cape mathematics 2012 paper 2.pdf

8/9/2019 cape mathematics 2012 paper 2.pdf

1/6

(a)

.

-2-

SECTIONA(Module

t)

AnswerBOTE

questioas

(i)

Given

lte

cuwey

:i

;,'

**

=

r", ,,.

*

^'

o

o-

a)

-/b)

find

di

d,

and

&

dt

LY-

'-(3*

""',

t

r

rJ

r

fr-*

,n

+

:,

,

o.L

dz,'

:

^:

^,

;

t,J

+

J.r-

n

Li

rr-T.oo

find

the

x-coordlnates

of

the

ponots

r,

*ni"l

g

:

;

:l( l::

r*

;

ar'l

+

i)

1,ol,r,

r

"'

,

[{-marks]

1i

' ,

I

(b)

dr

--d'

find

thex-coordinates

of

thepoints

at

which

{}=

drl_:0

[2

marks]

(ii)

Hence

,

determine

if

rhe

coordrnal3

iaefnd

in

(r)

b)

ana

c)

above

are

at

the

aximaminimaorpointsofinflection

of

y=Sn,

t7marksl

A

curve

is

&finedby

&e

parame&ic

equafioas

x

:

si#

ff,

y

:

f

_

Zt-

Find

(i)

the

gradienr

of

a

taogeat

to

fte

cur-e

at

the

point

with

paraureter

r

[6

marksl

(ti)

tfis

slrarirrnof

th

t?ilgntat

.Le

pointwherc

,:

+.

B

marks]

Ibtrl25

marks

o2234020tcAPE20t2 GO

GT

TO

TIIE

NEXT

PAGE


2/6

(a)

-3-

(i)

Express

f

-3{

in oartial

fractions.

(x-l)(t'+l)

(ir) Hence,

find

[ .

l=-3-r

*.

l

t-t+x-1

---

(r)

b)

/.$\

(iii)

Give,n

tbat

sinl

cos

B

-

cos

Z

iin

B

:

sin

(A

-

Blshow

that

cos3xsinx:sin3rcoer-sin2r'

:

If

/.:

J*rsin3rdx

and

c

J*:

lcof

r

sin

2rdr,

prc\.e

ftat

(m

+

3)

I.

:

mI

*r-cos'

r cos

3x.

j

i

"'"

Hence,

by

putting

fl=

l,Prove

that

alE

+Ji

"or.r

sin

3x *

:

II

fr,

Evaluate ["ti"

2xdx.

Jo

[7

marksl

[5

marksl

[2

marksl

F

marksl

[2

mrrtsl

p

-"rk;

Ibtrl25

mrrks

sin2r*

*

+.

(lY)

4- t{ 3\

L-(n-(

)( )

/

,

L-|,a,

*3)"

^

r

,

u'


3/6

-4-

SECTION

B

(Motute

2)

AnswerH}THqrestiom.

(a)

For

a

particular

G.P,

z,

:

486

d ,r,

:

llt

09g,

urte,re

u"

is

tbe

n&

tertrr.

(i)

Caledale

thc

first

kq

q

aud

&e

@q

#io,

r.

(ii)

Hence,

calculate

z

if,$:

177

146,.

{

i.

(i)

d

'L

u,

J-tL

i'v.r*

q

'.- .,.

'

;.i

1r;

n

J

Lr

/

L,i

t

*)

--

Gt

Al-Surm

g

+t1.rll

tL

f*rtrtz-\

l=t

.,1

-,t

Tt,

\lrtuq

1r'u\

-Ftt

X

,(vrr)

--

(*\

3.

fSmart*s4

[4

marksl

(b)

Thsfirstfourterms

efasequeneeme

l

xft)&1:

i

4

x

6.

Express,

in

terms

ofr,

the f

ierm,

a,

of the

*r"".

Prove,

by

mathematical

induction,

that

:

|

"

(r+

1)

(2n

+

71,\12

e

N.

(c)

(r)

(ii1

r

Zu,

'0lc

is

hv"e-

,lrrltr

r

r)

la

tcr

r)

?it

t'

ri'hur'

r7

i',ri

r)

(ir

rr)

{.:.(

*,-;.,r

=

1, ti(r,)

ii(

+:-i

l,"lr

:

$)

,6,_

'f'vIvtr)tOra+t

=

,y

it

Lrr r)

[rKr

t)

t

lL

"'

LEj

.I)

[-

rr-

ta

lc

r

r

)

rcL

'

l-

^

,

:

-

%,i

k

{ ')i-

;

'-'.

--i

i.

bL

[2

marksl

[7

marksl

Use

Maclaurin's

Thwrcm

to

findthe

firstthree

non-zero

terms

in

the power

series

expansion

of

cos 2r.

t5

markst

Henee,

or

otherwise,

obtain

the

first'trmo

no[-zero

terms

in

the power

series

expansionof

sin2r.

[2

marksl

'onr,

J

v

s

)

3/.5

)

.r

x

g

Ibtal25

merks

1it\osrh

*r

\a

t'-

7oo

tn

+ tJ

Ll'r

rr)

tY?*+r^G..ft* F

G++


4/6

(a)

(0

-5-

Express

[ I

a tflns

of factoriars.

LTJ

Hence,showthat

[ 'l:

f I

L

,J

_

V_,}

/'t

)B-'

(

' 3/*)'

[3

marksl

nl.

1jf[lfr'

)

t,

[l

mark

I

(ii)

n--B

Y:x.

t

(

^)t

a , ,.,8-, /-ctl'

a>L

(iii)

Findthecoefficientorrain

Lr-j]'

' ,

(^'l;

'r

9'4',:rJ .rrl

^a,

7_

.

)(

+

c3-,

+

cl,

wberec.:

[

;]

[8

marksl

(b)

lntf

(x):2f +

3*

-

qx

-

I

:0.

(i)

'Use

the int.emtediat

value

theorem

to

determine

whether

the

equation,f(x)

has

any toots

in

&e

interval

[0.2,

2].

[2

marksl

(ii)

{Jsingx,:0.6

as

afirstapproximationofaroot

T

of/(.r),

executejsB-iterations

of the

Newton-R4hson

method io

obtain

a second

approximatioff,

of

T.

,

[6

marksl

Ibtal25

marks

ri

tn

il{1'

u

\4.

'C\.b

ar

(

0fi1

16*jr

=

lb=

tbr

/J

tt

x'

(

r

t

r)'

g\,P

ld.-'\

[':]=

t

-.,r

q

.

(t,l)'(r{-r)

-

1

.r

l-

r'l(-

td(

1,.

ln-i

-o

s

J

1/

>(-

'rl

{os

7&

,, ,

.,ifr

'

'

t''.;ki

Irliil

L

GO

ON

TO

THE

NE}OPAGE


5/6

-6-

SECTION C

(Module

3)

I

Ansuer*o"**mr*

at_r, .t

,_,

,-,

,r..

I

/

t' , ,'

1:)

\ 1;

\ii

(a)

,

How

many

4digrt even

mrmbe*s

etn

be fmed fue.&p d*its

i,2,3,

+

6,7,8.

(i) ifeachdigitappearsatrmost'oce?

6'+0

+

+'57

[4

marksl

(iil

if

tbere

is

so

resci$ign

gF

eesdcr

sf,time+a

digt say appear.?

p

marko[

Z4ol

'

\\'lL

:

(b)

A

committee

of

gy5,is

to bc

frrmad

trom

among

six

Jamaicans,

two Tobagonians

and

three

Guyanese.

(i)

Fiad

the

pobability

fra$E-gmirre

consists

entirely of

Jamaicans.

(p.

,',

[3

marksl

'L1"

(ii) Find

the numbs of-rayr

in

yAich

the committee can

be forme4

givea

the

following

resfiirirm

fue

qe

as

,rwty:Tobaryrytqns

on

the committesgs there

.

are

&ryanese-

a\

/,

t


6/6

-7-

(a)

(,

DravrthepointsAandB

on anAgganddiagrsm,

whereA-+=andB=g.

,

FindALL

cornplax

aumbers,

z,

such

thzt

*

:

i-

(c)

Use

i

|.,'

I

r'

''':

r'

:

i

t,

*'l

i'r'

j

, .

t

'

*

,

'.

i

(i1)

Hence,

or

otherwise,

shcw

that

the

argument

"ti*Jf}

is

EXACTLY

*

.

[6

mark*]

[5

marksl

[3

marksl

.J

, t.

t

".

,i

I

"]

i

t

I"-l

4.

:

."-a

Hence,findAllcomprexrootsofther$:"o

r

s,

-_

-

br4

**(3+5i)z-(4-?i)=6- q$:4-''l' -.lo

t5marks|

de

Moivre's

theoremto

shot'ihat

cos

6

O: cos6

g-

15

cosa

Asir2

0

+

15 cos2

g

sina

6-

sin6 I

.

[6

marksl

Totel25

mrrks

J

J

'l

:

lt

1--

'')': -

(\-r''

r'-'

-

r

4-A"

[

'

.,."-':|

-,

-'-;--

Jt

I];

c

{l

;

:

t'i

t,

ENI} OFTSST

nnl

l

.-J

IFYOUFII{ISH

BETORE

TIME

IS

CALLEtr),

CTIECKYOT

RWORK

ON

TIIIS

TEST:

cape mathematics 2012 paper 2.pdf

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