238852477-Pure-Maths-Unit-1-Paper-1-2013

l:EST CODE 02134010 FORM TP 201323 MAY/JUNE 2013

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CARIBBEAN EXAMINATIONS COUNCIL

CARIBBEAN ADVANCED PROFICIENCY EXAMINATION®

PURE MATHEMATICS

ALGEBRA, GEOMETRY AND CALCULUS

Unit 1- Paper 01

1 hour 30 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 the following statements is/are correct?

I. -p>-q II. pz > pq III. p - I< q- I

(A) (B) (C)

(D)

I only II only I and III only

II and III only

Two roots of the cubic equation 2x3 + 3x2

- 5x- 6 are -I and -2. The THIRD root is

(A)

(B)

(C)

-3 2

I

2

3

2

(D) 3

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4. R . I. . J2 -I at10na Ismg ~ gives

(A)

(B)

(C)

(D)

v2 +I

I-2J2

3-2J2

I+Ji

I+2J2

5 . If a remainder of 7 is obtained when x3

- 3x + k is divided by x- 3, then k equals

6.

7.

(A) -II (B) -IO (C) IO (D) II

Which of the following are factors of

4 x4 +8x3 -2x2 -6x-4?

I. X+ I II. X- I III. X+ 2 IV. X- 2

(A) I and II only (B) II and III only (C) I and III only (D) I and IV only

(A) (a- b)(ct- a3b + a2b2- ab3 + b4

)

(B) (a- b)(a4 + a3b + a2b2 + ab3 + b4)

(C) (a+ b)(a4- a3b + a2b2 - ab3 + b4

)

(D) (a+ b)(a4 + a3b + a2b2 + ab3 + b4)

GO ON TO THE NEXT PAGE

8.

- 3 -

Which of the following mapping diagrams does NOT represent a function?

y

(A)

y

(B)

X

y

(C)

X

y

(D)

X

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9. If g(x) is the inverse of.f(x) then the correct diagram is

(A)

(B) t__ r:

(C)

~ r

(D)

L ~


10.

11.

12.

- 4 -

Which of the following is true if a., fi and y are roots of the cubic equation 3x3

- 4x2 -7x- 10 = 0?

(A) 4 -7

a+ fi + r =-, afi + fir + ra = -3 3

- 3 -7 a+ fi+r=-, afi+ fir+ra =-

4 3 (B)

(C) 3 7 a + fi + r = - ' afi + fir + ra = -

4 3

4 7 a + f3 + r = -' af3 + f3r + ra = -

3 3 (D)

The annual growth, g(x), (in thousands) of the population over x years is represented by g(x) = 2x. Over how many years will an annual growth of 32 thousand be achieved?

(A) 5 (B) 16 (C) log

216

(D) log2

30

1 4 logl5 - log6+- log-=

2 25

I 36 (A) -log-

2 25

25 (B) log-

4

(C) 0

(D) 1

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13.

14.

15.

The values of x that satisfy the inequality l2x- al > I x I, a> 0, are

(A)

(B)

(C)

(D)

a x <-or x >a

3

-a x < -or x >a

3

a x > - a andx <-

3

a x <a andx > -

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)

pA-q p :::::> -q -pA-q -p:::::>-q


- 5 -

16. A vector equation ts gtven as

s[ -~)+ tG) =[ -n. The 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 J3 . A - cosA - -sm 2 2

(B) 1 J3 . A - cosA + -sm 2 2

(C) J3 1 . A - cosA + -sm 2 2

(D) J3 1 . A - cosA -- sm 2 2

18. 2 sin e cos ~ is equivalent to

(A) sin (8 + ~) + sin (8- ~)

(B) sin(8+~)-sin(8-~)

(C) cos(8+~)+cos(8-~)

(D) cos(8+~)-cos(8-~)

19. The equation of the circle whose centre has coordinates ( 4, I) and whose radius is 7 units is

(A) x2 + y + 8x + y- 49 = 0 (B) x2 + y- 8x- 2y- 32 = 0 (C) x2 + y - 8x- y + 49 = 0 (D) x2 + y + 8x + 2y + 66 = 0

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20.

21.

22.

If~ is an acute angle and cos ~ = 2._ , then 13

sec~=

(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

x2 + y- 1 Ox + 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 sin[;+ p) is

(A) - sinp (B) - cosp (C) sinp (D) cosp


- 6 -

23. What value of e, 0 :S e :S n, satisfies the 27. Ifp = 2i+ j andq = /.. i+6j are perpendicular equation 2 cos2 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

{ ff

2nJC +-' . 6

(A) B= 5JC

(2n+1)-24. With respect to an ongm 0, A has 16

coordinates (3, -2). The position vector

of3 OA is {M+~ (B) B= 12

(A) (3 , -6) 5JC nJC+ -

12

(B) (9, - 2)

{ ff ( -~J nJC +-

(C) (C) B= 12 5JC

(2n7r) -

(_:) 12

(D)

{ ff

nJC+ -(D) B= 6

5JC 25. The expression sin 6A + sin 4A may be (n+1)6

written as

29. The cosine of the 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 + cos4A - sin4A = (B) 1

J2 (A) 1 +cos 4A (B) 2cos2A (C) cos2A (C)

- 5

J2 (D) 2 cos2 A sin2 A

(D) 6

J2

GO ON TO THE NEXT PAGE 0213401 0/CAPE 2013

-- --- - -----------------------:r-------------------------;~

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< 0

lim X2 -9.

--IS x~3 x - 3

(A) (B) (C) (D)

0 6

00

Given that lim sin x = 1 , where x is meas-x-+O

X • 3 . . Jim Sin X • ured In radians, then x---+0 ~ IS

(A) . 3

sm -2

(B) sin3x

2x

(C) 2

3

(D) 3

2

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33.

34.

35.

36.

~(x3 sin x) may be expressed as dx

(A) x2 (cos x + 3 sin x) (B) x2 (x cos x- 3 sin x) (C) x2 (3 cos x + sin x) (D) x2 (x cos x + 3 sin x)

The function g is defined as

{3x + 5 for x < 3

g(x)= px+2 for x~3

For the function to be continuous at x = 3, the value of 'p' should be

(A) -3 (B) -1 (C) 4 (D) 12

If y = x - 6 then dy is 3-4x dx

-21 (A)

(3 -4x)2

(B) 21

(3 -4x)2

(C) 27-8x

(3-4xf

(D) -27 -8x

(3 -4x)2

If y = -J2x + 1 then d2

Y is dx3

1 (A)

(2x+ 1)( -J2x+ 1)

(B) -1

(2x + 1)( -J2x + 1)

(C)

(D) (2x + 1)


37.

38.

39.

40.

If y =tan 6x then dy is dx

(A) 6 tan2 6x (B) sec2 6x (C) 6 sec2 6x (D) sec 6x tan 6x

If dy =cos x then dx

(A) y = sin x + k (B) y =cos X+ k (C) y = - COS X + k (D) y =-sin x + k

- 8-

If f"(x) = 6x, then given that f'(O) = 0, and cis a constant,j(x) =

(A) 3x2 + x + c (B) x3 + x + c (C) 3x2 + c (D) x 3 + c

The path of an object is given parametrically as x = sin t + 2, y = cos t + I . The slope of

1t the tangent at t = - is

4

(A) -I (B) 0 (C) (D) undefined

0213401 0/CAPE 2013

41.

42.

Given that J: 4f(x)dx = 9 , the value of

J: 3f(x)d;c is

(A)

(B)

(C)

(D)

4

3

4

9,

4

27 4

The gradient of the normal to the curve y = 3x2 - 2x + 1 at x = 1 is

(A)

(B)

(C) . -I

(D) 4

1

4

2


43.

- 9 -

Water is leaking from a tank. The rate of change in volume of the water in the tank with respect to time, t, is inversely proportional to the volume, V, of water in the tank. If k is a positive constant of proportionality, then the equation that models this situation lS

(A) -k

V =-.Ji

(B) dV - k ---dt v

(C) dV =-k.JV dt

(D) V =-kt

0213401 0/CAPE 2013

44. Given dy = 2x, then possible sketches of dx

the graph of y are •

I. y

II. y

III. y

IV. y

-----+--~--+---~~X

-1 0 1

(A) I and II only (B) III and IV only (C) I, III and IV only (D) II, III and IV only


- 10-

45. The radius of a circle is increasing at a rate of O.lcm s-1_ At the instant when the radius is 3 em, the rate of increase of the area in cm2 s- 1 is

(A)

(B)

(C)

(D)

2 -Jr 5

3 -Jr 5

2n

47t

END OF TEST

IF YOU FINISH BEFORE TIME IS CALLED, CHECK YOUR WORK ON THIS TEST.

0213401 0/CAPE 2013

238852477-Pure-Maths-Unit-1-Paper-1-2013

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