2014

-1-N

S-K

ING

GE

OR

GE

Sec

tion

A [4

5 m

arks]

An

swer a

llqu

estions in

this sectio

n.

1.

Th

e fun

ction f is d

efined

by

�:���,�

�0

(a) S

tate the ran

ge o

f f.

[1

mark

s]

(b)

Fin

d �

�.

[2

mark

s]

(c) S

ketch

the g

raph

of f an

d �

�.

[3

mark

s]

2.

Fin

d th

e exp

ansio

n o

f 9����� as far as term

in �

�an

d state th

e range o

f valu

es of x

for

wh

ich th

e exp

ansio

n is v

alid. H

ence, o

btain

the v

alue o

f √9.05

correct to

fou

r decim

al

p

laces.

[6

mark

s]

3.

Fin

d th

e inv

erse of m

atrix A

by u

sing elem

entary

row

operatio

ns

A =

29

8

61

3

20

1

[4 m

arks]

Hen

ce, solv

e the sim

ultan

eou

s equ

ation

s

x +

2z =

1

3x

+ y

+6

z = 2

8x

+9

y +

2z =

-1

[4

mark

s]

4.

Fin

d th

e roo

ts, den

oted

by �

and �

, of th

e equ

ation

��� 2�

�4�0

in p

olar fo

rm.

Usin

g d

e Mo

ivre’s th

eorem

, sho

w th

at ��� �

�.

[1

0 m

arks]

5.

Th

e equatio

n o

f an ellip

se is 4������8�

�12

�0.

(a) O

btain

the stan

dard

form

of th

e equ

ation

of th

e ellipse.

[2

mark

s]

(b)

Fin

d th

e centre, fo

ci and

vertices o

f the ellip

se.

[4

mark

s]

(c) S

ketch

the ellip

se.

[2

mark

s]

6.

Th

e vecto

r v =

ai + b

j + c

k is p

erpen

dicu

lar to th

e vecto

rs i – 2

k an

d 2

i + j –

k.

(a) F

ind a an

d b

in term

s of c.

[4

mark

s]

(b)

Giv

en th

at |# | = √56

, and

that c is p

ositiv

e, find

c.

[3

mark

s]

Sec

tion

B [1

5 m

arks]

An

swer a

ny o

ne q

uestio

n in

this sectio

n.

7.

Giv

en th

at ,

)(

f2

3q

xp

xx

x+

++

=w

here

pan

d q

are con

stants, is d

ivisib

le by

).1

(−

x

Wh

en d

ivid

ed b

y),

2(

−x

the rem

aind

er is 17

. Fin

d th

e valu

es of %

and

&.

[4 m

arks]

(a)Sho

w th

at 0

)(

f=

xh

as on

ly o

ne real ro

ot. F

ind

the set o

f valu

es of �

such

that

0)

(f

>x

.

[6

mark

s]

(b)E

xp

ress )

(f

91

1

x

x+

in p

artial fraction

s.

[5

mark

s]

8.

Th

e straigh

t line y

= m

x –

2 in

tersects the cu

rve �

��4�

at two

differen

t poin

ts, P(�

�, , ��)

and Q

(��, , �

� �. Sh

ow

that

(a) m '

0 an

d m

>���

(b) �

�����

( )

*��

)�

(c) ������

()

If the p

oin

t O is th

e orig

in an

d p

oin

t T is a p

oin

t such

that O

PT

Q is a p

arallelogram

, pro

ve

that w

hen

m ch

anges, th

e equ

ation

of th

e locu

s of T

is ���4�

�4�

. [1

5 m

arks]