Cape Unit 1 Pure Math Past Exam Papers (2003)

FORM TP 23240 CARIBBEAN

EXAMINATIONS

COUNCIL

ADVANCED PROFICIENCY EXAMINATION MATHEMATICS. 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 be given exactly OR to three significant figures as appropriate.

Mathematical formulae and tables Electronic calculator Ruler 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 within a 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 the point (0, 1).

x x

8.

Express sin

(J-

+1

cos

>

(J in

=

3x + 2 and passes through [3 marks]

o.

the form R sine (J - a), where a is acute, and hence find ALL the solutions of

E xpress t h e comp Iex num b er ---4 - 2i

1- 3i

. t h e fiorm In

0f

a + b·I

Show that the argument of the complex number in (a) above is

..!!....

•

4

(i) (ii)

~ the unit vector in the direction of OB ~ the position vector of the point C on OB produced such that ~

~

lOCI = IOAI.

r+x-2

lim x~

(b)

-2

r + 5x+

6

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

=

(I

X

Ix I 12 -

9)

Find the gradient of the tangent to the curve y

=

2x3 at the point where y

= 16.

x

f(x)

= x2 + 7 ' and

CARIBBEAN

EXAMINATIONS

COUNCIL

ADVANCED PROFICIENCY EXAMINATION MATHEMATICS 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 be given exactly OR to three significant figures as appropriate.

Mathematical formulae and tables Electronic calculator Ruler and graph paper

Copyright © 2002 Caribbean Examinations Council. All rights reserved.

The diagram below, not drawn to scale, shows the graph of y point at (2, -2).

A B

(iii)

(v)

= =

= f(x)

which has a minimum

{x:O,:Sx,:S4} {x:O,:Sx,:S8}.

By considering the solutions of the equation f(x)

=

8, show thatfis

Find the range of values of y for which the equationf(x)

=y

NOT onto. [4 marks]

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 the x-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

5

Given that sin A

A2

= 12 and sin B = ~, 13

5

where A and B are acute angles,

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

Given that j(x)

(ii)

If y

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

i-,

1].

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

= --,

1

T+2

d2y show that --dX-

=

2(3x2

-

2)y .

In the y=(l

diagram

given

below,

not

drawn

to scale,

the

area

under

the

+xr , O~x~ 1, is approximated by a set ofn rectangular strips each of width l

-*

curve units.

~/

~~/ ~~~~

/:

~~~~

/

//// 0

1 n

2 n

3 n

4

n

n -1 n --n n

x

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

n

Show that for

f (x) = --,2

n+2

n

x

x +4

12 - 3x2 (r

(ii)

+ 4)

2

dx.

Find the volume obtained by rotating the portion of the curve between x

= 1 through

21t

radians about they axis.

x

=

0 and

[7 marks]