Pure
Mathematics
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
© 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 AER
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 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
FORM TP 2005253
MAY /JUNE 2005CARIBBEAN EXAMINATIONS COUNCIL
ADVANCED
PROFICIENCY EXAMINATION
PURE MATHEMATICS
UN
IT
1 - PAPER
02
2 hours
( 25 MAY 2005 (p.m.) )
This examination paper consists of THREE sections: Modulel, 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 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.
Examination Materials
Mathematical formulae and tables Electronic calculator
Graph paper
02134020/CAPE 2005
Copyright© 2004 Caribbean Examinations Council All rights reserved.
-
2-Section A (Module 1) Answer BOTH questions.
1. (a) (i) Complete the table below for the function
I
f(x)I,
wheref(x)=
x (2-x).l
l
;l
ll
-2I
-1I
0 I 1 I 2 I 3I
4I
8 0 8 [ 2 marks] (ii) Sketch the graph ofI
f(x)I
for -2 ::=:; x ::=:; 4. [ 4 marks] (b) Find the value(s) of the real number, k, for which the equation k(x2 + 5) = 6 + 12x- x2has equal roots. [ 6 marks]
(c) (i) [ 4 marks]
(ii) Without using calculators or tables, evaluate
[ 4 marks] Total 20 marks
GO ON TO THE NEXT PAGE 02134020/CAPE 2005
2. (a) Prove, by Mathematical Induction, that !On - 1 ts divisible by 9 for all pos1t1ve
integers n. [ 9 marks]
(b) A pair of simultaneous equations is given by
(c)
px + 2y
=
8 - 4x+ p2y=
16 where p E R.(i) Find the value of p for which the system has an infinite number of solutions. [ 3 marks] (ii) 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 marks] Total 20 marks
Section B (Module 2) Answer BOTH questions.
3. The equation of the circle, Q, with centre 0 is x2 + y2- 2x + 2y
=
23.(a) Express the equation of Q in the form (x- a)2 + (y-b
i
=
c. [ 5 marks] (b) Hence, or otherwise, state(i) the coordinates of the centre of Q 2 marks]
(ii) the radius of Q. [ 1 mark]
(c) Show that the point A(4, 3) lies on Q. 3 marks]
(d) Find the equation of the tangent to Q at the point A. [ 5 marks] (e) The centre of Q is the midpoint of its diameter AB. Find the coordinates of B.
[ 4 marks] Total 20 marks
GO ON TO THE NEXT PAGE 02134020/CAPE 2005
-4-4. The diagrams shown below, not drawn to scale, represent
(a)
(b)
(c)
a sector, OABC, of a circle with centre at 0 and a radius of 7 em, where angle AOC
n
d.measures-ra tans. 3
a right circular 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.
A
0
B A,C
B
(i) Express the arc length ABC in terms of n. [ 1 mark ] (ii) Hence, show that
a) r=-7
6 3 marks]
b) if hem is the height of the cone, then the exact value of his 7 -{35 6 [ 2 marks] (i) Show that cos 3
e
=
4 cos3e
-
3 cose.
[ 5 marks] (ii) The position vectors of two points A and B relative to the origin 0 area = 4cos28i+(6cos8-l)j b
=
2 cose
i - j.By using the identity in (b) (i) above, find the value of 8, 0::; 8::; ~ , such that
a and b are perpendicular. [ 5 marks]
Find the modulus of the complex number
z
=
25 (2 + 3i)4 + 3i [ 4 marks]
Total 20 marks GO ON TO THE NEXT PAGE 02134020/CAPE 2005
5. (a)
(b)
Section C (Module 3) Answer BOTH questions.
(i) State the value of lim 0 sin u
u~ u [ 1 mark]
(ii) By means of the substitution u
=
3x, show that lim sin 3x=
3.x~O x
(iii) Hence, evaluate lim sin 3x x ~ 0 sin Sx
A
If y
= -
+ Bx, where A and B are constants, show that X 2 d2y ely X dx2 + X dx=
y. 4 marks] [ 4 marks] [ 4 marks](c) The diagram below, not drawn to scale, shows part of the curve y2 = 4x. Pis the point
on the curve at which the line y
=
2x cuts the curve.X
Find
(i) the coordinates of P [ 3 marks]
(ii) the volume of the solid generated by rotating the shaded area through 2rc radians
about the x-axis. [ 4 marks]
Total 20 marks
GO ON TO THE NEXT PAGE 02134020/CAPE 2005
-- -
6-6. (a) Differentiate, with respect to x,
(x2 +
7i
+ sin 3x. [ 6 marks] (b) Determine the values of x for which the function y=
x3- 9x2 + 15x + 4(i) has stationary points [ 3 marks]
(ii) IS mcreasmg 2 marks]
(iii) is decreasing. [ 2 marks]
(c) (i) USe the SUbstitution t
=
a-X tO ShOW thatr
j (x) cJx=
fOa j (a-X) dx.[ 4 marks]
(ii) If
r
f(x) dx=
12, use the substitution t =X- 1 to evaluateJl
5 3f(x-1) dx. [ 3 marks] Total 20 marks ENDOFTEST 02134020/CAPE 2005
-TEST CODE
02134032
FORM TP 2005254
MAY /JUNE 2005CARIBBEAN EXAMINATIONS COUNCIL
ADVANCED
PROFICIENCY EXAMINATION
PURE MATHEMATICSUNIT 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. 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 significant figures.
Examination materials
Mathematical formulae and tables Electronic calculator
Graph paper
02134032/CAPE 2005
Copyright © 2004 Caribbean Examinations Council All rights reserved.
1.
(a)(b)
-2
-Section A (Module 1)
Answer this question.
Given that 2x2 + 8x + 11
=
2(x + h)2 + k for all values of x, find the value of EACH ofthe constants h and k. [ 5 marks]
(i) If p, q, r, s E R, use the fact that (p-q)2 ;::: 0 to show that p2 + q2 ;::: 2 pq. [ 2 marks]
(ii) Deduce that if p2 + q2
=
1, then pq :::; ~- [ 1 mark](c) A club bakes and sellsxcakes, making a profit, in dollars, that is modelled by the function
f(x)
=
x2 - lOx.(i) Sketch the graph of the function f (x)
=
x?-
1 Ox. [ 8 marks] (ii) From your graph, determinea) the LEAST number of cakes sold for which a profit is realised
[ 2 marks]
b) the GREATEST possible loss in dollars [ 1 mark] c) the number of cakes for which the GREATEST possible loss occurs.
[ 1 mark]
Total 20 marks
GO ON TO THE NEXT PAGE
2. (a)
Section B (Module 2)
Answer this question.
The straight line through the point P (4, 3) is perpendicular to 3x + 2y
=
5 and meets the given line at N.Find
(i) the coordinates of N 6 marks]
(ii) the length of the line-segment PN. 2 marks] (b) The table below presents data collected on the movement of the tide at various times
after midnight on a particular day.
Tide Movement Time After Midnight Height
(t hours) (h metres)
High 0 12
Low 6 2
High 12 12
Low 18 2
The height, h metres, can be modelled by a function of the form h
=
p cos (qt/ + 7where t is the time in hours after midnight. Use the data from the table to find the
values of p and q. [12 marks]
Total 20 marks
GO ON TO THE NEXT PAGE 02134032/CAPE 2005
3. (a) (i) Find lim x--71
-
4
-Section C (Module 3)
Answer this question.
(ii) Determine the real values of x for which the function
f() - 3x- l X - x2- x- 2
is continuous.
[ 3 marks]
3 marks]
(b) Differentiate with respect to x, from first principles, the function x2 + 2x.
[ 5 marks]
(c) Initially, the depth of water in a tank is 32 m. Water drains from the tank through a
hole cut in the bottom. At t minutes after the water begins draining, the depth of water
in the tank is x metres. The depth of the water changes, with respect to time t, at the rate
equal to (-2t- 4).
(i) Find an expression for x in terms oft. [ 5 marks]
(ii) Hence, determine how long it takes for the water to drain completely from the
tank. [ 4 marks]
Total 20 marks
END OF TEST
FORM TP 2005256
MAY /JUNE 2005CARIBBEAN EXAMINATIONS COUNCIL
ADVANCED PROFICIENCY EXAMINATION
PURE
MATHEMATICS
UNIT
2 -
PAPER
02
2 hours
(ot
JUNE 2005 (p.m.0
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. 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
02234020/CAPE 2005
Copyright© 2004 Caribbean Examinations Council All rights reserved.
1. (a)
-2
-Section A (Module 1) Answer BOTH questions.
The 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) Describe clearly the relationship betweenf(x)=
ex and g(x)=
ln x.(iii) Using a calculator, find the value of a) b) r p. [ 3 marks] 1 mark] 2 marks] (b) Given that (c) log
0 (be)= x, Iogb (ea) = y, logc(ab) = z and a-:~; b-:~; e, show that axl.lcZ
= (abe l
Find the values of x E R for which eX + 3e-x
=
4.3 marks] 8 marks] Total 20 marks
GO ON TO THE NEXT PAGE
2. (a)
(b)
3. (a)
(b)
(c)
A curve is given parametrically by x = (3-2t)2, y = t3 - 2t. Find
(i)
1x
in terms oft 4 marks](ii) the gradient of the normal to the curve at the point t
=
2. 2 marks](i) Express A
B
2
2x + 1 in the form - +
x (x + 1) x x2 + x
____f_
+ 1,
where A, B and [ 7 C are comarks] n-stants.J
2 2x + 1 Hence, evaluate 2( 1.) dx· X X+ I [ 7 marks] (ii) Total 20 marks Section B (Module 2) Answer BOTH questions.(i) Use the fact that-1 - - -1 I
1
=
(
l) to show thatr r+ r r +
n
s
n=L
r=
I(
r(r1 )=1
+ I)--
n +1
I . [ 5 marks] (ii) Deduce, that as n ~ oo, S11 ~ I. [ 1 mark] Th e common ratiO. , r, o a geometnf . c sen. es I. S g1 . ven b y r=
- -
Sx 2 . p· 10 d ALL h t e va ues Iof x for which the series converges. 4 + x [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 (i)L
"c,.
=
2" r=O 11 (ii)L
"c,.
(-I/ =o.
r=O [ 2 marks] [ 2 marks] Total 20 marksGO ON TO THE NEXT PAGE 02234020/CAPE 2005
--4
-4. The function,f, is given by f(x)
=
6 - 4x - x3.5.
(a) Show that
(i)
f
is everywhere strictly decreasing 4 marks] (ii) the equationf(x)=
0 has a real root, a, in the closed interval [1, 2][ 4 marks]
(iii)
a
is the only real root of the equation fix)= 0. [ 4 marks] (b) If xn is the nth approximation to a, use the Newton-Raphson method to show that the(n + l)st approximation x11 + 1 is given by (a)
2x;':
+ 6 XII+ I=
_3_2_4_ . x, + Section C (Module 3)Answer BOTH questions.
[ 8 marks]
Total 20 marks
On a particular day, a certain fuel service station offered 100 customers who purchased
prentium or regular gasoline, a free check of the engine oil or brake fluid in their vehicles. The services required by these customers were as follows:
15% of the customers purchased premium 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 customers 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
51 Premium gasoline Regular gasoline [ 3 marks]
GO ON TO THE NEXT PAGE 02234020/CAPE 2005
6.
(ii) Find the probability that a customer chosen at random
a) who had purchased premium gasoline requested a check for engine oil 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] (b) A bag contains 12 red balls, 8 blue balls and 4 white balls. Three balls are drawn from
the bag at random without replacement.
(a)
Calculate
(i) 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]
Total 20 marks
Find the values of x for which X 1 2 1 X 1 2 2
=
0. x [10 marks](b) Twelve hundred people visited an exhibition on its opening day. Thereafter, the attendance fell each day by 4% of the number on the previous day.
(i) Obtain an expression for the number of visitors on the n1h day. 2 marks] (ii) 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
during the period for which it was opened. [ 3 marks]
(iv) If the exhibition had been kept opened indefinitely, what would be the maximum
number of visitors? [ 2 marks]
Total 20 marks
END OF TEST
TEST CODE
02234032
FORM TP 2005257
MAY /JUNE 2005CARIBBEAN 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. 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
02234032/CAPE 2005
Copyright © 2004 Caribbean Examinations Council All rights reserved.
Section 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 andy are related by the formula, y
=
bxn.(a) (i) By taking logarithms to base 10 of both sides, convert y
=
bxn to the formY
=
nX + d where n and d are constants. [ 4 marks](ii) Hence, express
a) Y in terms of y b) X in terms of x
c) din terms of b. [ 3 marks]
(b) Use the data from Table 1 to complete Table 2.
log x 10 log y 10 1.30 Table 2 3.21 1.60 3.57 [ 2 marks] (c) In the graph on page 3, log10x is plotted against log10y for 1.3 :S x :S 1.7.
(i) 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 marks]
c) the value of each of the constants, nand d, in Part (a) (i) above.
[ 2 marks] (ii) Using the graph, or otherwise, estimate the value of x for which y is 1800.
[ 3 marks]
Total 20 marks
GO ON TO THE NEXT PAGE
0 N N w ~ w
?5
~
N 0 0 VIa
0 0z
....,
0~
~
'"tl~
L"I
01 X ~OJ 17"1 9.1s·1
wSection B (Module 2) Answer this question.
2. Mr John Slick takes out an investment with an investment company which requires
making a fixed payment of $A at the beginning of each year. At the end of the invest
-ment period, John expects to receive a payout sum of money which is equal to the total payments made, together with interest added at the end of EACH year at a rate of r % per
annum of the total sum in the fund.
The table below shows information on Mr Slick's investment for the first three years. Amount at
Year Beginning of Year ($) Interest ($) Payout Sum$
1 A A X 100 r A+(Ax_r) 100 = A(1+-r) 100 =AR 2 A+ AR (A+ AR) X _r_
100 (A+ AR) + [(A+ AR) X
1~0
J
= (A + AR) (1 + I
~O)
=(A+ AR) R = AR+ AR2 3 A+AR+AR2 2 r (A+ AR + AR2 ) R (A+ AR + AR) x lOO = AR+AR2+AR3(a) Write expressions for
(i) the amounts at the beginning of Years 4 and 5 2 marks]
(ii) 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 [ 7 marks]
(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]
Total 20 marks
GO ON TO THE NEXT PAGE
- 5
-Section 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, where
(
1 2 A= 2 4 3 5
(a) Show that A is non-singular. 5 marks]
(b) -I
Show that X= A Y. 3 marks]
(c) Find A-1• [ 9 marks]
(d) Find the input matrix X corresponding to the output matrix Y
~
(!~
)
.
3 marks] Total 20 marks
END OF TEST
FORM TP 2006257
MAY /JUNE 2006CARIBBEAN
EXAMINATIONS
COUNCIL
ADVANCED PROFICIENCY EXAMINATION
PURE MATHEMATICS
UNIT
1 - PAPER 01
2 hours
c
19 MAY 2006 (p.m.))This 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 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.
Examination materials
Mathematical formulae and tables Electronic calculator
Graph paper
0213401 0/CAPE 2006
Copyright © 2005 Caribbean Examinations Council® All rights reserved.
-2
-Section A (Module I)
Answer ALL questions.
1. (a) The functionf(x) is given by f(x) =
.0 -
(p + 1)x2 + p, p E N.(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 n
(b) Given that
L
r = ..!!_(n + 1), show thatL
(3r + 1) =.!. n(3n + 5).r= l 2 r=l 2
[ 4 marks]
Total 8 marks
2. (a) LetA= {x: 2:::; x:::; 7} and B = {x:
I
x-41:::; h}, hE R.Find the LARGEST value of h for which B c A. [ 6 marks]
Find the value of k. [ 3 marks]
Total 9 marks
3. (a) Find a, b E R such that - -3x ax+ b
1 - 2
=
-
-
1 , where x ':1; -1.X+ X+
(i) [ 2 marks]
(ii) Hence, find the range of values of x E R for which ~ > 2.
X+ 1 [ 4 marks]
(b)
2
4 4
-Without the use of calculators or tables, show that = 2
(
..J
2 ).12
X 8-1/J[ 4 marks] Total tO marks
GO ON TO THE NEXT PAGE 0213401 0/CAPE 2006
4. The diagram below (not drawn to scale) represents the graph of the function f(x)
=
x2 + 1, -1 ~ x ~ 1 and p, q E R. f(x) (-l,p) (q, 2) ---r---~---,,---~ X -1 0 +1 (a) Find (i) the value of p and of q(ii) the range of the functionj(x) for the given domain. (b) Determine whether f(x)
(i) is surjective (onto)
(ii) is injective (one-to-one) (iii) has an inverse.
5. Find the values of m, n E R for which the system of equations
(a)
(b)
(c)
X+ 2y
=
12x+ my= n possesses a unique solution is inconsistent
possesses infinitely many solutions.
[ 2 marks] [ 1 mark] 1 mark] 1 mark] [ 1 mark] Total 6 marks [ 3 marks] [ 2 marks] [ 2 marks] Total 7 marks GO ON TO THE NEXT PAGE 02134010/CAPE 2006
-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 [ 2 marks]
(b) the coordinates of the point Q 3 marks]
(c) the EXACT length of the line segment PQ. 2 marks]
Total 7 marks
GO ON TO THE NEXT PAGE 0213401 0/CAPE 2006
7. In the diagram below (not drawn to scale), AC
=
BC, AD=
7 units, DC=
8 units, angle A CB=
~
radians and angle ADC=
2; radians.8.
9.
B
D Find the EXACT length of
(a) (b) (a) (b) (a) (b) AC [ 5 marks] AB. [ 3 marks] Total 8 marks
Solve the equation 4 cos28-4 sin 8-1
=
0 for 0 ~ 8 ~n.
[ 5 marks] 1- cos 2.x - 2Show that 1 +cos 2.x = tan x. [ 3 marks]
Total 8 marks
The roots of the quadratic equation x2 + 6x + k
=
0 are -3 + 2i and -3-2i .Find the value of the constant k. [ 2 marks]
Find the real numbers u and v such that u + 2i
=
1 +vi.3 - 4i 6 marks]
Total 8 marks
10. Given the vectors p
=
2i + 3j and q=
3i- 2j,(a) find x, y E R such that xp + yq
=
-3i- 11j [ 7 marks](b) show that p and q are perpendicular. [ 2 marks]
Total 9 marks GO ON TO THE NEXT PAGE 0213401 0/CAPE 2006
11.
12.
(a) Find lim X---7
-6
-Section C (Module 3)
Answer ALL questions.
(b) Find the values of x E R such that the function j(x)
=
9 -xz(x2-3)(JxJ-3)
is discontinuous.
2-x
(a) The functionj(x) is defined by j(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)
=
sin2(x2).[ 3 marks] [ 4 marks] Total 7 marks 6 marks] 3 marks] Total 9 marks
13. The diagram below (not drawn to scale) is a sketch of the section of the function j(x)
=
x (x2 - 12) which passes through the origin 0. A and B are the stationary points on thecurve. y (A) f(x) =x(x2 - 12) (B) Find
the coordinates of each of the stationary points, A and B
(a) (b) the equation of the normal to the curvej(x)
=
x (x2 - 12) at the origin. 5 marks] 4 marks] Total 9 marks GO ON TO THE NEXT PAGE 02134010/CAPE 200616 14. The diagram below (not drawn to scale) shows the shaded area, A, bounded by the curve y = x2
1 and the lines y
=
-x- 1 , x=
2 and x = 3. 2 y I y= 2x- l 16 Y = x2 X (a) (b)Express the shaded area, A, as the difference of two definite integrals. Hence, show that A= 16
f
2 3 x-2 dx - ;f
X d.x+
r
d.x. 1 mark] 2 marks] (c) Find the value of A. 3 marks] Total 6 marks15. Use the result
I
:
f(x)dx=
r
f(a-x)dx, a> 0 , to show that (a)f
n
x sin x dx=
f
n
(n -x) sin x dx. 0 0 [ 2 marks] (b) Hence, show that (i)r
X Sin X dx=
'!T; fsin X dx- J:x Sin X cfx [ 2 marks] (ii) J:xsinxd.x=n. [ 5 marks] Total 9 marks END OF TEST 0213401 0/CAPE 2006TEST CODE
02134020
FORM TP 2006258
MAY/JUNE 2006CARIBBEAN 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. 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
02134020/CAPE 2006
Copyright© 2005 Caribbean Examinations Council®
1.
2.
(a)
Section A (Module 1) Answer BOTH questions.
Solve the simultaneous equations x2+xy=6
X -3y + 1
=
0. [ 8 marks](b) The roots of the equation x2 + 4x + 1
=
0 are a and13.
Without solving the equation,(i) state the values of
a+
13
andal3
[ 2 marks](ii) find the value of
a
2 +13
2 [ 3 marks](iii) find the equation whose roots are 1 +
~
and 1 +i-·
[ 7 marks] Total 20 marksII
(a) Prove, by Mathematical Induction, that r~
1
r=
~n (n + 1). [10 marks](b) Express, in terms of nand in the SIMPLEST form, 2n (i)
Lr
r = 1 [ 2 marks] 2n (ii)L
r. [ 4 marks] r= n + 1 2n (c) Find n ifL
r=
100. [ 4 marks] r=n+J Total 20 marksGO ON TO THE NEXT PAGE 02134020/CAPE 2006
3. 4. (a) (b) (a) (i) - 3-Section B (Module 2)
Answer BOTH questions.
Find the coordinates of the centre and radius of the circle x2 + 2x + y2 - 4y = 4. [ 4 marks] (ii) By writing x + 1 = 3 sin 8, show that the parametric equations of this circle are
X= -1 + 3 Sin 8, y
=
2 + 3 COS 8. [ 5 marks)(iii) Show that the x-coordinates of the points of intersection of this circle with
the line x + y = 1 are x = -1
±
~
f2.
[
4 marks]Find the general solutions of the equation cos 8 = 2 sin28- 1. [ 7 marks] Total 20 marks
Given that 4 sin x-cos x = R sin (x-a), R > 0 and 0° <a< 90°,
(i) find the values of R and a correct to one decimal place [ 7 marks]
(ii) hence, find ONE value of x between 0° and 360° for which the curve y = 4 sin x-cos x has a stationary point. [ 2 marks]
(b) Letz1=2-3iandz2=3+4i.
(i) Find in the form a + bi, a, b E R,
a) 1 mark]
b) [ 3 marks]
c) [ 5 marks]
(ii) Find the quadratic equation whose roots are z1 and z2• [ 2 marks] Total 20 marks
GO ON TO THE NEXT PAGE 02134020/CAPE 2006
5. (a)
(b)
(i)
Section C (Module 3) Answer BOTH questions.
State the value of lim ox~O
sin
ox
ox .
[ 1 mark](ii) Given that sin 2(x + 8x)-sin 2x
=
2 cos A sin B, find A and Bin terms of xand/or 8x. [ 2 marks]
(iii) Hence, or otherwise, differentiate with respect to x, from first principles,
the function y
=
sin 2x. [ 7 marks]The curve y Find
=
hx2 +..!!:....passes through the point P (1,1) and has a gradient of 5 at P.X
(i) (ii)
the values of the constants hand k
the equation of the tangent to the curve at the point where x
=
_!.__2
[ 5 marks]
5 marks] Total 20 marks
GO ON TO THE NEXT PAGE
6. (a)
(b)
(c)
-5
-In the diagram given below (not drawn to scale), the areaS under the line y
=
x, for 0 ~ x ~ 1, is divided into a set of n rectangular strips each of width.!. units.II
y
(i) Show that the areaS is approximately
1 2 3 11-1
--z+--z
+----z+ ....+-2-n 11 11 11 [ 6 marks]
n-1
(ii) Given that
L
r=..!.
11 (n- 1), show that S"" ..!. (1 - _!_).r=! 2 2 n [ 2 marks]
(i) Show that forf(x)
=
2x j'( ) 8-2x2x2 + 4, x
=
(x2 + 4)2 [ 4 marks] (ii) H 1f
1 24 - 6x2 d ence, eva uate 0 (x2 + 4?
x. [ 3 marks]f
2u 1 7 Find the value of u > 0 if 11 x4 dx = 192 . [ 5 marks] Total 20 marks ENDOFTEST 02134020/CAPE 2006FORM TP 2006259
MAY/JUNE 2006CARIBBEAN
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 question.
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 significant figures.
Examination materials
Mathematical formulae and tables Electronic calculator
Graph paper
02134032/CAPE 2006
Copyright © 2005 Caribbean Examinations Council®
1.
-2
-Section A (Module 1)
Answer this question.
(a) Solve, for x, the equations
(i) lx+41=1 2x-11 [ 7 marks]
(ii) [ 7 marks]
(b) A coach of an athletic club has five athletes, u, v, w, x andy, in his training camp. He makes an assignment, f, of athletes u, v, x andy to physical activities 1, 2, 3 and 4 according to the diagram below in which A
= {
u, v, w, x, y}, B = { 1, 2, 3, 4} andf = {(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 function g: A --7 Bin (ii) above as a set of ordered pairs.
[ 3 marks] Total 20 marks
GO ON TO THE NEXT PAGE 02134032/CAPE 2006
2. (a)
(b)
(c)
Section B (Module 2) Answer this question.
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 hen weighed 500 grams and 25 days later it weighed 1 500 grams.
(i) Copy and complete the table below.
d (days) 25
w (gms) 500
[ 1 mark] (ii) Determine
a) the linear function,/, such thatf(d)
=
w [ 3 marks] b) the expected weight of a hen 10 days after the diet began.[ 2 marks] (iii) After how many days is the hen expected to weigh 2 180 grams? [ 2 marks]
(i)
(ii)
Show that (tan
e
_ sec 9)2=
sin29- 2 sin 9 +1 cos29Hence show that 1 -sin
e
=
(tan 8-sec 8)2 I +sin 8Given the complex number
z
=
{3"
+.!. i, find 2 2 (i)I
z
I
(ii) arg (z) (iii) zZ. [ 3 marks] [ 4 marks] 1 mark] [ 2 marks] 2 marks] Total 20 marksGO ON TO THE NEXT PAGE 02134032/CAPE 2006
3
.
-4-Section C (Module 3) Answer this question.
(a) (i) By expressing X-4 as
c-r-;
+ 2)c
.Y~-
2), find limG-
2x-74 x-4 [ 3 marks] (ii) Hence, find lim
.y-;-
2x -7 4 x2 - Sx + 4 [ 3 marks]
(b) Given that
J
:J(x)
dx=
10, findJ
:
[f(
x)
+
4]
dx+
J:
f(x)
dx. [ 7 marks] (c) A bowl is formed by rotating the area between the curves y = x2 andy= x2 - 1 for x ~ 0and 0 ~ y ~ 1 through 2n radians around they-axis. Calculate
(i) the capacity of the bowl, that is, the amount of liquid it can hold [ 3 marks] (ii) the volume of material in the bowl. [ 4 marks] Total 20 marks
END OF TEST
FORM TP 2006260
MAY/JUNE 2006CARIBBEAN EXAMINATIONS COUNCIL
ADVANCED PROFICIENCY EXAMINATION
PURE MATHEMATICS
UNIT
2 -PAPER 01
2 hours
( 22
MAY
2006 (a.m.>)
This 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 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.
Examination Materials
Mathematical formulae and tables Electronic calculator
Graph paper
0223401 0/CAPE 2006
Copyright©
20
0
5
Caribbean Examinations Council® All rights reserved.-
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
=
e.,
a+ SID . X [ 3 marks](b) y
=
tan 3x + In (x2 + 4) [ 4 marks]Total 7 marks
3. (a) Find the gradient of the curve
x2
+ xy=
2y2 at the point P (-2, 1). [ 5 marks](b) Hence, find the equation of the normal to the curve at P. [ 3 marks] Total 8 marks
4. If y
=
sin 2x + cos 2x ,(a) find
ax
dy [ 3 marks](b) [ 4 marks]
Total 7 marks
5. Use the substitution indicated in EACH case to find the following integrals:
(a)
J
sin8x cos x dx ; u=
sin x 4 marks](b)
J
xV
2x + 1 dx ; u2=
2x + 1 [ 6 marks]Total 10 marks
GO ON TO THE NEXT PAGE 0223401 0/CAPE 2006
Section B (Module 2) Answer ALL questions.
6. A sequence { U11} of real numbers satisfies U11 + 1 U11
=
3( -1)" ; u1=
1.7.
9
.
(a) Show that
(i) u11 + 2 = -u11 [ 3 marks]
(ii) un + 4
=
un. [ 1 mark ](b) Write the FIRST FOUR terms of this sequence. [ 3 marks] Total 7 marks
(a) Verify that the sum, S11, of the series -2
1
+ _!_3 + 1_
5 + ... , ton terms, is 511
=
~ (1--4-).
2 2 . 3 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 315. Find the value of
(i) X [ 2 marks]
(ii) the common difference d. [ 4 marks]
TotallO marks
(a) show that x2 - 5x-14 = 0 4 marks]
(b) find x. 2 marks]
Total 6 marks
(a) Expand (1 + ux) (2- x)3 in powers of x up to the term in x2, u E R. 6 marks]
(b) Given that the coefficient of the term in
x2
is zero, find the value of u. [ 2 marks]Total 8 marks
GO ON TO THE NEXT PAGE 02234010/CAPE 2006
10.
-4-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 marks
GO ON TO THE NEXT PAGE
02234010/CAPE 2006
-Section 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.
12.
Determine the number of ways in which the committee may be formed if it is to contain
(a) (b) (c) (a) (b) NO females [ 2 marks]
EXACTLY one female 3 marks]
AT LEAST one female. 4 marks]
Total 9 marks
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) the first letter is a consonant? 3 marks]
(ii) 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. The determinant!:::,. is given by 14. 1 a b+c !:::,.
=
1 b c+a 1 c a+bShow that!:::,.
=
0 for any a, b and c E R.(a)
(b)
Write the following system of equations in the form AX= D.
(i) (ii) (iii) x+y-z=2 2x-y+z= 1 3x + 2z
=
1Find the matrix B, the matrix of cofactors of the matrix A.
Calculate B T A. Deduce the value of
I
AI·
[ 6 marks] Total 6 marks 2 marks] [ 5 marks] [ 2 marks] [ 1 mark] Total10 marksGO ON TO THE NEXT PAGE
6
-15. 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.
(a) Write down a differential equation for r with respect to time t sees. [ 1 mark ]
(b) Find, in terms of 1r, the rate of increase with respect to time t of the total surface area, A, of the cylinder when the radius is 4 em and the height is 10 em.
[ 6 marks] [A
=
27rr2 + 21rrh]Total 7 marks
END OF TEST
FORM TP 2006261
MAY /JUNE 2006CAR
IBBEAN
EXAMINAT
IONS
COUNCIL
ADVANCED
PROFICIENCY EXAMINATION
PURE MATHEMATICS
UNIT
2 -
PAPER
02
2 hours
c
31 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. 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
02234020/CAPE 2006
Copyright© 2005 Caribbean Examinations Council® All rights reserved.
1.
2.
-
2-Section A (Module 1)
Answer BOTH questions.
(a) If f(x)
=
_0 ln2 X, show that(i) f'(x)
=
x2ln 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,
(a)
(b)
N
=
800 k E R r E Rl+ke-rr' ' '
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. [ 6 marks]
(iii) How many members will there be in the club 3 years after its formation?
(i) Express 1 + x in partial fractions.
(x - 1) (x2 + 1)
(ii) Hence, find
f
1 +X 2 dx. (x- 1) (x + 1)I
Given that I,
=f
x'
e
dx, where n E N.0 (i) Evaluate /1• (ii) Show that 111
=
e-n/11_ 1• [ 2 marks] Total 20 marks [ 6 marks] [ 3 marks] [ 4 marks] 4 marks](iii) Hence, or otherwise, evaluate /3, writing your answer in terms of e.
[ 3 marks]
Total 20 marks
GO ON TO THE NEXT PAGE 02234020/CAPE 2006
3.
4. (a) (i) (ii) (iii) Section B (Module 2) Answer BOTH questions.Show that the terms of m
L
ln 3r r=lare in arithmetic progression.
Find the sum of the first 20 terms of this series.
2m Hence, show that
L
ln 3r = (2m2+ m) ln 3. r= 1 [ 3 marks] [ 4 marks] [ 3 marks](b) The sequence of positive terms, {x11} , is defined by x,1 + 1 =
x;
+i
·
x1 <k·
(a)
(i)
(ii)
Show, by mathematical induction, or otherwise, that x <
.!.
for all positiveII 2
integers n. [ 7 marks]
By considering x11 + 1- X11, or otherwise, show that X
11 < X11 + 1• [ 3 marks] Total 20 marks
Sketch the functions y = sin x andy =
x2
on the SAME axes. [ 5 marks] (b) Deduce that the functionf(x) =sin x-x2
has EXACTLY two real roots.[ 3 marks] (c) Find the interval in which the non-zero root
a
ofj(x) lies. [ 4 marks](d) Starting with a first approximation of a at x1 = 0.7, use one iteration of the New ton-Raphson method to obtain a better approximation of a to 3 decimal places.
[ 8 marks] Total 20 marks
GO ON TO THE NEXT PAGE 02234020/CAPE 2006
5.
6.
(a) (i)
-4-Section C (Module 3) Answer BOTH questions.
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] (ii) 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) failures is represented
by nCxr q"-x, n > 0. Apply this model to the following problem.
(a)
The probability that John will hit the target at a firing practice is ~. He fires 9 shots. Calculate the probability that he will hit the target 6
(i) AT LEAST 8 times
(ii) NO MORE than seven times.
If A= ( -: 2 2 -2
-~
) andB = (~
(i) find AB (ii) deduce A-1• -1 1 1 [ 7 marks] [ 3 marks] Total 20 marks~
)
.
1 3 marks] 3 marks]GO ON TO THE NEXT PAGE 02234020/CAPE 2006
(b) A nursery sells 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
Grass Seed (Kilograms)
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
bA 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 ]
(ii) 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 kilograms 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 exists, write X in terms of M-1 and D. 3 marks]
(v) Given that M-1
=
0.35 0.1 -0.15 ,(
-0.2 -0.2 0.3 )
-0.05 0.2 -0.05
calculate how many bags of EACH brand, P, Q, and R, are required to
produce a blend containing 30 kilograms of C-grass, 30 kilograms of Z-grass and 50 kilograms of B-grass. [ 4 marks] Total 20 marks
END OF TEST
TEST CODE
02234032
FORM TP 2006262
MAY /JUNE 2006CARIBBEAN 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 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 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 significant figures.
Examination materials
Mathematical formulae and tables Electronic calculator
Graph paper
02234032/CAPE 2006
Copyright © 2005 Caribbean Examinations Council® All rights reserved.
Section A (Module 1)
Answer this question.
1. The rate of increase of the number of algae with respect to time, t days, is equal to k timesf(t),
wheref(t) is the number of algae at any given timet and k E R.
(a) Obtain a differential equation involvingfi:t) which may be used to model this situation. [ 1 mark] (b) Given that
the number of algae at the beginning is 106 the number of algae doubles every 2 days,
(i) determine the values ofj(O) andf(2) [ 2 marks]
(ii) show that a) k
=
12
zn
2 [10 marks]b) fit)
=
1 06(21'2) [ 5 marks](iii) determine the approximate number of algae present after 7 days.
[ 2 marks] Total 20 marks
GO ON TO THE NEXT PAGE 02234032/CAPE 2006
2. (a)
-3
-Section B (Module 2) Answer this question.
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 (1
-.!
),
q E N.q
(i) Copy and complete the table below showing the value of the car for the first
five years after purchase.
Year 1 Year2 Year 3 Year4 YearS Value at the Beginning p P(l--) 1 P(l-_1__)2 of Year q q ($) Value at the
(1-+>[p
(1-t)J
End of Year 1 P(l--)($)
q 1 2=
P(l--) q [ 3 marks] (ii) Describe FULLY the sequence shown in the table. [ 2 marks] (iii) Determine, in terms of P and q, the value of the car n years after purchase.[ 1 mark]
(b) 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) the value of q [ 5 marks]
(ii) the estimated value of the car after five years [ 2 marks]
(iii) the LEAST integral value of n, the number of years after purchase, for which the estimated value of the car falls below $500.00. [ 7 marks] Total 20 marks
GO ON TO THE NEXT PAGE 02234032/CAPE 2006
3.
(a)Section C (Module 3) Answer this question.
A box contains 8 green balls and 6 red balls. Five balls are selected at random. Find the probability that
(i) ALL 5 balls are green [ 4 marks]
(ii) EXACTLY 3 of the five balls are red 4 marks]
(iii) at LEAST ONE of the five balls is red. 3 marks]
(b) Use the method of row reduction to echelon form on the augmented matrix for the following system of equations to show that the system is inconsistent. [ 9 marks]
x + 2y + 4z
=
6 y+2z=3 x + y + 2z=
1 END OF TEST Total 20 marks 0223403 2/CAPE 2006a\
TEST CODE02134020
FORM TP 2008240
\5/
MAY/JUNE 2008CARIBBEAN
EXAMINATIONS
COUNCIL
ADVANCED PROFICIENCY EXAMINATION
PURE MATHEMATICS
UNIT
1-PAPER 02
ALGEBRA,GEOMETRYANDCALCULUS
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 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.
SECTION A (Module 1) Answer BOTH questions.
1. (a) The roots o f 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 m arks]
(b) Without using calculators or tables, show that
. 2 + v t
■ f s -
VT
[5 marks]
(ii) V T + V 6~ - V 2
VT- V T
■VT + V T
4. [5 marks](c) (i) Show that E r (r + 1) = ~ n (n + 1) (n + 2), n e N.
r» 1 [5 marks]
(ii) Hence, or otherwise, evaluate
50
Z
r ( r +1).
f = 3t[3 marks] 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
a) a 2 + p2
b) a 3 + p3
(iii) find a quadratic equation whose roots are a 3 and p3.
[2 marks]
[2 marks] [4 marks] [4 marks]
GO ON TO THE NEXT PAGE
02134020/CAPE 2008 n V~6~ - V T -< 2
flT+
(i)2 + V T
(b)
3. (a)
(b)
(c)
-3
-(i) Solve for x the equation x113 - 4x-113 = 3. [5 marks]
(ii) Find x such that logs (x + 3) + logs (x- 1)
=
1. [5 marks)(iii) Without the use of calculators or tables, evaluate
log10 (+) + log10
c;)
+ log10 (!) + ... + log10 (~)
+ log10 ( {0).[3 marks)
Total25 marks
SECTION B (Module 2)
Answer BOTH questions.
The lines y = 3x + 4 and 4y = 3x + 5 are inclined at angles a. and j3 respectively to the x-ax1s. (i) (ii) (i) (ii) (iii) (i)
State the values of tan a. and tan j3. [2 marks]
Without using tables or calculators, find the tangent of the angle between the two
lines. [4 marks]
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]
Hence show, without using tables or calculators, that tan 22.5°
=
-{2 - 1.Given that A, B and Care the angles of a triangle, prove that
a) b)
sin A+2 B
=
cos2
c
A
sin B + sin C = 2 cos
T
cos B-C2
[4 marks]
[3 marks] [2 marks]
(ii) Hence, show that
A B C
sin A
+
sin B+
sin C = 4 cosT
cos2
cos2
.
[5 marks) Total 25 marksGO ON TO THE NEXT PAGE 02134020/CAPE 2008