Mathcad - CAPE - 2004 - Math Unit 2 - Paper 01

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CAPE - Mathematics - Unit 2 - 2004Paper 01

Section A (Module 2.1)

1 In the diagram below (not drawn to scale) the line y = 3 cuts the curve y e2 xx

at the

point (a, b)

Calculate the values of a and b 4 marks

e2 x

3e2 x

x1

2ln 3.

1

2ln a

1

2ln 3.

1

2ln b = 3

2 Differentiate with respect to x

a( ) x ln 3 x2. x x 0 2 marks

b( ) y sin2

x cos. x.x x 3 marks

a( )1

3 x2

6 x( )2

x

1

3 x2

6 x( )x

b( ) 2 sin x. cos. x. cos x.( ). cos x. sin2 x. 2 sin x. cos2. x sin3 x2 sin x. cos. x. cos x.( ). cos x. sin2 x. x x x

1



3 a( ) Find the gradient at the point (1, 1) on the curve 2 xy y2

3 02 xy y2

3 5 marks

b( ) Solve for x the equation e2 x

3 ex

4 0e2 x

3 ex

4 3 marks

a( ) 2 xdy

dx. 2 y 2 y

dy

dx. 02 x

dy

dx. 2 y 2 y

dy

dx.

dy

dx

2 y

2 x 2 y

dy

dx

y

x y

dy

dx

y

x y

dy

dx

y

x y

dy

dx1 1,

1

2

dy

dx

b( ) ex

1 ex

4 0ex

1 ex

4 x 2 ln 2.2 ln

4 a( ) Express in partial fractionsx 1

x x 2( ).5 marks

b( ) Hence find xx 1

x x 2( ).d x > 0 3 marks

a( ) x 1 A x 2( ). B x( ).x 2( ) x( )x B x A1

2B

1

2

1

2 x

1

2 x 2( )

b( ) x1

2 xd x

1

2 x 2( ).d

1

2ln x x 2( ).( ). Cx x 2( ).( ) Cx x

1

2ln x.

1

2ln x 2( ). ln K.

5 Find xx2

ln x. d 5 marks

Total 30 marks

1

3x

3ln x. x

1

3x

3 1

xd

1

3x

3ln x.

1

9x

3C

1

9x

3Cx x x

1

9x

33 ln x. 1( ) C

2



Section B (Module 2.2)

6 Find the term independent of x in the expansion of 3 x1

2 x2

9

5 marks

9

r3 x( )

9 r 1

2 x2

r 9 r 2 r9 r r r = 39

33( )

6 1

2

315309

2

9

33( )

6 1

2

3

(7654.5)

7 The first four terms of an AP are 2, 5, (2x + y + 7) and (2x - 3y) respectively where

x and y are constants. Find the value of x and the value of y 8 marks

2x + y + 7 = 8 2x - 3y = 11

2 x 3 1 2 x( ) 11 02 x 3 1 2 x( ) 11 has solution(s) x7

4y

5

2

8 a( ) Find the sum to n terms of the geometric series

4 2 11

2... 5 marks

b( ) Deduce the sum to infinity of the series 2 marks

a( ) Sn

4 11

2

n

11

2

n

nS

n8 1

1

2nnn

b( ) S∞

∞n

8 11

2n

lim∞

yields S∞

8∞

3



9 If 1 ax( )n

1 6 x 16 x2

...... find the value of the constant a and the value of the

constant n 6 marks

1 n ax( ).n n 1( ).

2a

2x

2... 1 6 x 16 x

21 n ax( ).

n n 1( ).

2a

2x

2...

na = 6n n 1( ).

2a

216

n n 1( ).

2a

2 n n 1( )

2

6

n

2

16 0n n 1( )

2

6

n

2

16 n = 9 a2

3

10 A craftsman estimated the side of a square tile to be 16 cm but the actual measurement was16.14 cm. Calculate to two decimal places the percentage error in the actual measurementof the area of the tile

4 marks

Total 30 marks

15.5 side 16.5< areamin

15.52

minarea

min240.25

min

areamax

16.52

maxarea

max272.25

max

maximum error. 272.25 240.25maximum error.

oercentage error =272.25 240.25

16.142

100( ) 12.284=

4



Section C (Module 2.3)

11 A bag contains two red balls R1

and R2

one green ball G and two black balls B1

and B2

Randomly two balls are drawn together from the bag

a( ) Describe the sample space 2 marks

b( ) Determine the probability that

i( ) Both balls are the same colour 2 marks

ii( ) at least one ball is black 3 marks

a( ) S R1 R2 B1 B2,

R1 G,

R2 G,

B1 G,

B2 G,

R1 B1,

R1 R2 B1 B2,

R1 G,

R2 G,

B1 G,

B2 G,

R1 B1,

R2

B1

R1

B2

, R2

B2

,

b( ) i( ) P R1

R2

. or P. B1

B2

.1

5

1

5

1

5

1

5P R

1R

2. or P. B

1B

2.

2

25

ii( ) P(one black ball or two black balls)

B1 G or B2 G.

or B1 R1.

or B1 R2.

or B2 R1.

or B2 R2.

or B1 B2.

71

5

1

5

7

257

1

5

1

5

5



12 Only three horses A, B and C are in a race. The probability that A wins the race is twice theprobability that B wins. The probability that B wins is twice the probability that C wins.Find the probability of winning for each of the horses.

5 marks

let x be the random variable "horse wins"

x A B C

P X x( ). 4 k 2 k k k 1

7

4

7

2

7

1

7

13 Let A and B be the events such that P ( A U B ) =3

4P ( A' ) =

2

3P ( A ^ B ) =

1

4

Find

a( ) P ( A ) 1 mark

b( ) P ( B ) 3 marks

c( ) P ( A ^ B' ) 3 marks

a( ) P A( ).1

3P A( ).

b( ) P ( B ) =3

4

1

3

1

4

2

3

3

4

1

3

1

4

c( ) P ( A ) - P ( A ^ B ) =1

3

1

4

1

12

1

3

1

4

6



14 a( ) Which one of the following situations describes a mutually exclusive event?

i( ) Selecting either an even or a prime number from the set of realnumbers

ii( ) Selecting either a negative integer or a perfect square from the set of integers

iii( ) Selecting either a perfect square or an odd number from the set of numbers 1 to 100

1 mark

b( ) The probability that A hits a target is1

4and the probability that B hits the

same target is2

5The event that A hits the target is independent of the

event that B hits the target. What is the probability that both A and B hit the target

5 marks

a( ) ii( )

b( ) P ( A ^ B ) =1

4

2

5

1

10

1

4

2

5

15 A marksman shoots at a target and he either hits the target (H) or misses it (M). Theprobability of H is 0.4. He shoots at the target 3 times

Determine

a( ) the elements of the event A associated with the marksman hitting the targetexactly twice

2 marks

b( ) the probability that the marksman hits the target at least once 3 marks

Total 30 marks

a( ) HHM( ) HMH( ) MHH( )

b( ) 1 - P (MMM) = 1 0.6( )3

0.7841 0.6( )3

7

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Mathcad - CAPE - 2004 - Math Unit 2 - Paper 01