Copyright reserved Please turn over

T900(E)(N24)T

NOVEMBER EXAMINATION

NATIONAL CERTIFICATE

MATHEMATICS N6

(16030186)

24 November 2016 (X-Paper) 09:00–12:00

Calculators may be used.

This question paper consists of 5 pages and a formula sheet of 7 pages.

(16030186) -2- T900(E)(N24)T

Copyright reserved Please turn over

DEPARTMENT OF HIGHER EDUCATION AND TRAINING REPUBLIC OF SOUTH AFRICA

NATIONAL CERTIFICATE MATHEMATICS N6

TIME: 3 HOURS MARKS: 100

INSTRUCTIONS AND INFORMATION 1. 2. 3. 4. 5. 6. 7. 8.

Answer ALL the questions. Read ALL the questions carefully. Number the answers according to the numbering system used in this question paper. Questions may be answered in any order, but subsections of questions must be kept together. Show ALL the intermediate steps. ALL the formulae used must be written down. Use only BLUE or BLACK ink. Write neatly and legibly.

(16030186) -3- T900(E)(N24)T

Copyright reserved Please turn over

QUESTION 1 1.1 If z = calculate the following:

1.1.1 (2)

1.1.2 (1)

1.2 The parametric equations of a function are given as and .

Calculate the value of

(3) [6]

QUESTION 2 Determine if:

2.1 = (2)

2.2 = (4)

2.3 = (4)

2.4 = (4)

2.5 = (4) [18]

QUESTION 3 Use partial fractions to calculate the following integrals:

3.1

(6)

3.2

(6) [12]

)(cosec)tan( 223 xyyx +

xz¶¶

yz¶¶

2tx = 52ty =

2

2

dxyd

ò dxy

y x3arcsin

yx2sec

24

y24124

12 ++ xx

y xx 4cos.4cosec 35

y xx 2sin.3

ò -++-)1()12(133

2

2

xxxx dx

ò +-+xx

xx3

24 2 dx

(16030186) -4- T900(E)(N24)T

Copyright reserved Please turn over

QUESTION 4 4.1 Calculate the particular solution of

, if when .

(5)

4.2 Calculate the particular solution of

+ 2 = , if when and when .

(7) [12]

QUESTION 5 5.1 5.1.1 Make a neat sketch of the graph . Show the representative

strip/element that you will use to calculate the volume generated if the area bounded by the graph, the line , the -axis and the y-axis rotates about the x-axis.

(2)

5.1.2 Calculate the volume generated if the area described in

QUESTION 5.1.1 rotates about the -axis.

(5) 5.2 5.2.1 Calculate the points of intersection of the two curves = and

. Make a neat sketch of the two curves and show the area bounded by the curves in the second quadrant. Show the representative strip/element PARALLEL to the x-axis that you will use to calculate the area bounded by the curves.

(3)

5.2.2 Calculate the area bounded by the two curves in the second quadrant

described in QUESTION 5.2.1.

(3) 5.2.3 Calculate the second moment of area when the area bounded by the two

curves in the second quadrant described in QUESTION 5.2.1 is rotated about the x-axis.

(4)

5.2.4 Express the answer in QUESTION 5.2.3 in terms of the area. (1)

5.3

5.3.1 Make a neat sketch of the graph and show the representative strip/element PERPENDICULAR to the x-axis that you will use to calculate the volume of the solid generated when the area bounded by the curve for rotates about the x-axis.

(2) 5.3.2 Calculate the volume described in QUESTION 5.3.1. (3) 5.3.3 Calculate the moment of inertia of the solid obtained when the area

described in QUESTION 5.3.1 rotates about the x-axis.

(5)

12 2 +-=- ttydtdyt 1=t

21

-=y

2

2

dxyd

dxdy2+ y 26

x

e 1=y 0=x 1=dxdy 0=x

xy ln2=

2=y x

x

y 6+x8-=xy

3649 22 =+ yx

20 ££ x

(16030186) -5- T900(E)(N24)T

Copyright reserved

5.4 5.4.1 A water canal in the shape of a parabola is 5 m deep, 10 m wide at the top

and full of water. A vertical retaining wall is placed in the canal with its top 1 m below the water surface. Sketch the retaining wall and show the representative strip/element that you will use to calculate the area moment of the wall about the water level. Calculate the relation between the two variables and .

(4)

5.4.2 Calculate, by using integration, the area moment of the retaining wall

about the water level in QUESTION 5.4.1.

(3) 5.4.3 Calculate, by using integration, the second moment of area of the

retaining wall described in QUESTION 5.4.1, about the water level as well as the depth of the centre of pressure on the retaining wall.

(5) [40]

QUESTION 6 6.1 Calculate the length of the curve between and . (5) 6.2 Calculate the surface area of revolution generated by rotating the two curves,

and about the x-axis, between and .

(7) [12]

TOTAL: 100

x y

42 2 -= xy 0=x 2=x

q3cosax = q3sinay =2pq =

2pq -=

(16030186) -1- T900(E)(N24)T

Copyright reserved Please turn over

MATHEMATICS N6 FORMULA SHEET Any applicable formula may also be used. Trigonometry sin2 x + cos2 x = 1 1 + tan2 x = sec2 x 1 + cot2 x = cosec2 x sin 2A = 2 sin A cos A cos 2A = cos2A - sin2A

tan 2A =

sin2 A = ½ - ½ cos 2A cos2 A = ½ + ½ cos 2A sin (A ± B) = sin A cos B ± sin B cos A cos (A ± B) = cos A cos B sin A sin B

tan (A ± B) =

sin A cos B = ½ [sin (A + B) + sin (A - B)] cos A sin B = ½ [sin (A + B) - sin (A - B)] cos A cos B = ½ [cos (A + B) + cos (A - B)] sin A sin B = ½ [cos (A - B) - cos (A + B)]

Atan1Atan22-

!

BABA

tantan1tantan

!

±

xx

xx

xxx

sec1cos;

cosec1sin;

cossintan ===

(16030186) -2- T900(E)(N24)T

Copyright reserved Please turn over

_________________________________________________________________

f(x) f(x)dx

_________________________________________________________________

xn nxn - 1

axn a a xn dx

(ax + b)

(dx + e)

ln(ax) xln ax - x + C

-

-

ln f(x) -

sin ax a cos ax -

cos ax -a sin ax

tan ax a sec2 ax

cot ax -a cosec2 ax

sec ax a sec ax tan ax

cosec ax -a cosec ax cot ax

)(xfdxd

ò

Cnxn

++

+

1

11)-( =/n

nxdxd

ò

baxe +

dxde bax .+

( )C

baxdxde bax

++

+

edxa +

dxdaa edx .ln.+

( )C

edxdxda

a edx+

+

+

.ln

axdxd

ax.1

)(xfe )()( xfdxde xf

)(xfa )(.ln.)( xfdxdaa xf

)(.)(

1 xfdxd

xf

Caax

+cos

Caax

+sin

Caxa

+)]([secln1

Caxa

+)]([sinln1

Caxaxa

++ ]tan[secln1

Caxa

+úû

ùêë

é÷øö

çèæ2

tanln1

(16030186) -3- T900(E)(N24)T

Copyright reserved Please turn over

_________________________________________________________________

f (x) f (x) dx

_________________________________________________________________ sin f (x) cos f (x) . f '(x) -

cos f (x) -sin f (x) . f '(x) - tan f (x) sec2 f (x) . f '(x) - cot f (x) -cosec2f (x) . f '(x) - sec f (x) sec f (x) tan f (x) . f '(x) - cosec f (x) -cosec f (x) cot f (x) . f '(x) -

sin-1 f (x) -

cos-1 f (x) -

tan-1 f (x) -

cot-1 f (x) -

sec-1 f (x) -

cosec-1 f (x) -

sin2(ax) -

cos2(ax) -

tan2(ax) -

)(xfdxd

ò

2)]x(f[1

)x('f

-

2)]x(f[1

)x('f

-

-

1)]([)('2 +xfxf

1)]x(f[)x('f

2 +

-

[ ] 12)()(

)('

-xfxf

xf

1)]x(f[)x(f

)x('f2 -

-

Ca4)ax2sin(

2x

+-

Caaxx

++4

)2sin(2

Cx)ax(tana1

+-

(16030186) -4- T900(E)(N24)T

Copyright reserved Please turn over

_________________________________________________________________

f (x) f (x) dx

_________________________________________________________________

cot2 (ax) - -

f(x) g' (x) dx = f(x) g(x) - f ' (x) g(x) dx

Applications of integration AREAS

)(xfdxd

ò

Cx)ax(cota1

+-

ò ò

ò [ ] [ ] Cnxfdxxfxfn

n ++

=+

1)()(')(

11)-( =/n

Cxfdxxfxf

+= )(ln)()('∫

Cabxsin

b1

xba

dx 1222

+=-

-∫

Cabx

abxbadx 1- +=+

tan1∫ 222

Cxba2x

abxsin

b2adxxba 22212

222 +-+=- -∫

Cbxabxa

abxbadx

+÷÷ø

öççè

æ-+

=-

ln21∫ 222

[ ] Cbxxbbxxdxbx +±+±±=± 222

2222 ln22∫

[ ] Caxbbxbaxb

dx+±+=

±222

222ln1∫

( ) dxyyAydxAb

axb

ax ∫∫ 21; -==

( ) dyxxAxdyAb

ayb

ay ∫∫ 21; -==

(16030186) -5- T900(E)(N24)T

Copyright reserved Please turn over

VOLUMES

AREA MOMENTS

CENTROID

SECOND MOMENT OF AREA

VOLUME MOMENTS

CENTRE OF GRAVITY

MOMENTS OF INERTIA Mass = Density × volume M = rV DEFINITION: I = m r2

( ) ∫∫∫ 2;; 22

21

2 b

axb

axb

ax xydyVdxyyVdxyV ppp =-==

( ) ∫∫∫ 2;; 22

21

2 b

ayb

ayb

ay xydxVdyxxVdyxV ppp =-==

rdAArdAA ymxm == --

xA

rdA

AA

yA

rdA

AA

b

axmb

aym ∫∫ ; ==== --

dArIdArIb

ayb

ax ∫∫ 22 ; ==

∫∫ ;b

a

b

a ymxm rdVVrdVV == --

V

rdV

Vv

yV

rdV

Vv

x

b

axmb

aym ∫∫ ; ==== --

(16030186) -6- T900(E)(N24)T

Copyright reserved Please turn over

GENERAL:

CIRCULAR LAMINA

CENTRE OF FLUID PRESSURE

dVrdmrIb

a

b

a ∫∫ 22 r==

2

21mrIz =

dVrdmrIb

a

b

a ∫∫ 2221

21 r==

dyxIdxyIb

ayb

ax ∫∫ 4421

21 rprp ==

∫∫ 2

b

a

b

a

rdA

dAry =

nn baxZ

baxC

baxB

baxA

baxxf

)(...

)()()()(

32 ++

++

++

+=

+

=)+()+(

)(33 dcxbax

xf3232 )()()()()( dcx

Fdcx

Edcx

Dbax

Cbax

Bbax

A+

++

++

++

++

++

=+++ nedxcbxax

xf))((

)(2 nedx

Zedx

Cedx

Bcbxax

FAx)(

...)( 22 +

+++

++

+++

+

dxdxdyyA

b

ax ∫2

12 ÷øö

çèæ+= p

dydydxyA

c

dx ∫2

12 ÷÷ø

öççè

æ+= p

dxdxdyxA

b

ay ò ÷øö

çèæ+=

212p

dydydxxA

c

dy ò ÷÷ø

öççè

æ+=

2

12p

(16030186) -7- T900(E)(N24)T

Copyright reserved

dududy

dudxyA

u

ux

222

12 ÷

øö

çèæ+÷

øö

çèæ= ò p

dududy

dudxxA

u

uy

222

12 ÷

øö

çèæ+÷

øö

çèæ= ò p

dxdxdy1S

2ba ÷

øö

çèæ+= ∫

dydydx1S

2dc ÷÷

ø

öççè

æ+= ∫

dududy

dudxS

222u1u ÷

øö

çèæ+÷

øö

çèæ= ∫

dxQeyeQPydxdy PdxPdx ∫ ∫∫∴ ==+

211 ≠ rrBeAey xrxr 2+=

21)( rrBxAey rx =+=

ibarbxBbxAey ax ±=+= ]sincos[

dxd

dxdy

dd

dxyd q

q÷øö

çèæ=2

2