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1. (a) .(b)

(c) givenfunction f and fg ,findfunction g .

or givenfunction g and gf ,findfunction f .

givenfunction g and fg ,findfunction f .

or givenfunction f and gf ,findfunction g .

(d)

2. (a) 0

2=++ cbxax ,rootsofthequadraticequation =x ,

Hence,S.O.R. =a

b

S.O.P. =a

c

(b) 0NewNew2

=+ )..()..( ROPxROSx

(c) 2222 +=+ )(

(d)Factorisation, 02

=++ cbxax

Signfor For 1=a ,given qp >

b c

+ + ))(( qxpx ++

+ ))(( qxpx

+ ))(( qxpx +

))(( qxpx +

(e)(i)Two and roots means 042

> acb

(ii)Two and roots means 042

= acb

(iii)Two roots(specialcase) means 042

acb

(iv) roots means 042

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3.

(a) qpxay ++=2)(

(b)

(i) 02

>++= cbxaxy if 0>a ,therangeofx : x .

if 0

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Logarithm

(a) xNa =log x

aN= (interchange form)

(b) 01=alog , (c) 1=aalog

(d) If )__(log)__(log sideHandRightsideHandLeft aa = ,

Then )__()__( sideHandRightsideHandLeft = (Compare the values)

(e) If )__()__( sideHandRightsideHandLeft > ,

Then )__(log)__(log sideHandRightsideHandLeft aa >

6.

(a)Finding

),( 11 yxA

),(22 yxB

),(44 yxD

),( 33 yxC

Area= )()(14433221144332212

1xyyxxyxyyxyxyxyx ++++++

(b)Methodtofindtheequationofstraightline.

(i)Giventhe ofthestraightline,m and1 ),(11yxA

)(11xxmyy =

(ii)Given ),( 11 yxA and ),( 22 yxB 12

12

1

1

xx

yy

xx

yy

=

(iii)Given x =b and y =c 1=+cy

bx

(c)Theequationofstraightlinecanbewritteninthree (i) cmxy +=

(ii) 0=++ cbyax

(iii) 1=+c

y

b

x

(d)Iftwostraightlinesare ,then21

mm =

(e)Iftwostraightlinesare toeachother,then 121 =mm

Area=

1

1

4

4

3

3

2

2

1

1

2

1

y

x

y

x

y

x

y

x

y

x

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(f)Locusofpoint ),( yxP

Thegeneralformofanswerforlocusis

022

=++++ edycxbyax where =edcba ,,,, constant

(i)Distancefrompoint ),(11yxA isalwayskunits. kAP =

kyyxx =+ 212

1)()(

(ii)Equidistancefromtwofixedpoints ),(11yxA and ),(

22yxB BPAP =

2

2

2

2

2

1

2

1)()()()( yyxxyyxx +=+

(iii)Distancefromtwopoints ),(11yxA and ),(

22yxB alwaysintheratioof nm :

mBPnAPn

m

BP

AP==

2

2

2

2

2

1

2

1 )()()()( yyxxmyyxxn +=+ Squarebothsides,

])()[(])()[( 22

2

2

22

1

2

1

2yyxxmyyxxn +=+

7.

(a) , Cf

FNLm

m

)(

+=2

1

L -lowerboundaryofmedianclassN -totalfrequency, f

F -cumulativefrequencybeforemedianclassmf -frequencyofmedianclass

C-widthofmedianclass(b)Findthemodefroma

axisx -thelowerboundariesandupperboundariesofalltheclassesaxisy -thefrequencyofeachclass

5. (c)

axisx -upperboundariesofclassesincludingtheclassbeforethefirstclass.axisy -cumulativefrequenciesofclasses

(thecumulativefrequencyoftheclassbeforethefirstclassisZERO)

10.(d) Anewsetofdata hkuv =

Then,meanofv = k (meanofu ) h standarddeviationofv = k (standarddeviationofu )

varianceofv = 2

k (varianceofu )

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8.

(a)Length of chord AB =2

2

sinr , inunit(O)

(b)Area of triangle OAB sin2

2

1

r= , inunit(O

)

(c)Area of the segment ACB = )sin( 22

1j

9.

(a)Ifnaxy = ,then 1

=

naxn

dx

dy

(b)If

n

baxy )( += ,then abaxndx

dy n

+=

1

( )

(c)Forgraphofacurve,thegradientof tothecurveatthepoint ),( 11 yxA ,

1m =

dx

dy= )('

1xf

when1

xx = ,dx

dy=

1m

Thegradientofthenormaltocurveatpoint ),(11yxA ,

1

2

1

mm = because

121 =mm

(d)Maximumandminimumpoint

When 0=dx

dy,thevalueofx isthe x coordinatefor

- maximumpointif 02

2

dx

yd.

(e)Rateofchangedtdx

dxdy

dtdy =

Example,volumeofsphere,3

3

4rV = .then,

dt

dr

dr

dV

dt

dV=

(f)Smallchangesandapproximations

xdx

dyy

Where initialnew xxx = andthevalueofdx

dyiswhen initialxx =

yyy initialnew +=

r

B

A

O C

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10.

(a)AmbiguousCase

11.(a)FindingweighsIfacircleisgiven,theweightagesarethesimplestratiooftheangles.Example,

oooooo

1109060100360 =++= )(x

(b)Informationgiven(i)Thepriceincreasedby30%fromyear2003toyear2006means

Priceindex, 1301002003

2006==

P

PI

(ii)Thepricedecreasedby20%fromyear2003toyear2006means

Priceindex, 801002003

2006==

P

PI

(c)Changeofbasetime

Ifgiven 1201002003

2006

1==

P

PI and 90100

2003

2004

2==

P

PI

PriceIndexforyear2006basedonyear2004,

313310090

100

100

120100100

2004

2003

2003

2006

2004

2006 .====P

P

P

P

P

PI

'C

'' BCCCBC =

'BCBC= =BAC constant

B

DC

o

60

o

100

ox

Items Angle Weightage

A o100 10

B o60 6

C o90 9

D o110 11

B

C

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12.(a)Arithmetic Progression (A.P.).(i)MethodtoproveaseriesoftermsareArithmeticProgressionwhereexistsacommondifference,

11 + = nnnn TTTT example, 1223 TTTT =

(b)Geometry Progression (G.P.)(i)MethodtoproveaseriesoftermsareGeometryProgressionwhereexistsacommonratio,

1

1

+=

n

n

n

n

T

T

T

Texample,

1

2

2

3

T

T

T

T=

(c)A.P.andG.P.

(i) nnn TSS = 1 (ii)Thesumofthe 4th tothe 13th .

31313654 SSTTTT =++++ ...

13. Changethenon-linearequationtolinearform

cmXY += whereY axis newy

X axis newx m gradientofgraph

c Y intercept

14. (a)If )(xf

dx

dy= ,then

dxxfdxdx

dyy == )()(

(b) cna

baxdxbxa

nn

++

+=+

+

)()(

)(1

1

(c)

Ifgradientfunctionofacurve, )(xfdx

dy= ,

Thentheequationofthecurve, dxxfdxdx

dyy == )()(

(d)Additionalformulae

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(i) =b

a

a

bdxxfdxxf )()(

(ii) =+c

a

b

a

c

bdxxfdxxfdxxf )()()(

(iii) dxxfadxxfa )()( = example, dxxdxx 33 =

15.

(a)If~a parallelto

~b ,then

~~bka = wherekisaconstant.

(b)If BCkAB = ,then BA, andCarecollinear.

(c) OAOBAB =

(d)If nmBCAB :: = ,then BCn

mAB = .

If nmmACAB += :: ,then ACnm

mAB

+= .

(e)

=+=

y

xjyixr~~~

(f)If

=

1

1

y

xu~

and

=

2

2

y

xv~

,then

+

+=+

21

21

yy

xxvu~~

,

=

21

21

yy

xxvu~~

and

=

=

1

1

1

1

ky

kx

y

xkku

~

16.

(a)Quadrants

AS

CT

+= o1803

= o3604

= o1802 I II

III IV

III

III IV

B Cm

n

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(b)Graphsketchingoftrigonometricfunctions kos,sin and tan .

(c)Numberofsolutions17.

(a) Choosewitharrangementwhichmeans

arrangementdoesaffectthenumberofchoices

(b) Choosewithoutinvolvingarrangementwhichmeansarrangementdoesnotaffectthenumberofchoices

18.

(a)ConceptofComplement)')( AA P(1P =

where)(

)')'(

S

AA

n

n(P = and )()()' ASA nnn( =

(b)TreediagramTotalprobabilityofallthebranchesis119.

(a)Binomialdistribution(i)ConceptofComplement

)()()()()( 0P1P2P13P13P ==== v (v)OntheleftsideofpointO , 0

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21.

y notmorethanx xy

y notlessthanx xy

y atleastk timesofx kxy

y atmostktimesofx kxy

TheSumofx and y notlessthank kyx +

Minimumof y is k ky

Maximumofy is k ky

Valueof ymorethanx atleastk kxy

Ratioof y tox is kormore kx

y