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

CAPE Unit 1 P1 2008 ROR

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-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

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-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

~

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-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 a 3

(B)

-a x < - o r x >a 3

(C)

x > - a andx 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

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-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

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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

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-- ---

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)

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- 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

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-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

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- 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