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T950(E)(A)T
APRIL EXAMINATION
NATIONAL CERTIFICATE
MATHEMATICS N3
(16030143)
1 April 2016 (X-Paper)
09:00–12:00
This question paper consists of 6 pages and 1 formula sheet of 2 pages.
(16030143) -2- T950(E)(A1)T
Copyright reserved Please turn over
DEPARTMENT OF HIGHER EDUCATION AND TRAINING
REPUBLIC OF SOUTH AFRICA NATIONAL CERTIFICATE
MATHEMATICS N3
TIME: 3 HOURS
MARKS: 100
INSTRUCTIONS AND INFORMATION
1.
2.
3.
4.
5.
6.
7.
8.
9.
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 NOT be
separated.
Show ALL the calculations and intermediary steps.
ALL final answers must be accurately approximated to THREE decimal places.
ALL graph work must be done in the ANSWER BOOK. Graph paper is NOT
supplied.
Diagrams are NOT drawn to scale.
Write neatly and legibly.
(16030143) -3- T950(E)(A1)T
Copyright reserved Please turn over
QUESTION 1
1.1 Factorise the following expressions as far as possible in prime factors:
1.1.1 (3 2) (3 2)x x y y (4)
1.1.2 4 24 3 1p pn n (2)
1.2 Factorise the following expression completely: 3 22 5 2x x x
(5)
1.3 Simplify the following expression:
2
2
1 2 1 2 7 17
1 3 2 3
x x x x
x x x x
(6)
[17]
QUESTION 2
2.1 Simplify the following:
3
2
1
2
3
xx
x
(4)
2.2 Use logs to the base 2 and simplify the following WITHOUT using a calculator:
0,5log 128
(4)
2.3 Solve for x:
2.3.1 216 3 3 2x x
2.3.2 (log 2) log( 2) 0x x (2 4) (8)
[16]
(16030143) -4- T950(E)(A1)T
Copyright reserved Please turn over
QUESTION 3
3.1 Solve for x by completing the square: 24 48 2x x
(4)
3.2 Make 'b ' the subject of the formula:
x bD
x b
(4)
3.3 Make ' w ' the subject of the formula:
log log loge e et p w ds
(3)
3.4 Alex paid a deposit of 3R x for a computer. He paid the rest in 9 monthly
instalments. He paid a total of 33R x . What is the payment of each monthly
instalment in terms of x .
(4)
[15]
QUESTION 4
4.1 Consider FIGURE A below. ABC is an isosceles triangle with AB BC and
vertices A(2;1), B(4;5) and C(0;k).
FIGURE A
(16030143) -5- T950(E)(A1)T
4.1.1 Find the length of AB. (2)
4.1.2 Determine the value(s) of k. (4)
4.1.3 Show that AB is perpendicular to BC if k 7 . (3)
4.1.4 Calculate the area of ABC when 7k . (3)
4.2 P ( 2; 1) and Q (4;7) are points in the plane with M as the midpoint of PQ.
Determine the equation of the line parallel to the y-axis and passing through the
point M.
(4)
4.3 Consider FIGURE B below. The lines BA and CA with equations 2y x and
3 3y x respectively, intersect at A. Determine the size of ˆBAC where B and C
are the intercepts on the x-axis as shown.
FIGURE B
(5)
[21]
(16030143) -6- T950(E)(A1)T
Copyright reserved Please turn over
QUESTION 5
5.1 Draw the graph defined by the equation: 2733 22 yx (2)
5.2 Given : 3 26 9y x x x
5.2.1 Make use of differentiation to determine the coordinates of the turning
points of the given equation.
(5)
5.2.2 Draw the graph of the given function. Show ALL values at the points of
intersection with the system of axes and the co-ordinates of the turning
points.
(3)
5.3 Determine
dy
dx if
12y x
x . Leave the answers with positive indices and in
surd form.
(4)
[14]
QUESTION 6
6.1 Prove the following trigonometric identity:
2 2 2 2sin tan cos secA A A A
(4)
6.2 Calculate the value(s) of which will satisfy the equation if o o0 270 :
2sin 1 cos
(5)
6.3 Consider FIGURE C below. An observer, standing at a point A is watching the top
of a vertical tower BC . The angle of elevation of the top of the tower, BC, is o25
and the angle of depression of the foot of the tower is o20 . If the height of the
tower BC is known to be 30 m, determine the following:
6.3.1 The distance between the observer at A and the point B. (3)
6.3.2 The distance between the two towers. (3)
(16030143) -7- T950(E)(A1)T
Copyright reserved Please turn over
FIGURE C
6.4 Consider FIGURE D below. The sketch represents the graph of ( ) sinf x a px
where 0 x . Determine the values of a and p .
FIGURE D
(2)
[17]
TOTAL: 100
(16030143) -1- T950(E)(A1)T
Copyright reserved Please turn over
FORMULA SHEET
Any applicable formula may also be used.
1. Factors 2. Logarithms
a3 - b
3 = (a - b)(a
2 + ab + b
2)
a3 + b
3 = (a + b)(a
2 - ab + b
2)
blogalogablog
blogalogb
alog
3. Quadratic formula
blog
alogalog
c
cb
2a
4acbbx
2
alogmalog m
4. Parabola
blog
1alog
ab
cbxaxy 2
4a
b4acy
2
11∴ e n1aloga
meta m1ntloga ∴
2a
bx
5. Circle 6. Straight line
222 ryx
h4h
xD
2
24h4Dhx
)xm(xyy 11
Perpendicular: 1mm 21
Parallel lines: 21 mm
Distance: 212
212 )y(y)x(xD
Midpoint:
2
yy;
2
xxP 2121
Angle of inclination: mtanθ -1
(16030143) -2- T950(E)(A1)T
Copyright reserved
7. Differentiation
h
xfhxf
0h
lim
dx
dy
→
1-nn nxxdx
d
Max/Min
For turning points: 0xf '
8. Trigonometry
cosecθ
1
r
ysinθ
secθ
1
r
xcosθ
cotθ
1
x
ytanθ
1 θcosθsin 22
θsecθtan1 22
θcosecθcot1 22
cosθ
sinθtanθ
sinθ
cosθcotθ
c
sinC
b
sinB
a
sinA
cosA2bccba 222
Area of ∆ABC = ½ ac sin B
Copyright reserved Please turn over
NATIONAL CERTIFICATE
APRIL EXAMINATION
MATHEMATICS N3
1 APRIL 2016
This marking guideline consists of 10 pages.
MARKING GUIDELINE
MARKING GUIDELINE -2- T950(E)(A1)T
MATHEMATICS N3
Copyright reserved Please turn over
QUESTION 1
1.1 1.1.1
2 2
2 2
(3 2) (3 2)
3 2 3 2
3( ) 2( )
3( )( ) 2( )
( )[3( ) 2]
( )[3 3 2]
x x y y
x x y y
x y x y
x y x y x y
x y x y
x y x y
(4)
1.1.2 4 2
2 2
4 3 1
( 1)(4 1)
p p
p p
n n
n n
(2)
1.2 3 2
3 2
2
3 2
3 2
2
2
( ) 2 5 2
(1) 2(1) (1) 5(1) 2
=2 1 5 2
=0
-1 is a factor of f ( )
2 3 2
x-1 2 5 2
2 2
3 5
3 3
f x x x x
f
x x
x x
x x x
x x
x x
x x
2
- 2 2
-2 2
. .
( ) ( 1)(2 3 2)
= ( 1)(2 1)( 2)
x
x
f x x x x
x x x
(5)
MARKING GUIDELINE -3- T950(E)(A1)T
MATHEMATICS N3
Copyright reserved Please turn over
1.3
)3)(1(
1772
3
12
1
1 2
xx
xx
x
x
x
x
=)3)(1(
1772)1)(12()3)(1( 2
xx
xxxxxx
=)3)(1(
17721234 222
xx
xxxxxx
)3)(1(
)32(5 2
xx
xx
(6)
[17]
MARKING GUIDELINE -4- T950(E)(A1)T
MATHEMATICS N3
Copyright reserved Please turn over
QUESTION 2
2.1
3
2
3
21
2
1 3
2 2
2
1
2
3
2 13
2
2 1 1
2 3
2 1
6
xx
x
xx
x
x
x x
x
x
(4)
2.2 0,5
2
2
7
2
1
2
2
2
log 128
log 128
1log
2
log 2
log 2
7 log 2
1log 2
7
(4)
2.3 2.3.1
2
22
2 2
16 3 3 2
16 3 3 2
16 3 3 4.3 4
12 4.3
3 3
1
TEST : If 1 then LHS=RHS=5
x x
x x
x x x
x
x
x
x
(4)
2.3.2
0
2
(log 2) log( 2) 0
log 2 0 or log( 2) 0
log 2 2=10
10 2 1
100 3
x x
x x
x x
x x
x x
(4)
[16]
MARKING GUIDELINE -5- T950(E)(A1)T
MATHEMATICS N3
Copyright reserved Please turn over
QUESTION 3
3.1
(4)
3.2
bx
bxD
2
(4)
MARKING GUIDELINE -6- T950(E)(A1)T
MATHEMATICS N3
Copyright reserved Please turn over
3.3
dsp
twe log
dsep
tw
dset
pw
(3)
3.4 Deposit is R3x
Total price is R33x
Total amount for 9 instalments= R30x
Each monthly instalment =30 10
R9 3
x xR
(4)
[15]
MARKING GUIDELINE -7- T950(E)(A1)T
MATHEMATICS N3
Copyright reserved
QUESTION 4
4.1 4.1.1
2 2
2 1 2 1
2 2
AB
= 4 2 5 1
= 4 16
= 20
=2 5
x x y y
(2)
4.1.2
2 2
2
2
BC=2 5
0 4 5 20
16 5 = 20
5 = 4
5 = 2
5 2 5 2
7 3
k
k
k
k
k or k
k k
(4)
4.1.3 5 12
4 2
7 5 1
0 4 2
1
therefore CB AB
AB
CB
AB CB
M
M
M M
(3)
4.1.4
2
1Area of ΔABC = × base × height
2
1 =
2
1 = 20 20
2
=10 units
AB BC
(3)
MARKING GUIDELINE -8- T950(E)(A1)T
MATHEMATICS N3
Copyright reserved Please turn over
4.2
1 2 1 2coordinates of M ;2 2
2 4 1 7 = ;
2 2
=(1;3)
Equation =1
x x y y
x
(4)
4.3
o o
o o o
1 3
tan 1 tan =3
=45 =71,565
ˆBAC = = 71,565 45 = 26,565
AB ACM M
(5)
[21]
QUESTION 5
5.1 2733 22 yx
Y
3
-3 0 3 X
-3 (2)
MARKING GUIDELINE -9- T950(E)(A1)T
MATHEMATICS N3
Copyright reserved Please turn over
5.2.1
5.2.2
3 2
2
2
2
3 2
3 2
6 9
3 12 9
Let 0
3 12 9 0
3( 4 3) 0
( 1)( 3) 0
1 or 3
( 1) ( 1) 6( 1) 9( 1) 4
( 3) ( 3) 6( 3) 9( 3) 0
Turningpoints ( 3;0) and ( 1;4)
y x x x
dyx x
dx
y
x x
x x
x x
x x
f
f
(5)
(3)
5.3
1
1 2
1
2 2
2
12
2
2(0,5)
1 1
y xx
y x x
dyx x
dx
dy
dx x x
(4)
[14]
MARKING GUIDELINE -10- T950(E)(A1)T
MATHEMATICS N3
Copyright reserved Please turn over
QUESTION 6
6.1
(4)
6.2 2
2
2
o o
o
sin 1 cos
sin sin
sin sin 0
sin (sin 1) 0
sin 0 or sin 1
0 or 90
or 180
(5)
6.3 6.3.1
6.3.2
o o
o
o
o
o
o
:
30
sin 70 sin 45
30sin 70
sin 45
39,868
:
cos 25
cos 25
39,868 cos 25
36,133
In ABC
AB
AB
m
In ABD
AD
AB
AD AB
m
(3)
(3)
MARKING GUIDELINE -11- T950(E)(A1)T
MATHEMATICS N3
Copyright reserved Please turn over
6.4 2
2
a
p
(2)
[17]
TOTAL: 100
Website: www.ekurhulenitech.co.za Email: info@ekurhulenitech.co.za
REGISTERING WITH OUR COLLEGE
CONSIDER REGISTERING WITH OUR COLLEGE AND WE HAVE THE
FOLLOWING TYPES OF LEARNING:
ONLINE OR CORRESPONDENCE LEARNING. THIS IS THE MOST
PREFARED ONE BY THOUSANDS
PART-TIME CLASSES DURING SATURDAYS IF NEAR OUR
OFFICES IN KRUGERSDORP
FULL-TIME CLASSES IF NEAR OUR CLASSES IN
KRUGERSDORP
ABOUT EXTRA TEXTBOOKS
IF LOOKING FOR TEXBOOKS FOR CERTAIN SUBJECTS I N1-N6 ENGINEERING
STUDIES PLEASE SEND US AN EMAIL ON INFO@EKURHULENITECH.CO.ZA
ABOUT VIDEO MATERIAL
WE HAVE VIDEO MATERIAL FOR EXTRA UNDERSTANDING ON CERTAIN
ENGINEERING FOR A FEE. REQUEST A QUOTE. SEND US AN EMAIL ON
INFO@EKURHULENITECH.CO.ZA
EKURHULENI TECH COLLEGE. No. 3 Mogale Square, Krugersdorp.
Website: www. ekurhulenitech.co.za
Email: info@ekurhulenitech.co.za TEL: 011 040 7343 CELL: 073 770 3028/060 715 4529
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