FORM TP 23240

CARIBBEAN EXAMINATIONS COUNCIL

ADVANCED PROFICIENCY EXAMINATIONMATHEMATICS.

UNIT 1- PAPER 01

Each section consists of 5 questions.The maximum mark for each section is 30.The maximum mark for this examination is 90.This examination consists of 5 printed 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 calculatorRuler and graph paper

Copyright © 2002 Caribbean Examinations Council.All rights reserved.

(~)Solve, for x, the equation .L. ~.

27

4. The diagram below, not drawn to scale, shows a circular games field of radius 35 m enclosed withina circular road of radius 42 m. The field and the road have the same centre 0 and angle AOD is 30°.

Find the equation of the line that is perpendicular to the line y = 3x + 2 and passes throughthe point (0, 1). [3 marks]

x > o.x + 1

8. Express sin (J- cos (J in the form R sine (J - a), where a is acute, and hence find ALL the solutions of

E h I b 4 - 2i . h fi f b·xpress t e comp ex num er --- In t e orm 0 a + I1- 3i

Show that the argument of the complex number in (a) above is ..!!.... •4

~(i) the unit vector in the direction of OB

~(ii) the position vector of the point C on OB produced such that

~ ~lOCI = IOAI.

limx~ -2

r+x-2r + 5x+ 6

(b) Find the real values of x for which the function

Ix II(x) = (I X 12 - 9)

Find the gradient of the tangent to the curve y = 2x3 at the point where y = 16.

xf(x) = x2 + 7 ' and

CARIBBEAN EXAMINATIONS COUNCIL

ADVANCED PROFICIENCY EXAMINATIONMATHEMATICS

UNIT 1 - PAPER 02

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 6 printed 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 calculatorRuler and graph paper

Copyright © 2002 Caribbean Examinations Council.All rights reserved.

The diagram below, not drawn to scale, shows the graph of y =f(x) which has a minimumpoint at (2, -2).

A = {x:O,:Sx,:S4}B = {x:O,:Sx,:S8}.

(iii) By considering the solutions of the equation f(x) = 8, show thatfis NOT onto.[4 marks]

(v) Find the range of values of y for which the equationf(x) = y possesses a solution.[2 marks]

3. In the diagram shown below, not drawn to scale, the line 2x + 3y = 6 meets the y-axis at A and thex-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.

If cos A = -.L, find tan A-5 2

Given that sin A = 12 and sin B = ~, where A and B are acute angles,13 5

find cos (A - B) and sin (A + B).

Given that j(x) = Xl - 5T + 3x, show that j(x) = 0 possesses a root in the interval [ i-, 1].

(ii) the second derivative ofj(x), and hence, determine which stationary point is a localmaximum and which is a local minimum. [5 marks]

1 d2yIf y = --, show that --- = 2(3x2- 2)y .

T+2 dX-

In the diagram given below, not drawn to scale, the area under the curve

y=(l +xrl, O~x~ 1, is approximated by a set ofn rectangular strips each of width -* units.

~/

~~/~~~~

~~~~ /://// /

0 1 2 3 4 n -1 n x- - - - ---n n n n n n

Show that the sum, S , of the areas of the rectangular strips is _1 -1 + _1 ~ + ... + -21 .n n+ n+2 n

xShow that for f (x) = --,

x2 + 4

12 - 3x2

2 dx.(r + 4)

(ii) Find the volume obtained by rotating the portion of the curve between x = 0 and

x = 1 through 21t radians about they axis. [7 marks]