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l:EST CODE 2134 1

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2 1323

MAY/JUNE 2013

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CARIBBEAN

EXAMINATIONS

COUNC IL

CARIBBEAN ADVANCED

PROFICIENCY

EXAMINATION

PURE MATHEMATICS

ALGEBRA,

GEOMETRY

AND CALCULUS

Unit 1

Paper

01

hour

3 minutes

12 JUNE 2013 (p.m.))

READ

THE FOLLOWING

INSTRUCTIONS CAREFULLY.

1. This test consists

of

45 items. You will have 1 hour and 30 minutes to

answer

them.

2. In addition to this test booklet, you should have an answer sheet.

3. Do not be concerned that the answer sheet provides spaces for more answers than there are

items in this test.

4. Each item in this test has four suggested answers lettered A), B), C), D). Read each item

you are about to answer and decide which choice is best.

5. On your answer sheet, find the number which corresponds to your item and shade the space

having the same letter as the answer you have chosen. Look at the sample item below.

Sample Item

The expression 1 J3 )

2

is equivalent to

A)

B)

C)

D)

4

10

1+3.J3

4 +

2.J3

Sample

Answer

The

best answer to this item is

4

2 J3 ,so

answer

space D) has been shaded.

6. If you want to change your answer, be sure to erase it completely before you fill in your new

choice.

7. When you are told to begin, turn the page and work as quickly and as carefully as you can .

8.

9.

If

you cannot answer an item,

omit

it and go on to the next one. You can return later to the

item omitted. Your score will be the total number of correct answers.

You may do any rough

work

in this booklet.

The use of silent, non-programmable scientific calculators is allowed.

Examination Materials:

A list of mathematical formulae and tables. Revised 2012)

DO NOT TURN

THIS

PAGE UNTIL YOU ARE TOLD

TO

DO SO.

Copyright 2010 Caribbean Examinations Council

All rights reserved.

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1.

2.

3.

-2 -

J8

+

J32 .Jl62 can be simplified as

A)

-2J2

B)

C)

D)

If p

and

q are positive integers

such

that

p < q then which of he following statements

is/are correct?

I.

- p> - q

II.

pz

> pq

III. p -

I

I

x

I a> 0, are

A)

B)

C)

D)

a

x

< o r

x

>a

3

a

x

a

3

a

x > a

andx

< -

3

a

x

-

3

The statement p

v -

p is a

A) converse

B) tautology

C)

contradiction

D) contra positive

The statement - p v - p 1\ q)) is logically

equivalent to

A)

B)

C)

D)

p A q

p :::::> - q

p A q

-p:: : : :>-q

GO ON TO THE NEXT PAGE

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- 5 -

16

A vector equat ion ts

gtven

as

s[ tG

[ he values of

sand

tare,

respectively

A) -2 and -1

B) -2 and 1

C)

2 and 1

D) 2 and -1

17. sin

30-

A) is equal to

A)

1

J

.

- cosA - sm

2 2

B)

1 J

. A

- cosA + sm

2 2

C)

J

1 .

- cosA

+ - sm

2 2

D)

J 1 .

- cosA sm

2 2

18

2 sin e cos is equivalent to

A) sin 8

+

+ sin 8 -

B)

s i n 8 + ~ ) - s i n 8 - ~ )

C) c o s 8 + ~ ) + c o s 8 - ~ )

D)

c o s 8 + ~ ) - c o s 8 - ~ )

19

The

equation of

the

circle

whose

centre

has coordinates 4, I) and whose radius is

7 units is

A) x +

y

+

8x

+ y 49 = 0

B)

x

2

+y 8x y 32

=

0

C)

x

2

+y - 8x y + 49

= 0

D) x

2

+

y

+

8x

+

2y

+ 66 = 0

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2013

20

21.

22.

is an

acute angle and cos

=

2 _

, then

13

s e c ~ =

A)

5

13

B)

12

13

C)

13

-

12

D)

13

5 ,

The

point 2, 3) is at

one end

of a diameter

of the circle whose equation is

x

2

+

y 1 x

+

2y

+

1 = 0.

The

coordinates of the other end of the

diameter are

A) -12,

-5)

B) -12,

-1 )

C) 8, -5)

D)

8,

-1)

The value of s i n [ ;+ p is

A) - sinp

B) -

cosp

C) sinp

D) cosp

GO ON TO THE NEXT PAGE

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23.

What

value

of e

0 :S

e

S n, satisfies the

27. If p =

2i+

j andq = /.. i+6j are perpendicular

equation 2 cos

2

e

+ 3 cos

e-

2 =

0?

vectors, then the value of/.. is

7

(A)

-3

(A)

6

B)

-1

7

(C) 0

B)

4

D)

2

c)

7

3

28.

The general solution for sin

29

=

JC.

sm-ts

6

D)

7

2

{

2nJC

+ -

. 6

(A)

B=

JC

2n+1) -

24 With

respect

to

an ongm 0, A has

16

coordinates (3,

-2 .

The position vector

of3

O

is

{ M ~

B)

B= 12

(A)

3 , -6

JC

nJC -

12

B)

(9, - 2

{

(

nJC

(C)

(C)

B= 12

JC

2n7r) -

_:)

12

D)

{

nJC -

D)

B=

6

JC

25.

The expression sin

6A

+ sin 4A may be

n+1)

written as

29.

The cosine

of

he angle between the vectors

(A) sin lOA

-6 j and i + j is

B)

-2 cos 2A

(C)

2 cos

SA

sin A

-1

D)

2 sin 5AcosA

(A)

J2

26.

1 +

cos

4

A -

sin

4

A =

B)

1

J2

(A)

1

+cos 4A

B)

2cos

2

A

(C)

cos2A

(C)

- 5

J2

D)

2 cos

2

A sin

2

A

D)

6

J2

GO ON TO

THE

NEXT PAGE

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: r ; ~

30.

31.

32.

- 7 -

Item 30 refers to the diagram below.

y

l=x

In the diagram above showing

y

=

x,

y

is

NOT

defined for

A) X = 0

B) X ~ 0

C) x> 0

D) X