Mathcad - CAPE - 2005 - Math Unit 2 - Paper 02

7/29/2019 Mathcad - CAPE - 2005 - Math Unit 2 - Paper 02

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CAPE - 2005Mathematics Unit 2

Paper 02

Section A - Module 1

1 a( ) The diagram below, not drawn to scale, shows two points P (p, 0.368) and

R (3.5, r) on f (x) = e x for x R

y

f (x) = e x

R

P

x

O

i( ) Copy the diagram above and on the same axes sketch the graph ofg (x) = ln x

3 marks

ii( ) Describe clearly the relationship between f (x) = e x and g (x) = ln x

3 marks

iii( ) Using a calculator find the value of

a( ) r 1 mark

b( ) p 2 marks

b( ) Given than

loga

bc( ) xloga

bc( ) x logb

ca( ) ylogb

ca( ) y logc

ab( ) zlogc

ab( ) z and a b c

show that ax

by

cz

abc( )2

.. 3marks

c( ) Find the values of x R for which ex

3 ex

4ex

3 ex

8 marks

Total 20 marks

1


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a( ) i( )

et

ln t( )

t

t

5 0 5 10

5

ii( ) g (x) = f-1 (x) and f x( ). g1

x( )f x( ). x x > 0

iii( ) a( ) r = e 3.5 = 33.1 (3 s. f.) b( ) p = ln 0.368 = -1 (3 s. f.)

b( ) ax

bcax

bc by

caby

ca cz

abcz

ab ax

by

cz

bc( ) ca( ) ab( )ax

by

cz

bc ca ab

ax

by

cz

a2

b2

c2

ax

by

cz

a b c ax

by

cz

abc( )2

ax

by

cz

abc

c( ) ex 3

ex

4 0ex 3

ex

4 e2 x

4 ex

3 0e2 x

4 ex

3 ex

1 ex

3 0ex

1 ex

3

x = 0 x = ln 3

___________________________________________________________________________________________

2 a( ) A curve is given parametrically by x = (3 - 2t)2 y = t3 - 2t Find

i( )dy

dxin terms of t 4 marks

ii( ) the gradient of the normal to the curve at the point t = 2 2 marks

b( ) i( ) Express2 x 1

x2

x 1( )

in the formA

x

B

x2

C

x 1

where A, B and C are constants 7 marks

2


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iii( ) Hence evaluate

1

2

x2 x 1

x2

x 1( )

d 7 marks

a( ) i( )dy

dt3 t

22

dy

dtt

dx

dt4 3 2 t( )

dx

dtt

dy

dx

3 t2

2

4 3 2 t( )

dy

dx

t

t

ii( )dx

dt2

4 3 4( )

12 2

dx

dt2

grad2

5

b( ) i( )2 x 1

x2 x 1( )

expands in partial fractions to1

x2

1

x

1

x 1( )

ii( )

1

2

x2 x 1

x2

x 1( )

d yields1

22 ln 2( ). ln 3( )

1

2ln

4

3.

___________________________________________________________________________________________

Section B - Module 2

3 a( ) i( ) Use the fact that1

r

1

r 1

1

r r 1( )rto show that

Sn =

1

n

r

1

r r 1( )=

11

n 11

n

r

1

r r 1( )=

n5 marks

ii( ) Deduce that as n tends to Sn tends to 1 1 mark

b( ) The common ratio r of a geometric series is given by r5 x

4 x2

x

x

Find ALL the values of x for which the series converges 10 marks

3


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c( ) By substituting suitable values of x on both sides of the expansion of

( ) ==+

n

r

r

r

nn

xCx 01show that

=

=n

r

n

r

nC

0

2i( ) 2 marks

=

=n

r

r

r

nC

0

0)1(ii( ) 2 marks

Total 20 marks

a( ) i( )

1

r r 1( )

1

r

1

r 1

1

r r 1( ) r r

Sn

1

1

1

1 1

1

2

1

2 1...

1

n 1

1

n

1

n

1

n 1...

Sn

11

n 1nn

ii( )

n

11

n 1lim 1

n

11

n 1lim

b( )5 x

4 x2

1< has solution(s)

x 4

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Mathcad - CAPE - 2005 - Math Unit 2 - Paper 02