Qe py higher education
Le & training
| 3 Department:
\ 1) e) ’ — Higher Education and Training
SS REPUBLIC OF SOUTH AFRICA
SSZLELLLO?
(i
T1380(E)(N17)T
NOVEMBER 2011
NATIONAL CERTIFICATE
O MATHEMATICS N4
(16030164)
17 November (X-Paper)
09:00 — 12:00
Calculators may be used.
This question paper consists of 5 pages and a 1 page-formula sheet. t
0
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N4 MATHS SPOT ON R80 .00
N4 MATHS SPOT ON R80 .00, 90 pages. Read online or get the free PDF download.
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2011Study guide, 90 pages. Read online or download the PDF.
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ve
-2- T1380(E)(NI7)T
DEPARTMENT OF HIGHER EDUCATION AND TRAINING
REPUBLIC OF SOUTH AFRICA
NATIONAL CERTIFICATE
MATHEMATICS N4
TIME: 3 HOURS
MARKS: 100
INSTRUCTIONS AND INFORMATION
1,
2.
Answer ALL the questions.
Answer ALL FIVE questions in full. Show ALL the calculations and intermediary
steps. Simplify where possible.
ALL the graph work must be done in the ANSWER BOOK. Graph paper is NOT
supplied. The values of intercepts with the system of axes and the turning point(s)
must be shown on the graph.
ALL final answers must be accurately approximated to THREE decimal places.
Questions may be answered in any order, but subsections of questions must NOT be
separated.
A formula sheet is attached to this question paper. You are NOT compelled to use the
formulae and the list is NOT necessarily complete.
PTO
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-3- T1380(E\(NI7)T
QUESTION 1
1
Make v the subject of the formula if
ry
Iai -)
(4)
1.2 Given:
2x-y+3z=1
3x+y-z-2=0
x+2y+3z+6=0
Solve for y by using Cramer's rule. (10)
1.3 The sum of two integers is 5. The difference between their squares is 45. Determine
the numbers. @)
1.4 Given:
4 a 5x2
Solve forx. (3)
[20]
QUESTION 2
2.1 Sketch the graph of 4y? =36 - 4x”. (3)
2.2 2.2.1 Sketch the graph of y=2**. 3)
2.2.2 Is the graph of y = 2** continuous or discontinuous? Q)
2.3 Sketch the graph of y=2sinx; 0° <x<360°. re))
24 Solve for x if: P
x? -8x+41=0 \ (5)
i
2.5 Simplify: ' fi
30° 40° aan
6 | 20°
Leave the answer in a +jb form. (5)
[20]
PTO
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ood
4. T1380(E)(N17)T
QUESTION 3
3.1 Simplify:
a re
1+ tan? B cot? B+1 (3)
3.2 Prove that:
tan A cosec”A.cos” A = cot A (3)
3.3 Solve for A if:
6cos? A+7sin A~8=0 ; 0° < 4<180° (5)
3.4 If cos A= = andsin B= = ; and dA and B are acute angles, calculate the value of
cos(A — B) without using a calculator. (5)
3.5 Solve for A if
cos(4 +20°)=sin(4 - 20°); 0° <4<90° (4)
[20]
QUESTION 4
4.1 Determine the following limit:
tim ( 2x3 - 4)
x—>3\ x-3 (4)
42 Differentiate by the use of product rule if:
y = enx.cot x (4)
43 Differentiate the following with respect to x:
yas ao +2 —cos3x—32 (5)
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PTO
5 T1380(E)(N17)T
44 Given:
y=x?—x? -6x
Calculate, using differentiation, the co-ordinates of the turning points and the point of
inflection.
QUESTION 5
5.1 5.1.1 Sketch the graph of y=2x? —6x—8. Clearly indicate the area enclosed
between the graph and the x-axis; x=0 andx=4. Show also the
representative strip to be used to calculate the area indicated.
5.1.2 Calculate, using integration, the area enclosed.
5.2 Integrate:
2" ~ 0,67 + a+ 2 4+ sec x tan x + 2p Jax
e x
5.3 Simplify:
fv 1-cos? x dx
5.4 Determine:
87
1/5
TOTAL:
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(7)
[20}
(3)
(4)
”
@G)
(3)
[20]
100
FORMULA SHEET
a* =b & loga’ = log
énx = log.x
(lay =r" |n@ at+bj=ct+dji a=candb=d
sin(a + b) = sina cosb + sinb cosa
cos(a + b) = cosa cosb F sina sinb
tana + tanb
tan (a + b) = —
1¥ tana tanb
y ad
ax
n n-T
ax nax
ka* ka* tna
kenx k
x
sinx cos x
cos x -sin x
tanx sec*x
cot x -cosec*x
sec x sec x tan x
cosec x ~cosec x cotx
ntl
a
Jax" dx = +C
n+l]
a
Jede=atnxtc
x
Ska* dx = kat te
fna
Agy =f? vax
sin’x + cos’x = 1
1 +cot?x = cosec?x
1 + tan’x = sec*x
Y= ux) - v(x)
2 F< wave +0") v6)
= 4)
a v(x)
> wy - v(x)u'(x) — u(x)v'(x)
de [xP
D_Y du
dk du de
Jsinx de =-cosxte
Joosxde=sinx+c
Jtanx de =n secxte
Jsecx de = én (sec x + tan x) +¢
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< ‘suucation... .
wee * Department: , 3
a : Education
LS REPUBLIC OF SOUTH AFRICA
NATURAL SCIENCE _
MARKING GUIDELINES
November 2011
MATHEMATICS N4
16030164
November 2011
WISKUNDE N4
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iY Jen PO. aa) Toten WHAIOS 5/2.
i‘ I Cp ier 3. VF; + ‘a gp
QUESTION 1/ VRAAG 1
1.1
= jl
inet ~o{Z1) V
i
2. of 2 + 7 Vv
ct Tt
rated y
g&
1.2
|D|=
tl -] 3
2, | O-F
= 2 (5) (-)-1 (10) +3 (5)
|Dj=35 Vv
2 1 3
|Dy|=8 2 -4 ¥
1 -6 3
[Dy|= ap 5 | “i sf +f - vy
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Please turn over
(4)
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-3-
= 2(0)+1 (10)'#3 (20)
|Dy|=-70 V
v
Pal -70_
!
=—=-2 V
|p| 35
1.3
XH VS asec (1)
my? B45. (2)
fal) x=5-y
LA 2) (S- yy? =45
25-10y + y? =45
ye-2 V
Br) x-2=5
x=7 Vv
1.4
(2x- 1)én4 = (x+ 2)éns
2xén4 — fn4=xln5+20n5
2xln4—xén5=2n5+ ina ¥
1163x = 4,605
x=3,959 V
QUESTION 2/VRAAG 2
2.1
T777(E\(J12)T
(10)
(3)
[20]
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he an T777(E\(u12)T
4y? = 36 — 4x?
x-intercept ¥
y—intercept 1
@ x Shapes ¥
(3)
2.2 2.2.1
= 93x
Sys 2 Shapes V¥
x-intercept V
Label ¥
<_
(3)
2.2.2 Continuous
2.3
\ y= 2sinx
hlayiso.com
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j F Please turn over
i t x
gc t80° arc’ 36 ee
ij
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-5- . T777(E\J12)T
VShape
vLabel
(5)
2.4 _ 8+ ¥(-8) - 40X41)
x= a
8--100 8+ -100
1 and/or x, =——_——_
x =A and /or x, = Bt /N100 100 vy
2 2 (5)
x, =4-j5 and/or x, =44 j5 VV
2.5 z=(,667| sof
Z =134,856250°
Z =134,856(cos 250° + jsin 250°)
Z =-46,123 — j126,723 v
(5)
(20]
QUESTION 3/VRAAG 3
3.1 cos’ B+sin?B VV
(3)
11
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3.2
3.3
3.4
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sind |
‘>z7:C0s* A=cotd
cos 4 sin’ 4.
cos A
sin A
=cotd y
LHS=RHS ¥
i
Vv
6{l-sin? 4)+7sin4—8=0
6sin? 4-7sind4+2=0
7tvi
12
sin A=
sin A = 0,667 and/or sin A=0,5
A, = 41,836° ¥ A, =30° ¥
A, =138,164° ¥ A, =150 V
cos(A -B)=cos Acos B +sin Asin B
Fat VV
= 40. , 180
= Vv
221 221
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T777(E)(J12)T
(3)
(5)
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cos(4 - B) = = v
3.5 sin|90° -(4+20°)|=sin(4—20°)
sin(70° - 4)=sin(4-20)
70° -A=A-20° V
24 =90°
A=45° V
QUESTION 4 / VRAAG 4
41 tim (225%) _ fim 2x? -27) J
x—>3 x-3 x33 (x-3)
x73 (x-3)
ti
= 3 x? +3x+9)
= 237 +3(3)+9)
= 54 Vv
4.2 u=lnx v=cotx
ai Vv ® «_ cosectx V
dx x ax
x = (tnx\{- cos ec? x)+ (cot A) ay
x
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fim — 2{x-3)h? +3249)
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T777(E\(J12)T
[20]
(4)
(4)
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-8-. _—- T777(E\(J12)T
43 | et
2 -(E)be")+ (2* ¢n2)3)+ 2énx +3sin3x—0
V V V Vv Vv
4.4 = =3x? -2x-6
dx
3x? -2x-6=0
x, =1,786 V x,=-L119 V
y; = (1,786)’ — (1,786)? — 6(1,786)
= -8,208 V
y(Lusy -C1119) - 6(-1,119)
= 4,061 ¥
2
pot 22 6x-2
dx
6x-2=0
O89
QUESTION 5/ VRAAG 5
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(5)
[20]
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-tr a’ a ts T777(E\(J12)T
v Shown area (3)
V Strip
V Shape
5.1.2
4
Aox = f (2x? -6x~8)dx v
0
2x? 6x? 4
- = _ ox | 4
3 2 j
= [42,667 — 4g - 32]- fo]
=~37,333u2 V (4)
5.2 2x* 0,6" — @?
a ain06 1 gt lar tsecx+ 2px te (7)
V V Vovovoqviyv
5.3 fin xdx VV
—cosxte V ©)
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er rr T777(E\(J12)T
air. La a ae i 3”
=[7en8]-[7en1] V oo a,
= 14, 556-0
= 14,556 Vv (3)
[20]
[100]
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= uy higher education
ESS
— & training
3S f i. al 5
—_——= éfam\¢ Department:
— th: Higher Education and Training
=== Sd REPUBLIC OF SOUTH AFRICA
T1080(E)(M28)T
APRIL 2012
@ NATIONAL CERTIFICATE
MATHEMATICS N4
(16030164)
28 March (X-Paper)
09:00 — 12:00
Calculators may be used.
This question paper consists of 6 Pages and a 1-page formula sheet.
MASTER ,
COPY
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-2- T1080(E)(M28)T
DEPARTMENT OF HIGHER EDUCATION AND TRAINING
REPUBLIC OF SOUTH AFRICA
NATIONAL CERTIFICATE
MATHEMATICS N4
TIME: 3 HOURS
MARKS: 100
INSTRUCTIONS AND INFORMATION
1, Answer ALL FIVE the questions in full.
©. Show ALL the calculations and intermediary steps. Simplify where possible.
3. Read ALL the questions carefully.
4. All the graph work must be done in the ANSWER BOOK. Graph paper is NOT
supplied. Values of intercepts with the system of axes and the turning point(s) MUST
be shown on the graph.
5. ALL final answers must be accurately approximated to THREE decimal places.
6. Questions may be answered in any order but subsections of questions must NOT be
separated,
7. A formula sheet is attached to this question paper. You are NOT compelled to use the
formulae and the list is NOT necessarily complete.
@ Number the answers correctly according to the numbering system used in this
question paper.
9. Write neatly and legibly.
-
~~ "1 copy
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cans. [ MASTER
PTO
3. T1080(E)(M28)T
QUESTION 1
11 Solve for x and y if:
xe y= Ot P=)
(- 73) (5)
1.2 Given:
Z=-3+4 j2
1.2.1 Find Z (1)
1.2.2 Change Z into polar form. @ must be positive. Show ALL steps. @)
1.23 Represent all the calculated values in QUESTION 1.2.2, on the Argand O
diagram. (3)
13 Sketch the graph of y=secx ; 0°<x<360° @G)
14 1.4.1 Sketch the graph of y= 6 + 4x — 2x? ()
14.2 Is the graph of y=6+4x—2x? continuous or discontinuous? Q)
1.5 Sketch the graph of y = lnx (2)
[20]
QUESTION 2
2.1 Make P, the subject of the formula if:
H =100£og A (3)
P,
2.2 The hypotenuse of a right-angled triangle is 10 mm longer than the longest of the two
other sides. Calculate the lengths of the sides if the shortest side is 50 mm long. (5)
2.3 Solve for x if:
ye | copy
“
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-4- T1080(E)(M28)T
2.4 Given:
1 2 3
4 -1 -2
-3 -4 -5
2.4.1 Calculate the value of the determinant. (3)
2.4.2 What is the minor of -5? (1)
2.4.3 What is the cofactor of —-2? (1)
2.5 Solve for x by using determinants if:
x+y=10
@ ro yn? (20)
QUESTION 3
3.1 3.1.1 Derive a formula for cos if cos2.4 = 2cos? A-1. (4)
3.1.2 Use the formula derived in QUESTION 3.1.1 to calculate the value of
cos 15° WITHOUT using a calculator. (4)
3.2 Solve for @ if:
sec? 6+tan@-3=0 ;0°<'6 < 360° 4)
@,, Calculate the value of sin 120° WITHOUT the use of a calculator. (4)
3.4 Prove that:
tan (90° - B) = cot B (4)
. [20]
MASTER
COPY
+
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5. T1080(E\(M28)T
QUESTION 4
41 Expand (x + 6): to four terms only. “
4.2 Given:
_ sin 3x
4x3
Differentiate by using the quotient rule. (4)
43 Given:
y=x? —6x? +11x-6
Calculate with the aid of differentiation, the co-ordinates of the turning points and the
point of inflection. (7
4.4 Differentiate the following with respect to x:
2x 1 3
y=—3e™ — fog, — —cosec 3x + ap+7x (5)
* [20]
QUESTION 5
5.1 Integrate:
-4;3 -9° 4 14sin 7x —2+ sec? 4x-4 7
x (7)
5.2 5.2.1 Sketch and indicate the area enclosed by the graph y=3sinx, the
x-axis; x=0 and x= x Also, show the representative strip to be used
to calculate the area. 6. 23 G3)
5.2.2 Calculate, using. i
QUESTION 5.241*
COPY _
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tegration, the nhagnitude of the area indicated in
.* ' (4)
G3)
PTO
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3.4
6 71080(E)(M28)T
Determine:
2
o JG - 3 + 3 dee GB)
[20]
TOTAL: 100
| MASTER. |
CaPY |
ee
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a* =b & loga’® = logd
(|@)"=r" |n@
sin(a + b) = sina cosb + sinb cosa
cos(a + b) = cosa cosb ¥ sina sinb
T1080(E)(M28)T
MATHEMATICS N4
FORMULA SHEET
NEW SYLLABUS
én x = log.x
at+bj=ct+dj @ a=candb=d
sin’x + cos’x = 1
1+ cot’x = cosec’x
1 + tan?x = sec”x
tan (a+ b) = tana + tanb
1¥ tana tanb
y g®
a&
ax” nax™! Y= u(x) - v(x)
ka* ka* t ,
"a = ® cua) +0) (2)
kinx & a&
x
sin x cos x _ ux)
cos x -sin x w(x)
2,
‘anx “ex , BY _ v(eu'(x) ~ ue")
cotx -cosec’ x ak 7 [v (oe
secx sec x tan x
cosec x -cosec x cot x 2 = & x “
ntl
Jax nel re Ssin x de =-cosx +e
fe de=atnxte
x
x
Sha* de = fa’ +c
éna
b
Agy = 23a
Do
Jcosx dx =sinx+c
MASTER" «-2="
y Ise ax dx = fn (secx+tanx) +e
‘
ni OY, htayiso.com
re re
i he :
:
“ap “ rat
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higher education
& training
Department:
Higher Education and Training
REPUBLIC OF SOUTH AFRICA
MARKING GUIDELINE
NATIONAL CERTIFICATE
MATHEMATICS N4
APRIL 2012
This marking guideline consists of 12 Pages.
Copyright reserved Please turn over
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MARKING GUIDELINE 2- T...(ENMA)T
. MATHEMATICS N4
QUESTION 1
1.1 Vv
. 41—fll_ 14+ 73
i ar oe a a)
. 744 f1l2
x+ jy= 1 v
Vv .
zal) ond y= ll2
10 aaeT, (5)
1.2 12.1 Z=-3-j2 V (1)
12.2 r= JC3P+C2P
r= 413 =3,606 ¥
479
9 =180° + tan (2)
6=213,69° V
FL C}
Z =3,606|213,69 V (3)
1.2.3
f j- axis
¢ —> R - axis
Axis ¥
oN
v ZV (3)
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MARKING GUIDELINE -3- T...(EA)T
MATHEMATICS N4
1.3
A
( ' 1
i ! | !
| I | |
woe ES
' i] | i
a a
>
<— —? ob 480° 240” ado”
1 i i} |
a re
I ' ! 1
i} i} 1 1
| i}
v (3)
1401.44 %
6 free
@ >X
ShapeV¥
x-interceptV
| (2)
1.4.2 Continuous/kontinu ¥ (1)
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MARKING GUIDELINE -A- T...(E)(A)T
« MATHEMATICS N4
1.5 ay
\ yeenx
* oF +x 4 Shape
\ x-intercept
. (2)
[20]
QUESTION 2
21 7
=p io
P H
—1 =10100
BR %
P
Paz
101% (3)
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T...(EYA)T
MARKING GUIDELINE 5
; . MATHEMATICS N4
2.2
x x+10 V
50
x? +(50)? =(x +10)?» V
x? +2500=x? + 20x +100 v
20x — 2400 =0
x,=120 ¥
x, =120410
= 130 Vv
23 (x+1)én5 = (x-2)en?
xén5—xln2=-fn5—2en®? = y
1,609x ~0,693x = 1,609 1,386
0,916x = -2,955
x=-3,226V
(6)
(3)
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MARKING GUIDELINE -6- T..(EA)T
. MATHEMATICS N4
2.4 2.4.1
-1 -2
4 -5
4-2
=1 -3 —5
4 -1
+3 13 _4| V
2
=1(-3)- 2A 26)+3(—13)
=-3452.39 ¥
=10 Vv (3)
2.4.2 Minor of -5 / minor van -5:
l 2
4 -]
9 V
- (1)
. 2.4.3 Co-factor of -2 / ko-faktor van -2:
1 2
OL3 4
- Q
- 27 (1)
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MARKING GUIDELINE 7- T..AEWA)T
. MATHEMATICS N4
2.5
af i]
-12
-2
=6 (4)
[20]
QUESTION 3
3.1 3.1.1
2cos? A=cos2A +1
cos2A
V
2
1
cos’ A=—+
2
cos? 4-1, 20824) ,
2 2
2
1 s.A
siy+—* y
2
2A
2 2
2 V2" 2 (4)
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MARKING GUIDELINE s wean
; MATHEMATICS N4
3.1.2
° 0
costs* cos 0° = |, 90830
a
(Sy
_ [2+v3 V
4
_v2+v3 J
7 (4)
3.2 (1+ tan? 6)+4 tang =3=0 J
tan? + tan@-2=0_
(tan 0 +2)(tan0-1)=0
tan@=-2 and/or tan@=1
0=116,565° @=45°
0=296,565° ¥ 6=225° ¥ (4)
3.3 sin(60°+60")=sin120"
=sin 60° cos60° + cos 60° sin 60°
ay
2
le
44
2
ir
al
3
tt
N
3
4
ols
(4)
iia Please turn over
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MARKING GUIDELINE -9-
wy MATHEMATICS N4
3.4 in(90° —
tan(o0° + p)= S00 = 8) y
cos(90" — B
_ cosB
sin B
tan(90° - B)=cotB
RHS = RHS V
QUESTION 4
41 Jiz+— 3. —45_, 13,5
3x VGx) ¥Gx)’
Yovoqvo4¥
4.2 u=sin3x V=4x3
& = 300832 = =12x?¥
y_ (4x2) cos3x)— (sin 3x)(12x") W
a (a)
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T...(EMA)T
(4)
[20]
(4)
(4)
Please tum over
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MARKING GUIDELINE -10- T...(E)(A)T
MATHEMATICS N4
43 Bass -12x+11 ¥
3x? -12x+11=0
x,=2,577 and x, =1,422 VW
Y; = (2,577) — 62,577) +11(2,577)- 6
=-0,384 ¥
Y. = (1,422) — 6(1,422) +11(1,422)—6
= 0,384 ¥
2 =6x-12
6x~12=0
x=2 V
y=) - 62) +11(2)-6
F Ov (7)
44 o. (x) 22)+3(2) + (a)cosecx cot 3x)+0+21x?
YVooo4 v Voq roo)
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MARKING GUIDELINE -11- T...(E(A)T
MATHEMATICS N4
QUESTION 5
5.1 — 4x! 3x Iz
4x" __9 — \4sin —3énx + tanx-4x4+¢
4 ¢3)n9 #7 (7)
v v Voo4v Vo vv
5.2 5.2.1 ,
: ay
37
* go” 140" 270” apo” ==
V Shape
a sy \ Strip
V Shown area
§.2.2
%
AOX = [3sinxdk V
oO
=3cosx]¥ ¥
=]-F3]
=3u? “
5.3 fx—3)de
2
x
= 3
7 3xte (3)
Vovv
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T...(EMA)T
MARKING GUIDELINE -12-
‘ MATHEMATICS N4
5.4 Ite _ oh
=[9-27]-[o] v
=-18 ¥ (3)
[20]
TOTAL: 100
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higher education
& training
Department:
Higher Education and Training
REPUBLIC OF SOUTH AFRICA
Ele Lb i102
T1080(E)(A3)T
AUGUST 2012
oO NATIONAL CERTIFICATE
MATHEMATICS N4
(16030164)
3 August (X-Paper)
09:00 — 12:00
This question paper consists of 5 pages and a 1-page formula sheet.
@
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-2- T1080(E)(A3)T
DEPARTMENT OF HIGHER EDUCATION AND TRAINING
REPUBLIC OF SOUTH AFRICA
NATIONAL CERTIFICATE
MATHEMATICS N4
TIME: 3 HOURS
MARKS: 100
INSTRUCTIONS AND INFORMATION
1. Answer ALL the questions.
2. Answer ALL FIVE questions in full. Show ALL the calculations and intermediary
steps. Simplify where possible
3. Read ALL the questions carefully.
4. Number the answers correctly according to the numbering system used in this
question paper.
5. ALL the graph work must be done in the ANSWER BOOK. Graph paper is NOT
supplied. Values of intercepts with the system of axes and the turning point(s) must
be shown on the graph.
6. ALL final answers must be accurately approximated to THREE decimal places.
7. Questions can be answered in any order but subsections of questions must NOT be
separated.
8. A FORMULA SHEET is attached to this question paper. You are NOT compelled to
use the formulae and the list is NOT necessarily complete.
9. Write neatly and legibly.
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-3- T1080(E)(A3)T
QUESTION 1
11
The breadth of a rectangle is 3 m less than its length. Calculate the length and the
breadth of the rectangle if the area of the rectangle is 75 m?.
1.2 Make 'n' the subject of the formula if:
= alr - 1)
r-1
13 Solve for x if:
35" 1 = 52xt2
1.4 Given:
3y+2x=Z+1
3x+2Z =8-Sy
3Z -l=x-2Z
Determine the value of y by using Cramer's rule only.
QUESTION 2
2.1 2.1.1 Sketch the graph of xy = -10.
2.1.2 Is the graph of xy = -10 discontinuous?
x? y?
2.2 2.2.1 Sketch the graph of — +=—— =1.
16 9
x? y 2
2.2.2 Is the graph of eto” 1 asymmetric about the x-axis or not?
2.3 Solve for x if:
(x + x + 4) =-8
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(4)
(4)
4)
(8)
[20]
(3)
@
G3)
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4. T1080(E)(A3)T
2.4 Given:
Ye
Z= (24 86.3° ]
Use De Moivre's theorem to calculate for Z. Leave the answer in a + jb form.
2.5 Given:
Z=-3-)3
2.5.1 Find Z
2.5.2 Convert Z into polar form. 9 must be positive. Show ALL steps.
QUESTION 3
3.1 Simplify the following:
colA+t
tanA+1
3.2 Solve for x if:
3sec?x-5=4tanx ; O°<x<270°
3.3 Prove that:
ae =cosA-sinA
3.4 Calculate the value of sin- 75° without a calculator.
3.5 Derive a formula for cos2A in terms of sinA.
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(3)
(1)
G)
[20]
(4)
6)
@)
@)
3)
[20]
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-5- T1080(E)(A3)T
QUESTION 4
4.1 Differentiate from first principles if:
F(x)= V5
4.2 Given:
y=(2x'-7)
Differentiate by using the chain rule.
43 Differentiate with respect to x:
y=5x?- = -2p +2. tnax—cot 2x
3x x
4.4 Given:
yor 43x? -9x45
Calculate, with the aid of differentiation, the co-ordinates of the turning points.
QUESTION 5
5.1 S11 Sketch and clearly indicate the area bounded by the graph of
y=3x°-12, the x-axis; x=0 and x=2. Also, indicate the
representative to be used to calculate the area shown.
5.1.2 Calculate, using integration, the magnitude of the area indicated in
QUESTION 5.1.1.
5.2 Integrate:
3.14 -Tx
f ae /e-'* +16 cos 4x + cosec x cot x + 4m |dx
5.3 Determine:
f 2cos xdx ?
5.4 Simplify:
f cos” A~sin’ AMA
\
TOTAL:
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(4)
4)
(6)
(6)
[20]
(3)
(4)
©)
3)
(3)
[20]
100
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T1080(E)(A3)T
FORMULA SHEET
a* =b @ loga”® =logb én x = log.x
(roy =r" |n@ atbj=c+dj @ a=candb=d
sin(a + b) = sina cosb + sinb cosa sin*x + cosx = 1
cos(a + b) = cosa cosb ¥ sina sinb 1+ cot?x = cosec”x
1+ tan’x = sec”x
tana + tanb
tan (at b) = ——
1 tana tanb
y dy
ax. _
ax" nox"! y = u(x) - vx)
ka* ka* tna
| oe > wy u(x)v'(x) + u'(x) v(x)
kénx ik dx
x
sinx cos x - u(x)
cos x sin x v(x)
tan x | secx , dy — v(x)u'(x) - u(x)v'(x)
cot.x | -cosec "x mk hee
sec x sec x tanx
cosec x -cosec x cot x yy = wy x du
de du dk
ax”
n
Jax"de="" +c Ssin x dx =—cosx+c
x
Sek Jeos x dx = sinx+c
Po Re
Ska*ae ka* 3 at :
iene ‘Jtanx dx = lnsecxtc
ME
b * Ssecxdx = én (secx + tanx)+c
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bigher education
& training
Department:
Higher Education and Training
REPUBLIC OF SOUTH AFRICA
T1080(E)(N15)T
NOVEMBER 2012
NATIONAL CERTIFICATE
MATHEMATICS N4
(16030164)
15 November (X-Paper)
09:00 — 12:00
Calculators may be used.
This question paper consists of 5 pages and a 1-page formula sheet.
Copyright reserved NOV 2012 . Please turn over
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(16030164) -2- T1080(E)(N15)T
DEPARTMENT OF HIGHER EDUCATION AND TRAINING
REPUBLIC OF SOUTH AFRICA
NATIONAL CERTIFICATE
MATHEMATICS N4
TIME: 3 HOURS
MARKS: 100
INSTRUCTIONS AND INFORMATION
1.
2.
10.
Answer ALL the questions,
Read ALL the questions carefully.
Number the answers correctly 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 calculations and intermediate steps. Simplify where possible.
ALL graph work must be done in the ANSWER BOOK. Graph paper is NOT
supplied. The values of intercepts with the system of axes and the turning point(s)
must be shown on the graph.
ALL final answers must be accurately approximated to THREE decimal places.
Questions must be answered in BLUE or BLACK ink.
A formula sheet is attached to this question paper. You are NOT compelled to use the
formulae and the list is NOT necessarily complete.
Write neatly and legibly.
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(16030164) -3- T1080(E)(N15)T
QUESTION 1
1.1 Sketch the graph of y=tanx ; 0° <x<360°, (3)
12 1.2.1 Sketch the graph of y=—x? + 5x-4. (3)
1.2.2 Is the graph of y=-x? +5x-4 in QUESTION 1.2.1 continuous or
discontinuous? Q)
1.3 Sketch the graph of xy =—1. (2)
3
14 Given: Z,=20 |63° and Z, =4 [36° , aeteine{ 2) . Show ALL steps. Leave
2
the answer in rectangular form. (5)
1.5 Solve for x and y if:
@-4j) =
P (5)
[20]
QUESTION 2
2.1 Solve for x if:
3.3" =28 (4)
2.2 Make y the subject of the formula if:
q 29) oo 2 @)
2.3 Determine the length and the breadth of a rectangle if the length of a rectangle is 3 m
more than its breadth and the area of a rectangle is 54 m? , (4)
2.4 Given:
a+4b+8c=0
2a—5b=-2+ 6c
b-4c=-3
Solve for c by the use of Cramer’s rule. (8)
[20]
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(16030164) -4- T1080(E)(N15)T
QUESTION 3
3.1 Determine the value of cos 105° without the use of a calculator.
3.2 Prove that:
2siny+1
= = ery
sin2+cos y
33 Simplify:
cosecy + cot y
3.4 IfsinP == and cos 0-2 and both P and @ are acute angles, determine without
the use of a calculator, the value of cos(P +Q).
3.5 Solve for P if:
4sin? P-1=0, 0° <P <360°
QUESTION 4
4.1 Determine:
Lim x? -x
x72 |x°-1
4.2 Given:
y=-2x? 43x? 412x-5
Determine, using differentiation, the co-ordinates of the maximum and the minimum
turning points.
4.3 Given:
y=107 tanx
Differentiate by the use of the product rule.
44 Differentiate the following with respect to x:
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4)
(4)
@)
(4)
G)
[20]
4)
()
4)
6)
[20]
(16030164) -5- T1080(E)(N15)T
QUESTION 5
3.1 5.1.1 Sketch and indicate the area enclosed by the graph of y = 3x—x” and the
X- axis. Also, indicate the representative strip to be used to calculate the
value of the enclosed area. @)
5.1.2 Calculate, using integration, the value of the enclosed area in
QUESTION 5.1.1. (4)
3.2 Integrate the following in terms of x:
I(3* —2'* 4+ 4e% -_4___1, a a
cos ec4x x (7)
5.3 Determine:
je cos 2x)dx
0 (3)
5.4 Simplify:
foax* a&
(3)
[20]
TOTAL: 100
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(16030164)
MATHEMATICS N4
a* =b & loga’ =logb
T1080(E)(N15)T
én x= log.x
(Fla =r" |n0 a+ bj=ct+dj <> a=candb=d
sin(a + b) = sina cosb + sinb cosa
cos(a+ b) = cosa cosb ¥ sina sinb
+
tan (a+ B)= tana + tanb
IF tana tanb
y dy
dx
a na!
ka* ka* tna
kenx k
x
sin x cos x
cos x -sin x
tan x sec’x
cotx -cosec*x
sec x sec x tan x
cosec x ~cosec x cot x
nl
Jax" dr = +C
nt+1
sin’x + cos’x = 1
1+ cot’x = cosec”x
1 + tan’x = sec?x
> 2 = u(x)v'(x) + u'(x) v(x)
_ uc)
~ v(x)
= & _ Maw'(a) = ue)
de [GP
YY du
a& du &
Jsinx de =-cosx+e
Jcosx de=sinx +c
Jtanx de = énsecx+c
Jsecx de = én (secx+tanx)+e
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higher education
& training
Department:
Higher Education and Training
REPUBLIC OF SOUTH AFRICA
nw
Q
2
bard
ey
a
a
T1030(E)(M28)T
APRIL EXAMINATION
NATIONAL CERTIFICATE
MATHEMATICS N4
(16030164)
28 March 2013 (X-Paper)
09:00-12:00 .
Calculators may be used.
This question paper consists of 5 pages and a 1-page formula sheet.
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DHET -2- T1030(E)(M28)T
DEPARTMENT OF HIGHER EDUCATION AND TRAINING
REPUBLIC OF SOUTH AFRICA
NATIONAL CERTIFICATE
MATHEMATICS N4
TIME: 3 HOURS
MARKS: 100
INSTRUCTIONS AND INFORMATION
1. Answer ALL the questions.
2. Answer ALL FIVE questions in full. Show ALL the calculations and intermediary
steps. Simplify where possible.
3. ALL the graph work must be done in the ANSWER BOOK. Graph paper is NOT
supplied. The values of intercepts with the system of axes and the turning point(s)
must be shown on the graph.
4. ALL final answers must be accurately approximated to THREE decimal places.
5. Questions may be answered in any order, but subsections of questions must NOT be
separated.
6. A FORMULA SHEET is attached to this question paper. You are NOT compelled to
use the formulae and the list is NOT necessarily complete.
7. Write neatly and legibly.
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Copyright reserved Please turn over
DHET -3- T1030(E)(M28)T
QUESTION 1
11 1.1.1 Sketch the graph of 5 -Fek B)
1.1.2 What is the domain of the graph in QUESTION 1.1.1? (2)
1.1.3 Is the graph in QUESTION 1.1.1 symmetrical about both axes? qj)
1.14 Does the graph in QUESTION 1.1.1 represent a function or a relation? qd)
1.2 Sketch the graph of x =— 20-9? : re)
13 Solve for x if:
xtytyjt2j=87+7 (4)
1.4 Given:
Z, =1- 1,732
Z, =— 6,928 — j4
1.4.1 Convert Z, and Z, into polar form. The arguments may only be positive. Q)
1.4.2 Calculate Zz and leave the answer in a+ jb form.
Z, (4)
20]
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DHET 4 T1030(E)(M28)T
QUESTION 2
2.1 Given:
2x -2y-Z=3
4x+5y-2Z=-3
3x+4y-3Z=-7
2.1.1 Solve for the value of Z by using Cramer's rule. (8)
2.1.2 Determine the value of the cofactor of -3. (2)
2.2 Given:
t= tog + 2)
Make y the subject of the formula. (4)
2.3 Solve for x if:
2°? = 5") (3)
2.4 The sum of two numbers is 10. The difference of their squares is 50. Calculate the
TWO numbers. 3)
[20]
QUESTION 3
3.1 Prove that:
Sin2A—Cos2A +1 _
Sin2A + Cos2A +1 (5)
3.2 Solve for A if:
2cos? A=—CosA;0° < AS 360° (4)
3.3 3.3.1 Derive a formula for Cos2A. (2)
3.3.2 Use the derived formula in QUESTION 3.3.1 to determine the value of
Cos120° without the use of a calculator. (3)
3.4 If sin A=0,6 and sinB=0,4; and both A and B are acute angles, determine the
value of Cos(A + B) without the use of a calculator. 3)
3.5 Prove that:
sin(180° + 4)=-sin A @)
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DHET 5- T1030(E)(M28)T
QUESTION 4
41 Determine:
Lim ( x-2 )
x2 \x?-8
4.2 Given:
y=e**.cos3x. Differentiate by using the product rule.
4.3 Differentiate the following with respect to x:
y = 2.3 — 4cos2x — Sénx — 3x7
44 Given:
y=x? 11x? +32x-28
Determine, using differentiation, the maximum and the minimum turning points.
Also determine the point of inflection by using the second derivative.
QUESTION 5
5.1 Integrate the following in terms of x:
J@sin3x-e* —3x~ + 2.3** — cosec?x + 2) dx
5.2 Determine: fovax
1
5.3 5.3.1 Sketch and indicate the area enclosed by the graph of y=4 cosx, the
x-axis, x=Qand x='%. Also, indicate the representative strip to be
used to calculate the area of the enclosed area.
5.3.2 Calculate, using integration, the value of the enclosed area indicated in
QUESTION 5.3.1.
54 Simplify:
cos écx
TOTAL:
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4)
(4)
(4)
(8)
[20]
1)
@)
(3)
(4)
(3)
[20]
100
DHET
FORMULA SHEET
a” =b = loga’® =logb
T1030(E)(M28)T
én x = logex
(le =r |nd atbj=ct+d <> a=candb=d
sin(a + b) = sina cosb + sinb cosa
cos(a + b) = cosa cosb ¥ sina sinb
tana + tanb
tan (a + b) = ——————_
1¥ tana tanb
y dy
ax
n n-I
ax nax
ka* ka* lna
kenx k
x
sin x cos x
cos x -sin x
tan x sec*x
cot x -cosec?x
sec x sec x tanx
cosec x -cosec x cotx
ntl
ax
"dk = +C
bax nt+1
F dx =alnx+c
x
x
fata = 5 ¢
fna
Aox = $ ydx
sin’x + cos*x = 1
1+ cot’x = cosec’x
1+ tan’x = sec’x
y= utx) - v(x)
at = u(x)v'(x) + u'(x) v(x)
_ ue)
© v(x)
a WD _ vu) ~ u@)v'@)
de Ivo?
db _ dy du
a& du ax
Jsinx de =-cosxt+e
Jcosx de =sinx+e
Stanx de =énsecx+e
Jsecx dx = £n (sec x +tanx)+c
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