2 hours

(20 MAY 2005 (p.m.»)

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 6 pages.

3. Unless otherwise stated in the question, any numerical answer that isnot exact MUST be written correct to three significant figures.

Mathematical formulae and tablesElectronic calculatorGraph paper

Copyright © 2004 Caribbean Examinations CouncilAll rights reserved.

1. The diagram below, not drawn to scale, shows the graph ofj(x) = x3 + JU2 - 8x + k where h, kare constants.

x2 - 21 x 1-3 < O.

(b) Show that if x and y are real numbers such that x < y, then for any real number k < 0,kx > ky. [ 4 marks)

4

";11- ";"7

Given that x + _1 = 1, by considering (x + _1 )2x x

show that x2 + \- = -1.x

Hence, or otherwise, find the value of .x3 + ~.x

x - 2y = -3x2 + 3y = 7

6. In the diagram below (not drawn to scale),M is the mid-point of AB. MN is perpendicular tothe straight line through A, M and B.

(a) Find

(i)

[ 2 marks]

[ 2 marks]

(b) The point P on AB divides AB internally such that the ratio AP : PB is 3 : 1. Find thecoordinates of P. [ 2 marks]

Express j (fJ) = .../2 cos e - sin e in the form R cos (e + a).

(c) Determine the value of e, 0::; e::; 2n, at which the minimum value ofj(fJ) occurs.[ 2 marks]

Find the range of values of k for which the quadratic equation x2 + 2kx + 9 = 0 hascomplex roots. [ 4 marks]

Express the complex number 2 + 3~ in the form x + yi, where x and y are real numbers.3 + 41 [ 4 marks]

9. Three points, A, Band C, have coordinates (1,2), (2,5) and (0, - 4) respectively relative to theorigin O.

(a) Express the position vector of EACH of A, B and C in terms of i andj. 3 marks]~ ~

(b) If AB = CD, find the position vector of D in terms of i and j. [ 6 marks]

10. Find the values of e, 0 ::; e::; 2n, for which the vectors cos e i + ...J} j and ii + sin ej areparallel.

Total 7 marks

Jimh ~ 0

v;+h - Ii =h

1

2~

(b) Deduce, from first principles, the derivative with respect to x of y = -IX.[lmark]

X

x2 - 2x - 8

the value of the constant k

d2

the value of ~ at P

Find the coordinates of the stationary points of the functionf x ~ x3 - 3x2 - 9x + 6.[ 6 marks]

15. Three points, P, Q and R, on the curve y = x2 - 2x are shown in the diagram (not drawn toscale) below.

(b) Find the TOTAL area bounded by the curve shown above, the x-axis and the linesx = -1 and x = 2. [ 4 marks]

PURE MA THEMA TICS

UNIT 1 - PAPER 02

( 25 MAY 2005 (p.m.»)

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.

3. Unless otherwise stated in the question, any numerical answer that is notexact MUST be written correct to three significant figures.

Mathematical formulae and tablesElectronic calculatorGraph paper

Copyright © 2004 Caribbean Examinations CouncilAll rights reserved.

Complete the table below for the function I fix) I. where fix) = x (2 - x).

Sketch the graph of I fix) I for -2 ~ x ~ 4.

(b) Find the value(s) of the real number, k, for which the equation k(x2 + 5) = 6 + l2x - x2

has equal roots. [ 6 marks]

(ii) Without using calculators or tables, evaluate

(...{2+ 1)3 - (12- II

Prove, by Mathematical Induction, that IOn - 1 is divisible by 9 for all positiveintegers n. [ 9 marks]

px+ 2y = 8-4x + P7 = 16

(i) Find the value of p for which the system has an infinite number of solutions.[ 3 marks]

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

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

a sector, OABC, of a circle with centre at 0 and a radius of 7 cm, where angle AOC1t d·measures - ra lans.3

a right circular cone with vertex 0 and a circular base of radius r cm which is formed whenthe sector OABC is folded so that OA coincides with OC.

7r=-6

7f35if h cm is the height of the cone, then the exact value of h is 6

[ 2 marks]

By using the identity in (b) (i) above, find the value of e, 0 ~ e ~ : ' such that

a and b are perpendicular. [ 5 marks]

= 25 (2 + 3i)4 + 3i

lim sin uState the value of 0u~ u

By means of the substitution u = 3x, show that limx~O

sin 3xx

limHence, evaluate x~Osin 3xsin 5x

AIf y = - + Bx, where A and B are constants, show thatx

d2y dyx2 - + X - = y.dx2 . dx

(c) The diagram below, not drawn to scale, shows part of the curve y2 = 4x. P is the pointon the curve at which the line y = 2.x cuts the curve.

(ii) the volume of the solid generated by rotating the shaded area through 2n radiansabout the x-axis. [ 4 marks]

Use the substitution t = a -x to show that foaf(x) dx = fo

af(a -x) dx.

[ 4 marks]

(ii) If f0

4f(x) dx = 12, use the substitution t = x-I to evaluate r 3f(x - 1) dx.

[ 3 marks]