rEsr coDE 02134032MAY/JI.]NE2OO4FORM TP 2004247

CARIBBEAN EXAMINATIONS COUNCIL

ADVANCED PROFICIENCY EXAMINATIONMATIIE,MATICS

UMT 1 - PAPERO3/2

7Lhourc2

21 MAY 2fl)4 (p.n")

This examination paper consists of TIIREE sections: Module 1.1, Module 1.2 and Module 1.3.

Each section consists of I question.

The maximum mark for each section is 20.

The maximum mark for this examination is 60.

This examination consiss 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, all numerical answers MUST be

given exactly OR to three significant figures as appropriate'

Examination Materials

Mathematical formulae and tables

Electronic calculator

Graph PaPer

Copyrigtrt O 2003 Caribbean Examinations Council

o2r34032tcAPE2W4

All rights reserved'

1.

sEcTroNA (MODULE 1.1)

Answer this question.

(a) The expressions .f -7x + 6 and, f - * - 4x + 24 have the same remainder when dividedby x - p . Find the possible value(s) of the constant p. [7 marks]

The figure below (not drawn to scale) shows a piece of wire 40 cm long formed into theshape of a sector of a circle of radius r cm, and angle 0 radians.

(i) write down an expression for the perimeter of the sector in terms of r and e and

hence, show that t = *

=" . [3 marks]r

(ii) Show that the area, A cm2, of the sector is given by A = 20r _ p.

[2 marks]

(c) Solve the pair of simultaneous equations

f+ry=3y-3x = -l .

[8 marks]

Total 20 marks

(b)

U2BN3ACAPE2OU CO ON TO T}TE NEXT PAGE

)

-J-

sEcTroN B (MODI.ILE 1.2)

Answer this question.

(a) In the diagram below (not drawn to scale), the line I cuts the positive x and y axei at thepointsA and B respectively. The perpendicular from the origin O meets J at P, OP = {-2 unitsmd POA = 45o.

(b)

(i) Find the coordinates ofP'

(ii) Show that the equation of I is x + y = ) '

(D Express sin 49 in terms of sin 20 and cos 20'

(ii) Hence, solve the equation sin 4O = cos 20 for O 3 O < n'

[3 marks]

[3 marks]

[1 mark ]

[5 marks]

Total 20 marks

GO ON TO THE NEXTPAGE

(c) The roots of the quadratic equation f -4x + c = 0 are the complex numbers 2+iand2 - i ' Find the value of the constant c' [2 marks]

(d) The position vectors of twopointsA and ^B-ar€ ?i

*.:.J. "nd

3i - 8j respectively' Disthe

midpoint of AB ano,rr" poi,i r aivides oD in the ratio 2:3' Find th" pttititt t;;tff.lfk:i

o2134032tcAw2004

-4-

sEcTroN c (MoDr.lLE 1.3)

Answer this question.

3. (a)

(b)

(c)

Evaluate limx--> -2 .ri + 5x + 6

Differentiate, with respect to t, the function

(Given that Jo fl.x) dx = 6, evaluate

(Jo(4x -frxDdx.

The rate of change ofP with respect to r i. f; + r.

(i) Write down the differential equation for p in terms of l.

ta-2f+l---E-'

[6 marks]

[4 marks]

[3 marks]

[1 mark ]

[3 marks]

[3 marks]

Total 20 marks

(d)

(ii)

(iiD

Fidd P in terms of r.

Find the change in the value ofP when , increases from 2 to 4.

END OF TEST

02134032/CAPE 2$4