solutions of the equation cos 8 = 2 sin28- 1.

[ 7 marks] Total 20 marks

4.

(a)

(b)

Given that 4 sin x- cos x = R sin (x- a), R > 0 and 0° 0 if

= 1

f

2x

x2 + 4,

j'( ) x

=

24 - 6x2 d (x2 + ? x. 4 2u 1

11

x4

7 dx = 192 .

2

8- 2x (x2 + 4)2

-

_!_). n

[ 2 marks]

[ 4 marks]

[ 3 marks]

[ 5 marks] Total 20 marks

ENDOFTEST

02134020/CAPE 2006

TEST CODE 02134032

FORM TP 2006259

MAY/JUNE 2006

CARIBBEAN EXAMINATIONS

COUNCIL

ADVANCED PROFICIENCY EXAMINATION PURE MATHEMATICS UNIT 1 - PAPER 03/B lf hours

( 19 MAY 2006 (p.m.))

This examination paper consists of THREE sections: Module 1, Module 2, and Module 3. Each section consists of 1 questio n. The maximum mark for each section is 20. The maximum mark for this examination is 60. This examination paper consists of 4 pages.

INSTRUCTIONS TO CANDIDATES 1.

DO NOT open this examination paper until instructed to do so.

2.

Answer ALL questions from the THREE sections.

3.

Unless otherwise stated in the question, any numerical answer that is not exact MUST be written correct to three sig nificant fi gures.

Examination materials M athematical formul ae and tables Electronic calculator Graph paper

Copyright © 2005 Caribbean Examinations Council ® 02134032/CAPE 2006

All rights reserved.

-2Section A (Module 1) Answer this question.

1.

(a)

Solve, for x, the equations (i)

[ 7 marks]

l x+4 1=1 2x -11

[ 7 marks]

(ii)

(b)

A coach of an athletic club has five athletes, u, v, w, x and y, in his training camp. He makes a n assignment, f, of athletes u, v, x and y to physical activities 1, 2, 3 and 4 according to the di agram below in whic h A = {u, v, w , x, y}, B = { 1, 2, 3, 4} and f = {(u, 1), (v, 1), (v, 3), (x, 2), (y, 4)}.

A

___!_

B

(i)

State ONE reason why the assignment/from A to B is not a function. [ 1 mark]

(ii)

State TWO changes that the coach would need to make so that the assignment, f, becomes a function g: A --7 B. [ 2 marks]

(iii)

Express the functi on g: A

--7

B in (ii) above as a set of ordered pairs. [ 3 marks] Total 20 marks

02134032/CAPE 2006

GO ON TO THE NEXT PAGE

-3Section B (Module 2) Answer this question.

2.

(a)

In an experiment, the live weight, w grams, of a hen was found to be a linear function,/, of the number of days, d, after the hen was placed on a special diet, where 0 ::::; d::::; 50. At the beginning of the experiment, the he n weighed 500 grams and 25 days later it weighed 1 500 grams. (i)

Copy and complete the table below. d (days) w (gms)

25 500

[ 1 mark] (ii)

Determine

=w

[ 3 marks]

a)

the linear function, /, s uch thatf(d)

b)

the expected weight of a hen 10 days after the diet began. [ 2 marks]

(iii)

(b)

(c)

After how many days is the hen expected to weigh 2 180 grams? [ 2 marks]

e _ sec 9)2 = s in 29 -

( i)

Show that (tan

(ii)

Hence show that 1 - s in e I +sin 8

2 sin 9 + 1 cos29

= (tan 8- sec 8)

2

[ 3 marks]

[ 4 marks]

Given the complex number z = {3" +.!. i, find 2 2 (i) (ii)

(iii )

Iz I arg (z) zZ.

1 mark] [ 2 marks] 2 marks] Total 20 marks

GO ON TO THE NEXT PAGE 021 34032/CAPE 2006

-4-

Section C (Module 3) Answer this question.

3.

(a)

(i)

(ii)

By expressing X- 4 as

c-r-; + 2) c.Y~- 2), find

lim x-74

G- 2 x- 4

.y-;-

2 Hence, find lim 2 x -7 4 x - Sx + 4

[ 3 marks] [ 3 marks]

=10, find J: [f(x) + 4] dx + J: f(x) dx.

[ 7 marks]

(b)

Given that J:J(x) dx

(c)

A bowl is formed by rotating the area between the curves y = x 2 and y = x 2 - 1 for x ~ 0 and 0 ~ y ~ 1 through 2n radians around the y-axis. Calculate (i) (ii)

the capacity of the bowl, that is, the amount of liquid it can hold

[ 3 marks]

the volume of material in the bowl.

[ 4 marks] Total 20 marks

END OF TEST

021 34032/CAPE 2006

TEST CODE

FORM TP 2006260 CARIBBEAN

02234010

MAY/JUNE 2006

EXAMINATIONS

COUNCIL

ADVANCED PROFICIENCY EXAMINATION PURE MATHEMATICS UNIT 2 - PAPER 01 2 hours

( 22 MAY 2006 (a.m.>)

Thi s examination paper consists of THREE sections: Module 1, Module 2 and M odule 3. Each section consists of 5 questions. The maximum mark for each section is 40. The maximum mark for this examination is 120. Thi s exami nation consists of 6 pages.

INSTRUCTIONS TO CANDIDATES 1.

DO NOT open this examination paper until instructed to do so.

2.

Answer ALL questions from the THREE sections.

3.

Unless otherwise stated in the question, any numerical answer that is not exact MUST be written correct to three significant figures.

Examinatio n Materials Mathematical formul ae and tables Electronic calcul ator Graph paper

Copyright © 2005 Caribbean Examinations Counc il® A ll rights reserved. 0223401 0/CAPE 2006

- 2-

Section A (Module 1) Answer ALL questions.

1.

Solve, for x, the equations (a)

[ 5 marks]

(b)

[ 3 marks] Total 8 marks

2.

Differentiate with respect to x the following: (a)

y

., = ea+

(b)

y

=tan 3x + In (x2 + 4)

.

[ 3 marks]

SID X

[ 4 marks] Total 7 marks

3.

= 2y2 at the point P (-2, 1).

(a)

Find the gradient of the curve x2 + xy

(b)

He nce, find the equation of the normal to the curve at P.

[ 5 marks] [ 3 marks] Total 8 marks

4.

If y (a)

= sin 2x + cos 2x , find

dy ax

[ 3 marks] [ 4 marks]

(b)

Total 7 marks

5.

Use the substitution indicated in EACH case to find the following integrals: (a)

Jsin 8x cos x dx ; u = sin x

(b)

Jx V2x + 1 dx ; u 2 = 2x + 1

4 marks] [ 6 marks] Total 10 marks GO ON TO THE NEXT PAGE

0223401 0/CAPE 2006

-3-

Section B (Module 2) Answer ALL questions.

6.

A sequence {U11 } of real numbers satisfies (a)

(b)

U 11

+ 1 U 11

= 3(- 1)" ; u 1 = 1.

Show that (i)

u11 + 2 = -u11

[ 3 marks]

(ii)

un

= un.

[ 1 mark ]

+4

[ 3 marks]

Write the FIRST FOUR terms of this seque nce.

Total 7 marks

7.

(a)

-4-).

1

Verify that the sum, S 11 , of the series - + _!_3 + 1_ + ... , ton terms, is 5 11 =~ (1 2 25 . 3 2 2

I

[ 4 marks] (b)

Three consecutive terms, x- d, x and x + d, d > 0, of an arithmetic series have sum 21 and product 3 15. Find the value of (i) (ii)

X

[ 2 marks]

the common difference d.

[ 4 marks] TotallO marks

(a)

show that x 2 - 5x- 14 = 0

4 marks]

(b)

find x.

2 ma rks] Total 6 marks

9.

(a)

Expand (1 + ux) (2- x) 3 in powers of x up to the term in x 2 , u

(b)

Given that the coefficient of the term in x2 is zero, find the value of u.

E

R.

6 marks] [ 2 marks]

Total 8 marks

GO ON TO THE NEXT PAGE 02234010/CAPE 2006

-4-

10.

The diagram above (not drawn to scale) shows the graphs of the two functions

y=e

and

y =-x .

(a)

State the equation in x that is satisfied at B (a, [3), the point of intersection of the two graphs. [ 2 marks]

(b)

Show thal a lies in Lhe closed inlerval [-1 , 0] .

[ 7 marks]

Total 9 m arks

GO ON TO THE NEXT PAGE 02234010/CAPE 2006

-5Section C (Module 3) Answer ALL questions. 11.

A committee of 4 people is to be selected from a group consisting of 8 males and 4 females. Determine the number of ways in which the committee may be formed if it is to contain

[ 2 marks]

(a)

NO females

(b)

EXACTLY one female

3 marks]

(c)

AT LEAST one female.

4 marks] Total 9 marks

12.

(a)

The letters H, R, D , S and T are consonants. In how many ways can the letters of the word HARDEST be arranged so that (i)

(ii) (b)

the first letter is a consonant?

3 marks]

the first and last letters are consonants?

3 marks]

Find the probability that the event in (a) (i) above occurs.

2 marks] Total 8 marks

13.

T he determinant!:::,. is given by

!:::,.

=

1 1 1

a b c

S how that!:::,. = 0 for any a, b and c

b+c c+a a+b E

[ 6 marks]

R.

Total 6 marks 14.

(a)

Write the following system of equations in the form AX= D. x+y- z =2 2x-y+ z = 1 3x + 2z = 1

(b)

(i)

(ii) (iii)

2 marks]

Find the matrix B, the matrix of cofactors of the matrix A.

[ 5 marks]

Calcul ate B T A.

[ 2 marks]

I I·

Deduce the value of A

[ 1 mark] Total10 marks

GO ON TO THE NEXT PAGE

0223401 0/CAPE 2006

-615.

A closed cylinder has a fixed height, h em, but its radius, r em, is increasing at the rate of 1.5 em per second.

[ 1 mark ]

(a)

Write down a differential equation for r with respect to time t sees.

(b)

Find, in terms of 1r, the rate of increase with respect to time t of the total surface area, A, of the cylinder whe n the radius is 4 em and the height is 10 em. [ 6 marks] [A

= 27rr2 + 21rrh] Total 7 marks

END OF TEST

02234010/CAPE 2006

TEST CODE

FORM TP 2006261 CAR IBBEAN

02234020

MAY/JUNE 2006

EXAMINAT IONS

COUNCIL

ADVANCED PROFICIENCY EXAMINATION PURE MATHEMATICS UNIT 2 - PAPER 02

c

2 hours

31 MAY 2006 (p.m) )

This e xamination paper consists of THREE sections: Modu le 1, Module 2 and Module 3. Each section consists of 2 questions. The maximum mark for each section is 40. The maximum mark for this examination is 120. This examination consists of 5 pages.

INSTRUCTIONS TO CANDIDATES 1.

DO NOT ope n this examination paper until instructed to do so.

2.

Answer ALL questions from the THREE sections.

3.

U nless otherw ise stated in the question , any numerical answer that is not exact MUST be writte n correct to three significant fi gures.

Examination Materia ls M athematical formulae and tables Electronic calc ul ator Graph paper

Copyright © 2005 Caribbean Examinations Counc il® All rights reserved. 02234020/CAPE 2006

- 2-

Section A (Module 1) Answer BOTH questions. 1.

(a)

If f(x)

= _0 ln2 X, show that

(i)

f'(x)

=x 2 ln x(3 ln x + 2) [ 5 marks]

(ii)

f"(x)

= 6x ln2 x + lOx ln X+ 2x. [ 5 marks]

(b)

The enrolment pattern of membership of a country club follows an exponential logistic function N, N

=

800

l+ke-rr'

k E R

'

r E R

'

where N is the number of members enrolled t years after the formation of the club. The initial membership was 50 persons and after one year, there are 200 persons enrolled in the club. (i)

What is the LARGEST number reached by the membership of the club? [ 2 marks]

(ii)

Calculate the EXACT value of k and of r.

(iii)

[ 6 marks]

How many members will there be in the club 3 years after its formation? [ 2 marks] Total 20 marks

2.

(a)

(i)

(ii)

Express

1+x

in partial fractio ns.

(x - 1) (x 2 + 1)

Hence, find

f

1 +X (x- 1) (x2 + 1)

dx.

[ 6 marks]

[ 3 marks]

I

(b)

Given that I,

=f0 x' e dx, where n

(i)

Evaluate / 1•

(ii)

Show that 111

(iii)

E

N.

[ 4 marks]

= e- n/

11

_

1•

4 marks]

Hence, or otherwise, evaluate / 3 , writing your answer in terms of e. [ 3 marks] Total 20 marks

02234020/CAPE 2006

GO ON TO THE NEXT PAGE

-3Section B (Module 2) Answer BOTH questions.

3.

(a)

(i)

Show that the terms of m ln 3r r=l

L

(ii)

are in arithmetic progression.

[ 3 marks]

Find the sum of the first 20 terms of this series.

[ 4 marks]

2m

(iii)

Hence, show that

L

ln 3r = (2m2 + m ) ln 3.

[ 3 marks]

r= 1

(b)

The sequence of positive terms, {x

11

(i)

(ii)

},

is defined by x,1 + 1 =

x; + i· x < k· 1

Show, by mathematical induc tion, or otherwise, that x < .!. for all positive 2 II [ 7 marks] integers n. By considering x11 + 1 -

X 11 ,

or otherwise, show that X 11 <

X + • 1 11

[ 3 marks]

Total 20 marks

4.

(a)

Sketch the functions y = sin x and y = x2 on the SAME axes.

(b)

Deduce that the functionf(x) =sin x- x2 has EXACTLY two real roots.

[ 5 marks]

[ 3 marks] (c)

Find the interval in which the non-zero root a ofj(x ) lies.

(d)

Starting with a first approximation of a at x 1 = 0.7, use one ite ration of the NewtonRaphson method to obtai n a better approximation of a to 3 dec imal pl aces. [ 8 marks]

[ 4 marks]

Total 20 marks

02234020/CAPE 2006

GO ON TO THE NEXT PAGE

-4-

Section C (Module 3) Answer BOTH questions. 5.

(a)

(i)

(ii)

How many numbers lying between 3 000 and 6 000 can be formed from the digits, 1, 2, 3, 4 , 5, 6, if no digit is used more than once in forming the number? [ 5 marks] Determine the probability that a number in 5 (a) (i) above is even.

[ 5 marks] (b)

In an experiment, pis the probability of success and q is the probability of failure in a single trial. For n trials, the probability of x successes and (n- x) fai lures is represented by nCxr q"-x, n > 0. Apply this model to the following problem. The probability that John will hit the target at a firing practice is ~. He fires 9 shots. 6 Calcu late the probability that he will hit the target (i)

AT LEAST 8 times

[ 7 marks]

(ii)

NO MORE than seven times.

[ 3 marks]

Total 20 marks

2

6.

(a)

If A= ( -:

2

-2

-~

) andB = (

~

-1

1 1

~ ). 1

(i)

find AB

3 marks]

(ii)

deduce A- 1•

3 marks]

GO ON TO THE NEXT PAGE 02234020/CAPE 2006

- 5-

(b)

A nursery sell s three brands of grass-seed mix, P, Q and R. Each brand is made from three types of grass, C, Z and B. The number of kilograms of each type of grass in a bag of each brand is summarised in the table below. Type of Grass (Kilograms)

Grass Seed Mix

C-grass

Z-grass

B-grass

Brand P

2

2

6

Brand Q

4

2

4

Brand R

0

6

4

Blend

c

z

b

A blend is produced by mixing p bags of Brand P, q bags of Brand Q and r bags of Brand R.

(i)

Write down an expression in terms of p, q and r, for the number of kilograms of Z-grass in the blend. [ 1 mark ]

(i i)

Let c, z and b represent the number of kilograms of C-grass, Z-grass and B-grass respectively in the blend. Write down a set of THREE equations in p, q, r, to represent the number of kilogram s of EACH type of grass in the blend . [ 3 marks]

(iii)

Rewrite the set of THREE equations in (b) (ii) above in the matrix form MX = D where M is a 3 by 3 matrix, X and D are column matrices. [ 3 marks]

(iv)

Given that M - 1 ex ists, write X in terms of M - 1 and D.

(v)

Given that M -

1

=

(

- 0.2 -0.2 0.35 0.1

0.3 ) -0. 15 ,

-0.05

-0.05

0.2

3 marks]

calculate how many bags of EACH brand, P , Q, and R, are required to produce a blend containing 30 ki lograms of C-grass, 30 kilograms of Z-grass and 50 ki lograms of B-grass. [ 4 marks] Total 20 marks

END OF TEST

02234020/CAPE 2006

TEST CODE

FORM TP 2006262 CARIBBEAN

02234032

MAY/JUNE 2006

EXAMINATIONS

COUNCIL

ADVANCED PROFICIENCY EXAMINATION PURE MATHEMATICS UNIT 2 - PAPER 03/B 1~ hours ( 22 MAY 2006 (p.m.))

This examination paper consists of THREE sections: Module 1, Module 2, and Modu le 3. Each section consists of 1 question. The maximum mark for each secti on is 20. The maximum mark for this examination is 60. This examination paper consists of 4 pages.

INSTRUCTIONS TO CANDIDATES

1.

DO NOT open thi s examination paper until instructed to do so.

2.

Answer ALL questions from the THREE sections.

3.

Unless otherwise stated in the question, any numerical answer that is not exact MUST be written correct to three signi fican t figures.

Examination materials Mathematical formulae a nd tables Electronic calcu lator Graph paper

Copyright © 2005 Caribbean Examinations Council® All rights reserved. 02234032/CAPE 2006

-2 -

Section A (Module 1) Answer this question.

1.

T he rate of inc rease of the number of algae with respect to time, t days, is equal to k timesf(t), where f (t) is the number of algae at any given time t and k E R. (a)

Obtain a differe nti al equation involvingfi:t) which may be used to model this situation. [ 1 mark]

(b)

Given that 6

the number of algae at the beginning is 10 the number of algae doubles every 2 days, (i)

determine the values ofj(O) andf(2)

(ii)

show that

(iii)

[ 2 marks]

a)

k=

2 zn 2

[10 marks]

b)

fit)

= 106(21'2)

[ 5 marks]

1

determine the approximate number of algae present after 7 days.

[ 2 marks] Total 20 marks

GO ON TO THE NEXT PAGE 02234032/CAPE 2006

-3-

Section B (Module 2) Answer this question.

2.

(a)

A car was purchased at the beginning of the year, for P dollars. The value of a car at the

end of each year is estimated to be the value at the beginning of the year multiplied by -.! ), q E N.

(1

q

(i)

Value at the Beginning of Year

Copy and complete the table below showing the value of the car for the first five years after purchase.

Year 1

Year2

Year 3

p

1 P(l--) q

P(l- _1__) q

Year4

YearS

2

($)

Value at the End of Year

($)

1 P(l--) q

(1-+>[p (1-t)J 1 2 = P(l--) q

[ 3 marks] (ii) (iii)

(b)

Describe FULLY the sequence shown in the table.

[ 2 marks]

Determine, in terms of P and q, the value of the car n years after purchase. [ 1 mark]

If the original value of the car was $20 000.00 and the value at the end of the fourth year

was $8 192.00, find (i)

(ii) (iii)

the value of q

[ 5 marks]

the estimated value of the car after five years

[ 2 marks]

the LEAST integral value of n, the number of years after purchase, for which the estimated value of the car fal ls below $500.00. [ 7 marks]

Total 20 marks

GO ON TO THE NEXT PAGE 02234032/CAPE 2006

- 4-

Section C (Module 3) Answer this question.

3.

(a)

A box contains 8 green balls and 6 red balls. Five balls are selected at random. Find the probability that (i)

(ii) (iii)

(b)

ALL 5 balls are green

[ 4 marks]

EXACTLY 3 of the five balls are red

4 marks]

at LEAST ONE of the five balls is red.

3 marks]

Use the method of row reduction to echelon form on the augmented matrix for the [ 9 marks] following system of equations to show that the system is inconsistent. x + 2y + 4 z = 6 y +2z =3 x + y + 2z = 1

Total 20 marks

END OF TEST

0223403 2/CAPE 2006

FORM TP 200824 0 CARIB BEAN

a\ \5/ EXAMI NATIO NS

TEST CODE

021340 20

MAY/JUNE 2008

COUNC IL

ADVA NCED PROF ICIEN CY EXAM INATI ON PURE MATHEM ATICS UNIT 1 -PAPER 02 ALGEBR A,GEOM ETRYAN DCALCU LUS 2 Y2 hours

( 21 MAY 2008 .(p.m.))

This examination paper consists of THREE sections: Module 1, Module 2 and Module 3. Each section consists of 2 questions. The maximum mark for each Module is 50. The maximum mark for this examinatio n is 150. This examination consists of 5 printed pages.

INSTRUC TIONS TO CANDIDA TES 1.

DO NOT open this examination paper until instructed to do so.

2.

Answer ALL questions from the THREE sections.

3.

Write your solutions, with full working, in the answer booklet provided.

4.

Unless otherwise stated in the question, any numerical answer that is not exact MUST be written correct to three significant figures.

Examinati on Materials Permitted Graph paper (provided) Mathematical formulae and tables (provided) - Revised 2008 Mathematical instruments Silent, non-programmable, electronic calculator

Copyright © 2007 Caribbean Examinations Council ®. All rights reserved. 02134020/C APE 2008

-

2

-

SECTION A (Module 1) Answer BOTH questions.

1.

(a)

The roots of the cubic equation x3 + 3px1 + qx + r = 0 are 1,-1 and 3. Find the values of the real constants p, q and r. (7 marks]

(b)

Without using calculators or tables, show that

(i)

V~6~- < 2 V fTs-- VT ■

.

(ii)

flT+ V T

+

VT- V T (c)

v t 22 ++ V T

[5 marks]

V 6~- V 2

(i)

n Show that E r (r + 1) = ~ r» 1

(ii)

Hence, or otherwise, evaluate

[5 marks]

4.

■VT+ V T

n (n + 1) (n + 2),

n e N.

[5 marks]

50

Z

r(r+

1).

[3 marks]

f = 3t

Total 25 marks

2.

(a)

The roots o f the quadratic equation 2x2 + 4x + 5 = 0 are a and p . Without solving the equation (i)

write down the values of a + p and af3

(ii)

calculate

(iii)

[2 marks]

a)

a 2 + p2

[2 marks]

b)

a 3 + p3

[4 marks]

find a quadratic equation whose roots are a 3 and p3.

[4 marks]

GO ON TO THE NEXT PAGE 02134020/CAPE 2008

-3(b)

(i) (ii) (iii)

Solve for x the equation x 113

-

4x- 113 = 3.

[5 marks]

Find x such that logs (x + 3) + logs (x- 1) = 1.

[5 marks)

Without the use of calculators or tables, evaluate log 10 (+) + log 10

c;) +

log 10 (!) + ... + log 10

(

~)

+ log 10

( {

0

).

[3 marks)

Total25 marks

SECTION B (Module 2) Answer BOTH questions. 3.

(a)

The lines y = 3x + 4 and 4y = 3x + 5 are inclined at angles a. and j3 respectively to the x-ax1s. (i)

(b)

[2 marks]

(ii)

Without using tables or calculators, find the tangent of the angle between the two lines. [4 marks]

(i)

Prove that sin 28 - tan 8 cos 28 = tan 8.

[3 marks]

Express tan 8 in terms of sin 28 and cos 28.

[2 marks]

(ii)

(c)

State the values of tan a. and tan j3.

(iii)

Hence show, without using tables or calculators, that tan 22.5 ° = -{2 - 1. [4 marks]

(i)

Given that A, B and Care the angles of a triangle, prove that

(ii)

A +B 2

= cos

c

a)

sin

b)

sin B + sin C = 2 cos TA cos

2

[3 marks]

B-C

[2 marks]

2

Hence, show that A sin A + sin B + sin C = 4 cos T cos B cos C

2

2

.

[5 marks) Total 25 marks

GO ON TO THE NEXT PAGE 02134020/CAPE 2008

- 4-

4.

(a)

(b)

In the Cartesian plane with origin 0, the coordinates of points P and Q are (-2, 0) and (8, 8) respectively. The midpoint of PQ is M. (i)

Find the equation of the line which passes through M and is perpendicular to PQ. [8 marks)

(ii)

Hence, or otherwise, find the coordinates of the centre of the circle through P, 0 [9 marks] and Q.

(i)

Prove that the line y = x + 1 is a tangent to the circle x 2 + y2 + 1Ox - 12y + 11 = 0. [6 marks)

(ii)

Find the coordinates of the point of contact of this tangent to the circle. [2 marks)

Total 25 marks

SECTION C (Module 3) Answer BOTH questions.

5.

(a)

Find

lim X~

(b)

r

3

xl- 27

[4 marks]

+x- 12

A chemical process is controlled by the function u

P = -

t

+ vt 2 , where u and v are constants.

Given that P = - 1 when t = 1 and the rate of change of P with respect to t is - 5 when t =+, find the values ofu and v.

(c)

[6 marks]

The curve C passes through the point (- 1, 0) and its gradient at any point (x, y) is given by ~ = 3x2 - 6x. (i) (ii)

(iii)

Find the equation of C.

[3 marks]

Find the coordinates of the stationary points of C and determine the nature of [7 marks) EACH point. Sketch the graph of C and label the x-intercepts.

[5 marks] Total 25 marks

GO ON TO THE NEXT PAGE 02134020/CAPE 2008

-5-

6.

(a)

Differentiate with respect to x

"'2x - 1

[3 marks]

(ii)

sin2 (x3 + 4).

[4 marks]

(i)

Given that

(i)

(b)

X

J 6 I

(ii)

J 6

j(x) dx = 7, evaluate

[3 marks]

[2 - j(x)] dx.

I

The area under the curve y = x 2 + kx- 5, above the x-axis and bounded by the lines x = 1 and x = 3, is 14

~ units 2.

Find the value of the constant k. (c)

[4 marks]

The diagram below (not drawn to scale) represents a can in the shape of a closed cylinder with a hemisphere at one end. The can has a volume of 45 1t units3 .

(i)

Taking r units as the radius of the cylinder and h units as its height, show that, a)

b)

(ii)

[3 marks]

A

_ 51tr 3

+

901t

- r- , where A units is the external surface area of the can.

[3 marks]

Hence, find the value of r for which A is a minimum and the corresponding minimum value of A. [5 marks] 4

[Volume of a sphere = T 1t r3, surface area of a sphere = 4 1t r2. ] [Volume of a cylinder= 1t r2 h , curved surface area of a cylinder= 2 1t r h.] Total 25 marks

END OF TEST

02134020/CAPE 2008

FORM TP 2008241 CARIBBE AN

a\

\3)

EXAMIN ATIONS

TEST CODE 0213403 2 MAY/JUNE 2008

COUNCI L

ADVAN CED PROFIC IENCY EXAMI NATION PURE MATHEMA TICS UNIT 1 -PAPER 03/B ALGEBRA ,GEOMETR YANDCAL CULUS 1 % hours

( 16 MAY 2008 (p.m.))

This examination paper consists of THREE sections: Module 1, Module 2 and Module 3. Each section consists of 1 question. The maximum mark for each Module is 20. The maximum mark for this examination is 60. This examination consists of 3 printed pages.

INSTRUCTI ONS TO CANDIDATE S 1.

DO NOT open this examination paper until instructed to do so.

2.

Answer ALL questions from the THREE sections.

3.

Write your solutions, with full working, in the answer booklet provided.

4.

Unless otherwise stated in the question, any numerical answer that is not exact MUST be written correct to three significant figures.

Examination Materials Permitted Graph paper (provided) Mathematical formulae and tables (provided) - Revised 2008 Mathematical instruments Silent, non-programmable, electronic calculator

Copyright © 2007 Caribbean Examinations Council ®. All rights reserved. 02134032/CA PE 2008

-2-

SECTION A (Module 1) Answer this question.

1.

(a)

One root of the quadratic equation x 2 + 12x + k constant.

=

0 is three times the other and k is a

Find

(b)

(i)

the roots of the equation

[3 marks]

(ii)

the value of k.

[2 marks]

(i)

The functionj{.x) has the property that

f(2x + 3) = 2j(x) + 3, x e R. If JCO) (ii) (c)

=

6, find the values of f(3) andf(9).

[4 marks]

sx - sx- 2 = 15 000.

[4 marks]

Solve for X the equation

A computer manufacturer finds that when x million dollars are spent on research, the profit, P(x), in millions of dollars, is given by P(x) = 20 + 5 log3 (x + 3).

(i)

What is the profit if 6 million dollars are spent on research?

(ii)

How much should be spent on research to make a profit of 40 million dollars? [4 marks]

[3 marks]

Total 20 marks

SECTION B (Module 2) Answer this question.

2.

(a)

Find the values of x in the range 0 :=:: x :=:: 2n such that 4 cos3 x + 2 cos x - 5 sin 2x

(b)

(i)

=

0.

[10 marks]

Determine the value of the real number t such that the vectors p = 4i + 5j and q = 3i - tj are perpendicular. [2 marks]

GO ON TO THE NEXT PAGE 02134032/CAPE 2008

-3(ii)

Given that vectors u = 2i + 3j and v = i + Sj, find the acute angle 8 between u and v. [4 marks]

(iii)

Given that the vector u in (b) (ii) above represents a force F with respect to the origin 0, and axes Ox and Oy, calculate a)

the magnitude ofF

[2 marks]

b)

the angle

[2 marks]



of inclination ofF to Ox.

Total 20 marks

SECTION C (Module 3) Answer this question.

3.

(a)

A curve has equation y

=x + ~ . X

J

d2

d

+x ~

(i)

Show that x 2

(ii)

Find the equation of the normal to the curve at the point where x

[5 marks]

= y.

= 4. [5 marks]

f x2 +xx4 - 1 dx.

(b)

Find

(c)

The volume of the liquid in a container is V cm3 • The liquid leaks from the container at the rate of 30t cms3 per sec, where t is the time in seconds. (i) (ii)

[4 marks]

Write down a differential equation for Vwith respect to timet sees.

[2 marks]

Find the amount of liquid lost in the 3n1 second.

[4 marks] Total 20 marks

END OF TEST

02134032/CAPE 2008

FORM TP 2008243 CARIBBEAN

a\

TEST CODE

~ EXAMINATIONS

02234020

MAY/JUNE 2008

COUNCIL

ADVANCED PROFICIENCY EXAMINATION PURE MATHEMATICS UNIT 2 - PAPER 02 ANALYSIS, MATRICES AND COMPLEX NUMBERS 2 ¥2 hours

( 28 MAY 2008

(p.m.))

This examination paper consists of THREE sections: Module 1, Module 2 and Module 3. Each section consists of 2 questions. The maximum mark for each Module is 50. The maximum mark for this examination is 150. This examination consists of 5 printed pages.

INSTRUCTIONS TO CANDIDATES 1.

DO NOT open this examination paper until instructed to do so.

2.

Answer ALL questions from the THREE sections.

3.

Write your solutions, with full working, in the answer booklet provided.

4.

Unless otherwise stated in the question, any numerical answer that is not exact MUST be written correct to three significant figures.

Examination Materials Permitted

Graph paper (provided) Mathematical formulae and tables (provided) - Revised 2008 Mathematical instruments Silent, non-programmable, electronic calculator

Copyright © 2007 Caribbean Examinations Council ®. All rights reserved. 02234020/CAPE 2008

-

2

-

SECTION A (Module 1) Answer BOTH questions. 1.

(a)

Differentiate with respect to r (i)

(ii) (b)

eAx COS

In

[4 marks]

7LT

+ -.

[4 marks]

r r

Given y = 3 show, by using logarithms, that = —3~x In 3. dr dr 4y

(c)

(i)

[5 marks]

Express in partial fractions 2x2 - 3r + 4

[7 marks]

( r - l ) ( r 2+ l ) ‘

(ii)

Hence, find 2x2 - 3x + 4 dr. I ( r - l ) ( r 2+ l)

[5 marks] Total 25 marks

2.

(a)

Solve the differential equation dy_ dx

(b)

+ y = e1*.

[5 marks]

The gradient at the point (r, y ) on a curve is given by dy dr = e4x.

Given that the curve passes through the point (0, 1), find its equation. (c)

e

Evaluate J, x2 In r dr, writing your answer in terms of e.

[5 marks] [7 marks]

GO ON TO THE NEXT PAGE 02234020/CAPE 2008

- 3-

(d)

(i)

Use the substitution v = 1 - u to find du

f (ii)

(3 marks]

~·

Hence, or otherwise, use the substitution u = sin x to evaluate trfl

J 0

--.J

1 + sin x

dx.

(5 marks]

Total 25 marks

SECTION B (Module 2) Answer BOTH questions.

3.

(a)

A sequence {un} is defined by the recurrence relation

u n + 1 = u n + n, u 1 = 3,

(c)

E

N.

(i)

State the first FOUR terms of the sequence.

(ii)

Prove by mathematical induction, or otherwise, that un =

(b)

n

n2 - n

[3 marks]

+6

(8 marks]

2

A GP with first term a and common ratio r has sum to infinity 8 1 and the sum of the first [6 marks] four terms is 65. Find the values of a and r.

(i)

Write down the first FIVE terms in the power series expansion of In (1 + x), [3 marks] stating the range of values of for which the series is valid.

(ii)

a)

x

b)

Using the result from (c) (i) above, obtain a similar expansion for ln (1 - x) . [2 marks] Hence, prove that

1 +x -] In [1- x

=

2 (x + - 1 Y? + -1 .... 5 xs+) 3

[3 marks] Total25 marks

..

GO ON TO THE NEXT PAGE

02234020/CAPE 2008

-

4*

(a)

4

-

(i)

Sh(>w that the function J{x) = x? - 3x + 1 has! a root a in 2], i the closed sed interval [1, [1,2]. [3 marks]

(ii)

Use the Newton-Raphson method to show that hat if jt, is a first approximation jproximation to a imation tc in the interval [1,2], then a second approximation to a in the interval [1, 2] is given by 2xJ- 1 x2 : • [5 marks] 3x 1 —3 ~

(b)

(i)

(ii) (iii)

Use the binomial theorem or Maclaurin’s theorem to expand (1 +xY'A x )'A inascending in ascending powers o f x as far as the term in x3, stating the values of* for which the expansion tich is valid. [4 marks] marks] [4 Obtain a similar expansion for (1 -~x)'A.

Prove that if x is so small that x3 and higher powers o f x can be neglected, then I 1 —x

.

,

1 ,

•J i +x * ~ x Y x (iv)

marks] [4[4marks]

'

[5ESmarks] m ark s^

iking j:x == -^-,, show, without vi Hence, by taking using calculators or tables, ables,that thatVVT "2~isis 17 approximately ly equal to i f | § - .

m arksj [4[4marks] Total 25 m marks arks

SECTION C (Module 3) Answer BOTH questions. 5.

(a)

A cricket selection committee of 4 members is to be chosen from 5 former batsmen and 3 former bowlers. In how many ways can this committee be selected so that the committee includes AT LEAST (i)

ONE ONEformer formerbatsman? batsman?

(ii)ONE ONE batsman batsman andand ONE ONE bowler? bowler?

[8 marks]

[3 marks]

GO ON TO THE NEXT PAGE 02234020/CAPE 2008

-

(b)

5

-

Given the matrices

r 3 1 0^ ii 00 1 , 00 -1 00 v. J r 3

A =

1

V.

(i)

(ii) (iii)

1

r 00i -i00- 1r

-1

-1

^

-1 -1

-1 3 -1

r o0 rn 00 00--3 3 ~1 3 -1 V J

f ii

-1

B -

and

M =

J

-

-1

-1

determine EACH o f the following matrices: a)

A - B

[2 marks]

b)

AM

[3 marks]

deduce from (i) b) above the inverse A-1 o f the matrix A

[3 marks]

find the matrix X such that AX + B = A.

[6 marks] Total 25 marks

6.

(a)

(i)

Express the complex number —— ^ ~ 5 - zi

(ii)

(iii)

(b)

in the form X (1 —*).

[4 marks]

State the value of X.

[1 mark ]

^ 2 - 33/1 T • Verify that :hatf— is a real number and state its value. 1 55--i' J 1

[5 marks]

The complex number z is represented by the point T in an Argand diagram.

Given that z = show that

1 3 + it

where t is a variable and z denotes the complex conjugate o f z,

CO

z + z = 6 zz

(ii)

as t varies, 7Ties on a circle, and state the coordinates of the centre o f this circle. ]8 marks]

[7 marks]

Total 25 marks

END OF TEST

02234020/CAPE 2008

FORM TP 200824 4 CARIB BEAN

~

ruSTCO DE0223 4032

~

MAY/JUNE 2008

EXAMI NATIO NS

COUNC IL

ADVANCED PROF ICIEN CY EXAMINATION PURE MATHEM ATICS

UNIT 2 - PAPER 03/B ANALYSIS, MATRIC ES AND COMPLE X NUMBER S

c

1 Y2 hours

19 MAY 2008 (p.m.))

This examination paper consists of THREE sections: Module 1, Module 2 and Module 3. Each section consists of 1 question. The maximum mark for each Module is 20. The maximum mark for this examination is 60. This examination consists of 4 printed pages.

INSTRUC TIONS TO CANDIDA TES 1.

DO NOT open this examination paper until instructed to do so.

2.

Answer ALL questions from the THREE sections.

3.

Write your solutions, with full working, in the answer booklet provided.

4.

Unless otherwise stated in the question, any numerical answer that is not exact MUST be written correct to three significant figures.

Examinati on Materials Permitted Graph paper (provided) Mathematical formulae and tables (provided) - Revised 2008 Mathematical instruments Silent, non-programmable, electronic calculator

Copyright © 2007 Caribbean Examinations Council®. All rights reserved. 02234032/CAJPE 2008

-2-

SECTION A (Module 1) Answer this question.

1.

(a)

(b)

The parametric equations of a curve are given by x

=

3t2 andy = 6t.

at the point P on the curve where y

(i)

Find the value of :

(ii)

Find the equation of the normal to the curve at P.

= 18.

[5 marks] [3 marks]

In an experiment it was discovered that the volume, V cm3, of a certain substance in a room after t seconds may be determined by the equation V

= 60 e 0·041•

(i)

. terms o f t. . d -dV m Fm dt

(ii)

Determine the rate at which the volume

[3 marks]

a)

increases after 10 seconds

[1 mark]

b)

is increasing when it is 180 cm3•

[3 marks]

(iii)

Sketch the graph of V = 60 e 0 ·04 ' showing the point(s) of intersection, where they [5 marks] exist, with the axes.

Total 20 marks

GO ON TO THE NEXT PAGE

02234032/CAPE 2008

-3SECTION B (Module 2) Answer this question.

2.

(a)

Matthew started a savings account at a local bank by depositing $5 in the first week. In each succeeding week after the first, he added twice the amount deposited in the previous week. (i)

Derive an expression for a)

the amount deposited in the rth week, in terms of r

b)

the TOTAL amount in the account after n weeks, in terms of 11.

[3 marks]

[3 marks]

(ii) (b)

Calculate the MINIMUM number, n, of weeks it would take for the amount in the account to exceed $1000.00 if no withdrawal is made. [3 marks]

The series S is given by

s = (i)

(c)

1-

1 2

+ 3 1- + 4

5-1 8

+ 7-1 + 16

Express S as the sum of an AP and a GP.

[3 marks]

(ii)

Find the sum of the first n terms of S.

[3 marks]

(i)

Use the binomial theorem to expand ~ as a power series in y as far as the term iny". Y [2 marks]

(ii)

Given that the Maclaurin series expansion for cos x is

cosx = 1 -

-

;x2

2!

+ -

x4

4!

- -

x6

6!

+

find the first THREE non-zero terms in the power series expansion of sec x. [3 marks] Total 20 marks

GO ON TO THE NEXT PAGE 02234032/CA PE 2008

-

4

-

SECTION C (Module 3) Answer this question.

3.

(a)

(i)

By considering the augmented matrix for the following system of equations, determine the value o f k for which the system is consistent. x + 3y + 5z = 2 x+ + 4y —z = 1 z == kk y -—66z

(ii)

(b)

[5 marks]

Find ALL the solutions to the system for the value of A: obtained in (i) above. [4 marks]

The probability that a person selected at random — — —

owns a car is 0.25 is self-employed is 0.40 is self-employed OR owns aacar carisis 0.6.

0.6.

(i)

personselected selectedat at randomowns owns a car ANDis is Determine the probability that aaperson random a car AND self-employed. [4 marks]

(ii)

Stating a reason in EACH case, determine whether the events ‘owns a car’ and ‘is self-employed’ are a)

independent events

[4 marks]

b)

mutually exclusive events,

[3 marks] Total 20 marks

END OF TEST

02234032/CAPE 2008

~

FORM TP 2008240 CARIBBEAN

\3) EXAMINATIONS

TEST CODE

22134020

MAY/JUNE 2008

COUNCIL

ADVANCED PROFICIENCY EXAMINATION PURE MATHEMATICS UNIT 1 -PAPER 02 ALGEBRA,GEOMETRYANDCALCULUS 2% hours

(3o JUNE 2008 (a.m.))

This examination paper consists of THREE sections: Module 1, Module 2 and Module 3. Each section consists of 2 questions. The maximum mark for each Module is 50. The maximum mark for this examination is 150. This examination consists of 6 printed pages.

INSTRUCTIONS TO CANDIDATES 1.

DO NOT open this examination paper until instructed to do so.

2.

Answer ALL questions from the THREE sections.

3.

Write your solutions, with full working, in the answer booklet provided.

4.

Unless otherwise stated in the question, any numerical answer that is not exact MUST be written correct to three significant figures.

Examination Materials Permitted Graph paper (provided) Mathematical formulae and tables (provided) - Revised 2008 Mathematical instruments Silent, non-programmable, electronic calculator

Copyright © 2008 Caribbean Examinations Council ®. All rights reserved. 22134020/CAPE 2008

-2SECTION A (Module 1) Answer BOTH questions.

1.

(a)

(i)

Determine the values of the real number h for which the roots of the quadratic equation 4.x2 - 2hx + (8 - h) = 0 are real. [8 marks]

(ii)

The roots of the cubic equation ,il - 15.x2

+ px- 105 = 0

are 5 - k, 5 and 5 + k. Find the values of the constants p and k. (b)

(i)

[7 marks]

Copy the table below and complete by inserting the values for the functions j{x) = I x + 2 I and g(x) = 2 I x - 1 1.

X

-3

fl.x)

1

g(x)

8

-2

-1

0

2

3

3

1

6

1

2

5

4

6 2 [4 marks]

(ii)

Using a scale of 1 em to 1 unit on both axes, draw on the same graph j{x) and g(x) for - 3 S x S 5.

(iii)

[4 marks]

Using the graphs, fmd the values of x for which.f(x) = g(x).

[2 marks] Total 25 marks

GO ON TO THE NEXT PAGE 22134020/CAPE 2008

-

2.

(a)

3

-

Without using calculators or tables, evaluate 2710 + 910 274 + 911

(b)

(i)

(ii)

(c)

[8 marks]

Prove that log m = log,,=10 W , for m, n e N. l°g10"

[4 marks]

Hence, given that y = (log, 3) (log 4) (log 5) ... (log 32), calculate the exact value ofy. [6 marks]

Prove, by the principle of mathematical induction, that fin) = 7" —1 is divisible by 6, for all n e N.

[7 marks] Total 25 m arks

SECTION B (Module 2) Answer BOTH questions.

3.

(a)

Let p = i - j. If q = X,i + 2j, find values of X such that (i)

q is parallel to p

f1 m ark ]

(ii)

q is perpendicular to p

[2 marks]

(iii)

(b)

(c)

the angle between p and q is

71

[5 marks]

T

ou — cos 2A + sin 2A — = tan , A.. Show j-u that* ™1— ;-----———:— 1 + cos IA + sin 2A (i)

(ii) (iii)

[6 marks]

Using the formula for sinyl + sin B, show that if t = 2 cos 0 then sin (n + 1) 0= t sin n6 —sin (n - 1) 6

[2 marks]

Hence, show that sin 30 = (t2 - 1) sin 9.

[2 marks]

Using (c) (ii) above, or otherwise, find ALL solutions o f sin 3 9 = sin 0, 0 < 0 < n. [7 marks] Total 25 m arks

GO ON TO THE NEXT PAGE 22134020/CAPE 2008

.

4.

(a)

(i)

4

-

The line x - 2 y + 4 = 0 cuts the circle, x1+y2- 2 x - 2 0 y + 51 = 0 with centre P, at the points A and B. Find the coordinates of P, A and B.

(ii)

[6 marks)

The equation of any circle through A and B is of the form jc2 + y 2 —2x - 20y + 51 + A (x - 2j/ + 4) = 0 where A, is a parameter. A new circle C with centre Q passes through P, A and B, Find

(b)

a)

the value of X

[2 marksl

b)

the equation of circle C

[2 marks]

c)

between the the centres the distance, | PQ j,I, between centres

[3 marks]

d)

ifPQ PQ cuts cuts AB AB at at M.M. the distance 1PM || if

[4 marks]

A curve is given by the parametric equations x = 2 + 3 sin t, y = 3 + 4 cos t. Show that (i)

the Cartesian equation of the curve is (* -2 )> + Cv^3)2 = j 9 16

(ii)

[3 marks]

every point on the curve lies within or on the circle ( x - 2 ) 2 + ( y - 3)2 = 25.

(5 marks] Total 25 marks

GO ON TO THE NEXT PAGE 22134020/CAPE 2008

-5SECTION C (Module 3) Answer BOTH questions.

5.

(a) (b)

Use L'Hopital's rule to obtain

(i)

(ii)

(c)

Given thaty =

lim

x

-7

sin 4x 0 sin 5x

[3 marks]

x l - 4x

a)

fmd dy dx

b)

show that r

[4 marks]

~ =y2•

[2 marks]

d2

d

Hence, or otherwise, show that r ~ + 2 (x - y) ~ = 0.

[3 marks]

A rectangular box without a lid is made from thin cardboard. The sides of the base are 2x em and 3x em, and its height is h em. The total surface area of the box is 200 cm2 •

(i)

20 3x Show that h = - - - . X 5

(ii)

Find the height of the box for which its volume V cm3 is a maximum.

[4 marks]

[9 marks] Total 25 marks

GO ON TO THE NEXT PAGE 22134020/CAPE 2008

-

6.

(a)

6

-

Use the substitution u = 3x1 + I to find J

xk dx dr x2 +

(b)

[6 m arks] i'

’

A curve C passes through the point (3, -1 ) and has gradient x2 - 4x +■3 at the point (x, 7 ) on C. [4 m arks]

Find the equation o f C.

(c)

The figure below (not d raw n to scale) shows part o f the line y + 2x = 5 and part of the curve y = x ( 4 - x ) which meet at A. The line meets Oy at B and the curve cuts Ox at C,

y

A

B s y v \y

O

C

X

(i)

Find the coordinates o f A, B and C.

[6 m arks]

(ii)

Hence find the exact value o f the area of the shaded region.

[9 m arks] Total 25 m arks

END OF TEST

22134020/CAPE 2008

FORM TP 2008241 CARIBBEAN

a\

TESTCODE 22134032

~ EXAMINATIONS

MAY/JUNE 2008

COUNCIL

ADVANCED PROFICIENCY EXAMINATION PURE MATHEMATICS UNIT 1 - PAPER 03/B ALGEBRA,GEOMETRYANDCALCULUS 1 % hours

(26 JUNE 2008 (a.m.))

This examination paper consists of THREE sections: Module 1, Module 2 and Module 3. Each section consists of 1 question. The maximum mark for each Module is 20. The maximum mark for this examination is 60. This examination consists of 4 printed pages.

INSTRUCTIONS TO CANDIDATES 1.

DO NOT open this examination paper until instructed to do so.

2.

Answer ALL questions from the THREE sections.

3.

Write your solutions, with full working, in the answer booklet provided.

4.

Unless otherwise stated in the question, any numerical answer that is not exact MUST be written correct to three significant figures.

Examination Materials Permitted Graph paper (provided) Mathematical formulae and tables (provided) - Revised 2008 Mathematical instruments Silent, non-programmable, electronic calculator

Copyright © 2008 Caribbean Examinations Council ® . All rights reserved. 22134032/CAPE 2008

-2

-

SECTION A (M odule 1) Answer this question.

1.

(a)

(b)

(i)

Write log2 2P in terms o fp only.

[2 marks]

(ii)

Solve for x the equation log2 [logj {2x - 2)] = 2.

[3 marks]

The diagram below (not drawn to scale) shows the graph o f the function fix ) = 3.x- + hx2 + he + m which touches the jc-axis at x = —1.

/M

HA

-r-i,oT

( §3 '°> ’»> <

(0,-2),

(i)

Determine the values o f the constants h, k and m.

(ii)

[7 marks]

State the range o f values ofx in (-- 00. 0] for \vhich/(x) is a decreasing function. [2 marks] 100

(c)

Evaluate X (3 r + 2). r =

1

[6 marks] Total 20 marks

GO ON TO THE NEXT PAGE 22134032/CAPE 2008

-

3

-

SECTION B (M odule 2) Answer this question.

2.

The diagram below shows the path o f a com et around the sun S. The path is described by the parametric equation x = at2 and y = 2at, where a > 0 is a constant. y/ p

Q

R

s

x

p7 (a)

Show that the Cartesian equation for the path is y'1 = 4ax.

(b)

2a Given that the gradient m o f the tangent at any point on the path satisfies m■= -------,, yy (i) show that the equation o f the tangent at (•*,> (jc,, y j is y y 1l == 2a *,) in Cartesian 2 a ((x x + jtj) form and ty = jc + at2 in parametric form [5 marks] (ii) (iii)

(iv)

[2 marks]

find the equation o f the normal at the point P with parameter ■tl *1

[3 marks]

f, t2 r2 + + 22 = show that *,2 + + tx = 00 if the normal in (ii) above intersects the path again at the point P" with param eter f [6 marks] find the distance | QR | if the tangent at Pmeets meets the x-axis at R.

the x-axis at Q and the normal [4 marks] Total 20 marks

GO ON TO THE NEXT PAGE 22134032/CAPE 2008

-4-

SECTION C (Module 3) Answer this question.

3.

(a)

By expressing X

(ii)

Hence, fmd

(i)

Find the value of u if

-

lim rx-3 crx + 3) crx- 3), fmd x-79 - 9

9 as

(i)

lim x-79

rx-3

[4 marks]

:x?- - 1Ox+ 9 2u

(b)

[3 marks]

X

J

II

~ dx = X

!.

[3 marks]

4

(ii)

G iven that

J j{x) dx = 7, evaluate 1

L (c)

L 4

2

(f{x) + 1] dx +

(f{x) - 2] dx.

[5 marks]

The figure below (not drawn to scale) shows a hemispherical bow l which contains liquid.

Th e volume V cm3 of liquid is given by

v = -31

1t

h2 (24 - h)

where his the greatest depth of the liquid in em. Liquid is poured into the bowl at the rateof 100 cm3 per second.

(i)

Find dV in terms of h. dt

(ii)

Calculate the rate at which his increasing when h terms of rt.)

[3 marks]

= 2 em. (Leave your answer in [2 marks] Total 20 marks

END OF TEST

22134032/CAPE 2008

~

FORM TP 2008243 CARIBBEAN

\3) EXAMINATIONS

TEST CODE

22234020

MAY/JUNE 2008

COUNCIL

ADVANCED PROFICIENCY EXAMINATION \

PURE MATHEMATICS

.,.. ~

UNIT 2- PAPER 02 ANALYSIS, MATRICES AND COMPLEX NUMBERS 2% hours

( 15 JULY 2008 (p.m.))

This examination paper consists of THREE sections: Module 1, Module 2 and Module 3. Each section consists of 2 questions. The maximum mark for each Module is 50. The maximum mark for this examination is 150. This examination consists of 5 printed pages.

INSTRUCTIONS TO CANDIDATES 1.

DO NOT open this examination paper until instructed to do so.

2.

Answer ALL questions from the THREE sections.

3.

Write your solutions, with full working, in the answer booklet provided.

4.

Unless otherwise stated in the question, any numerical answer that is not exact MUST be written correct to three significant figures.

Examination Materials Permitted Graph paper (provided) Mathematical formulae and tables (provided) - Revised 2008 Mathematical instruments Silent, non-programmable, electronic calculator

Copyright © 2008 Caribbean Examinations Council®. All rights reserved. 22234020/CAPE 2008

-

4.

(a)

-

The sequence {an} o f positive numbers is defined by __ 3

_ 4 (1 + +a)

_

an+ \i

(b)

4

4 + an

_

, aa '

’

_

2 '

(i)

Find a2 and ay

[2 marks]

(ii)

Express an+ , - 2 in terms o f an.

[2 marks]

(iii)

that Given that an < 2for all n, show that a)

an+l < 2

[3 marks]

b)

an < a n+l.

[6 marks]

Find the term independent o f x in the binomial expansion o f (x2 — ^-)15. [You may leave your answer in the form of factorials and powers, for example, 151x8*.] x 85.] 2! ]6 marks]

(c)

Use the binomial theorem to find the difference between 2 10 and (2.002)10 correct to 5 decimal places. [6 marks] Total 25 marks

SECTION C (Module 3) A nswer BOTH questions.

5.

(a)

Four-digit, numbers are formed from the digits 1, 2, 3, 4, 7, 9. (i)

(ii)

(b)

How many 4-digit numbers can be formed if a)

the digits, 1, 2, 3, 4, 7, 9, can all be repeated?

[2 marks]

b)

none o f the digits, 1, 2, 3, 4, 7, 9, can be repeated?

[2 marks]

Calculate the probability that a 4-digit number in (a) (i) b) above is even. [3 marks]

A father and son practise shooting at basketball, and score when the ball hits the basket. The son scores 75% o f the time and the father scores 4 out o f 7 tries. If EACH takes one shot at the basket, calculate the probability that only ONE o f them scores. [6 marks]

GO ON TO THE NEXT PAGE 22234020/CAPE 2008

- 5(c)

(i)

Find the values of h, k

E

R such that 3 + 4i is a root of the quadratic equation

z2 + hz + k= 0. (ii)

[6 marks]

Use De Moivre's theorem for (cos cos 38

= 4 cos3 e-

3 cos

e + i sin 8)3 to show that

e.

[6 marks] Total 25 marks

6.

(a)

Solve for x the equation 1

x3 (b)

1 1 1

1 2 8

X

=0. [12 marks]

The Popular Taxi Service in a certain city provides transportation for tours of the city using cars, coaches and buses. Selection of vehicles for tours of distances (in km) is as follows:

x cars, 2y coaches and 3z buses cover 34 lan tours. 2x cars, 3y coaches and 4z buses cover 49 lan tours. 3x cars, 4y coaches and 6z buses cover 71 km tours. (i)

Express the information above as a matrix equation AX=Y where A is 3 x 3 matrix, X and Y are 3 x 1 matrices with

x~ (ii)

(iii)

LetB

~

[H

[3 marks]

[1 ~J 0 6 -4

a)

Calculate AB.

[3 marks]

b)

Deduce the inverse A - I of A.

[3 marks]

Hence, or otherwise, determine the number of cars and buses used in the 34lan tours. [4 marks] Total 25 marks

END OF TEST 22234020/CAPE 2008

FORM TP 2008244 CARIBBE AN

a\

\.3) EXAMIN ATIONS

TEST CODE 2223403 2 MAY/JUNE 2008

COUNCI L

ADVAN CED PROFIC IENCY EXAMI NATION PURE MATHEMA TICS UNIT 2- PAPER 03/B ANALYSIS, MATRICES AND CO:MPLEX NUMBERS 1 % hours

(27 JUNE 2008 (a.m.))

This examination paper consists of THREE sections: Module 1, Module 2 and Module 3. Each section consists of 1 question. The maximum mark for each Module is 20. The maximum mark for this examination is 60. This examination consists of 4 printed pages.

INSTRUCTI ONS TO CANDIDATE S 1.

DO NOT open this examination paper until instructed to do so.

2.

Answer ALL questions from the THREE sections.

3.

Write your solutions, with full working, in the answer booklet provided.

4.

Unless otherwise stated in the question, any numerical answer that is not exact MUST be written correct to three significant figures.

Examination Materials Permitted

Graph paper (provided) Mathematical formulae and tables (provided) - Revised 2008 Mathematical instruments Silent, non-programmable, electronic calculator

Copyright © 2008 Caribbean Examinations Council ®. All rights reserved. 22234032/CPJPE 2008

-

2

-

SECTION A (Module 1) Answer this question.

1

.

(a)

Given that x = In [y + V (y2 - 1)], y > 1, express y in terms of x.

(b)

Use the substitution u = sin x to find

J (c)

[5 marks]

[6 marks]

cos3x dx.

Engine oil at temperature T °C cools according to the model T = 60 e-*' + 10 where t is the time in minutes from the moment the engine is switched off. (i)

Determine the initial temperature of the oil when the engine is first switched off. [2 marks]

(ii)

If the oil cools to 32°C after three minutes, determine how long it will take for the oil to cool to a temperature of 15°C. [7 marks] Total 20 marks

SECTION B (Module 2) Answer this question.

2.

(a)

(i)

Write the general term of the series whose first four terms are 1 1x3

+ — .— + — .— + — -— + ... 3x5 5x7 7x9

[2 marks]

(ii)

Use the method of differences to find the sum of the first n terms.

[5 marks]

(iii)

Show that the series converges and fmd its sum to infinity.

[3 marks]

GO ON TO THE NEXT PAGE 22234032/CAPE 2008

-3

(b)

-

The diagram below (not drawn to scale) shows part o f the suspension o f a bridge A support cable POQ, is in the shape of a curve with equation

y-

1 10

X3' 2

+ c, where c is a constant.

Starting at P, through O and finishing at Q, 51 vertical cables are bolted 1 metre apart to the roadway and to the support cable POQ. The shortest vertical cable OAhas a length of 5 metres, where O is the lowest point of the support cable. The cost, in dollars, o f installing the cable LH at a horizontal distance of r metres from OA is S i00 plus-S $ h V~r^ w where h is the height of the point L above O.

P

0

L,

Support Cable

Vertical Cables

O

^Roadway A

H

(i)

ofr,r, the th cost o f installing the cable LH. Find, in termsis of

[4 marks]

(ii)

Hence, obtain the total cost of installing the 5 1 vertical cables.

[6 marks) Total 20 marks

GO ON TO THE NEXT PAGE 22234032/CAPE 2008

-

4

-

SECTION C (Module 3) Answer this question.

3.

w (a)

(b)

L c t:L t

( l - 2 i ) a + i) (1 + i)’

•

(i)

Express z in the form a + b\, where a, b e R.

[5 marks]

(ii)

Calculate the exact value o f j z |.

[3 marks]

Two 3 x 1 matrices X and Y satisfy the equation X = AY, where the matrix f l A = V.

-1

3 4

2 1

n 4 6J

is non-singular.

Find (i)

A“!

(ii)

Y, when X =

[8 marks]

6 [4 marks]

4 11

Total 20 marks

END OF TEST

22234032/CAPE 2008

TEST CODE

FORM TP 2009234 CARIBBEAN

02134020

MAY/JUNE 2009

EXAMINATIONS

COUNCIL

ADVANCED PROFICIENCY EXAMINATION PURE MATHEMATICS UNIT 1 - PAPER 02 ALGEBRA, GEOMETRY AND CALCULUS

c

2% hours

20 MAY 2009 (p.m.) )

This examination paper consists of THREE sections: Module 1, Module 2 and Module 3. Each section consists of 2 questions. The maximum mark for each Module is 50. The maximum mark for this examination is 150. This examination consists of 7 printed pages.

INSTRUCTIONS TO CANDIDATES 1.

DO NOT open this examination paper until instructed to do so.

2.

Answer ALL questions from the THREE sections.

3.

Write your solutions, with full working, in the answer booklet provided.

4.

Unless otherwise stated in the question, any numerical answer that is not exact MUST be written correct to three significant figures.

Examination Materials Permitted Graph paper (provided) Mathematical formulae and tables (provided) - Revised 2009 Mathematical instruments Silent, non-programmable, electronic calculator

Copyright © 2009 Caribbean Examinations Council ®. All rights reserved. 02134020/CAPE 2009

-

2

-

SECTION A (Module 1) Answer BO TH questions.

1.

(a)

Without the use of tables or a calculator, simplify V~28 ++• V 343 in. in the form k 'T l', where k is an integer. [5 marks]

(b)

Let x andy be positive real numbers such that x ^ y .

(i)

(ii)

Simplify

JC4 - /

[6 marks]

x-y

Hence, or otherwise, show that

4_yt + 1)3 + ^ + ^2^ + ^ + i ) y 2 + y i (y+ + 1i )) 4/ = = (( yy+ l ) 3 + ( y + l ) 2y + 0 + l ) j 2 + y .

(y

(iii) (c)

Deduce that (y + l ) 4 —y 4 < 4 (y + l) 3 .

Solve the equation log4 x = 1 + log2 2x, x > 0.

[4 marks] [2 marks] [8 marks] Total 25 marks

2.

(a)

The roots of the quadratic equation 2X2 + 4x + 5 = 0 are a and (3 . ? 2 Without solving the equation, find a quadratic equation with roots — and — .

[6 marks]

GO ON TO THE NEXT PAGE 02134020/CAPE 2009

-

(b)

3

-

The coach of an athletic club trains six athletes, u, v, w, x, y and z, in his training camp. He makes an assignment,/, of athletes u, v, x, y and z to physical activities 1, 2, 3 and 4 according to the diagram below in which A = {u, v, w, x, y, zj and B = {1, 2, 3, 4}. A

/

B

u V

1

w

2

X

3

y.

►4

2

(c)

(i)

Express / as a set of ordered pairs.

[4 marks]

(ii)

a)

State TWO reasons why / is NOT a function.

[2

b)

Hence, with MINIMUM changes to / construct a function g : A —> B as a set of ordered pairs. [4 marks]

c)

Determine how many different functions are possible for g in (ii) b) above. [2 marks]

marks]

The function /o n R is defined by Ax) =

x - 3 if jc < 3 x if x>3. 4

{

Find the value of /1/(20)]

[3 marks]

(ii) /[/(§ )] (hi) f [ f (3)].

[2 marks]

(i)

[2 marks] Total 25 marks

02134020/CAPE 2009

GO ON TO THE NEXT PAGE

-

4

.

SECTION B (Module 2) Answer BOTH questions.

3,

Answers to this question obtained by accurate drawing wili not be accepted. (a)

The circle C has equation (jc - 3)2 + (y - 4)2 = 25. (i)

State the radius and the coordinates o f the centre o f C.

[2 marks]

(ii)

Find the equation of the tangent at the point (6, 8) on C.

[4 marks]

(iii)

(b)

Calculate the coordinates o f the points o f intersection o f C with the straight line y = 2x + 3. [7marks]

The points P and Q have position vectors relative to the origin O given respectively p = - i + 6j and q = 3i + 8j. (i)

(ii)

by

a)

Calculate, in degrees, the size o f the acute angle G between p and q. [5 marks]

b)

Hence, calculate the area o f triangle POQ.

[2 marks]

Find, in terms of i and j, the position vector of a)

M, where M is the m idpoint o f PQ

[2 marks]

b)

R, where R is such that PQRO, labelled clockwise, forms a parallelogram. [3 marks] Total 25 marks

GO ON TO THE NEXT PAGE 02134020/CAPE 2009

-

4.

(a)

5

-

The diagram below, which is n o t d ra w n to scale, shows a quadrilateral ABCD in which AB = 4 cm, BC = 9 cm, AD = x cm and z. BAD = z BCD = 0 and ^ CZM is a nght-angle.

4 cm

A

e

B

9 cm

x cm

e C

D

(i)

Show that x = 4 cos 0 + 9 sin 0.

(ii)

By expressing x in the form r cos (9 - a), where r is positive and 0 < a < ~ n, find the MAXIMUM possible value o f x.

(b)

(c)

[4 marks]

[6 marks]

Given that A and B are acute angles such that sin A = — and cos B —---- -, find, w ithout using tables or calculators, the EXACT values of 5 1^ (i)

sin (,(A + B) sin

[3 marks]

(ii)

cos cos ((A - B)

[3 marks]

(lii)

cos 1cos 2A.

[2 marks]

Prove that

tan

X

2

+

71

4

sec x + tan x.

[7 marks]

Total 25 m arks

GO ON TO THE NEXT PAGE 02134020/CAPE 2009

-

6

-

SECTION C (Module 3) Answer BOTH questions.

5.

lim x —>2

x3- 8 __________ x2 - 6x + 8

(a)

Find

(b)

The function/on R is defined by ftx) =

Sketch the graph offlx) for the domain —1 < x < 2.

(ii)

Find

b)

(iii)

(d)

f 3 —x i f x > l |_1 + x if x < 1.

(i)

a)

(c)

[5 marks]

x

[2 marks]

[2 marks]

hm Ax) 1+

[2

/i>” i fix), x —=>I-

marks]

[3 marks]

Deduce that/[x) is continuous at x = 1. .

J_

'

X 2

Differentiate from first principles, with respect to x, the function v =

.

[6 marks]

'

The function /f,x) is such that / (x) = 3x2 + 6x + k where k is a constant. Given that/fO) = - 6 and/ ( l ) = -3 , find the function j(x).

[5 marks] Total 25 marks

GO ON TO THE NEXT PAGE 02134020/CAPE 2009

.- . '

- 76.

(a)

Given that y

= sin 2x + cos 2x, show that

d2y

-- + d.x2

4y

= 0.

a

(b)

(c)

[6 marks]

a

Given that J o (x + 1) dx

= 3 J (x- 1) dx, a> 0, fmd the value of the constant a. o

[6 marks]

The diagram below (not drawn to scale) represents a piece of thin cardboard 16 em by 10 em. Shaded squares, each of side x em, are removed from each comer. The remainder is folded to form a tray.

X

X

10 em

16 em (i)

Show that the volume, V cm3, of the tray is given by V

(ii)

= 4 (r - 13x2 + 40x).

[5 marks]

Hence, fmd a possible value of x such that Vis a maximum.

[8 marks] Total25 marks

END OF TEST 02 134020/CAPE 2009

TEST CODE

FORM TP 2009235 CARIBBEA N

02134032

MAY/JUNE 2009

EXAMI NA TIONS

CO UNC IL

ADVANCED PROFICIENCY EXAMINATION PURE MATHEMATICS UNIT 1 - PAPER 03/B ALGEBRA, GE OMETRY ANDCALCULUS 1

~ hours

( 10 J UNE 2009

(p.m.) )

This examination paper consists of THREE sections: Module 1, Module 2 and Module 3. Each section consists of 1 question. The maximum mark for each Module is 20. The maximum mark for this examination is 60. This examination consists of 3 printed pages.

I NSTRUCTIONS T O CANDIDATES

(

1.

DO NO T open this examination paper until instructed to do so.

2.

Answer ALL questions from the T HREE sections.

3.

Write your solutions, with full working, in the answer booklet provided.

4.

Unless otherwise stated in the question, any numerical answer that is not exact M UST be written correct to three significant figures.

Examination M ater ials Permitted Graph paper (provided) Mathematical formulae and tables (provided) - R evised 2009 Mathematical instruments Silent, non-programmable, electronic calculator

Copyright © 2009 Caribbean Examinations Council ®. All rights reserved. 02134032/CAPE 2009

-

2

-

SECTION A (Module 1) Answer this question.

1.

(a)

Find the set of real values of x for which [ x - 1 | > | 2x + 1 | .

(b)

[6 marks]

A packaging company makes crates for special purposes. The company finds that the unit costy(x), in thousands of dollars, of producing crates with a square base of x metres is f{x) = (x2 - Ax)2 + 2x2 - 8x. Using the substitution;-1= x2 - 4x, find the sizes of the crates for which the unit cost is three thousand dollars. [7 marks]

(c)

(i)

By taking logarithms, show that for any positive integers p and x, pte%x - x.

(ii)

[4 marks]

Hence, without using calculators or tables, find the EXACT value of « (logj 6 + log, 1 5 - 2 logj3)

2

.

[3 marks] Total 20 marks

SECTION B (Module 2) Answer this question.

2.

(a)

(i)

Show that the equation of the tangent to the circle x2 + y2 + 8x + 14 = 0 at the point (p, q) is (p + 4) (x - p) + q {y - q) = 0.

(ii)

Show that the equation of the tangent can also be written as px + qy + 4 (jc + p ) + 14 = 0.

(iii)

[3 marks]

[2 marks]

If the tangent at (p, q) on the circle passes through thepoint (-3, 3), find the values o fp and q. [7 marks]

GO ON TO THE NEXT PAGE 02134032/CAPE 2009

-

(b)

3

-

A point moves so that at time t its distances from the coordinate axes are given by x = 2 + 3 cos t and y = 4 + 4 sin t. (i)

Find the maximum and minimum values o f x and y. Find the maxii

[4 marks]

(ii)

Find the Cartesian equation o f the curve traced by the point.

[4 marks] Total 20 marks

SECTION C (Module 3) Answer this question.

J U + 3t5f - 1

[5 marks]

(a)

Find

(b)

The point P (-1, 5) is a point o f inflexion on the curve y = X s + bx2 + c, where b and c are constants.

dt.

Find (i)

the values b and ocf b and c theo fvalues

to curve the curve (ii)the equation o f the normal tolalthe at P.at P.

(c)

[5 marks] [3 marks]

Scientists on an experimental station released a spherical balloon into the atmosphere. The volume of air in the balloon is increased or decreased as required. (i)

The radius, r, of the balloon is increasing at the constant rate o f 0.02 cm/s. Find the rate at which the volume, V cm3, is increasing when r = 3 cm. Express your answer in terms o f n. [2 marks]

(ii)

The volume, V cm3, o f the balloon decreases by 6% when the radius decreases by p%. Find p. [5 marks] Total 20 marks

END OF TEST 02134032/CAPE 2009

TEST CODE

FORM TP 2009237 CARIBBEAN

02234020

MAY/JUNE 2009

EXAMINAT IONS

COUNCIL

ADVANCED PROFICIE NCY EXAMINATION PURE MATHEMATIC S UNIT 2- PAPER 02 ANALYSIS, MATRICES AND COMPLEX NUMBERS

c

2 Yz hours

27 MAY 2009 (p.m.) )

This examination paper consists of THREE sections: Module 1, Module 2 and Module 3. Each section consists of 2 questions. The maximum mark for each Module is 50. The maximum mark for this examination is 150. This examination consists of 5 printed pages.

INSTRUCTIONS T O CANDIDATES 1.

DO NOT open this examination paper until instructed to do so.

2.

Answer ALL questions from the THREE sections.

3.

Write your solutions, with full working, in the answer booklet provided.

4.

Unless otherwise stated in the question, any numerical answer that is not exact MUST be written correct to three significant figures.

Examination Materials Permitted Graph pape~ (provided) Mathematical formulae and tables (provided) - Revised 2009 Mathematical instruments Silent, non-programmable, electronic calculator

Copyright © 2009 Caribbean Examinations Council ® . All rights reserved. 02234020/CAPE 2009

-2SECTION A (Module 1) Answer BOTH questions.

1.

(a)

. d - dy 1"f Fm dx y = sin2 5x + sin2 3x + cos2 3x

[3 marks]

(ii)

y = ~ cos x 2

[4 marks]

(iii)

y =XX.

[4 marks]

(i)

(b)

(i)

[Note: cos-'x (ii)

-----;=1~~ 2

Given that y = cos-' x, where 0 :::; cos-' x :::; n , prove that ddx y = -

~ l -x

=arc cos x]

[7 marks]

The parametric equations of a curve are defined in terms of a parameter t by

y = ~ and x=cos-' t, w here O:=;t< 1.

a)

Show that -

b)

Hence, find

dy -f1+t = dx 2

~

[4 marks]

in terms oft, giving your answer in simplified form. (3 marks]

Total 25 marks

I

2.

(a)

Sketch the region whose area is defined by the integral

J~ 1 - x

2

dx .

[3 marks]

0

1

(b)

Using FIVE vertical strips, apply the trapezium rule to show that

J ~ l -x

2

0

dx ;::: 0. 759.

[6 marks]

(c)

(i)

Use integration by parts to show that, if I =

I

= x

~ 1- x

2

-

I +

J----;:::=1=

f~2 dx, then

::::- dx .

~ l - x2

[9 marks]

GO ON TO THE NEXT PAGE 02234020/CAPE 2009

_

(li)

Deduce that I = integration.

X

3

-

V 1 —x 2 + sin~' x - + cc, where c is an arbitrary constant o f 2

[Note: cos-1* = arc cos jc]

[2 m arksj

i

iii)

(iv)

Hence, find

V 1 - x 1 dx Jo

[3 m arks]

Use the results in Parts (b) and (c) (iii) above to find an approximation to n. [2 m arks]

Total 25 m arks SECTIO N B (M odule 2) A nswer B O T H questions.

3.

(a)

A sequence {tn} is defined by the recurrence relation tn + = *tn . for all n se N. K +> 5’5 (i f\] = 11 11 tn+i 5> + Ii = «n + 5, (i) (ii)

Determine termine tv titi and and tA t .,

[3 m arks]

Express press tn t in in terms terms of of nn.n.

[5 m arks]

(b)

Find the range o f values o f jt for which the common ratio r o f a convergent geometric 2 jc -3 series is [8 m arks] x+4 '

(c)

Let J[r) j{r) =

(i) (ii)

} ■, r e N. r+ 1

Express j{ f) - f i r + 1) in terms o f r.

[3 m arks]

Hence, or otherwise, find «n . s = xZ _____ - ______ S " ,-i (r+l)(r+2) ■ '

(iii)

[4 m arks]

Deduce the sum to infinity o f the series in (c) (ii) above.

[2 m arks]

Total 25 m arks GO ON TO THE NEXT PAGE 02234020/CAPE 2009

- 4-

4.

(a)

N such that

sene) = 2(n+ CJ 2

(i)

Find n

(ii)

The coefficient of x 2 in the expansion of

E

[5 marks]

(1 + 2x) 5 (1 + px) 4 is -26. Find the possible values of the real number p. (b)

[7 marks]

(i)

Write down the first FOUR non-zero terms of the power series expansion of ln (1 + 2x), stating the range of values of x for which the series is valid. [2 marks]

(ii)

Use Maclaurin's theorem to obtain the first THREE non-zero terms in the power [7 marks] series expansion in x of sin 2x.

(iii)

Hence, or otherwise, obtain the first THREE non-zero terms in the power series expansion in x of 1n (1 + sin 2x).

[4 marks]

Total25 marks

SECTION C (Module 3) Answer BOTH questions.

5.

(a)

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. Calculate the number of ways in which the committee of 4 persons can be formed. [5 marks]

(b)

Two balls are drawn without replacement from a bag containing 12 balls numbered 1 to 12. Find the probability that (i)

the numbers on BOTH balls are even

(ii)

the number on one ball is odd and the number on the other ball is even. [4 marks]

[4 marks]

GO ON TO THE NEXT PAGE 02234020/CAPE 2009

-

(c)

(i)

5

-

Find complex numbers u = x + iy such that x and y are real numbers and u2 = -1 5 + 8i.

(ii)

|7 marks]

Hence, or otherwise, solve for z the equation z2 - (3 + 2i) z + (5 + i) = 0.

[5 marks] Total 25 m arks

6.

(a)

Solve for x the equation x

-

3

1

1 -1

(b)

(i)

x

~

-1

5 1

= 0.

1 * -3

[10 m arks]

Given the matrices r A =

1 1 1

-1 -2 3

I 4 9

,

30 - 1 2 5 -8 -5 4

B -

2^ 3 1

V

(ii)

a)

find AB

]4 marks]

b)

hence deduce the inverse A-1 of the matrix A.

[3 m arks]

A system o f equations is given by x —y + z — 1 x — 2y + 4z = 5 x + 3y + 9z = 25. a)

Express the system in the form Aa' = b , where A is a matrix and x and b are column vectors.

b)

Hence, or otherwise, solve the system o f equations.

[5 marks] Total 25 m arks

END O F TEST

02234020/CAPE 2009

TEST CODE

FORM TP 2009238 CARIBBEAN

02234032

MAY/JUNE 2009

EXAMINATIONS

COUNCIL

ADVANCED PROFICIENCY EXAMINATION PURE MATHEMATICS UNIT 2- PAPER 03/B

ANALYSIS, MATRICES AND COMPLEX NUMBERS 1% hours ( 03 JUNE 2009 (a.m.) ) This examination paper consists of THREE sections: Module 1, Module 2 and Module 3 . Each section consists of 1 question. The maximum mark for each Module is 20. The maximum mark for this examination is 60. This examination consists of 3 printed pages.

INSTRVCTIONS TO CANPIDATES 1.

DO NOT open this examination paper until instructed to do so.

2.

Answer ALL questions from the THREE sections.

3.

Write your solutions, with full working, in the answer booklet provided.

4.

Unless otherwise stated in the question, any numerical answer that is not exact MUST be written correct to three significant figures.

Examination Materials Permitted

Graph paper (provided) Mathematical formulae and tables (provided) - Revised 2009 Mathematical instruments Silent, non-programmable, electronic calculator

Copyright © 2009 Caribbean Examinations Council ®. All rights reserved. 02234032/CPlPE 2009

-

2

-

SECTION A (Module 1) Answer this question.

1.

(a)

Solve the differential equation

* r+ , (/+1) = 0. x+1

(d)

[8 marks]

[5 marks]

By using y = 2x, or otherwise, solve 4x – 3 (2x + 1) + 8 = 0.

[7 marks] Total 25 marks

02134020/CAPE 2010

GO ON TO THE NEXT PAGE

-3-

2.

(a)

(i)

n 1 Use the fact that Sn = Σ r = — n (n + 1) to express 2 r=1 2n

[2 marks]

S2n = Σ r in terms of n. r=1

(ii)

Find constants p and q such that [5 marks]

S2n – Sn = pn2 + qn. (iii)

Hence, or otherwise, find n such that [5 marks]

S2n – Sn = 260. (b)

The diagram below (not drawn to scale) shows the graph of y = x2 (3 – x). The coordinates of points P and Q are (2, 4) and (3, 0) respectively. y P 0

1

2

Q x

3

[4 marks]

(i)

Write down the solution set of the inequality x2 (3 – x) < 0.

(ii)

Given that the equation x2 (3 – x) = k has three real solutions for x, write down the set of possible values for k. [3 marks]

(iii)

The functions f and g are defined as follows: f : x → x2 (3 – x), 0 < x < 2 g : x → x2 (3 – x), 0 < x < 3 By using (b) (ii) above, or otherwise, show that a)

f has an inverse

b)

g does NOT have an inverse.

[6 marks] Total 25 marks

02134020/CAPE 2010

GO ON TO THE NEXT PAGE

-4SECTION B (Module 2) Answer BOTH questions. 3.

(a)

The vectors p and q are given by p = 6i + 4j q = –8i – 9j.

(b)

(i)

Calculate, in degrees, the angle between p and q.

(ii)

a)

Find a non-zero vector v such that p.v = 0.

b)

State the relationship between p and v.

[5 marks]

[5 marks]

The circle C1 has (–3, 4) and (1, 2) as endpoints of a diameter. [6 marks]

(i)

Show that the equation of C1 is x2 + y2 + 2x – 6y + 5 = 0.

(ii)

The circle C2 has equation x2 + y2 + x – 5y = 0. Calculate the coordinates of the points of intersection of C1 and C2. [9 marks] Total 25 marks

02134020/CAPE 2010

GO ON TO THE NEXT PAGE

-

4.

(a)

(i) (11) (iii)

(b)

5

-

Solve the equation cos 3A = 0.5 for 0 < A < n.

[4 marks]

Show that cos 3A = 4 cos3A - 3 cos A.

[6 marks]

The THREE roots o f the equation 4p3- 3p - 0.5 = 0 all lie between -1 and 1. Use the results in (a) (i) and (ii) to find these roots. [4 marks]

The following diagram, not drawn to scale, represents a painting o f height, h metres, that is fastened to a vertical wall at a height o f d metres above, and x metres away from, the level o f an observer, O.

Painting hm /\

dm a O

x m

v

The viewing angle o f the painting is ( a - P), where a and p are respectively the angles of inclination, in radians, from the level o f the observer to the top and base o f the painting.

(i)

(ii)

hx Show that tan ( a - P) = —— — d~—— — . x I d (d I h)

[6 marks]

The viewing angle o f the painting, ( a - P), is at a maximum when x = V h (d + h). Calculate the maximum viewing angle, in radians, when d = 3h. [5 marks] Total 25 marks

GO ON TO THE NEXT PAGE 02134020/CAPE 2010

-6SECTION C (Module 3) Answer BOTH questions. 5.

(a)

(b)

Find (i)

lim x→3

x2 – 9 ———– x3 – 27

[4 marks]

(ii)

lim x→0

tan x – 5x ————— . sin 2x – 4x

[5 marks]

The function f on R is defined by

f(x) =

(i)

(ii) (c)

3x – 7, if x > 4 1 + 2x, if x < 4.

Find a)

lim f(x) x → 4+

[2 marks]

b)

lim f(x). x → 4–

[2 marks] [2 marks]

Deduce that f(x) is discontinuous at x = 4.

∫

1

1 x–— x

2

[6 marks]

(i)

Evaluate

(ii)

Using the substitution u = x2 + 4, or otherwise, find

–1

∫ x √x

2

+ 4 dx.

dx.

[4 marks] Total 25 marks

02134020/CAPE 2010

GO ON TO THE NEXT PAGE

-

6.

(a)

7

-

Differentiate with respect to x (i)

(ii)

y = sin (3x + 2) + tan 5x

y =

[3 marks]

x2 + 1 x3 - 1

[4 marks]

,

4

(b)

The function f(x) satisfies f f x ) dx = 7. 4

(i)

Findf

[3 f x ) + 4] dx.

[4 marks] 3

(ii)

Using the substitution u = x + 1, evaluate

f 2f

(x + 1) dx.

[4 marks]

0

(c)

In the diagram below (not drawn to scale), the line x + y = 2 intersects the curve y = x2 at the points P and Q. 2 /y = X

y/

p

XT o‘

X

x +y = 2

(i) (ii)

Find the coordinates o f the points P and Q.

[5 marks]

Calculate the area o f the shaded portion o f the diagram bounded by the curve and the straight line. [5 marks] Total 25 marks

END OF TEST

02134020/CAPE 2010

TEST CODE

FORM TP 2010228 CARIBBEAN

02134032

MAY/JUNE 2010

E XAM I NAT I O N S

COUNCIL

ADVANCED PROFICIENCY EXAMINATION PURE MATHEMATICS UNIT 1 – PAPER 03/B ALGEBRA, GEOMETRY AND CALCULUS 1 ½ hours

09 JUNE 2010 (p.m.)

This examination paper consists of THREE sections: Module 1, Module 2 and Module 3. Each section consists of 1 question. The maximum mark for each Module is 20. The maximum mark for this examination is 60. This examination consists of 4 printed pages.

INSTRUCTIONS TO CANDIDATES 1.

DO NOT open this examination paper until instructed to do so.

2.

Answer ALL questions from the THREE sections.

3.

Write your solutions, with full working, in the answer booklet provided.

4.

Unless otherwise stated in the question, any numerical answer that is not exact MUST be written correct to three significant figures.

Examination Materials Permitted Graph paper (provided) Mathematical formulae and tables (provided) – Revised 2009 Mathematical instruments Silent, non-programmable, electronic calculator

Copyright © 2009 Caribbean Examinations Council ®. All rights reserved. 02134032/CAPE 2010

-2SECTION A (Module 1) Answer this questions. 1.

(a)

(b)

The roots of the cubic equation x3 + px2 + qx + 48 = 0 are α, 2α and 3α. Find (i)

the value of α

[2 marks]

(ii)

the values of the constants p and q.

[4 marks]

The function f on R is given by f: x → 3x – 2. (i)

Show that f is one-to-one.

(ii)

Find the value of x for which f (f (x)) = f (x + 3).

(c)

[3 marks]

[4 marks]

Prove by mathematical induction that 9n – 1 is divisible by 8 for all n ∈ N.

[7 marks] Total 20 marks

02134032/CAPE 2010

GO ON TO THE NEXT PAGE

-3SECTION B (Module 2) Answer this questions. 2.

A surveyor models the boundaries and extent of a triangular plot of land on a Cartesian plane as shown in the diagram below (not drawn to scale). The line 2x + 3y = 6 meets the y-axis at A and the x-axis at B. C is the point on the line 2x + 3y = 6 such that AB = BC. CD is drawn perpendicular to AC to meet the line through A parallel to 5x + y = 7 at D. 2x

+

y 3y =

6

A 0

B

x C

D

(a)

Find (i)

the coordinates of A, B and C

[6 marks]

(ii)

the equations of the lines CD and AD.

[5 marks]

(b)

Show that the point D has coordinates (2, –8).

[4 marks]

(c)

Calculate the area of triangle ACD.

[5 marks] Total 20 marks

02134032/CAPE 2010

GO ON TO THE NEXT PAGE

-4SECTION C (Module 3) Answer this questions. 3.

∫ (cos 5x + tan

[4 marks]

x) dx.

(a)

Find

(b)

Part of the curve y = x (x – 1) (x – 2) is shown in the figure below (not drawn to scale).

2

y

0

(c)

q

p

x

[2 marks]

(i)

Find the values of p and q.

(ii)

Hence find the area of the region enclosed by the curve and the x-axis from [5 marks] x = 0 to x = q.

A piece of wire, 60 cm long, is bent to form the shape shown in the figure below, (not drawn to scale). The shape consists of a semi-circular arc of radius r cm and three sides of a rectangle of height x cm. m

rc

x cm

(i)

Express x in terms of r.

(ii)

Show that the enclosed area A cm2 is given by π . A = 60r – 2r2 1 + — 4

(iii)

Find the exact value of r for the stationary point of A.

[3 marks]

[3 marks] [3 marks] Total 20 marks

END OF TEST 02134032/CAPE 2010

TEST CODE

FORM TP 2010230 CARIBBEAN

02234020

MAY/JUNE 2010

E XAM I NAT I O N S

COUNCIL

ADVANCED PROFICIENCY EXAMINATION PURE MATHEMATICS UNIT 2 – PAPER 02 ANALYSIS, MATRICES AND COMPLEX NUMBERS 2 ½ hours

26 MAY 2010 (p.m.)

This examination paper consists of THREE sections: Module 1, Module 2 and Module 3. Each section consists of 2 questions. The maximum mark for each Module is 50. The maximum mark for this examination is 150. This examination consists of 6 printed pages.

INSTRUCTIONS TO CANDIDATES 1.

DO NOT open this examination paper until instructed to do so.

2.

Answer ALL questions from the THREE sections.

3.

Write your solutions, with full working, in the answer booklet provided.

4.

Unless otherwise stated in the question, any numerical answer that is not exact MUST be written correct to three significant figures.

Examination Materials Permitted Graph paper (provided) Mathematical formulae and tables (provided) – Revised 2009 Mathematical instruments Silent, non-programmable, electronic calculator

Copyright © 2009 Caribbean Examinations Council ®. All rights reserved. 02234020/CAPE 2010

-2SECTION A (Module 1) Answer BOTH questions. 1.

(a)

(b)

The temperature of water, x° C, in an insulated tank at time, t hours, may be modelled by the equation x = 65 + 8e–0.02t. Determine the [2 marks]

(i)

initial temperature of the water in the tank

(ii)

temperature at which the water in the tank will eventually stabilize

(iii)

time when the temperature of the water in the tank is 70° C.

(i)

Given that y = etan

[4 marks]

1 1 π, show that , where – — π < tan–1 (2x) < — 2 2

–1(2x)

dy (1 + 4x2) —– = 2y. dx (ii) (c)

[4 marks]

d 2y Hence, show that (1 + 4x2)2 —–2 = 4y (1 – 4x). dx

Determine

[2 marks]

[4 marks]

4

dx ∫ ——— e +1 x

[6 marks]

(i)

by using the substitution u = ex

(ii)

4 by first multiplying both the numerator and denominator of the integrand ——— ex + 1 [3 marks]

by e–x before integrating.

Total 25 marks 2.

(a)

(i)

d Given that n is a positive integer, find —– [x (ln x)n]. dx

(ii)

Hence, or otherwise, derive the reduction formula In = x (ln x)n – nIn – 1, where In =

(iii)

02234020/CAPE 2010

∫ (ln x)

n

[4 marks]

[4 marks]

dx.

∫

Use the reduction formula in (a) (ii) above to determine (ln x)3 dx.

[6 marks]

GO ON TO THE NEXT PAGE

-3(b)

The amount of salt, y kg, that dissolves in a tank of water at time t minutes satisfies the 2y dy differential equation —– + ——— = 3. t + 10 dt (i)

Using a suitable integrating factor, show that the general solution of this differential c equation is y = t + 10 + ————, where c is an arbitrary constant. (t + 10)2

(ii)

[7 marks]

Given that the tank initially contains 5 kg of salt in the liquid, calculate the amount of salt that dissolves in the tank of water at t = 15. [4 marks] Total 25 marks

SECTION B (Module 2) Answer BOTH questions. 3.

(a)

The first four terms of a sequence are 2 x 3,

(b) (c)

5 x 5,

8 x 7,

11 x 9. [2 marks]

(i)

Express, in terms of r, the r th term of the sequence.

(ii)

If Sn denotes the series formed by summing the first n terms of the sequence, find Sn in terms of n. [5 marks]

The 9th term of an A.P. is three times the 3rd term and the sum of the first 10 terms is 110. Find the first term a and the common difference d. [6 marks] (i)

(ii) (iii)

Use the binominal theorem to expand (1 + 2x)½ as far as the term in x3, stating the values of x for which the expansion is valid. [5 marks] x 1 Prove that ——————— = — (1 + x – √ 1 + 2x) for x > 0. x 1 + x + √ 1 + 2x

[4 marks]

Hence, or otherwise, show that, if x is small so that the term in x3 and higher powers of x can be neglected, the expansion in (c) (ii) above is approximately equal to 1 — x (1 – x). 2

[3 marks] Total 25 marks

02234020/CAPE 2010

GO ON TO THE NEXT PAGE

-44.

(a)

(i)

By expressing nCr and nCr – 1 in terms of factorials, prove that nCr + nCr – 1 = n + 1Cr. [6 marks]

(ii)

a)

1 Given that r is a positive integer and f(r) = —, show that r! r f(r) – f(r + 1) = ——— (r + 1)!

b)

Hence, or otherwise, find the sum n

Sn = Σ

r=1

c) (b)

[3 marks]

r ——— . (r + 1)!

[5 marks]

Deduce the sum to infinity of Sn in (ii) b) above.

[2 marks]

(i)

Show that the function f(x) = x3 – 6x + 4 has a root x in the closed interval [0, 1]. [5 marks]

(ii)

By taking 0.6 as a first approximation of x1 in the interval [0, 1], use the NewtonRaphson method to obtain a second approximation x2 in the interval [0, 1]. [4 marks] Total 25 marks

02234020/CAPE 2010

GO ON TO THE NEXT PAGE

-5SECTION C (Module 3) Answer BOTH questions. 5.

(a)

(b)

(c)

Calculate (i)

the number of different permutations of the 8 letters of the word SYLLABUS [3 marks]

(ii)

the number of different selections of 5 letters which can be made from the letters of the word SYLLABUS. [5 marks]

The events A and B are such that P(A) = 0.4, P(B) = 0.45 and P(A

[3 marks]

(i)

Find P(A

(ii)

Stating a reason in each case, determine whether or not the events A and B are

(i) (ii)

B).

B) = 0.68.

a)

mutually exclusive

[3 marks]

b)

independent.

[3 marks]

i–1 Express the complex number (2 + 3i) + —— in the form a + ib, where a and b are i+1 both real numbers. [4 marks] Given that 1 – i is the root of the equation z3 + z2 – 4z + 6 = 0, find the remaining roots. [4 marks] Total 25 marks

02234020/CAPE 2010

GO ON TO THE NEXT PAGE

-66.

(a)

A system of equations is given by x+y+z=0 2x + y – z = –1 x + 2y + 4z = k where k is a real number.

(b)

(i)

Write the augmented matrix of the system.

[2 marks]

(ii)

Reduce the augmented matrix to echelon form.

[3 marks]

(iii)

Deduce the value of k for which the system is consistent.

[2 marks]

(iv)

Find ALL solutions corresponding to the value of k obtained in (iii) above. [4 marks]

Given A =

(i)

0 –1 –1 0 1 1

1 1 1

.

Find a)

A2

[4 marks]

b)

B = 3I + A – A2

[4 marks]

(ii)

Calculate AB.

[4 marks]

(iii)

Deduce the inverse, A–1, of the matrix A.

[2 marks] Total 25 marks

END OF TEST

02234020/CAPE 2010

TEST CODE

FORM TP 2010231 CARIBBEAN

02234032

MAY/JUNE 2010

E XAM I NAT I O N S

COUNCIL

ADVANCED PROFICIENCY EXAMINATION PURE MATHEMATICS UNIT 2 – PAPER 03/B ANALYSIS, MATRICES AND COMPLEX NUMBERS 1 ½ hours

02 JUNE 2010 (a.m.)

This examination paper consists of THREE sections: Module 1, Module 2 and Module 3. Each section consists of 1 question. The maximum mark for each Module is 20. The maximum mark for this examination is 60. This examination consists of 4 printed pages.

INSTRUCTIONS TO CANDIDATES 1.

DO NOT open this examination paper until instructed to do so.

2.

Answer ALL questions from the THREE sections.

3.

Write your solutions, with full working, in the answer booklet provided.

4.

Unless otherwise stated in the question, any numerical answer that is not exact MUST be written correct to three significant figures.

Examination Materials Permitted Graph paper (provided) Mathematical formulae and tables (provided) – Revised 2009 Mathematical instruments Silent, non-programmable, electronic calculator

Copyright © 2009 Caribbean Examinations Council ®. All rights reserved. 02234032/CAPE 2010

-2SECTION A (Module 1) Answer this questions. 1.

(a)

Express in partial fractions 1 – x2 . –––––––— x (x2 + 1)

(b)

[7 marks]

The rate of change of a population of bugs is modelled by the differential equation dy —– – ky = 0, where y is the size of the population at time, t, given in days, and k is the dt constant. Initially, the population is y0 and it doubles in size in 3 days. (i)

(ii)

Show that a)

y = y0 ekt

[7 marks]

b)

1 k = — ln 2. 3

[3 marks]

Find the proportional increase in population at the end of the second day. [3 marks] Total 20 marks

02234032/CAPE 2010

GO ON TO THE NEXT PAGE

-3SECTION B (Module 2) Answer this questions. 2.

(a)

The sum to infinity of a convergent geometric series is equal to six times the first term. Find the common ratio of the series. [5 marks]

(b)

Find the sum to infinity of the series Σ ar whose r th term ar is

∞

r=1

2r + 1 ——— . r! (c)

[8 marks]

1 A truck bought for $15 000 depreciates at the rate of 12 — % each year. Calculate the 2 value of the truck (i)

after 1 year

[2 marks]

(ii)

after t years

[2 marks]

(iii)

when its value FIRST falls below $5 000.

[3 marks] Total 20 marks

02234032/CAPE 2010

GO ON TO THE NEXT PAGE

-4SECTION C (Module 3) Answer this questions. 3.

(a)

Find the number of integers between 300 and 1 000 which can be formed by using the digits 1, 3, 5, 7 and 9 (i)

if NO digit can be repeated

[3 marks]

(ii)

if ANY digit can be repeated.

[2 marks] [3 marks]

(b)

Find the probability that a number in (a) (ii) above ends with the digit 9.

(c)

A farmer made three separate visits to the chicken farm to purchase chickens. On each visit he paid $ x for each grade A chicken, $ y for each grade B chicken and $ z for each grade C. His calculations are summarised in the table below. Number of Visits

Number of Chickens Bought Grade A

Grade B

Grade C

Total Spent $

1

20

40

60

1 120

2

40

60

80

1 720

3

60

80

120

2 480

(i)

Use the information above to form a system of linear equations in x, y and z. [3 marks]

(ii)

Express the system of equations in the form Ax = b.

[2 marks]

(iii)

Solve the equations to find x, y and z.

[7 marks] Total 20 marks

END OF TEST

02234032/CAPE 2010

TEST CODE

FORM TP 2011231 CARIBBEAN

02134020

MAY/JUNE 2011

E XAM I NAT I O N S

COUNCIL

ADVANCED PROFICIENCY EXAMINATION PURE MATHEMATICS UNIT 1 – PAPER 02 ALGEBRA, GEOMETRY AND CALCULUS 2 ½ hours

10 MAY 2011 (p.m.)

This examination paper consists of THREE sections: Module 1, Module 2 and Module 3. Each section consists of 2 questions. The maximum mark for each Module is 50. The maximum mark for this examination is 150. This examination consists of 7 printed pages.

INSTRUCTIONS TO CANDIDATES 1.

DO NOT open this examination paper until instructed to do so.

2.

Answer ALL questions from the THREE sections.

3.

Write your solutions, with full working, in the answer booklet provided.

4.

Unless otherwise stated in the question, any numerical answer that is not exact MUST be written correct to three significant figures.

Examination Materials Permitted Graph paper (provided) Mathematical formulae and tables (provided) – Revised 2010 Mathematical instruments Silent, non-programmable, electronic calculator

Copyright © 2010 Caribbean Examinations Council All rights reserved. 02134020/CAPE 2011

-2SECTION A (Module 1) Answer BOTH questions. 1.

(a)

Without using calculators, find the exact value of (i) (ii)

(b)

2

(√75 + √12 ) – (√75 – √12 ) 27

–14

x 9

–38

2

[3 marks]

–18

[3 marks]

x 81 .

The diagram below, not drawn to scale, represents a segment of the graph of the function f(x) = x3 + mx2 + nx + p where m, n and p are constants. f(x) (0,4)

Q

0

1

x

2

Find

(c)

(i)

the value of p

[2 marks]

(ii)

the values of m and n

[4 marks]

(iii)

the x-coordinate of the point Q.

[2 marks]

(i)

By substituting y = log2x, or otherwise, solve, for x, the equation √ log2x = log2 √ x .

(ii)

[6 marks]

Solve, for real values of x , the inequality x2 – | x | – 12 < 0.

[5 marks] Total 25 marks

02134020/CAPE 2011

GO ON TO THE NEXT PAGE

-32.

(a)

The quadratic equation x2 – px + 24 = 0, p ∈ R, has roots a and b. (i)

(ii) (b)

Express in terms of p a)

a + b

[1 mark ]

b)

a2 + b2.

[4 marks]

Given that a2 + b2 = 33, find the possible values of p.

[3 marks]

The function f(x) has the property that f(2x + 3) = 2f(x) + 3, x ∈ R. If f(0) = 6, find the value of (i)

f(3)

[4 marks]

(ii)

f(9)

[2 marks]

(iii)

f(–3).

[3 marks]

(c)

Prove that the product of any two consecutive integers k and k + 1 is an even integer. [2 marks]

(d)

Prove, by mathematical induction, that n (n2 + 5) is divisible by 6 for all positive integers n. [6 marks] Total 25 marks

02134020/CAPE 2011

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-4SECTION B (Module 2) Answer BOTH questions. 3.

(a)

(b)

(i)

Let a = a1i + a2j and b = b1i + b2j with | a | = 13 and | b | = 10. Find the value of (a + b) . (a – b). [5 marks]

(ii)

If 2b – a = 11i, determine the possible values of a and b.

[5 marks]

The line L has equation x – y + 1 = 0 and the circle C has equation x2 + y2 – 2y – 15 = 0. (i)

Show that L passes through the centre of C.

[2 marks]

(ii)

If L intersects C at P and Q, determine the coordinates of P and Q.

[3 marks]

(iii)

Find the constants a, b and c such that x = b + a cos θ and y = c + a sin θ are parametric equations (in parameter θ) of C. [3 marks]

(iv)

Another circle C2, with the same radius as C, touches L at the centre of C. Find the possible equations of C2. [7 marks] Total 25 marks

4.

(a)

By using x = cos2θ, or otherwise, find all values of the angle θ such that [6 marks]

8 cos4 θ – 10 cos2 θ + 3 = 0, 0 < θ < p. (b)

The diagram below, not drawn to scale, shows a rectangle PQRS with sides 6 cm and 8 cm inscribed in another rectangle ABCD. A

Q

B

6 cm

P

R 8 cm

D

02134020/CAPE 2011

S

θ

C

GO ON TO THE NEXT PAGE

-5-

(c)

(i)

The angle that SR makes with DC is θ. Find, in terms of θ, the length of the side [2 marks] BC.

(ii)

Find the value of θ if | BC | = 7 cm.

[5 marks]

(iii)

Is 15 a possible value for | BC |? Give a reason for your answer.

[2 marks]

1 – cos 2 θ Show that ———— = tan θ.

[3 marks]

(i) (ii)

(iii)

sin 2 θ

Hence, show that a)

1 – cos 4 θ ———— = tan 2 θ.

[3 marks]

b)

1 – cos 6 θ ———— = tan 3 θ.

[2 marks]

sin 4 θ

sin 6 θ

Using the results in (c) (i) and (ii) above, evaluate n

S (tan r θ sin 2r θ + cos 2r θ) r=1 where n is a positive integer.

[2 marks] Total 25 marks

02134020/CAPE 2011

GO ON TO THE NEXT PAGE

-6SECTION C (Module 3) Answer BOTH questions.

5.

lim x → –2

x2 + 5x + 6 ————— . x2 – x – 6

(a)

Find

(b)

The function f on R is defined by f(x) =

[4 marks]

x2 + 1 if x > 2 bx + 1 if x < 2.

Determine

(c)

(i)

f(2)

[2 marks]

(ii)

lim f(x) x → 2+

[2 marks]

(iii)

lim f(x) in terms of the constant b x → 2–

[2 marks]

(iv)

the value of b such that f is continuous at x = 2.

[4 marks]

The curve y = px3 + qx2 + 3x + 2 passes through the point T (1, 2) and its gradient at T is 7. The line x = 1 cuts the x-axis at M, and the normal to the curve at T cuts the x-axis at N. Find (i)

the values of the constants p and q

[6 marks]

(ii)

the equation of the normal to the curve at T

[3 marks]

(iii)

the length of MN.

[2 marks] Total 25 marks

02134020/CAPE 2011

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-

6

.

(a)

7

-

The diagram below, not drawn to scale, is a sketch o f the section o f the function f x ) = x (x2 - 12) which passes through the origin O. A and B are stationary points on the curve.

y

A f(x) = x(x? - 1 1 ) -

o

X

B

Find (i)

(b)

the coordinates o f each o f the stationary points A and B

[8 marks]

(ii)

the equation o f the normal to the curve f x ) = x (x2 - 12) at the origin, O [2 marks]

(iii)

the area between the curve and the positive x-axis.

(i)

Use the result a

f

I* a

f(x) dx = I" f (a - x) dx, a > 0,

^0

0

rn fn to show that I x sin x dx = I (n - x) sin x dx. 0

0

(ii)

[6 marks]

[2 marks]

Hence, show that a)

b)

•n

I

x sin x dx = n f n sin x dx - f n x sin x dx

n

0

0

[2 marks]

0

x sin x dx = n.

[5 marks] Total 25 marks

END OF TEST

02134020/CAPE 2011

TEST CODE

FORM TP 2011232 CARIBBEAN

02134032

MAY/JUNE 2011

E XAM I NAT I O N S

COUNCIL

ADVANCED PROFICIENCY EXAMINATION PURE MATHEMATICS UNIT 1 – PAPER 03/B ALGEBRA, GEOMETRY AND CALCULUS 1 ½ hours

08 JUNE 2011 (p.m.)

This examination paper consists of THREE sections: Module 1, Module 2 and Module 3. Each section consists of 1 question. The maximum mark for each Module is 20. The maximum mark for this examination is 60. This examination consists of 3 printed pages.

INSTRUCTIONS TO CANDIDATES 1.

DO NOT open this examination paper until instructed to do so.

2.

Answer ALL questions from the THREE sections.

3.

Write your solutions, with full working, in the answer booklet provided.

4.

Unless otherwise stated in the question, any numerical answer that is not exact MUST be written correct to three significant figures.

Examination Materials Permitted Graph paper (provided) Mathematical formulae and tables (provided) – Revised 2010 Mathematical instruments Silent, non-programmable, electronic calculator

Copyright © 2010 Caribbean Examinations Council All rights reserved. 02134032/CAPE 2011

-2SECTION A (Module 1) Answer this question. 1.

(a)

Solve, for x, the equation [5 marks]

2x + 22–x = 5. (b)

The functions f and g are defined on R by f: x → 3x + 5 and g: x → x – 7.

(c)

(i)

Show that f is one-to-one.

[3 marks]

(ii)

Solve, for x, the equation f (g (2x + 1)) = f (3x – 2).

[4 marks]

A car manufacturer finds that when x million dollars are spent on research, the profit, P(x), in millions of dollars, is given by P(x) = 15 + 10 log4 (x + 4). (i)

What is the expected profit if 12 million dollars are spent on research? [3 marks]

(ii)

How much money should be spent on research to make a profit of 30 million dollars? [5 marks] Total 20 marks SECTION B (Module 2) Answer this question.

2.

(a)

L1 and L2 are lines with equations 2x – y = 5 and x – 2y = 1, respectively. C is a circle with equation x2 + y2 – 12x + 6y + 20 = 0. (i)

Show that L1 and L2 intersect at a point P on C.

[3 marks]

(ii)

Find the point Q, other than P, at which the line L1 intersects C.

[4 marks]

(iii)

Find the equation of the tangent to C at P.

[4 marks]

02134032/CAPE 2011

GO ON TO THE NEXT PAGE

-3(b)

[5 marks]

(i)

Show that sin 3A = 3 sin A – 4 sin3A.

(ii)

Given the vectors u = 2 sin θi + cos 2θj and υ = cos2 θi + sin θj, 0 < θ < p, find the values of θ for which u and υ are perpendicular. [4 marks] Total 20 marks

SECTION C (Module 3) Answer this question. 3.

(a)

Find

(b)

(i)

lim x3 – 4x ——— . x→2 x–2

[4 marks]

Differentiate, with respect to x, x ——— . 3x + 4

(ii)

[4 marks]

Hence, or otherwise, find 16 dx. ∫ ———— (3x + 4)

[3 marks]

2

(c)

A packaging company wishes to make a closed cylindrical container of thin material to hold a volume, V, of 10 cm3. The outside surface of the container is S cm2, the radius is r cm and the height is h cm. (i)

20 Show that S = 2p r2 + —– . r

(ii)

Hence, find the exact value of r for which S has a MINIMUM value. [V = p r2 h, S = 2p r2 + 2p rh]

[3 marks]

[6 marks] Total 20 marks

END OF TEST

02134032/CAPE 2011

TEST CODE

FORM TP 2011234 CARIBBEAN

02234020

MAY/JUNE 2011

E XAM I NAT I O N S

COUNCIL

ADVANCED PROFICIENCY EXAMINATION PURE MATHEMATICS UNIT 2 – PAPER 02 ANALYSIS, MATRICES AND COMPLEX NUMBERS 2 ½ hours

25 MAY 2011 (p.m.)

This examination paper consists of THREE sections: Module 1, Module 2 and Module 3. Each section consists of 2 questions. The maximum mark for each Module is 50. The maximum mark for this examination is 150. This examination consists of 7 printed pages.

INSTRUCTIONS TO CANDIDATES 1.

DO NOT open this examination paper until instructed to do so.

2.

Answer ALL questions from the THREE sections.

3.

Write your solutions, with full working, in the answer booklet provided.

4.

Unless otherwise stated in the question, any numerical answer that is not exact MUST be written correct to three significant figures.

Examination Materials Permitted Graph paper (provided) Mathematical formulae and tables (provided) – Revised 2010 Mathematical instruments Silent, non-programmable, electronic calculator

Copyright © 2010 Caribbean Examinations Council All rights reserved. 02234020/CAPE 2011

-2SECTION A (Module 1) Answer BOTH questions. 1.

(a)

(b)

dy Find —– if dx (i)

x2 + y2 – 2x + 2y – 14 = 0

[3 marks]

(ii)

y = ecos x

[3 marks]

(iii)

y = cos2 6x + sin2 8x.

[3 marks]

1 Let y = x sin — , x ≠ 0. x Show that

(c)

(i)

1 dy x —– = y – cos (—) x dx

[3 marks]

(ii)

d 2y x4 —–2 + y = 0. dx

[3 marks]

1 . A curve is given by the parametric equations x = √ t , y – t = —— √t (i)

Find the gradient of the tangent to the curve at the point where t = 4.

(ii)

Find the equation of the tangent to the curve at the point where t = 4.

[7 marks] [3 marks]

Total 25 marks

02234020/CAPE 2011

GO ON TO THE NEXT PAGE

-3-

2.

(a)

1 Let Fn(x) = — n!

∫

x 0

tn e–t dt .

(i)

Find F0(x) and Fn(0), given that 0! = 1.

[3 marks]

(ii)

1 Show that Fn (x) = Fn – 1 (x) – — xn e–x. n!

[7 marks]

(iii)

Hence, show that if M is an integer greater than 1, then x2 x3 xM ) + (ex – 1). ex FM (x) = – ( x + — + — + ... + —– 2! 3! M!

(b)

(i)

2x2 + 3 Express ——— in partial fractions. (x2 + 1)2

(ii)

Hence, find

∫

2x2 + 3 ——— dx. (x2 + 1)2

[4 marks] [5 marks] [6 marks] Total 25 marks

02234020/CAPE 2011

GO ON TO THE NEXT PAGE

-4SECTION B (Module 2) Answer BOTH questions. 3.

(a)

(b)

1 1 The sequence of positive terms, {xn}, is defined by xn + 1 = x2n + — , x1 < — , n > 1. 4 2 (i)

1 for all positive integers n. Show, by mathematical induction, that xn < — 2 [5 marks]

(ii)

By considering xn + 1 – xn , show that xn < xn + 1.

(i)

Find the constants A and B such that

[3 marks]

A 2 – 3x B —————— ≡ —— + ——– . 1 – x (1 – x) (1 – 2x) 1 – 2x

[3 marks]

(ii)

Obtain the first FOUR non-zero terms of the expansion of each of (1 – x)–1 and (1 – 2x)–1 as power series of x in ascending order. [4 marks]

(iii)

Find a)

the range of values of x for which the series expansion of 2 – 3x —————— (1 – x) (1 – 2x)

b) (iv)

is valid

[2 marks]

the coefficient of xn in (iii) a) above.

[2 marks]

The sum, Sn, of the first n terms of a series is given by Sn = n (3n – 4). Show that the series is an Arithmetic Progression (A.P.) with common difference 6. [6 marks] Total 25 marks

02234020/CAPE 2011

GO ON TO THE NEXT PAGE

-54.

(a)

A Geometric Progression (G.P.) with first term a and common ratio r, 0 < r < 1, is such that the sum of the first three terms is 26 — and their product is 8. 3

(b)

(i)

1 13 Show that r + 1 + — = —– . r 3

(ii)

Hence, find

[4 marks]

a)

the value of r

[4 marks]

b)

the value of a

[1 mark ]

c)

the sum to infinity of the G.P.

[2 marks]

Expand 2 ——— , |x| 1, 4 + px, x < 1.

Find a)

lim x → 1+

b)

the value of the constant p such that

[2 marks]

f (x) lim f (x) exists. x→1

[4 marks]

Hence, determine the value of f (1) for f to be continuous at the point x = 1. [1 mark ]

A chemical process in a manufacturing plant is controlled by the function v M = ut2 + — t2 where u and v are constants. 35 Given that M = –1 when t = 1 and that the rate of change of M with respect to t is –— when 4 t = 2, find the values of u and v. [8 marks] Total 25 marks

02134020/CAPE 2012

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-6-

6.

(a)

dy dx

(i)

Given that y = √ 4x2 – 7, show that y —– = 4x.

(ii)

Hence, or otherwise, show that

d2y dx

dy dx

y —–2 + —– (b)

2

= 4.

[3 marks]

[3 marks]

The curve, C, passes through the point (–1, 0) and its gradient at the point (x, y) is given by

dy dx

—– = 3x2 – 6x. (i)

Find the equation of C.

[4 marks]

(ii)

Find the coordinates of the stationary points of C.

[3 marks]

(iii)

Determine the nature of EACH stationary point.

[3 marks]

(iv)

Find the coordinates of the points P and Q at which the curve C meets the x-axis. [5 marks]

(v)

Hence, sketch the curve C, showing a)

the stationary points

b)

the points P and Q.

[4 marks] Total 25 marks

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

02134020/CAPE 2012

TEST CODE

FORM TP 2012232 CARIBBEAN

02134032

MAY/JUNE 2012

E XAM I NAT I O N S

COUNCIL

ADVANCED PROFICIENCY EXAMINATION PURE MATHEMATICS UNIT 1 – Paper 032 ALGEBRA, GEOMETRY AND CALCULUS 1 hour 30 minutes 08 JUNE 2012 (p.m.)

This examination paper consists of THREE sections: Module 1, Module 2 and Module 3. Each section consists of 1 question. The maximum mark for each Module is 20. The maximum mark for this examination is 60. This examination consists of 4 printed pages.

READ THE FOLLOWING INSTRUCTIONS CAREFULLY. 1.

DO NOT open this examination paper until instructed to do so.

2.

Answer ALL questions from the THREE sections.

3.

Write your solutions, with full working, in the answer booklet provided.

4.

Unless otherwise stated in the question, any numerical answer that is not exact MUST be written correct to three significant figures.

Examination Materials Permitted Graph paper (provided) 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 © 2011 Caribbean Examinations Council All rights reserved. 02134032/CAPE 2012

-2SECTION A (Module 1) Answer this question. 1.

(a)

The roots of the cubic equation x3 – px – 48 = 0 are α, 2α and –3α. Find

(b)

(i)

the value of α

[3 marks]

(ii)

the value of the constant p.

[4 marks]

Prove by mathematical induction that 9n – 1 is divisible by 8 for all integers n > 1.

(c)

[6 marks]

Let m and n be positive integers. (i)

1 Prove that lognm = ——– . logmn

(ii)

Hence, solve for x, the equation log2 x + 2 logx2 = 3.

[3 marks]

[4 marks] Total 20 marks

02134032/CAPE 2012

GO ON TO THE NEXT PAGE

-

3

-

SECTION B (Module 2) Answer this question.

2.

(a)

The diagram below (not drawn to scale) shows the graph o f the circle,C, whose equation is x2 + y2 - 6x + 2y - 15 = 0.

y

.P(7,2)

o

>x

Q'

(i)

(b)

Determine the radius and the coordinates o f the centre o f C.

[3 marks]

(ii)

Find the equation o f the tangent to the circle at the point P (7, 2).

[5 marks]

(iii)

Find the coordinates o f the point Q (Q / P) at which the diameter through P cuts the circle. [2 marks]

(i)

Express f(9) = 3 V3~ cos 0 - 3 sin 0 in the form R cos (0 + a) where R > 0 and 00 is acute. [4 marks]

(ii) (c)

Hence, obtain the maximum value off(d).

[2 marks]

The vector PQ = i - 3j is parallel to the vector OR with |OR| = V 5. Find scalars a and b such that OR = a i + bj.

[4 marks] Total 20 marks

GO ON TO THE NEXT PAGE 02134032/CAPE 2012

-4SECTION C (Module 3) Answer this question. 3.

(a)

(i)

By expressing x – 4 as (√ x + 2) (√ x – 2), find lim x→4

(ii)

√x–2 ——— . x–4

[3 marks]

Hence, or otherwise, find lim x→4

√x–2 ————— . x2 – 5x + 4

[5 marks]

(b)

Find the gradient of the curve y = 2x3 at the point P on the curve at which y = 16. [3 marks]

(c)

The diagram below (not drawn to scale) represents an empty vessel in the shape of a right circular cone of semi-vertical angle 45°. Water is poured into the vessel at the rate of 10 cm3 per second. At time, t, seconds after the start of the pouring of water, the height of the water in the vessel is x cm and its volume is V cm3.

(i)

Express V in terms of t only.

[1 mark ]

(ii)

Express V in terms of x only.

[2 marks]

(iii)

Find, correct to 2 decimal places, the rate at which the water level is rising after 5 seconds. [6 marks] Total 20 marks 1 3

[The volume of a right circular cone of height h and radius of base r is V = — p r2 h.] END OF TEST IF YOU FINISH BEFORE TIME IS CALLED, CHECK YOUR WORK ON THIS TEST. 02134032/CAPE 2012

TEST CODE

FORM TP 2012234 CARIBBEAN

02234020

MAY/JUNE 2012

E XAM I NAT I O N S

COUNCIL

ADVANCED PROFICIENCY EXAMINATION PURE MATHEMATICS UNIT 2 – Paper 02 ANALYSIS, MATRICES AND COMPLEX NUMBERS 2 hours 30 minutes 25 MAY 2012 (p.m.)

This examination paper consists of THREE sections: Module 1, Module 2 and Module 3. Each section consists of 2 questions. The maximum mark for each Module is 50. The maximum mark for this examination is 150. This examination consists of 7 printed pages.

READ THE FOLLOWING INSTRUCTIONS CAREFULLY. 1.

DO NOT open this examination paper until instructed to do so.

2.

Answer ALL questions from the THREE sections.

3.

Write your solutions, with full working, in the answer booklet provided.

4.

Unless otherwise stated in the question, any numerical answer that is not exact MUST be written correct to three significant figures.

Examination Materials Permitted Graph paper (provided) 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 © 2011 Caribbean Examinations Council All rights reserved. 02234020/CAPE 2012

-2SECTION A (Module 1) Answer BOTH questions. 1.

(a)

(i)

(ii) (b)

Given the curve y = x2 ex, a)

dy d2y find —– and —— dx dx2

[5 marks]

b)

dy find the x-coordinates of the points at which —– = 0 dx

[2 marks]

c)

d2y find the x-coordinates of the points at which —— =0 dx2

[2 marks]

Hence , determine if the coordinates identified in (i) b) and c) above are at the [7 marks] maxima, minima or points of inflection of y = x2 ex.

A curve is defined by the parametric equations x = sin–1 √ t , y = t2 – 2t. Find (i)

the gradient of a tangent to the curve at the point with parameter t

[6 marks]

(ii)

1 the equation of the tangent at the point where t = —. 2

[3 marks] Total 25 marks

02234020/CAPE 2012

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-32.

(a)

(i)

Express x2 – 3x —————— in partial fractions. (x – 1) (x2 + 1)

(ii)

Hence, find

∫ (b)

(i)

[7 marks]

x2 – 3x —————— dx. 3 x – x2 + x – 1

[5 marks]

Given that sin A cos B – cos A sin B = sin (A – B) show that [2 marks]

cos 3x sin x = sin 3x cos x – sin 2x. (ii)

If Im =

∫ cos

Jm =

∫ cos

x sin 3x dx and

m

m

x sin 2x dx, [7 marks]

prove that (m + 3) Im = mJm–1 – cosm x cos 3x. (iii)

Hence, by putting m = 1, prove that 4

(iv)

Evaluate

∫ ∫

–�4 0

–�4 0

cos x sin 3x dx =

sin 2x dx.

∫

–�4 0

3 sin 2x dx + — . 2

[2 marks]

[2 marks] Total 25 marks

02234020/CAPE 2012

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-4SECTION B (Module 2) Answer BOTH questions. (a)

(b)

For a particular G.P., u6 = 486 and u11 = 118 098, where un is the nth term. (i)

Calculate the first term, a, and the common ratio, r.

[5 marks]

(ii)

Hence, calculate n if Sn = 177 146.

[4 marks]

The first four terms of a sequence are 1 × 3, 2 × 4, 3 × 5, 4 × 6. (i)

Express, in terms of r, the rth term, ur, of the sequence.

(ii)

Prove, by mathematical induction, that n 1 n (n + 1) (2n + 7), ∑ ur = — 6 r=1

(c)

n ∈ N.

A

3.

[2 marks]

[7 marks]

(i)

Use Maclaurin’s Theorem to find the first three non-zero terms in the power series expansion of cos 2x. [5 marks]

(ii)

Hence, or otherwise, obtain the first two non-zero terms in the power series expansion of sin2 x. [2 marks] Total 25 marks

02234020/CAPE 2012

GO ON TO THE NEXT PAGE

-5-

4.

(a)

n in terms of factorials. r

[1 mark ]

(i)

Express

(ii)

Hence, show that

(iii)

Find the coefficient of x in

(iv)

Using the identity (1 + x)2n = (1 + x)n (1 + x)n, show that

n n = n–r . r

4

2n n

2

2

3 x –— x 2

2

[3 marks]

8

[5 marks]

.

2

2

= c 0 + c 1 + c 3 + . . . + c n – 1 + c n , where cr =

n r . [8 marks]

(b)

Let f (x) = 2x3 + 3x2 – 4x – 1 = 0. (i)

Use the intermediate value theorem to determine whether the equation f (x) has any roots in the interval [0.2, 2]. [2 marks]

(ii)

Using x1 = 0.6 as a first approximation of a root T of f (x), execute FOUR iterations of the Newton–Raphson method to obtain a second approximation, x2, of T. [6 marks] Total 25 marks

02234020/CAPE 2012

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-6SECTION C (Module 3) Answer BOTH questions. 5.

(a)

(b)

(c)

How many 4-digit even numbers can be formed from the digits 1, 2, 3, 4, 6, 7, 8 (i)

if each digit appears at most once?

[4 marks]

(ii)

if there is no restriction on the number of times a digit may appear?

[3 marks]

A committee of five is to be formed from among six Jamaicans, two Tobagonians and three Guyanese. (i)

Find the probability that the committee consists entirely of Jamaicans. [3 marks]

(ii)

Find the number of ways in which the committee can be formed, given the following restriction: There are as many Tobagonians on the committee as there are Guyanese. [6 marks]

Let A be the matrix

1 2 1

0 1 –1

3 –1 1

.

(i)

Find the matrix B, where B = A2 – 3A – I.

[3 marks]

(ii)

Show that AB = –9I.

[1 mark ]

(iii)

Hence, find the inverse, A–1, of A.

[2 marks]

(iv)

Solve the system of linear equations B

x y z

=

3 –1 2

.

[3 marks] Total 25 marks

02234020/CAPE 2012

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-76.

(a)

(i)

Draw the points A and B on an Argand diagram, √2 1+i where A = —— and B = —— . 1–i 1–i

(ii)

[6 marks]

(1 + √ 2 + i) 3� . Hence, or otherwise, show that the argument of ———–—— is EXACTLY —– 1–i 8 [5 marks]

(b)

(i)

Find ALL complex numbers, z, such that z2 = i.

(ii)

Hence, find ALL complex roots of the equation

[3 marks]

[5 marks]

z2 – (3 + 5i) z – (4 – 7i) = 0. (c)

Use de Moivre’s theorem to show that cos 6 θ = cos6 θ – 15 cos4 θ sin2 θ + 15 cos2 θ sin4 θ – sin6 θ .

[6 marks] Total 25 marks

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

02234020/CAPE 2012

TEST CODE

FORM TP 2012235 CARIBBEAN

02234032

MAY/JUNE 2012

E XAM I NAT I O N S

COUNCIL

ADVANCED PROFICIENCY EXAMINATION PURE MATHEMATICS UNIT 2 – Paper 032 ANALYSIS, MATRICES AND COMPLEX NUMBERS 1 hour 30 minutes 01 JUNE 2012 (a.m.)

This examination paper consists of THREE sections: Module 1, Module 2 and Module 3. Each section consists of 1 question. The maximum mark for each Module is 20. The maximum mark for this examination is 60. This examination consists of 4 printed pages.

READ THE FOLLOWING INSTRUCTIONS CAREFULLY. 1.

DO NOT open this examination paper until instructed to do so.

2.

Answer ALL questions from the THREE sections.

3.

Write your solutions, with full working, in the answer booklet provided.

4.

Unless otherwise stated in the question, any numerical answer that is not exact MUST be written correct to three significant figures.

Examination Materials Permitted Graph paper (provided) 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 © 2011 Caribbean Examinations Council All rights reserved. 02234032/CAPE 2012

-2SECTION A (Module 1) Answer this question.

1.

(a)

–1

x (x – 1) 3 Given that y = ———— , 1 + sin3 x dy by taking logarithms of both sides, or otherwise, find —– in terms of x. dx

(b)

(i)

1 Sketch the curve y = √ 1 + x3 , for values of x between – — and 1. 2

(ii)

Using the trapezium rule, with 5 intervals, find an approximation to

∫ (c)

(i)

[3 marks]

1 0

√ 1 + x3 dx.

[5 marks]

Use integration by parts to find

∫x (ii)

[4 marks]

2

cos x dx.

[6 marks]

� Hence, find the area under the curve y = x2 cos x, between x = 0 and x = — . 2 [2 marks] Total 20 marks

02234032/CAPE 2012

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-3SECTION B (Module 2) Answer this question. 2.

(a)

(b)

5

(i)

1 Write down the binomial expansion of 1 + — x 4

(ii)

Hence, calculate (1.025)5 correct to three decimal places.

.

[4 marks] [4 marks]

Let f (x) = x2 – 5x + 3 and g(x) = ex be two functions. (i)

Sketch the graphs for f (x) and g(x) on the same coordinate axes for the domain –1 < x < 2. [4 marks]

(ii)

Using x1 = 0.3 as an initial approximation to the root x of f (x) – g(x) = 0, execute TWO iterations of the Newton-Raphson method to obtain a better approximation, x3, of x correct to four decimal places. [6 marks]

(iii)

Assuming that x3 is the true root of f (x) – g(x) = 0, calculate the relative error of x 1. [2 marks] Total 20 marks

02234032/CAPE 2012

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-4SECTION C (Module 3) Answer this question. 3.

(a)

A computer programmer is trying to break into a company’s code. His program generates a list of all permutations of any set of letters that it is given, without regard for duplicates. For example, given the letters TTA, it will generate a list of six 3-letter permutations (words). If the program generates a list of all 8-letter permutations of TELESTEL, without regard for duplicates,

(b)

(i)

how many times will any given word be repeated in the list?

[5 marks]

(ii)

in how many words will the first four letters be all different?

[5 marks]

(i)

Find the inverse of the matrix 1 1 1

A =

(ii)

1 3 6

.

[5 marks]

Find a 3 × 1 matrix, Y, such that

A

(iii)

1 2 3

3 –1 2

= Y.

[2 marks]

Hence, find a 3 × 3 matrix B such that

BY =

6 –2 4

.

[3 marks] Total 20 marks

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

02234032/CAPE 2012

TEST CODE

FORM TP 2013233 CARIBBEAN

02134020

MAY/JUNE 2013

E XAM I NAT I O N S

COUNCIL

CARIBBEAN ADVANCED PROFICIENCY EXAMINATION® PURE MATHEMATICS UNIT 1 – Paper 02 ALGEBRA, GEOMETRY AND CALCULUS 2 hours 30 minutes 14 MAY 2013 (p.m.)

This examination paper consists of THREE sections: Module 1, Module 2 and Module 3. Each section consists of 2 questions. The maximum mark for each Module is 50. The maximum mark for this examination is 150. This examination consists of 6 printed pages.

READ THE FOLLOWING INSTRUCTIONS CAREFULLY. 1.

DO NOT open this examination paper until instructed to do so.

2.

Answer ALL questions from the THREE sections.

3.

Write your solutions, with full working, in the answer booklet provided.

4.

Unless otherwise stated in the question, any numerical answer that is not exact MUST be written correct to three significant figures.

Examination Materials Permitted Graph paper (provided) 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 © 2012 Caribbean Examinations Council All rights reserved. 02134020/CAPE 2013

-2SECTION A (Module 1) Answer BOTH questions. 1.

(a)

(b)

Let p and q be two propositions. Construct a truth table for the statements (i)

p→q

(ii)

~ (p ˄ q).

[1 mark] [2 marks]

A binary operator ⊕ is defined on a set of positive real numbers by y ⊕ x = y2 + x2 + 2y + x – 5xy. [5 marks]

Solve the equation 2 ⊕ x = 0. (c)

Use mathematical induction to prove that 5n + 3 is divisible by 2 for all values of n ∈ N. [8 marks]

(d)

Let f(x) = x3 – 9x2 + px + 16. (i)

Given that (x + 1) is a factor of f(x), show that p = 6.

[2 marks]

(ii)

Factorise f(x) completely.

[4 marks]

(iii)

Hence, or otherwise, solve f(x) = 0.

[3 marks] Total 25 marks

02134020/CAPE 2013

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-32.

(a)

Let A = {x : x ∈ R, x > 1}. A function f : A → R is defined as f(x) = x2 – x. Show that f is one to one.

(b)

Let f(x) = 3x + 2 and g(x) = e2x. (i)

(ii) (c)

[7 marks]

Find a)

f –1(x) and g–1(x)

b)

f֯֯ [g(x)] (or f֯֯ ◦ g(x)).

[4 marks] [1 mark] [5 marks]

Show that (f֯֯ ◦ g)–1 (x) = g–1 (x) ◦ f –1 (x).

Solve the following: (i)

3x2 + 4x +1 < 5

[4 marks]

(ii)

| x + 2 | = 3x + 5

[4 marks] Total 25 marks

SECTION B (Module 2) Answer BOTH questions. 3.

(a)

(b)

(i)

2 tan θ Show that sin 2θ = ————. 1 + tan2 θ

[4 marks]

(ii)

Hence, or otherwise, solve sin 2θ – tan θ = 0 for 0 < θ < 2π.

[8 marks]

(i)

Express f (θ) = 3 cos θ – 4 sin θ in the form r cos (θ + α) where π r > 0 and 0° < α < —. 2

(ii)

(iii)

[4 marks]

Hence, find a)

the maximum value of f (θ)

[2 marks]

b)

1 the minimum value of ——— . 8 + f (θ)

[2 marks]

Given that the sum of the angles A, B and C of a triangle is π radians, show that ]3 marks]

a)

sin A = sin (B + C)

b)

sin A + sin B + sin C = sin (A + B) + sin (B + C) + sin (A + C). [2 marks] Total 25 marks

02134020/CAPE 2013

GO ON TO THE NEXT PAGE

-44.

(a)

(b)

A circle C is defined by the equation x2 + y2 – 6x – 4y + 4 = 0. (i)

Show that the centre and the radius of the circle, C, are (3, 2) and 3, respectively. [3 marks]

(ii)

a)

Find the equation of the normal to the circle C at the point (6, 2). [3 marks]

b)

Show that the tangent to the circle at the point (6, 2) is parallel to the y-axis. [3 marks]

Show that the Cartesian equation of the curve that has the parametric equations x = t2 + t, y = 2t – 4 [4 marks]

is 4x = y2 + 10y + 24. (c)

The points A (3, –1, 2), B (1, 2, –4) and C (–1, 1, –2) are three vertices of a parallelogram ABCD.

→

→

[3 marks]

(i)

Express the vectors AB and BC in the form xi + yj + zk.

(ii)

Show that the vector r = – 16j – 8k is perpendicular to the plane through A, B [5 marks] and C.

(iii)

Hence, find the Cartesian equation of the plane through A, B and C.

[4 marks]

Total 25 marks

02134020/CAPE 2013

GO ON TO THE NEXT PAGE

-

5

-

SECTION C (Module 3) Answer BOTH questions.

5.

(a)

X+ 2 X< 2 A functionf(x) is defined as f(x) = -j x2 ’ x > 2 [ x2, x>2 (1)

(ii)

'

Find lim f x ) . x^ 2

[4 marks]

Determine whether f(x) is continuous at x = 2. Give a reason for your answer. [2 marks]

T . x2 + 2x + 3 , dy -4x3 - 10x2 - 14x + 4 (b)Let y = — —5— —=3—. Show that —— = --------------------------— -— ------v7 * (x2 + 2)3 dx (x2 + 2)4 dx (x + 2) (c)

[5 marks]

The equation of an ellipse is given by x = 1 - 3 cos 0, y = 2 sin 0, 0 < 0 < 2n. dy Find —jxr in terms of 0.

(d)

[5 marks]

The diagram below (not drawn to scale) shows the curve y = x2 + 3 and the line y = 4x.

y

Q

p o

(i)

(ii)

X

Determine the coordinates of the points P and Q at which the curve and the line intersect. [4 marks] Calculate the area of the shaded region.

[5 marks] Total 25 marks

GO ON TO THE NEXT PAGE 02134020/CAPE 2013

-

6.

(a)

(i)

(ii)

-

By using the substitution u = 1 - x, find

J x (1 - x)2 dx.

[5 marks]

Given that f t ) = 2 cos t, g(t) = 4 sin 5t + 3 cos t,

show that

(b)

6

J f(t) + g(t)] dt = J f(t) dt + J g(t) dt.

[4 marks]

A sports association is planning to construct a running track in the shape o f a rectangle surmounted by a semicircle, as shown in the diagram below. The letter x represents the length o f the rectangular section and r represents the radius o f the semicircle. -

X

-

r

The perimeter o f the track must be 600 metres.

(i)

(ii)

(c)

(i)

Show that r =

600 - 2x 2 +n

[2 marks]

Hence, determine the length, x, that maximises the area enclosed by the track. [6 marks] Let y = -x sin x - 2 cos x + A x + B, where A and B are constants. Show that y " = x sin x.

(ii)

[4 marks]

Hence, determine the specific solution o f the differential equation y ” = x sin x, given that when x = 0, y = 1 and when x = n, y = 6.

[4 marks] Total 25 marks

END OF TEST

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

02134020/CAPE 2013

TEST CODE

FORM TP 2013234 CARIBBEAN

02134032

MAY/JUNE 2013

E XAM I NAT I O N S

COUNCIL

CARIBBEAN ADVANCED PROFICIENCY EXAMINATION® PURE MATHEMATICS UNIT 1 – Paper 032 ALGEBRA, GEOMETRY AND CALCULUS 1 hour 30 minutes 12 JUNE 2013 (p.m.)

This examination paper consists of THREE sections: Module 1, Module 2 and Module 3. Each section consists of 1 question. The maximum mark for each Module is 20. The maximum mark for this examination is 60. This examination consists of 5 printed pages.

READ THE FOLLOWING INSTRUCTIONS CAREFULLY. 1.

DO NOT open this examination paper until instructed to do so.

2.

Answer ALL questions from the THREE sections.

3.

Write your solutions, with full working, in the answer booklet provided.

4.

Unless otherwise stated in the question, any numerical answer that is not exact MUST be written correct to three significant figures.

Examination Materials Permitted Graph paper (provided) 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 © 2012 Caribbean Examinations Council All rights reserved. 02134032/CAPE 2013

-2SECTION A (Module 1) Answer this question. 1.

(a)

Let p and q be two propositions. [1 mark]

(i)

State the converse of (p ˄ q) → (q ˅ ~ p).

(ii)

Show that the contrapositive of the inverse of (p ˄ q) → (q ˅ ~ p) is the converse of (p ˄ q) → (q ˅ ~ p). [3 marks] [5 marks]

(b)

Solve the equation log2 (x + 3) = 3 – log2 (x + 2).

(c)

The amount of impurity, A, present in a chemical depends on the time it takes to purify. It is known that A = 3e4t – 7e2t – 6 at any time t minutes. Find the time at which the chemical is free of impurity (that is when A = 0). [6 marks]

(d)

On the same axes, sketch the graphs of f(x) = 2x + 3 and g(x) = |2x + 3|. Show clearly ALL intercepts that may be present.

[5 marks] Total 20 marks

02134032/CAPE 2013

GO ON TO THE NEXT PAGE

-

3

-

SECTION B (Module 2) Answer this question.

2.

(a)

4 3 A is an acute angle and B is an obtuse angle, where sin (A) =and cos (B) = - =- y TO Without finding the values of angles A and B, calculate cos (3A).

.

[5 marks]

(b)

Solve the equation 4 cos 20 - 14 sin 0 = 7 for values of 0 between 0 and 2n radians. [8 marks]

(c)

An engineer is asked to build a table in the shape of two circles C and C2 which intersect each other, as shown in the diagram below (not drawn to scale).

Cl

R

Q

Cl

The equations of Cx and C2 arex2+ y2 + 4x + 6y - 3 = 0 and x2+ y2 + 4x + 2y - 7 = 0 respectively. A leg of the table is attached at EACH of the points Q and R where the circles intersect. Determine the coordinates of the positions of the legs of the table.

[7 marks] Total 20 marks

GO ON TO THE NEXT PAGE 02134032/CAPE 2013

-

4

-

SECTION C (Module 3) Answer this question.

3.

(a)

The diagram below shows the graph of a function, fx ).

f(x)

-3

-2

,1

0

(i)

(ii)

1

2\

3

x

Determine for the function li"

a)

xx ^^ o0 f x )

b) b)

xxx ^ 22 f(x).

li"

[1 mark]

[2 marks]

State whether f is continuous State whether at x =f 2. is continuous at x = 2. Justify your answer.[2marks]

GO ON TO THE NEXT PAGE 02134032/CAPE 2013

-5-

(b)

1 Differentiate f(x) = —— from first principles. √ 2x

(c)

Find the x-coordinates of the maximum and minimum points of the curve f(x) = 4x3 + 7x2 – 6x.

(d)

[5 marks]

[7 marks]

y2 x2 A water tank is made by rotating the curve with equation — + —– = 1 about the x-axis 25 4 between x = 0 and x = 2. Find the volume of water that the tank can hold.

[3 marks] Total 20 marks

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

02134032/CAPE 2013

TEST CODE

FORM TP 2013236 CARIBBEAN

02234020

MAY/JUNE 2013

E XAM I NAT I O N S

COUNCIL

CARIBBEAN ADVANCED PROFICIENCY EXAMINATION® PURE MATHEMATICS UNIT 2 – Paper 02 ANALYSIS, MATRICES AND COMPLEX NUMBERS 2 hours 30 minutes 29 MAY 2013 (p.m.)

This examination paper consists of THREE sections: Module 1, Module 2 and Module 3. Each section consists of 2 questions. The maximum mark for each Module is 50. The maximum mark for this examination is 150. This examination consists of 6 printed pages.

READ THE FOLLOWING INSTRUCTIONS CAREFULLY. 1.

DO NOT open this examination paper until instructed to do so.

2.

Answer ALL questions from the THREE sections.

3.

Write your solutions, with full working, in the answer booklet provided.

4.

Unless otherwise stated in the question, any numerical answer that is not exact MUST be written correct to three significant figures.

Examination Materials Permitted Graph paper (provided) 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 © 2012 Caribbean Examinations Council All rights reserved. 02234020/CAPE 2013

-2SECTION A (Module 1) Answer BOTH questions. 1.

(a)

Determine the derivative, with respect to x, of the function ln (x2y) – sin y = 3x – 2y. [5 marks]

(b)

Let f (x, y, z) = 3 yz2 – e4x cos 4z –3y2 – 4. ∂f / ∂y ∂z ∂z Given that —– = – ———, determine —– in terms of x, y and z. ∂f / ∂z ∂y ∂y

(c)

Use de Moivre’s theorem to prove that [6 marks]

cos 5θ = 16 cos5 θ – 20 cos3 θ + 5 cos θ. (d)

[5 marks]

(i)

Write the complex number z = (–1 + i)7 in the form reiθ, where r = | z | and θ = arg z. [3 marks]

(ii)

Hence, prove that (–1 + i)7 = –8(1 + i).

[6 marks] Total 25 marks

2.

(a)

(b)

∫ sin x cos 2x dx.

(i)

Determine

(ii)

Hence, calculate

Let f(x) = x | x | =

∫

–�2

[5 marks] [2 marks]

sin x cos 2x dx.

0

x2 ; x > 0 . –x2 ; x < 0

Use the trapezium rule with four intervals to calculate the area between f(x) and the x-axis for the domain –0.75 < x < 2.25. [5 marks] (c)

(i)

2 4 2x2 + 4 = ——— Show that ———– – ———– . x2 + 4 (x2 + 4)2 (x2 + 4)2

[6 marks]

(ii)

2x2 + 4 dx. Use the substitution x = 2 tan θ. Hence, find ———– (x2 + 4)2

[7 marks]

∫

Total 25 marks

02234020/CAPE 2013

GO ON TO THE NEXT PAGE

-3SECTION B (Module 2) Answer BOTH questions. 3.

(a)

3

The sequence{an} is defined by a1 = 1, an+1 = 4 + 2 √ an . Use mathematical induction to prove that 1 < an < 8 for all n in the set of positive integers. [6 marks]

(b)

1 Let k > 0 and let f(k) = —. k2 (i)

(iii)

Show that a)

2k + 1 . f(k) – f(k + 1) = ———— 2 k (k + 1)2

b)

∑

n

k=1

1 – ——— 1 — 2 k (k + 1)2

1 = 1 – ——— . (n + 1)2

(i)

(ii)

[5 marks]

Hence, or otherwise, prove that ∞ 2k + 1 = 1. ∑ ———— 2 k (k + 1)2 k=1

(c)

[3 marks]

[3 marks]

Obtain the first four non-zero terms of the Taylor Series expansion of cos x in π ascending powers of (x – —). 4

[5 marks]

π Hence, calculate an approximation to cos (—–). 16

[3 marks] Total 25 marks

02234020/CAPE 2013

GO ON TO THE NEXT PAGE

-44.

(a)

(i)

Obtain the binomial expansion of 4

4

√ (1 + x) + √ (1 – x) [4 marks]

up to the term containing x2. (ii)

(b) (c)

1 compute an approximation of √4 17 + √4 15 to four Hence, by letting x = —–, 16 decimal places. [4 marks]

Show that the coefficient of the x5 term of the product (x + 2)5 (x – 2)4 is 96.

[7 marks]

(i)

Use the Intermediate Value Theorem to prove that x3 = 25 has at least one root in the interval [2, 3]. [3 marks]

(ii)

The table below shows the results of the first four iterations in the estimation of the root of f(x) = x3 – 25 = 0 using interval bisection. The procedure used a = 2 and b = 3 as the starting points and pn is the estimate of the root for the nth iteration. n

an

bn

pn

f(pn)

1

2

3

2.5

–9.375

2

2.5

3

2.75

–4.2031

3

2.75

3

2.875

–1.2363

4

2.875

3

2.9375

0.3474

5 6 ...... ...... Complete the table to obtain an approximation of the root of the equation x3 = 25 correct to 2 decimal places. [7 marks] Total 25 marks

02234020/CAPE 2013

GO ON TO THE NEXT PAGE

-5SECTION C (Module 3) Answer BOTH questions. 5.

(a)

Three letters from the word BRIDGE are selected one after the other without replacement. When a letter is selected, it is classified as either a vowel (V) or a consonant (C). Use a tree diagram to show the possible outcomes (vowel or consonant) of the THREE selections. Show all probabilities on the diagram. [7 marks]

(b)

The augmented matrix for a system of three linear equations with variables x, y and z respectively is

(i)

A=

1 –5 1

1 1 –5

–1 1 3

1 2 3

By reducing the augmented matrix to echelon form, determine whether or not the system of linear equations is consistent. [5 marks] (ii)

The augmented matrix for another system is formed by replacing the THIRD row of A in (i) above with (1 –5 5 | 3). Determine whether the solution of the new system is unique. Give a reason for your answer. [5 marks]

(c)

A country, X, has three airports (A, B, C). The percentage of travellers that use each of the airports is 45%, 30% and 25% respectively. Given that a traveller has a weapon in his/ her possession, the probability of being caught is, 0.7, 0.9 and 0.85 for airports A, B, and C respectively. Let the event that: the traveller is caught be denoted by D, and the event that airport A, B, or C is used be denoted by A, B, and C respectively.

• • (i)

What is the probability that a traveller using an airport in Country X is caught with a weapon? [5 marks]

(ii)

On a particular day, a traveller was caught carrying a weapon at an airport in Country X. What is the probability that the traveller used airport C? [3 marks] Total 25 marks

02234020/CAPE 2013

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-66.

(a)

(i)

Obtain the general solution of the differential equation

dy dx

cos x —– + y sin x = 2x cos2 x. (ii) (b)

[7 marks]

15 √ 2 π2 π Hence, given that y = ———— , when x = —, determine the constant of the 32 4 integration. [5 marks]

The general solution of the differential equation y" + 2y' + 5y = 4 sin 2t is y = CF + PI, where CF is the complementary function and PI is a particular integral. (i)

a)

Calculate the roots of λ2 + 2λ + 5 = 0, the auxiliary equation.

b)

Hence, obtain the complementary function (CF), the general solution of y" + 2y' + 5y = 0.

(ii)

[2 marks]

[3 marks]

Given that the form of the particular integral (PI) is up(t) = A cos 2t + B sin 2t, 16 4 Show that A = – —– and B = —–. 17 17

(iii)

[3 marks]

Given that y(0) = 0.04 and y'(0) = 0, obtain the general solution of the differential equation. [5 marks] Total 25 marks

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

02234020/CAPE 2013

TEST CODE

FORM TP 2013237 CARIBBEAN

02234032

MAY/JUNE 2013

E XAM I NAT I O N S

COUNCIL

CARIBBEAN ADVANCED PROFICIENCY EXAMINATION® PURE MATHEMATICS UNIT 2 – Paper 032 ANALYSIS, MATRICES AND COMPLEX NUMBERS 1 hour 30 minutes 05 JUNE 2013 (a.m.)

This examination paper consists of THREE sections: Module 1, Module 2 and Module 3. Each section consists of 1 question. The maximum mark for each Module is 20. The maximum mark for this examination is 60. This examination consists of 4 printed pages.

READ THE FOLLOWING INSTRUCTIONS CAREFULLY. 1.

DO NOT open this examination paper until instructed to do so.

2.

Answer ALL questions from the THREE sections.

3.

Write your solutions, with full working, in the answer booklet provided.

4.

Unless otherwise stated in the question, any numerical answer that is not exact MUST be written correct to three significant figures.

Examination Materials Permitted Graph paper (provided) 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 © 2012 Caribbean Examinations Council All rights reserved. 02234032/CAPE 2013

-2SECTION A (Module 1) Answer this question. 1.

(a)

A firm measures production by the Cobb-Douglas production function

–1

P(k(t), l(t)) = 20k 4 l

–34

where k is the capital (in millions of dollars) and l is the labour force (in thousands of workers). Let l = 3 and k = 4. Assume that the capital is DECREASING at a rate of $200 000 per year and that the labour force is INCREASING at a rate of 60 workers per year. dP ∂P dk ∂P dl dP Given that —– = —– • —– + —– • —– , calculate —– . dt ∂k dt ∂l dt dt (b)

[6 marks]

∫

Let Fn(x) = cosn x dx. By rewriting cosn x as cos x cosn–1 x or otherwise, prove that

(c)

1 n–1 Fn(x) = — cosn–1 x sin x + ——— Fn – 2 (x). n n

[6 marks]

Find the square root of the complex number z = 2 + i.

[8 marks] Total 20 marks

02234032/CAPE 2013

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-3SECTION B (Module 2) Answer this question.

2.

(a)

(i)

1 Show that the binomial expansion of 1 + — x 2

4

is

3 x2 + — 1 x3 + —– 1 x4 . 1 + 2x + — 2 2 16

(b)

(c)

[4 marks] [4 marks]

(ii)

Hence, compute 1.3774 correct to two decimal places.

(i)

Derive the first three non-zero terms in the Maclaurin expansion of ln (1 + x). [4 marks]

(ii)

Hence, express the Maclaurin expansion of ln (1 + x) in sigma notation. [2 marks]

A geometric series is given by x2 + — x3 + — x4 + . . . x + — 2 4 8 (i)

Determine the values of x for which the series is convergent.

[3 marks]

(ii)

Hence, or otherwise, if the series is convergent, show that S2 < 4.

[3 marks]

Total 20 marks

02234032/CAPE 2013

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-4SECTION C (Module 3) Answer this question. 3.

(a)

A system of equations Ax = b is given by 1 1 –1 2 –1 3 1 –2 –2

(b)

(i)

Calculate | A |.

(ii)

Let the matrix C =

x y z

=

6 –9 3 [3 marks]

8 7 –3 4 –1 3 2 –5 –3

a)

Show that CTA – 18I = 0.

[4 marks]

b)

Hence or otherwise, obtain A–1.

[2 marks]

c)

Solve the given system of equations for x, y and z.

[4 marks]

To make new words, three letters are selected without replacement from the word TRAVEL and are written down in the order in which they are selected. [2 marks]

(i)

How many three-letter words may be formed?

(ii)

For a three-letter word to be legal, it must have at least one vowel (that is a, e, i, o or u). What is the probability that a legal word is formed on a single attempt? [5 marks] Total 20 marks

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

02234032/CAPE 2013

TEST CODE

FORM TP 2014240 CARIBBEAN

02134020

MAY/JUNE 2014

E XAM I NAT I O N S

COUNCIL

CARIBBEAN ADVANCED PROFICIENCY EXAMINATION® PURE MATHEMATICS UNIT 1 – Paper 02 ALGEBRA, GEOMETRY AND CALCULUS 2 hours 30 minutes 13 MAY 2014 (p.m.)

READ THE FOLLOWING INSTRUCTIONS CAREFULLY. 1.

This examination paper consists of THREE sections.

2.

Answer ALL questions from the THREE sections.

3.

Each section consists of TWO questions.

4.

Write your solutions, with full working, in the answer booklet provided.

5.

Unless otherwise stated in the question, any numerical answer that is not exact MUST be written correct to three significant figures.

Examination Materials Permitted Graph paper (provided) 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 © 2013 Caribbean Examinations Council All rights reserved. 02134020/CAPE 2014

-2SECTION A Module 1 Answer BOTH questions. 1. (a) Let p, q and r be three propositions. Construct a truth table for the statement (p → q) ˄ (r → q). [5 marks]

(b)

A binary operator ⊕ is defined on a set of positive real numbers by

y ⊕ x = y3 + x3 + ay2 + ax2 – 5y – 5x + 16 where a is a real number.

(i)

State, giving a reason for your answer, if ⊕ is commutative in R. [3 marks]



(ii)

Given that f(x) = 2 ⊕ x and (x – 1) is a factor of f(x),



a)

find the value of a [4 marks]



b)

factorize f(x) completely.



(c)

[3 marks]

Use mathematical induction to prove that

n 12 + 32 + 52 + ..... + (2n – 1)2 = — (4n2 – 1) for n ∈ N. 3

[10 marks]



Total 25 marks

02134020/CAPE 2014

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-32.

(a)

The functions f and g are defined as follows:

f(x) = 2x2 + 1 x–1 g(x) = —— √ 2

(i)

where 1 < x < ∞, x ∈ R.

Determine, in terms of x,

a) f 2(x)

[3 marks]

b) f֯֯ [g(x)].

[3 marks]



(ii)



Hence, or otherwise, state the relationship between f֯֯ and g.

[1 mark]



(b)

a+b Given that a3 + b3 + 3a2b = 5ab2, show that 3 log ——– = log a + 2 log b. [5 marks] 2



(c)

Solve EACH of the following equations:



(i)

1 ex + — – 2 = 0 ex

[4 marks]



(ii) log2 (x + 1) – log2 (3x + 1) = 2

[4 marks]



(d)

Without the use of a calculator, show that



√3+1 √2–1 √2+1 √3–1 ——— + ——— + ——— + ——— = 10. √3–1 √2+1 √2–1 √3+1

02134020/CAPE 2014

[5 marks] Total 25 marks

GO ON TO THE NEXT PAGE

-4SECTION B Module 2 Answer BOTH questions.

3.

(a)



(i)

cot y – cot x sin (x – y) Prove that –————— = ————– . cot x + cot y sin (x + y)

(ii)

Hence, or otherwise, find the possible values for y in the trigonometric equation



(b)

cot y – cot x –————— = 1, 0 < y < 2π, cot x + cot y

1 ,0 3



(i)

Find the value of a if f(x) is continuous at x = 3.



(ii)

x2 + 2 Let g(x) = ————— . 2 bx + x + 4



(b)



Given that

dy 1 . Using first principles, find —– (i) Let y = —— . dx √x

(c)

[5 marks] [8 marks]

dy x (ii) If y = ——— , determine an expression for —– . dx √ 1 +x



lim lim 2g (x) = g(x), find the value of b. x→1 x→0

[4 marks]

Simplify the answer FULLY.

[4 marks]

The parametric equations of a curve are given by

x = cos θ, y = sin θ, 0 < θ < 2π.

dy dx

Find —– in terms of θ.

Simplify the answer as far as possible. [4 marks] Total 25 marks

02134020/CAPE 2014

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-76.

(a)

The gradient of a curve which passes through the point (–1, –4) is given by

dy dx

—– = 3x2 – 4x + 1. (i) Find



a)

the equation of the curve

[4 marks]

b) the coordinates of the stationary points and determine their nature. [8 marks]

(b)

(ii)

Sketch the curve in (a) (i) a) above, clearly marking ALL stationary points and intercepts. [4 marks]

The equation of a curve is given by f(x) = 2x √ 1 + x2 . 3

∫

(i) Evaluate f(x) dx. [5 marks] 0

(ii)

Find the volume generated by rotating the area bounded by the curve in (b) (i) above, the x-axis, and the lines x = 0 and x = 2 about the x-axis. [4 marks]



Total 25 marks

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

02134020/CAPE 2014

TEST CODE

FORM TP 2014241 CARIBBEAN

02134032

MAY/JUNE 2014

E XAM I NAT I O N S

COUNCIL

CARIBBEAN ADVANCED PROFICIENCY EXAMINATION® PURE MATHEMATICS UNIT 1 – Paper 032 ALGEBRA, GEOMETRY AND CALCULUS 1 hour 30 minutes 11 JUNE 2014 (p.m.)

READ THE FOLLOWING INSTRUCTIONS CAREFULLY. 1.

This examination paper consists of THREE sections.

2.

Answer ALL questions from the THREE sections.

3.

Each section consists of ONE question.

4.

Write your solutions, with full working, in the answer booklet provided.

5.

Unless otherwise stated in the question, any numerical answer that is not exact MUST be written correct to three significant figures.

Examination Materials Permitted Graph paper (provided) 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 © 2013 Caribbean Examinations Council All rights reserved. 02134032/CAPE 2014

-2SECTION A Module 1 Answer this question. 1.

(a)

The binary operation * is defined on a set {e, a, b, c, d, f} as shown in the table below. For example, a * b = d.



* e a b c d f



e e a b c d f



a a e d f b c



b b f e d c a



c c d f e a b



d d c a b f e



f f b c a e d



(i)

State, giving a reason, if * is commutative.

[2 marks]



(ii)

Name the identity element for the operation *.

[1 mark]



(iii)

Determine the inverse of

a) d [1 mark]

b) c. [1 mark]

(b) Let α, β and γ be the roots of the equation 2x3 + 4x2 + 3x – 1 = 0.



(i)

Calculate EACH of the following:



a)

1 1 1 —– + —– + —– α2 β2 γ2

[5 marks]



b)

1 + —— 1 + —— 1 —— α2 β2 β2 γ2 γ2 α2

[2 marks]

1 1 1 (ii) Hence, or otherwise, find the equation whose roots are —– , —–, —– . α2 β2 γ2 [2 marks]

02134032/CAPE 2014

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-3 (c) An answer sheet is provided for this question.

1 The diagram below shows the graph of the function g(x) = —— . x–1

4 3 2 1 _4

_3

_2

_1

1 _ _ _ _

2

3

4

x

1 2 3 4

(i) On the answer sheet provided, sketch the graphs of |g(x)| and f(x) = x – 1, showing clearly the intercepts and the asymptotes. [5 marks] (ii)

Hence, or otherwise, obtain the value of x such that f(x) = g(x).



02134032/CAPE 2014

[1 mark] Total 20 marks

GO ON TO THE NEXT PAGE

-4SECTION B Module 2 Answer this question. 2. (a) P (1, 3, 2), Q (–1, 2, 3) and R (1, 3, 5) are the vertices of a triangle.



→

→



(i)

Find the displacement vectors PQ and PR.



(ii)

Hence, determine



→

→

a) | PQ | and | PR |



→

→

[4 marks]

[3 marks] [4 marks]



b)

the cosine of the acute angle between PQ and PR



c)

the area of triangle PQR. [4 marks]



(b)



π π π – —, Given that — = — show without the use of a calculator, that the EXACT value 12 3 4 π is 2 – √ 3 . of tan — [5 marks] 12



02134032/CAPE 2014

Total 20 marks

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-5SECTION C Module 3 Answer this question. sin 8x lim ——— . x → 0 2x

3.

(a)

Evaluate



(b)

The equation of a curve is given by

[4 marks]

y = x3 + x2 + 2. (i) Determine the coordinates of the points on the curve where the gradient is 1. [6 marks] (ii) Determine the equation of the normal which intersects the curve at (–1, 2). [4 marks]

(c)

The diagram below (not drawn to scale) shows the design of a petal drawn on a square tile of length 1 metre. y 1

y = √x

Petal

y = x2

0

1

x



The design may be modelled by the finite region enclosed by the curves y = √ x and y = x2 where x and y are lengths measured in metres.



Calculate the area of the petal.



[6 marks]



Total 20 marks

END OF TEST IF YOU FINISH BEFORE TIME IS CALLED, CHECK YOUR WORK ON THIS TEST. 02134032/CAPE 2014

TEST CODE

FORM TP 2014241 CARIBBEAN

02134032

MAY/JUNE 2014

E XAM I NAT I O N S

COUNCIL

CARIBBEAN ADVANCED PROFICIENCY EXAMINATION® PURE MATHEMATICS UNIT 1 – Paper 032

Answer Sheet for Question 1 (c)

Candidate Number .............................................

y 4 3 2 1 -4

-3

-2

-1

0

1

2

3

4

x

-1 -2 -3 -4

ATTACH THIS ANSWER SHEET TO YOUR ANSWER BOOKLET

02134032/CAPE 2014

TEST CODE

FORM TP 2014243 CARIBBEAN

02234020

MAY/JUNE 2014

E XAM I NAT I O N S

COUNCIL

CARIBBEAN ADVANCED PROFICIENCY EXAMINATION® PURE MATHEMATICS UNIT 2 – Paper 02 ANALYSIS, MATRICES AND COMPLEX NUMBERS 2 hours 30 minutes 28 MAY 2014 (p.m.)

READ THE FOLLOWING INSTRUCTIONS CAREFULLY. 1.

This examination paper consists of THREE sections.

2.

Answer ALL questions from the THREE sections.

3.

Each section consists of TWO questions.

4.

Write your solutions, with full working, in the answer booklet provided.

5.

Unless otherwise stated in the question, any numerical answer that is not exact MUST be written correct to three significant figures.

Examination Materials Permitted Graph paper (provided) 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 © 2013 Caribbean Examinations Council All rights reserved. 02234020/CAPE 2014

-2SECTION A Module 1 Answer BOTH questions. 1.

(a)

(i)

Differentiate, with respect to x,

x y = ln (x2 + 4) – x tan–1 ( — ). 2

(ii)

A curve is defined parametrically as



(b)

x = a cos3 t, y = a sin3 t.

Show that the tangent at the point P (x, y) is the line



[5 marks]

y cos t + x sin t = a sin t cos t. [7 marks]

Let the roots of the quadratic equation x2 + 3x + 9 = 0 be α and β. Determine the nature of the roots of the equation.

[2 marks]



(i)



(ii) Express α and β in the form reiθ, where r is the modulus and θ is the argument, where –π < θ < π. [4 marks]



(iii)

Using de Moivre’s theorem, or otherwise, compute α3 + β3.

[4 marks]

(iv) Hence, or otherwise, obtain the quadratic equation whose roots are α3 and β3. [3 marks] Total 25 marks

02234020/CAPE 2014

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-3-

∫

2. (a) Let Fn (x) = (ln x)n dx.



(i)

Show that Fn (x) = x (ln x)n – n Fn – 1(x). [3 marks]



(ii)

Hence, or otherwise, show that



(b)

(i)

F3 (2) – F3 (1) = 2 (ln 2)3 – 6 (ln 2)2 + 12 ln 2 – 6.

y2 + 2y + 1 By decomposing ————— into partial fractions, show that y4 + 2y2 + 1



1 2y y2 + 2y + 1 ————— = ——— + ———– . y2 + 1 (y2 + 1)2 y4 + 2y2 + 1 1



∫

y2 + 2y + 1 dy. ————— y4 + 2y2 + 1

[7 marks]



[8 marks]



Total 25 marks

(ii)

Hence, find

0



[7 marks]

02234020/CAPE 2014

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-4SECTION B Module 2 Answer BOTH questions. 3.

(a)

(i)

Prove, by mathematical induction, that for n ∈ N



(b)

(ii)

1 1 1 1 1 Sn = 1 + — + —2 + —3 + ...... + —— n – 1 . n – 1 = 2 – —— 2 2 2 2 2

Hence, or otherwise, find

[8 marks]

lim S . [3 marks] n→∞ n

Find the Maclaurin expansion for

f(x) = (1 + x)2 sin x

up to and including the term in x3.

[14 marks]

Total 25 marks 4.

(a)

(i)

For the binomial expansion of (2x + 3)20, show that the ratio of the term in x6 to



3 the term in x7 is — . 4x

(ii)

a)

Determine the FIRST THREE terms of the binomial expansion of (1 + 2x)10.



b)

Hence, obtain an estimate for (1.01)10.



(b)

(n + 1) ! n! n! Show that ————– + ————————– = ——————. (n – r + 1) !r! (n – r) !r! (n – r + 1) ! (r – 1) !

[5 marks]

[7 marks] [6 marks]

(c) (i) Show that the function f(x) = –x3 + 3x + 4 has a root in the interval [1, 3]. [3 marks]

(ii)

By taking x1 = 2.1 as a first approximation of the root in the interval [1, 3], use the Newton–Raphson method to obtain a second approximation, x2, in the interval [1, 3]. [4 marks]

Total 25 marks

02234020/CAPE 2014

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-5SECTION C Module 3 Answer BOTH questions. 5.

(a)

(i)

Five teams are to meet at a round table. Each team consists of two members AND one leader. How many seating arrangements are possible if each team sits together with the leader of the team in the middle? [7 marks]

(ii)

In an experiment, individuals were asked to colour a shape by selecting from two available colours, red and blue. The individuals chose one colour, two colours or no colour.



In total, 80% of the individuals used colours and 600 individuals used no colour.



a)



b)



Given that 40% of the individuals used red and 50% used blue, calculate the probability that an individual used BOTH colours. [4 marks] Determine the TOTAL number of individuals that participated in the experiment. [2 marks]

(b) A and B are the two matrices given below. 1 x –1 1 2 5 A = 3 0 2 B = 2 3 4 2 1 0 1 1 2

(i)

Determine the range of values of x for which A–1 exists.

[4 marks]



(ii)

Given that det (AB) = –21, show that x = 3.

[4 marks]



(iii)

Hence, obtain A–1. [4 marks]

Total 25 marks

02234020/CAPE 2014

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-66.

(a)

(i)

Show that the general solution of the differential equation

y' + y tan x = sec x

is (ii)

y = sin x + C cos x. [10 marks]

π 2 and x = —. Hence, obtain the particular solution where y = —— 4 √2

[4 marks]

A differential equation is given as y" – 5y' = xe5x. Given that a particular solution is [11 marks] yp(x) = Ax2 e5x + Bxe5x, solve the differential equation.



(b)





Total 25 marks

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

02234020/CAPE 2014

TEST CODE

FORM TP 2014244 CARIBBEAN

02234032

MAY/JUNE 2014

E XAM I NAT I O N S

COUNCIL

CARIBBEAN ADVANCED PROFICIENCY EXAMINATION® PURE MATHEMATICS UNIT 2 – Paper 032 ANALYSIS, MATRICES AND COMPLEX NUMBERS 1 hour 30 minutes 04 JUNE 2014 (a.m.)

READ THE FOLLOWING INSTRUCTIONS CAREFULLY. 1.

This examination paper consists of THREE sections.

2.

Answer ALL questions from the THREE sections.

3.

Each section consists of ONE question.

4.

Write your solutions, with full working, in the answer booklet provided.

5.

Unless otherwise stated in the question, any numerical answer that is not exact MUST be written correct to three significant figures.

Examination Materials Permitted Graph paper (provided) 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 © 2013 Caribbean Examinations Council All rights reserved. 02234032/CAPE 2014

-2SECTION A Module 1 Answer this question. 1

1. (a) Let l =

∫

–1

1 ——— dx. 1 + e–x



(i)

Use the trapezium rule with two trapezia of equal width to obtain an estimate of [3 marks] l.



(ii)

Evaluate the integral l by means of the substitution u = ex.



(b)



The diagram below (not drawn to scale) shows an open rectangular box with a partition in the middle.

The dimensions of the box, measured in centimetres, are x, y, and z. The volume of the box is 384 cm3. (i)

The pieces from which the box is assembled are cut from a flat plank of wood. Show that the TOTAL area of the pieces cut from the plank, A cm2, is given by



[7 marks]

(ii)

768 1152 A = xy + —— + —— . y x

[5 marks]

∂A ∂A The minimum value of A occurs where —– = 0 and —– = 0 simultaneously. ∂x ∂y



a)

∂A ∂A Determine —– and —– . ∂x ∂y



b)

∂A = 0 and —– ∂A = 0 are both satisfied by Hence, show that the equations —– ∂x ∂y x = 12, y = 8. [2 marks]



02234032/CAPE 2014

[3 marks]

Total 20 marks

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-3SECTION B Module 2 Answer this question.

2.

(a)

(i)

1 1 2 . Show that ——— – ——— = ——— 2r + 1 2r – 1 4r2 – 1

(ii)

2 2n . Hence, or otherwise, show that Σ ——— = ——— 2 – 1 4r 2n +1 r=1

n



(b)

[3 marks]

[5 marks]

An arithmetic progression is such that the fifth and tenth partial sums are S5 = 60 and S10 = 202 respectively.



(i)

Calculate the first term, a, and the common difference, d.

[5 marks]



(ii)

Hence, or otherwise, calculate the 15th term, u15.

[2 marks]



(i)

Show that the function f(x) = e–x – 2x + 3 has a root, α, in the closed interval [1, 2]. [2 marks]

(ii)

Apply linear interpolation ONCE in the interval [1, 2] to find an approximation to the root, α. [3 marks]

(c)





02234032/CAPE 2014

Total 20 marks

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-4SECTION C Module 3 Answer this question. 3.

(a)

A bag contains 2 red balls, 3 blue balls and 1 white ball. In an experiment, 2 balls are drawn at random from the bag without replacement.

(i) Use a tree diagram to show the possible events and their corresponding probabilities. [5 marks] (ii) Calculate the probability that the second ball drawn is blue. [4 marks]

(b)

The current flow in a particular circuit is defined by the differential equation

di L —– + Ri = V, dt where i is the current at time t, and V, R and L are constants representing the voltage, resistance and self-inductance respectively.

The switch in the circuit is closed at time t = 0 and i(0) = 0. (i)

By solving the differential equation using an appropriate integrating factor, verify

R t V L ). that i = — (1 – e– — [8 marks] R



(ii)



The steady-state current in the circuit is lim i. Use the result of (b) (i) above to t→∞ lim evaluate i. [3 marks] t→∞



Total 20 marks

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

02234032/CAPE 2014