n6 mathematics april 2016 memorandum - future managers · sin5x(1sin5x)5cos5xdx 5 = 1ò4-2 duxdx ux...

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NATIONAL CERTIFICATE

APRIL EXAMINATION

MATHEMATICS N6

6 APRIL 2016

This marking guideline consists of 17 pages.

MARKING GUIDELINE

MARKING GUIDELINE -2- T980(E)(A6)T MATHEMATICS N6


TOTAL:

NOTE: Do NOT subtract marks for incorrect units or units omitted. QUESTION 1 1.1 z =

(4) 1.2

= 1,9 A

(8) [12]

1002200

=

yxyyx 2423 35 +--

)3410( 233 xyyxxy

zx

+--¶¶

=÷÷ø

öççè

æ¶¶

¶¶

xyx 630 2 +-=

RVI =

1-=VRI

RRIV

VII D

¶¶

+D¶¶

=D

RVRVR D-D= -- 21

RRVV

RD-D= 2

1

)8()10(30)5(

)10(1

2 ---=

ü

ü

ü ü

ü ü

ü ü

ü ü

ü

ü



QUESTION 2 2.1 =

ü =

ü =

ü ü ü

ü

ü ü

=

ü ü

=

OR

=

OR

=

ü =

ü =

ü ü ü ü

=

ü ü ü ü

=

(10)

y dxxxò 5cos5sin 34

xdxxx 5cos.5cos5sin 24ò

xdxxx 5cos)5sin1(5sin 24 -òxdxxx 5cos5)5sin1(5sin

51 24 -= ò xdxdu

xu5cos5

5sin==

duuu )1(51 24 -= ò

duuu )51 64 -= ò

cuu+úû

ùêë

é-755

1 75

cxx+úû

ùêë

é-

75sin

55sin

51 75

cxx +- 5sin3515sin

251 75

y dxxxò 5cos5sin 34

xdxxx 5cos.5cos5sin 24ò

xdxxx 5cos)5sin1(5sin 24 -ò

xdxxxdxx 5cos.5sin5cos.5sin 64 òò -xdxdu

xu5cos5

5sin==

cxx+-

75sin.

51

55sin.

51 75



2.2 =

(6)

2.3 =

= ü

= ü

=

= ü

ü ü ü ü ü

=

=

(8)

y dxxxò - 216

1

cx

dxx

xx

xxxx

+-

=

--=

--=

---=

--=

+-=

-

ò

8)8(sin

)8(641)8(64

]64)8[(

)16(16

1

2

2

2

2

2

y ò mxdx4sin

dxmx 22 )(sinòdxmx 2)2cos

21

21(ò -

dxmxmx )2cos21

21)(2cos

21

21( --ò

dxmxmx )2cos412cos

21

41( 2+-ò

cmmxx

mmxx +úû

ùêëé ++-

84sin

241

22sin.

21

41

cmmxx

mmxx +++-

324sin

842sin

41

ü

ü

ü

ü

ü

ü



2.4 =

ü ü

ü ü

ü ü

ü

ü

ü

ü ü

OR

=

ü ü

ü

ü ü

ü ü

y ò xdxex

3cos.2

ò ò-= dxxexeydxxx

)33sin(

21)

33sin( 22

ò-= xdxexexx

3sin.613sin

31 22

2

2

21)('

)(x

x

exf

exf

=

=

33cos)(

3sin)('xxg

xxg

-=

=

úû

ùêë

é----= ò dxxexexe

xxx

33cos.

21

33cos.

613sin

31 222

xdxexexexxx

3cos.3613cos.

1813sin

31 222 ò-+=

IxexeIxx

3613cos.

1813sin

31 22 -+=

\ xexeIxx

3cos.1813sin

31

3637 22 +=

cxexeIxx

++= )3cos.1813sin

31(

3736 22

cxexexx

++= )3cos.054,03sin333,0(973,0 22

cxexexx

++= 3cos.054,03sin324,0 22

y ò xdxex

3cos.2

ò ò --= dxexxeydxxx22 2)3sin3(3cos2

ò+= dxexxexx22 .3sin63cos2

úû

ùêë

é-+= ò dxexexxe

xxx222 2.3cos32.3sin63cos2

dxexexxexxx

ò-+= 222 .3cos36.3sin123cos2

2

2

21)('

)(x

x

exf

exf

=

=

33sin)(

3cos)('xxg

xxg

=

=

ü

xxfxxf3sin3)('

3cos)(-=

=2

2

2

2

21)(

)('

xx

x

eexg

exg

==

=

xxf

xxf3cos3)('

3sin)(==

2

2

2)(

)('x

x

exg

exg

=

=



ü ü

ü ü ü

(12) [36]

QUESTION 3

3.1

(12)

IxexeIxx

363sin.123cos2 22 -+=

\ xexeIxx

3sin.123cos2.37 22 +=

cxexeIxx

++= )3sin.123cos2(371 22

cxexexx

++= 3sin324,03cos054,0 22

ò -++-2

2

)21(43

xxxx dx

cxx

x

cxxx

dxx

dxx

dxx

Cx

CxCxBxAxAxAxx

BxLet

AxLetxCxBxxAxx

xC

xB

xA

xxxx

+---

+=

+----

-=

-+

-+=

=

-+++-=++-

=\=

=\=-++-=++-

-+

-+=

-++-

-

òòò

)21ln(4

17)21(4

21ln4

)21ln(4

171

)21(.421ln4

)21(2

17

)21(221

4

)5,8(2

17: of coeff Equate

24443

)5,10(221;

21

4;0)21()21(43

)21()21()21(43

1

2

2

222

22

22

2

cxx

x +---

+= )21ln(444,1)21(

25,5ln444,0

ü

ü

ü ü

ü

ü

ü

ü ü ü

ü

ü



3.2

(12) [24]

QUESTION 4 4.1

ü

ü

ü

ü

ü

(10)

ò -+++)14)(12(1710

2

2

xxxx dx

dxx

dxx

dxxx

dxx

dxxx

BxEquateAx

Cxlet

CCxBAxBxAxxxxCxBAxxx

xC

xBAx

xxxx

òòò

òò

-+

++

+=

-+

++

=

==

=\=

++--+=++

++-+=++

-+

++

=-+++

143

122

12

143

122

2: 1: of coeff Equate

341

2441710)12()14)((1710

1412)14)(12(1710

22

2

2

222

22

22

2

cxxx +-+÷øö

çèæ++= )14ln(

432arctan

212)12ln(

41 2

xxxy

dxdyx

secsin2)2(sinsin2 =-

xxxy

dxdy

sec1

sin2)2sin2(=-

xxxxy

dxdy

sec1

sin2)cossin2(=-

xxydxdy coscos. =-

xdxxee sin.cos --

=ò

2c )1.(

cos..

0sin0sin

sin

sinsin

=\+-=

+-=

=

--

-

-- ò

ceece

xdxeyex

xx

2. sinsin +-= -- xx eye

ü

ü

ü ü

ü

ü

ü

ü

ü ü ü

ü ü

ü

ü

ü

ü



4.2 + 6 =

(14) [24]

2

2

dxyd

dxdy7- y 32 +x

91

981

98

31

98

31

)889,0(98367

)333,0(31

326670

0

yfind To

1;60)1)(6(067 :

6

2

2

p

6

2

=+\++=

+++=

+=\

=\=+-

=

+=++-

=

=

+=\+=

===--=+-

BABA

xBeAey

xy

DDC

C

xDCxCdxyd

Cdxdy

DCxyBeAey

mmmmmmy

xx

p

xxc

c

889,0333,02,0311,098

31

51

4514

51

4592,0

4514311,0

3162

316

6

6

6

++-=

++-=\

÷øö

çèæ ---=

÷øö

çèæ=\

++=

++=

xeey

xeey

orBand

A

BA

BeAedxdy

xx

xx

xx

ü

ü ü

ü

ü

ü

ü

ü

ü

ü

ü

ü

ü

ü



QUESTION 5 5.1 5.1.1

(6) 5.1.2

(10)

43+-= x

x

( )( )

( ) (1;3) and 1;3 3;11;3

0130342

\====

=--=+-

yyxx

xxxx

33

23

23

3

1

23

3

1

2

3

1

3

1 12

12

38278,8667,2

)1(3)1(23)1()3(3)3(2

3)3(2

324

32

)34(2

)34(2

)(2

)(2

unitsororunits

xxx

dxxx

dxx

xx

dxyyxV

xyyxV

y

y

pp

p

p

p

p

p

p

=

þýü

îíì

úû

ùêë

é-+--ú

û

ùêë

é-+-=

úû

ùêë

é-+-=

-+-=

-+-=

-=

D´-´=D

ò

ò

ò

ü

ü

ü

ü

ü

ü

ü

ü

ü

Incorrect limits: max 7 marks

ü

ü

ü

ü ü

ü

ü

x



5.2 5.2.1

(4)

5.2.2

(6)

[ ]

33

30

30

2

30

2

2

152,2685,0

)00(tan)33

(tan

)(tan

)(tan

unitsorunits

xx

dxx

dxyV

xyV

x

x

p

ppp

p

p

p

p

p

p

p

=

úûù

êëé ---=

-=

=

=

D=D

ò

ò

ü ü

ü

ü

ü

ü


x xD

ü

ü

ü

ü



5.2.3

(12)

5.3 5.3.1

(6)

[ ]

[ ]

units

x

uorunits

xxxxx

dxxxxxx

dxxx

dxxyM

xxyM

y

y

836,0152,2798,1

798,1572,0

2

)3(

3secln)

33(tan3

2secln)(tan

)(tan)(tan

)(tan

33

2

3

0

230

30

30

3

0

2

3

0

2

2

=

=

=

ïþ

ïý

ü

ïî

ïí

ì

úúú

û

ù

êêê

ë

é

--úûù

êëé -=

ïþ

ïýü

ïî

ïíì

úû

ùêë

é---=

þýü

îíì

---=

=

=\

´D=D

ò

ò

ò

p

pppppp

p

p

p

p

p

pp

pp

p

p

xx 32 2 =

( )

)29;

23( and 0;0)(

214;0

23;0

032032 2

\==

==

=-=-

yy

xx

xxxx

ü

ü

ü ü

ü

ü

ü

ü

ü

ü


1)('

)(==

xfxxf

xxxgxxg-=

=tan)(tan)(' 2

ü

ü

ü

ü

ü



5.3.2

(6)

5.3.3

(8) 5.3.4

(2)

( )

( )

( )

2

3232

1

21

32

1

21

2

1

21 12

12

542,0

3

)21(2

2

)21(3

3)1(2

2)1(3

32

23

23

units

xx

dxxx

dxyyA

xyyA

=

úúúú

û

ù

êêêê

ë

é

--úû

ùêë

é-=

úû

ùêë

é-=

-=

-=

D-=D

ò

ò

( )

( )

( )

4

5454

1

21

54

1

21

43

1

21

22

212

316,0

5

)21(2

4

)21(3

5)1(2

4)1(3

52

43

23

23

units

xx

dxxx

dxxxx

xxyyI y

=

úúú

û

ù

êêê

ë

é

--úû

ùêë

é-=

úû

ùêë

é-=

-=

-=

´D-=D\

ò

ò

A

I

553,0542,0316,0

=

=

ü

ü

ü

ü

ü

ü

ü

ü

ü

ü

ü

ü

ü

ü

ü





5.4 5.4.1

ü

ü or

(6)

5.4.2

ü ü ü =

ü

= ü

=

= 12 units3 ü

(6)

12

12

1

1

xxyy

xxyy

--

=--

2502

20

--

=--xy

)2(32

-= xy

223

+=\ yx dyydA )223(2 +=\ dyydA )43( +=\

ò2

0rdA

ò +-2

0)43)(2( dyyy

ò --+=2

0

2 )4386( dyyyy2

0

232

24

338

26

úû

ùêë

é--+yyyy

úû

ùêë

é--+

2)2(4

3)2(3)2(8

2)2(6 232


ü

ü

ü ü



5.4.3 ü ü =

ü

= ü

= ü

= 14,667 units4 ü

ü

ü

(8)

Or alternative method

5.4 5.4.1

ü

ü or

(6)

ò2

0

2dAr

ò +-2

0

2 )43()2( dyyy

ò +-++-=2

0

232 )4161631212( dyyyyyy2

0

32432

34

21616

43

312

212

úû

ùêë

é+-++-yyyyyy

úû

ùêë

é+-++- 32

432 )2(

34)2(8)2(16

4)2(3)2(4)2(6

12667,14

=y

units222,1=

12

12

1

1

xxyy

xxyy

--

=--

5202

50

---

=--xy

)5(32

-= xy

523

+=\ yx dyydA )523(2 +=\ dyydA )103( +=\


);( yx

ü

ü

ü

ü



5.4.2 ü ü ü =

= ü

= ü

= ü

= - 12 units3 ü

(6) 5.4.3 ü ü

=

= ü

= ü

= ü

= 14,667 units4 ü

ü

ü

(8) [80]

ò- +0

2)103( dyyy

ò- +0

2

2 )103( dyyy0

2

23

210

33

-úû

ùêë

é+

yy

úû

ùêë

é -+

--ú

û

ùêë

é+

2)2(10

3)2(3

2)0(10

3)0(3 2323

ò- +0

2

2 )103( dyyy

ò- +0

2

23 )103( dyyy0

2

34

310

43

-úû

ùêë

é+

yy

úû

ùêë

é -+

--ú

û

ùêë

é+

3)2(10

4)2(3

3)0(10

4)0(3 3434

12667,14

-=y

units222,1-=





QUESTION 6 6.1 and

ü ü

Or OR

ü ü

ü

ü ü

S =

= ü

= ü

= ü

= ü

= 2,5 units or 24,674 units ü

(12) 6.2

ü

ü

ü

ü

ü

ü

ü

)sin(cos5 tttx += )cos(sin5 ttty -=

)sincossin(5 ttttdtdx

++-= )cossin(cos5 ttttdtdy

-+=

ttttdtdx sin5cos5sin5 ++-= tttt

dtdy cos5sin5cos5 -+=

tt cos5= tt sin5=

22

)cos5( ttdtdx

=÷øö

çèæ 2

2

)sin5( ttdtdy

=÷øö

çèæ

ttttdtdy

dtdx 2222

22

sin25cos25 +=÷øö

çèæ+÷

øö

çèæ

)sin(cos25 222 ttt +=225t=

dtdtdy

dtdx

ò ÷øö

çèæ+÷

øö

çèæp

0

22

dttòp

0

225

dttòp

0.5

p

0

2

25 ú

û

ùêë

é t

[ ]225 p

2p

xy 16=

21

4xy =

21

2-

= xdxdy

12

4 -=÷øö

çèæ xdxdy

xdxdy 411

2

+=÷øö

çèæ+

xx 4+

=

dxxxyAx42

4

1

+= òp




Copyright reserved

ü

ü

ü

ü

= 61,051 units2 or 191,798 units2 ü

(12) [24]

TOTAL: 100

dxxxx 442

4

121 +

= òp

dxx 484

1+= òp

dxxò +=4

121

)4(8p

4

1

23

23)4(8úúú

û

ù

êêê

ë

é+

=xp

4

1

23

)4(316

úû

ùêë

é+= xp

4

1

23

23

)5()8(316

úû

ùêë

é-=

p

p

n6 mathematics april 2016 memorandum - future managers · sin5x(1sin5x)5cos5xdx 5 = 1ò4-2 duxdx ux...

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