CAPE pure math unit 1 2008-14

1 1.

In the real number system, the inverse of addition is represented by (A) (B) (C) (D)

2.

5.

x+0= x x + (− x) =0 0+x=x+0 x( y + z ) = xy + xz

If x += b x ; x, b ∈ N , then the value of x in terms of b is (A)

−b

(B)

−

(C)

b

(D)

b 2

Which of the following statements is true? n

(A)

n

∑ r = 2∑ r 2

r 1= r 1 =

n

(B)

 n r = r 1 1= 

∑

r =

2

∑

6.

 r  

2

n

(C)

n

∑ ( 2 + r ) =2 + ∑ r 2

2

r 1= r 1 = n

(D)

The polynomial P ( x) = 2 x 3 + x 2 − 13x + 6, when divided by ( x − 1), gives a remainder of (A)

−4

(B)

0

(C)

6

(D)

18

n

∑r = ∑r 2

2

b 2

r 1= r 0 =

n

3.

The (k + 1)th term in

∑ r (r − 1) is

7.

(4 x)3 − (4 y )3 can be expressed in the form (A)

(4 x − 4 y ) (16 x 2 − 16 y 2 )

(B)

(4 x − 4 y ) (16 x 2 + 16 y 2 )

(C)

(4 x − 4 y ) (16 x 2 − 16 xy − 16 y 2 )

(D)

(4 x − 4 y ) (16 x 2 + 16 xy + 16 y 2 )

r =1

4.

2

(A) (B)

k k +1

(C)

k (k + 1)

(D)

(k + 1) 2

The basic wage Wb and the overtime wage Wo of a shop attendant never differ by more than $100. An inequality representing this statement is (A)

Wo − Wb ≤ 100

(B)

Wo − Wb < 100

(C)

Wo − Wb ≥ 100

(D)

Wo − Wb > 100

8.

If α and β represent roots of the

equation x 2 − px + q = 0, then the value

of α 2 + β 2 is (A) (B)

p2 p−q

(C)

p 2 − 2q

(D)

p 2 + 2q

CAPE Unit 1 P1 2008 ROR

2

9.

10.

11.

 25  The exact value of    16  (A)

2 5

(B)

4 5

(C)

5 4

(D)

5 2

Rationalising

−

1 2

is

12.

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

y

x

2 −1 gives 2 +1

(A)

1− 2 2

(B)

1+

(C)

3+2 2

(D)

3−2 2

(B)

y

2 2 3

x

The expression 2 − 4 x + 3 x 2 can be written as

(C)

y

2

(A)

2 3  3 x −  − 3 2 

(B)

2 2  3 x −  − 3 3 

(C)

3 2  3 x −  + 2 3 

x

2

2

(D)

y

2

(D)

2 2  3 x −  + 3 3 

x


3 Item 13 refers to the diagram below.

15.

The sketch below shows a function y = f ( x).

The function y = f ( x) is represented by (A)

13.

The function f ( x) is decreasing for the range

14.

(A)

x<3

(B)

x≥5

(C)

3≤ x<4

(D)

4≤x<5

The general quadratic equation with roots α and β may be written as (A)

x 2 − (α + β ) x − αβ = 0

(B)

x 2 + (α + β ) x − αβ = 0

(C)

x 2 − (α + β ) x + αβ = 0

(D)

x 2 + (α + β ) x + αβ = 0

(B)

(C)

(D)


4 16.

If

a= 5i + j

and

= b λi + 5j

are

parallel vectors, then the value of λ is (A)

−5

(B)

1

(C)

5

(D)

25

20.

Which of the following sketches BEST represents the curve 1 = y cos x, (0 ≤ x ≤ 2π )? 2

(A)

Items 17 – 18 refer to a circle with general equation

x 2 − 2 x + y 2 + 4 y − 11 = 0. 17.

18.

The coordinates of the centre of the circle are (A)

(−1, − 2)

(B)

(−1, 2)

(C)

(1, − 2)

(D)

(1, 2)

(B)

The radius of the circle is (C)

19.

(A)

2 units

(B)

4 units

(C)

6 units

(D)

11 units

The function

π  sin  x +  2 

can be (D)

simplified to (A)

− cos x

(B)

− sin x

(C)

cos x

(D)

sin x


5

21.

Sin (30o − A) is equal to

24.

If 2cos θ + 9sin θ = r cos(θ − α ), where

π

r > 0 and 0 < α < , then the maximum 2 value of the expression is

(A)

1 3 cos A − sin A 2 2

(B)

1 3 cos A + sin A 2 2

(A)

11

(B)

85

3 1 cos A + sin A 2 2

(C)

11

(D)

85

(C)

(D)

3 1 cos A − sin A 2 2

Item 25 refers to the following diagram.

22.

If β is an acute angle and cos β =

5 , 13

then sec β =

23.

(A)

5 13

(B)

12 13

(C)

13 12

(D)

13 5

25.

2sin θ cos φ is equivalent to (A)

sin(θ + φ ) + sin(θ − φ )

(B)

sin(θ + φ ) − sin(θ − φ )

(C)

cos(θ + φ ) + cos(θ − φ )

(D)

cos(θ + φ ) − cos(θ − φ )

26.

Which of the following equations BEST represents the graph shown above? (A)

y = sin x

(B)

y = sin 2 x

(C)

y = 2sin x

(D)

1 y = sin x 2

The point A has coordinates (3, − 2).  The vector 3OA is (A)

(3, − 6)

(B)

(9, − 2)

(C) (D)

( −92) ( −96) CAPE Unit 1 P1 2008 ROR

6 Items 27 – 28 refer to the vectors

30.

= x 2= t , y t 2 has equation

x =− 3i 8 j, y =+ 6i 12 j, z= 6i + 4mj, m ∈ R. 27.

The value of the scalar product x  y is (A)

28.

29.

The curve with parametric representation (A)

xy 2 = 4

(B)

x2 y = 4

(C)

4x = y 2

(D)

4 y = x2

−78

(B)

78

(C)

−114

(D)

114

Item 31 refers to the diagram below.

The vectors y and z are perpendicular when m has a value of (A)

−3

(B)

−

y2 = x

3 4

(C)

3 4

(D)

3

31.

The line passing through the centre of the circle

( x − 3) + ( y + 2) = 25 2

2

and

parallel to the x-axis, has the equation (A)

y = −2

(B)

y=3

(C)

= y 2( x − 3)

(D)

= y 3( x − 2)

32.

In the diagram above showing the graph of y 2 = x, y is NOT defined for (A)

x=0

(B)

x<0

(C)

x>0

(D)

x≥0

The

function

f ( x) =

x2 − 2 x+2

is

discontinuous for the domain value of (A)

−2

(B)

− 2

(C)

2

(D)

2


7

33.

lim

x→3

x2 − 9 = is x−3

(A)

−∞

(B)

0

(C)

6

(D)

∞

Item 36 refers to the diagram below which shows the curve

x 2 + y 2= 4, 0 ≤ x ≤ 2. y

34.

sin x Given that lim = 1, where x is x→0 x measured in radians, then the value of sin 4 x is lim x→0 x (A)

2 x2 + y 2 = 4

x

2 36.

An expression for obtaining the volume generated by rotating the bounded, shaded region through 360o about the x-axis is

4 (A)

(B)

4x

(C)

sin 4x x

(B)

(D)

4sin x x

(C)

π

∫ ( 4 − y ) dx 2

2

0

π

∫ ( 4 − x ) dx 2

2

0

π

∫ ( 4 + y ) dx 2

2

0

(D)

π

∫ ( 4 + x ) dx 2

2

0

35.

Given that f (= x) (2 x + 1)3 , then f ′(2) equals (A)

37.

The first derivative of −

25 (A)

(B)

75

(C)

125

(D)

−

(B)

(

(

2

)

2 x −1 x

)

2 x −1

150

2x

(C)

(D)

x 2

1 is x −1 2

(x −

2

)

−1

2

2x

( x2 − 1)

2


8

38.

If

dy = cos x, then dx

Item 41 refers to the diagram below.

(A)

= y sin x + k

(B)

= y cos x + k

y

y = ln x

1

(C)

y= − cos x + k

(D)

y= − sin x + k

2

x

Item 39 refers to the following diagram 41. y y=

0.5

39.

40.

1 x

4

The gradient of the normal to the curve y = ln x at x = 2 is (A)

−2

(B)

−

x

The area of the finite region shaded in the diagram is

42.

1 2

(C)

1 2

(D)

2

∫

π 2

2cos5 x dx is

0

(A)

ln (4 − 0.5)

(B)

ln (0.5 − 4)

(A)

(C)

ln 0.5 − ln 4

(B)

(D)

ln 4 − ln 0.5

(C)

−10

(D)

10

−

2 5 2 5

The stationary point of the function

= y ( x − 1) 2 is (A)

(0, 1)

(B)

(−1, 0)

(C)

(1, 0)

(D)

(0, − 1) CAPE Unit 1 P1 2008 ROR

9

43.

dy = 2 x, a sketch of y versus x dx may be represented by

Given

44.

A curve is defined by the equation

y= − 5(2 x − 1) 2 . Given that x increases at a rate of 1 unit per second when x = 1, what is the

I.

corresponding rate of change for y?

1 0

(A)

−40

(B)

−20

(C)

20

(D)

40

II. Item 45 refers to the following diagram.

1 0 0

1

III.

1 0

45.

IV.

(A)

I and II only

(B)

III and IV only

(C)

I, II and IV only

(D)

II, III, IV only

From the diagram above, which of the following statements are true? I.

f ′(1) < 0

II.

f (1) > k

III.

f (2) = 0

IV.

f ′(2) = k

(A)

I and II only

(B)

I and III only

(C)

II and III only

(D)

II and IV only


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l:EST CODE

FORM TP 201323

02134010

MAY/JUNE 2013

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 + (A) (B)

4 10

(C)

1+3.J3 4 + 2.J3

(D)

.J3 ) is equivalent to 2

The best answer to this item is "4 + 2

---

i iiiiii

---==

Sample Answer

.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. 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.

8.

You may do any rough work in this booklet.

9.

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

iiiii

i iiiiii

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. 0213401 0/CAPE 2013

-2-

1.

.J8 + ..J32- .Jl62 can be simplified as (A)

4.

(C) (D) 5.

I. II. III.

-p>-q pz > pq p - I< q- I

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

I only II only I and III only II and III only

I-2J2

(B)

3-2J2

(C)

I+Ji

(D)

I+2J2

If a remainder of 7 is obtained when

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

6.

4 x 4 +8x3 -2x 2 -6x-4? I.

Two roots of the cubic equation 2x3 + 3x2 - 5x- 6 are -I and -2. The THIRD root is

(C)

(D)

-3 2

I

-II -IO IO II

Which of the following are factors of

IV.

X+ I X- I X+ 2 X- 2

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

I and II only II and III only I and III only I and IV only

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

(a- b)(ct- a 3 b + a 2 b 2 - ab 3 + b4 ) (a- b)(a4 + a 3 b + a 2 b 2 + ab 3 + b 4) (a+ b)(a4 - a 3 b + a 2 b2 - ab 3 + b4 ) (a+ b)(a4 + a 3 b + a 2 b 2 + ab 3 + b4 )

II.

(B)

gives

x3 - 3x + k is divided by x- 3, then k equals

If p and q are positive integers such that p < q, then which of the following statements is/are correct?

(A)

v2 +I

(A)

III. 3.

J2 -I

~

-2J2

(B)

2.

. I. . Rat10na Ismg

7.

2

3 2

3

GO ON TO THE NEXT PAGE 021340 I 0/CAPE 2013

-3-

8.

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

9.

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

(A)

y

t__

(B) (B) X

y

(C) (C) X

~

y

r: r

L

(D) (D) X

~

GO ON TO THE NEXT PAGE 02 13401 0/CAPE 2013

-410.

Which of the following is true if a., fi and y are roots of the cubic equation 3x3- 4x2 -7x- 10 = 0?

a+ fi + r

(B)

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

(C)

a + fi + r

=- '

a + f3 + r

4 = -'

=-,

3

afi + fir + ra = -

-3

-7

4

3

3 4 3

7 3

afi + fir + ra = af3 + f3r + ra

(D)

5 16 log2 16 log 2 30

1

4

2

25

(A)

I 36 -log-

(B)

log-

(C)

0

(D)

1

2

(A)

a x <-or x >a 3

(B)

-a x < - o r x >a 3

(C)

x > - a andx <-

a

3

7 =-

3

(D)

14.

a

x 3

The statement p v - p is a (A)

(B) (C) (D)

15.

logl5 - log6+- log-=

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

3

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) (B) (C)

12.

-7

(A)

(D)

11.

4

13.

converse tautology contradiction 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

25

25 4

GO ON TO THE NEXT PAGE 0213401 0/CAPE 2013

-516.

A

vector equation ts gtven as

s[ -~)+ tG) =[ -n.

20.

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

sec~=

The values of sand

tare, respectively

(A)

5 13

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

-2 -2 2 2

and -1 and 1 and 1 and -1

(B)

(C) (D)

17.

sin (30°- A) is equal to -1 cosA - -sm A

(B)

-1 cosA + -sm A

(D)

18.

2 sin (A) (B) (C) (D)

19.

J3 .

(A)

(C)

12 13 13 12 13 5 ,

2

2

21.

J3 .

2

J3

-

2

J3

-

2

x2 + y- 1Ox + 2y + 1 = 0.

2

The coordinates of the other end of the diameter are

1 . A cosA + -sm

2

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

1 sm . A cosA - -

2

e cos ~ is equivalent to sin (8 + ~) + sin (8- ~) sin(8+~)-sin(8-~) cos(8+~)+cos(8-~) cos(8+~)-cos(8-~)

The point (2, 3) is at one end of a diameter of the circle whose equation is

22.

(-12, -5) (-12, -1 ) (8, -5) (8, -1)

The value of sin[;+ (A) (B) (C) (D)

p)

is

- sinp - cosp sinp cosp

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

(C) (D)

x2 + y + 8x + y- 49 = 0 x 2 + y- 8x- 2y- 32 = 0 x 2 + y - 8x- y + 49 = 0 x 2 + y + 8x + 2y + 66 = 0

GO ON TO THE NEXT PAGE

0213401 0/CAPE 2013

-623.

What value of e, 0 :S e :S n, satisfies the equation 2 cos 2 e + 3 cos e - 2 = 0?

27.

(A) (B) (C)

7(

(A)

(B)

(c) (D)

Ifp = 2i+ j andq = /.. i+6j are perpendicular vectors, then the value of/.. is

6 7(

(D)

4

-3 • -1 0 2

7(

3

28.

7(

JC. The general solution for sin 29 = sm-ts 6

2 (A)

2nJC +. ff6 B= 5JC (2n+1)16

(B)

B=

'{

24.

With respect to an ongm 0, A has coordinates (3, -2). The position vector of3 OA is (A)

(3 , -6)

(B)

(9, - 2)

(C)

(-~J

(D)

25.

(B) (C)

(D) 26.

5JC nJC+ 12

B= {

(D)

nJC+ ff B= 6 5JC (n+1)

(_:)

sin lOA -2 cos 2A 2 cos SA sin A 2 sin 5AcosA

1 + cos 4A - sin 4A = (A) (B) (C)

(D)

1 +cos 4A 2cos2 A cos2A 2 cos 2 A sin 2 A

nJC +12 ff 5JC (2n7r) 12

(C)

The expression sin 6A + sin 4A may be written as (A)

{M+~12

{

6

29.

The cosine of the angle between the vectors -6 j and i + j is (A)

(B)

(C)

(D)

-1

J2 1

J2 -5

J2 6

J2


-- ---

r

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

-7Item 30 refers to the diagram below. y

33.

l=x

~(x 3 sin x) dx

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

34.

x2 x2 x2 x2

may be expressed as

(cos x + 3 sin x) (x cos x- 3 sin x) (3 cos x + sin x) (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 the value of 'p' should be

30.

In the diagram above showing NOT defined for (A) (B) (C) (D)

31.

lim x~3

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

y = x, y

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

is

0 X~ 0 x> 0 X< 0 X =

35.

2

(A)

-21 (3 -4x) 2

(B)

21 (3 -4x) 2

x- 3

(C)

Given that lim sin x = 1 , where x is measx-+O X • 3 . . Jim Sin X • ured In radians, then x---+0 ~ IS

27-8x (3-4xf

00

(D)

32.

-3 -1 4 12

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

X -9. --IS

0 6

= 3,

36.

If y

-27 -8x (3 -4x) 2

= -J2x + 1 then

2

d Y is dx 3

1 (A) (B)

(C) (D)

. 3 sm 2 sin3x 2x 2 3 3 2

(A)

(2x+

1)( -J2x+ 1) -1

(B)

(2x +

1)(-J2x + 1)

(C) (D)

(2x + 1)


- 837.

38.

If y =tan 6x then dy is dx 2 (A) 6 tan 6x (B) sec 2 6x (C) 6 sec 2 6x (D) sec 6x tan 6x

(A)

(B)

(B)

y = sin x + k y =cos X+ k

(C) (D)

y = - COS X + k y =-sin x + k

(C)

(D)

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

If

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

Given that

3x2 + x + c x3 + x + c 3x2 + c x3 + c

42.

3 4

9, 4

27 4

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

y

The path ofan object is given parametrically as x = sin t + 2, y = cos t + I . The slope of the tangent at t (A)

-I

(B) (C) (D)

0

is

4

1

(A) 40.

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

J: 3f(x)d;c

If dy =cos x then dx (A)

39.

41.

=-1t

4

4 (B)

2

is (C)

. -I

(D)

4

undefined


-943.

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

44.

Given dy = 2x, then possible sketches of dx the graph of y are

•

I.

y

II.

y

III.

y

IV.

y

lS

(A)

-k V =-

(B)

---

(C)

dV =-k.JV dt V =-kt

(D)

.Ji

dV dt

-k

v

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

-1

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

0

1

I and II only III and IV only I, III and IV only II, III and IV only


- 1045.

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 2

(A)

-Jr

(B)

-Jr

5 3

5

(C)

2n

(D)

47t

END OF TEST

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

0213401 0/CAPE 2013

CAPE Mathematics U1 P1 CAPE June 20142014 Pure Pure Mathematics U1 P1

CAPE pure math unit 1 2008-14

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