13

14

-1 S

MK

TIN

GG

I PO

RT

DIC

KS

ON

Sectio

n A

[45

mark

s]

An

swer a

ll qu

estion

s in th

is section

.

1.

Fu

nctio

ns f an

d g

, each w

ith d

om

ain R

, are defin

ed b

y f : x

→ x

+ 1

, and

g →

│x│

.

a) W

rite do

wn

the ex

pressio

n fo

r f -1(x), an

d state, g

ivin

g a reaso

n, w

heth

er g h

as an in

verse.

[ 2 m

arks ]

b)

Sk

etch th

e grap

h y

= g

of

[ 2 m

arks ]

c) F

ind

the so

lutio

n set o

f the eq

uatio

n g

of = fog

[ 2

ma

rks ]

2.

Th

e nth

term o

f an arith

metic p

rog

ression

is Tn .

a) S

ho

w th

at �� � �� ��2 �� �� �� is the n

th term

of a g

eom

etric pro

gressio

n.

[ 3

ma

rks]

b)

If Tn =

�� �17��14, evalu

ate ∑��

����.

[ 5

ma

rks ]

3.

a) Giv

en th

at zi =

5 +

i and

z2 =

– 2

+ 3

i

i) S

ho

w th

at │z

1 │2 =

2│z

2 │2

[ 1

ma

rk ]

ii) F

ind

arg (z

1 z2 ).

[ 3 m

arks ]

b)

Determ

ine th

e squ

are roo

ts of 1

6 –

30

i, in th

e form

of a

+ b

i.

[ 5

ma

rks ]

4.

a) Sh

ow

that th

e equ

ation

� 1 2 �3 2 6 �111 �2 7 �

� �� ! =

� "#$ !

has so

lutio

ns o

nly

if r + 2

q –

5p

= 0

. Describ

e the ty

pe o

f system

of eq

uatio

ns an

d so

lutio

ns.

[ 5 m

arks ]

b) H

ence, fin

d its so

lutio

ns if p

= r =

1 an

d q

= 2

.

[ 3

ma

rks ]

5.

a) Sk

etch th

e grap

h o

f the ellip

se with

equ

ation

of x

2 + 4

y2 =

1 [ 2

ma

rks ]

b) P

oin

t P lies o

n th

e ellipse an

d N

is the fo

ot o

f the p

erpen

dicu

lar from

P to

the lin

e x =

2.

Fin

d th

e equ

ation

of th

e locu

s of th

e mid

po

int o

f PN

wh

en P

mo

ves o

n th

e ellipse.

Describ

e the ty

pe o

f curv

e ob

tained

for th

e locu

s ob

tained

.

[8

ma

rks ]

6.

a) Fin

d th

e exp

ansio

n o

f ��%& ' (��)& *

in ascen

din

g p

ow

ers of x

un

til the term

s x3. [ 3

ma

rks ]

b)

If p=

– �+ , an

d q

lies in th

e interv

al [0, 9

], find

the larg

est po

ssible co

efficient o

f x3 .

[ 3

ma

rks ]

Sectio

n B

[15

mark

s]

An

swer an

y o

ne q

uestio

n in

this sectio

n.

7.

Giv

en th

at f(x) =

x3 +

px

2 + 7

x +

q, w

here p

, q are co

nstan

ts. Wh

en x

= –

1, f ’(x

) = 0

.

Wh

en f(x

) is div

ided

by (x

+ 1

), the rem

aind

er is –1

6 . F

ind

the v

alues o

f p an

d q

.

[ 4 m

arks ]

a) S

ho

w th

at f(x) =

0 h

as on

ly o

ne real ro

ot. F

ind

the set o

f x su

ch th

at f(x) >

0 . [ 7

ma

rks ]

b)

Ex

press &%,-�& in

partial fractio

ns.

[ 4

ma

rks ]

8.

Th

e po

sition

vecto

rs a, b

, and

c of th

ree po

ints A

, B an

d C

respectiv

ely are g

iven

by

a =

i + j +

k , b

= i +

2j +

3k

, c = i –

3j +

2k

.

a) Fin

d a u

nit v

ector p

arallel to a

+ b

+ c.

[3

mark

s]

b) C

alculate th

e acute an

gle b

etween

a an

d a

+ b

+ c. [3

mark

s]

c) Fin

d th

e vecto

r of th

e form

i + λ

j + µ

k p

erpen

dicu

lar to b

oth

a an

d b

. [2 m

arks]

d) D

etermin

e the p

ositio

n v

ector o

f the p

oin

t D w

hich

is such

that A

BC

D is a

parallelo

gram

hav

ing B

D as a d

iago

nal.

[3

mark

s]

e) Calcu

late the area o

f the p

arallelog

ram A

BC

D. [4

mark

s]