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 +±+±±=± 2222222 ln22∫
[ ] Caxbbxbaxb
dx+±+=
±222
222ln1∫
( ) dxyyAydxA bax
b
ax ∫∫ 21; -==
( ) dyxxAxdyA bay
b
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;; 22212 baxb
axb
axxydyVdxyyVdxyV ppp =-==
( ) ∫∫∫ 2;; 22212 bayb
ayb
ayxydxVdyxxVdyxV ppp =-==
rdAArdAA ymxm == --
xA
rdA
AA
yA
rdA
AA
b
axmb
aym ∫∫ ; ==== --
dArIdArIb
ayb
ax ∫∫22 ; ==
∫∫ ;b
a
b
a ymxmrdVVrdVV == --
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 ∫∫22
21
21 r==
dyxIdxyIb
ayb
ax ∫∫44
21
21 rprp ==
∫∫ 2b
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 ÷÷øö
ççè
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 ÷ø
öçèæ+= ∫
dydydx1S
2dc ÷
÷ø
öççè
æ+= ∫
dududy
dudxS
222u1u ÷ø
öçèæ+÷
øö
çèæ= ∫
dxQeyeQPydxdy PdxPdx ∫ ∫∫∴ ==+
211 ≠ rrBeAey xrxr 2+=
21)( rrBxAeyrx =+=
ibarbxBbxAey ax ±=+= ]sincos[
dxd
dxdy
dd
dxyd q
q÷øö
çèæ=2
2
