11- hours

( 21 MAY 2004 (p.m.) )

Each section consists of 5 questions.The maximum mark for each section is 30.The maximum mark for this examination is 90.This examination paper consists of 6 pages.

3. Unless otherwise stated in the question, all numerical answers MUSTbe given exactly OR to three significant figures as appropriate.

Mathematical formulae and tablesElectronic calculatorGraph paper

Copyright © 2003 Caribbean Examinations CouncilAll rights reserved.

The functionJ{x) = x3 - p2X2 + 2x - p has remainder - 5 when it is divided by x + 1.Find the possible values of the constant p.

Given that x> y, and k < 0 for the real numbers x, y and k, show that kx < Icy.[4 marks]

Show that ~ = 22-x.2

5. The figure below (not drawn to scale) represents a cross-section through a tunnel. The cross-section is part of a circle with radius 5 metres and centre O. The widthAB of the floorofthe tunnelis 8 metres.

6. Obtain the Cartesian equation of the curve whose parametric representation is x = 2t2 + 3,y = 3t4 + 2 in the form y = Y + Bx + C, where A, Band C are real numbers.

Find the range of values of x E R for which x - 23> 0, x *- -3.x +

sin 2AShow that = cot A, for cos 2A *- 1.

1- cos 2A

9. Given that a and f3 are the roots of the equation x2 - 3x - 1 = 0, find the equation whose roots are1 + a and 1 + f3. [5 marks]

~the unit vector in the direction of OP

~ ~the position vector of a point Q on OP produced such that I OQ I= 5

~the value of t such that the vector 3t i+ 4j is perpendicular to the vector OP.

G· th Jim {41l)} 5lven at x~ -2 J\x = ,Jim

evaluate x~ -2 {j{x) + 2x}.

j{x) = x3,with respect to x .

(b) find, in terms of rand s, the conditions under whichj{x) will have a maximum[3 marks]

14. The curve y = px3 + qx2 + 3x + 2 passes through the point T (1, 2) and its gradient at Tis 7.Find the values of the constants p and q. [5 marks]

15. The diagram below is a rough diagram of y = I x - 2 I for real values of x from x = 0 tox=4.

(b) Find the volume generated by rotating the triangle OAB shown above through 3600 aboutthe x-axis. [4 marks]

FORM TP 2004246

ADVANCED PROFICIENCY EXAMINATIONMATHEMATICS

UNIT 1 - PAPER 02

2thours

( 26 MAY 2004 (p.m.) )

Each section consists of 2 questions.The maximum mark for each section is 50.The maximum mark for this examination is 150.This examination consists of 5 pages.

3. Unless otherwise stated in the question, all numerical answers MUST begiven exactly OR to three significant figures as appropriate.

Mathematical formulae and tablesElectronic calculatorGraph paper

Copyright © 2003 Caribbean Examinations CouncilAll rights reserved.

Given that both (x -1) and (x - 2) are factors off(x) = x3 + mx + n, find the constants m andn, and the third factor off(x). [10marks]

Express f(x) = 12x- 2x2 in the formA + B(x +p)2 where A, B andp are real numbers,and find the maximum value of 12x - 2x2• [7 marks]

(ii) Hence, sketch the graph of f(x) = 12x - 2x2, showing clearly its main features.[5 marks]

Copy and complete the following table for the function fix) = sin x,o $x$ 2n.

Ix 1

0II(x)

------

3n"""2

(ii) On a separate diagram, sketch the graph of I(x) = I sin x I ,0 $ x $ 2n.[4 marks]

(iii) By comparing the diagrams in (b)(i) and (ii) above, determine the solution set of theequation sin x = I sin x I , 0 $ x $ 2n. [3 marks]

Answer BOTH questions.

In the diagram below (not drawn to scale), PQ is perpendicular to AQB.

y

(i) the equation of the line AB

(ii) the equation of the line PQ

(iii) the coordinates of the point Q.

Sol ve, for 0° ~ e ~ 180°, the equation

6 cos2 e + sin e = 4.

(c) Solve, for 0 ~ x ~ n, the equation

[4 marks]

[4 marks]

[4 marks]

[6 marks]

Total 25 marks

h 1 b z - 1. "1 fExpress t e comp ex num er, W = --2 ,Ill a SlInl ar orm.z+

(b) The argument of W is %'(i) Find the equation connecting x and y in the form

aY2 + bi + ex + dy + f = 0 where a, b, e, d,fare integers.

(c) The diagram below (not drawn to scale) shows a parallelogram OLMNwhose diagonals~OM and LN, intersect at P. The position vectors of Land N relative to the origin, 0, are-3i + 6j and 2i + 3j respectively.

1· r-2x-3Evaluate 1m ----x--t3 x? - 4x + 3

Determine the values of x E R for which the function eX + 21) is NOT continuous.

x x + [3 marks]

r-lGiven that y = ~ ,

r+ 1

find dy in terms of xdx

show that x(x2 + 1) dy _ 4y = _4_ .dx r+l

(d) By investigating the sign of r (x), determine the range of real values of x for whichr-5x + 3 is decreasing [8marks]

.Sketch the curve, fix) = x3 - 3x + 2, -2 $ x $ 2.