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T870(E)(N21)T
NOVEMBER EXAMINATION
NATIONAL CERTIFICATE
MATHEMATICS N3
(16030143)
21 November 2016 (X-Paper)
09:00–12:00
Nonprogrammable scientific calculators may be used.
This question paper consists of 6 pages and 1 formula sheet.
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(16030143) -2- T870(E)(N21)T
DEPARTMENT OF HIGHER EDUCATION AND TRAINING
REPUBLIC OF SOUTH AFRICA
NATIONAL CERTIFICATE
MATHEMATICS N3
TIME: 3 HOURS
MARKS: 100
INSTRUCTIONS AND INFORMATION
1. Answer ALL the questions.
2. Read ALL the questions carefully.
3. Number the answers according to the numbering system used in this question paper.
4. Questions may be answered in any order but subsections of questions must NOT be
separated.
5. Clearly show ALL calculations, diagrams, graphs, et cetera, which you have used in
determining the answers.
6. If necessary, answers should be rounded off to THREE decimals, unless stated
otherwise.
7. Diagrams are NOT drawn to scale.
8. Write neatly and legibly.
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(16030143) -3- T870(E)(N21)T
QUESTION 1
1.1 Simplify:
1.1.1 49 x 2 56 x 16 (2)
1.1.2 13
2
1 1
a a 3
(a a 2)
(7)
1.1.3 3x 2 7 x 2 x 2 x 6
2
2 x2 5x 3 x 9 (6)
1.2 If 16x 3 px 2 12x 3 is divided by 2x 1 , the remainder is –7. Determine the
value of p . (4)
[19]
QUESTION 2
2.1 Express 0,125 as a power of 4. (3)
2.2 Solve for x :
3 3
2.2.1 2log x + log log 2 x 0
4 4
Show All the steps. The use of a calculator is excluded. (6)
2.2.2 4 2x 3 5
x2 4 x 2
x2 (5)
[14]
QUESTION 3
3.1 Solve for x by completing the square: 18 x 2 9 x (5)
3.2 There are 302 residents in a town. There are 34 more women than men and 60 more
children than women. How many women are there? (3)
3.3 Given: PV n C
Calculate the value of n if P =2000, V =0,48 and C =18,3. (5)
[13]
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QUESTION 4
Consider FIGURE 1 below. ABCD is a parallelogram with vertices A(-2;1), B (2;2),
C(1;-2) and D ( x; y ) .
FIGURE 1
4.1 1 1
Prove that the midpoint M, of AC is equal to ( ; ) .
2 2 (2)
4.2 Use the co-ordinates of the midpoint, M to determine the co-ordinates of
the fourth vertex, D. (3)
4.3 Determine the length of BC. (2)
4.4 Calculate the angle of inclination, of the line BC. (2)
4.5 Determine the equation of the line in the form y mx c which is
perpendicular to BC and passes through A. (3)
4.6 Determine the coordinates of the point of intersection of the line through
A(-2;1), which is parallel to the y-axis and the line 2 y x 5 . (3)
[15]
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(16030143) -3- T870(E)(N21)T
QUESTION 5
5.1 Sketch the graph of y 36 x 2 in the ANSWER BOOK. ALL values at the points
of intersection with the axes must be shown. (3)
5.2 FIGURE 2 shows the graph of f ( x) x3 ax 2 bx c .
A and B(6;0) are turning points.
FIGURE 2
5.2.1 Find the values of a , b and c . Show ALL your calculations. (5)
5.2.2 If f ( x) x3 12 x 2 36 x , find the coordinates of A. (4)
1
dy 3
5.3 Determine if y x 2 2 by making use of the rules of differentiation. ALL
dx x
final answers must be with positive exponents and in surd form where applicable. (4)
[16]
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(16030143) -6- T870(E)(N21)T
QUESTION 6
6.1 Make use of trigonometric identities to prove the following:
cos 2 (90o x) 1 cos x
sin(90 x) 1 sin x
o 2
cos x (7)
6.2 To find the height of a tower, a surveyor sets up his theodolite 150 m from the base of
the tower. He finds that the angle of elevation to the top of the tower is 60°.
What is the height of the tower? (3)
6.3 Consider FIGURE 3 below. The crew in a boat, C noticed two lighthouses A and B,
one on either side of a beach. The two lighthouses are 0,67 km apart and one is
exactly due east of the other. The lighthouses tell how close a boat is by taking
bearings to the boat. The bearing from the lighthouse A to the boat C is 127° and
the bearing from the lighthouse B to the boat C is 255°. A bearing is an angle
measured clockwise from north.
Calculate how far the boat, C is from the lighthouse at B. (5)
FIGURE 3
6.4 6.4.1 Draw the graphs which are represented by f ( x) 2sin 2 x and g ( x) cos 2 x
on the same system of axes for 0 x 180 . ALL values at the points of
intersection with the system of axes and the coordinates of the turning points
must be shown. (6)
6.4.2 Use the graph to read off the value(s) of x if 2tan 2x 1 for 0 x 180 (2)
[23]
TOTAL: 100
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(16030143) T870(E)(N21)T
FORMULA SHEET
1. Factors 3. Quadratic formula
a3 b3 (a b)(a 2 ab b 2 ) b b 2 4ac
x
2a
a3 b3 (a b)(a 2 ab b 2 )
4. Parabola
2. Logarithms
y ax 2 bx c
log ab log a log b
4ac b 2
y
4a
a b
log log a log b x
b 2a
log a
logb a c
log c b
log a m m log a 5. Circle
1 x2 y 2 r 2
logb a
log a b
log a a 1 , ln e 1 x2
D h
4h
a loga t t , eln m m x 4 Dh 4h 2
6. Straight line 8. Trigonometry
y y1 m( x x1 ) y 1
sin
r cos ec
Perpendicular lines : m1 m2 1 x 1 cos
cos cot
Parallel lines : m1 m2 r sec sin
Distance : D ( x2 x1 ) 2 ( y2 y1 ) 2 y 1 sin
tan tan
x cot cos
x x y y sin 2 cos 2 1
Midpoint : M 1 2 ; 1 2
2 2
Angle of inclination : tan 1 m 1 tan 2 sec 2
1 cot 2 cosec 2
7. Differentiation sin A sin B sin C
a b c
dy f ( x h) f ( x ) a b c 2bc cos A
2 2 2
lim
dx h 0 h
d n 1
( x ) nx n 1 Area of ABC ac sin B
dx 2
For turning point : f '( x) 0
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MARKING GUIDELINE
NATIONAL CERTIFICATE
NOVEMBER EXAMINATION
MATHEMATICS N3
21 NOVEMBER 2016
This marking guideline consists of 10 pages.
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MARKING GUIDELINE -2- T870(E)(N21)H
MATHEMATICS N3
✓ = 1 Mark
QUESTION 1
1.1 1.1.1 49 x 2 56 x 16
7 x 4
2
7x 4 (2)
1.1.2 13
2
1 1 13
2
1 1
a a (a a 2)
3
a a (a a 2)
3
1 1 1 1 1
3 a (a 2) ✓ 1
a 3 2 (a 1 2) 1
a a
3 2
a
a3
3 a . 3 a 2 1 a 2 2a 1 1
a 1 a 2 2a 1 1
3 2 a 2
a
3 a
a 1 (a 1) 2
1 a
a 1 a
a a
3 2
2
3 a 12
a 1 a a
2
3 (a 1) 2
1
a a3
1
3 3
a 1
a a
or (7)
(a 1) a 1
1.1.3 3x 2 7 x 2 x 2 x 6
2
2 x2 5x 3 x 9
✓
(3 x 1)( x 2) ✓ ( x 3)( x 3) ✓
(2 x 1)( x 3) ( x 2)( x 3) ✓
3x 1
2x 1 ✓ (6)
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MARKING GUIDELINE -3- T870(E)(N21)H
MATHEMATICS N3
1.2 1
If f ( x) is divided by 2x+1, then the remainder is f
2
✓
1
3 2
1 1 1
Hence f 16 p 12 3 7 ✓
2 2 2 2
1 1
16 p 6 3 7
8 4
1
2 p 6 3 7
4
1
p 7 5 ✓
4
4
p 2
1
p8 ✓ (4)
[19]
QUESTION 2
2.1 0,125 0,125 4 x
1
1
1 2 ✓ 4x ✓
8
8
1 23 22 x ✓
1 2
3 ✓ 3
2 2 x
3 3
2
2 2
4 4
✓ x
3 ✓
4
3
0,125 4 4 (3)
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MARKING GUIDELINE -4- T870(E)(N21)H
MATHEMATICS N3
2.2 2.2.1 3 3 3 3
2 log x + log log 2 x 0 2 log x + log log 2 x 0
4 4 4 4
3 2 3 3
x log x 2 + log log 2 x ✓
log10 4 0 ✓ 4 4
3
2x 3x 2 3
4 log log 2 x ✓
3 2 4 4
x
3x 2 3
100 4 2x
3 ✓ 4 4 ✓
2x
4 3x 2 8 x 3 ✓
2
3x
1 ✓ 3x 2 8 x 3 0 ✓
8x 3
✓ (3x 1)( x 3) 0
3x 2 8 x 3
3x 1 0 x 3 0
3x 2 8 x 3 0 ✓
1
(3 x 1)( x 3) 0 x (n/a) x 3 ✓
3
3x 1 0 x 3 0
1
x (n/a) x 3 ✓
3
(6)
2.2.2 4 2x 3 5
x2 4 x 2
x2
4 2x 3 5
✓ ✓
x 2 x 2 ( x 2) x 2
4( x 2) 2 x 3 5( x 2)
✓
4 x 8 2 x 3 5 x 10
3 x 21 ✓
x7 ✓ (5)
[14]
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MARKING GUIDELINE -5- T870(E)(N21)H
MATHEMATICS N3
QUESTION 3
3.1 18 x 2 9 x
x 2 9 x 18 0 ✓
81 81
x2 9x 18 ✓
4 4
2
9
x 2, 25
2 ✓
9
x 2, 25 (5)
2
x 3 or x 6
✓ ✓
3.2 Let the number of women = x Let the number of men = x
∴ the number of men = x 34 let the number of women = x 34 ✓
✓
let the number of children =
and the number of children = x 60
x 34 60
x x 34 x 60 302 ✓
x x 34 x 94 302 ✓
3x 276
3 x 128 302
x 92
3 x 174
x 58
There are 92 women ✓
There are 92 women ✓ (3)
3.3 PV n C PV n C
Vn
C 2000 (0, 48) n 18,3 ✓
✓
P 18,3
C (0, 48) n ✓
logV n log ✓ 2000
P
18,3
n logV log C log P ✓ log(0, 48) n log ✓
2000
logC log P
n 18,3
logV n log(0, 48) log
log18,3 log 2000 ✓ 2000
n log18,3 log 2000
log 0, 48 n
log 0, 48 ✓
1, 262 3,301
n 1, 262 3,301
0,319 n
n 6,395 ✓ 0,319
n 6,395 ✓ (5)
[13]
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MARKING GUIDELINE -6- T870(E)(N21)H
MATHEMATICS N3
QUESTION 4
4.1 2 1 1 2
M ( xM ; yM ) ;
2 2 ✓
1 1
; ✓
2 2 (2)
4. 2 DM MB ..... diagonals of parm
1 1
Midpoint of DB M ( xM ; yM ) ;
2 2
x 2 y 2 1 1 ✓
M ( xM ; yM ) ; ;
2 2 2 2
x 2 1 y 2 1
and
2 2 2 2 ✓
x 3 y 3
Coordinates of D (3; 3) ✓ (3)
4.3 BC ( x2 x1 ) 2 ( y2 y1 ) 2
[1 2]2 [2 2]2
1 16 ✓
17 units ✓
(2)
4.4 2 (2)
mBC 4 ✓
2 1
tan 1 m tan 1 (4) 75,964o ✓ (2)
4.5 1
The gradient of a line perpendicular to BC
4
The equation of the line is
1
y y A ( x xA )
4
1
y 1 ( x 2) ✓ ✓
4
1 1
y 1 x
4 2
1 1
y x ✓
4 2 (3)
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MARKING GUIDELINE -7- T870(E)(N21)H
MATHEMATICS N3
4.6 The gradient is not defined when a line is parallel to the y-axis(that is, vertical). ALL
x co-ordinates are equal
x 2 is the equation of the line …………..equation 1 ✓
2y x 5 ……………………….equation 2
Solve equation 1 and 2 simultaneously
Substitute 1 into 2
2 y (2) 5
3
y ✓
2
3
Point of intersection ( 2; ) ✓
2 (3)
[15]
QUESTION 5
5.1 y 36 x 2 ✓ 1 mark for
scale
2 marks for
correct graph
✓ ✓
(3)
5.2 5.2.1 Since the zeros of f ( x ) are 0, 6 and 6 (the x-axis is tangent to the curve):
f ( x) x( x 6)( x 6) ✓ Coefficient of x3 is -1 as given.
f ( x) x3 12 x 2 36 x ✓
a 12, b 36 and c=0
✓ ✓ ✓
(5)
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MARKING GUIDELINE -8- T870(E)(N21)H
MATHEMATICS N3
5.2.2 f ' ( x) 3x 2 24 x 36 ✓
but at the turning point , f ' ( x ) 0
3x 2 24 x 36 0 ✓
x 2 8 x 12 0
✓
( x 6)( x 2) 0
x 6 or x 2
✓
If x 2 then f (2) (2)3 12(2) 2 36(2) 32
A(2; 32) (4)
5.3 1
3
yx 2
x2
1
y x 2 3 x 2 ✓
dy 1
6 x 3 ✓ ✓
dx 2 x
1 6
3
2 x x ✓ (4)
[16]
QUESTION 6
6.1 cos 2 (90o x) 1 cos x
sin(90 x) 1 sin x
o 2
cos x
cos 2 (90o x)
LHS
sin(90o x) 1 sin 2 x
sin 2 x ✓
cos x cos 2 x ✓ ✓
1 cos x2 ✓
cos x(1 cos x) ✓
1 cos x 1 cos x ✓
cos x(1 cos x)
1 cos x
✓
cos x
RHS (7)
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MARKING GUIDELINE -9- T870(E)(N21)H
MATHEMATICS N3
6.2
height of tower
tan 60o
150
height of tower 150 tan 60✓o 259,808 m ✓ ✓ (3)
6.3
6.3.1
Aˆ 127o 90o =37o
✓
Bˆ 270o 255o 15o
✓
Cˆ 180o 15o 37 o 128o
BC AB ✓
sin A sin C
BC 0, 67
sin 37 o
sin128o ✓
0, 67 sin 37 o
BC
sin128o
0,51 km ✓ (5)
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MARKING GUIDELINE -10- T870(E)(N21)H
MATHEMATICS N3
6.4 6.4.1
✓
✓shape
✓
✓shape
✓
✓
Shape(1 each = 2) Min TP (1 each = 2) Max TP (1 each = 2) = 6 (6)
6.4.2 13° and 103° ✓ ✓ (2)
[23]
TOTAL: 100
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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.
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(16030143) -2- T950(E)(A1)T
DEPARTMENT OF HIGHER EDUCATION AND TRAINING
REPUBLIC OF SOUTH AFRICA
NATIONAL CERTIFICATE
MATHEMATICS N3
TIME: 3 HOURS
MARKS: 100
INSTRUCTIONS AND INFORMATION
1. Answer ALL the questions.
2. Read ALL the questions carefully.
3. Number the answers according to the numbering system used in this question paper.
4. Questions may be answered in any order but subsections of questions must NOT be
separated.
5. Show ALL the calculations and intermediary steps.
6. ALL final answers must be accurately approximated to THREE decimal places.
7. ALL graph work must be done in the ANSWER BOOK. Graph paper is NOT
supplied.
8. Diagrams are NOT drawn to scale.
9. Write neatly and legibly.
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(16030143) -3- T950(E)(A1)T
QUESTION 1
1.1 Factorise the following expressions as far as possible in prime factors:
1.1.1 x(3x 2) y (3 y 2) (4)
1.1.2 4n 4 p 3n 2 p 1 (2)
1.2 Factorise the following expression completely:
2 x3 x 2 5 x 2 (5)
1.3 Simplify the following expression:
x 1 2 x 1 2 x 2 7 x 17
2
x 1 3 x x 2x 3 (6)
[17]
QUESTION 2
2.1 Simplify the following:
1
x
2 x
3
3x 2 (4)
2.2 Use logs to the base 2 and simplify the following WITHOUT using a calculator:
log0,5 128 (4)
2.3 Solve for x:
2.3.1 16 32 x 3x 2
2.3.2 (log x 2) log( x 2) 0 (2 4) (8)
[16]
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(16030143) -4- T950(E)(A1)T
QUESTION 3
3.1 Solve for x by completing the square:
4 x 48 2 x 2 (4)
3.2 Make ' b ' the subject of the formula:
xb
D
x b
(4)
3.3 Make ' w ' the subject of the formula:
log e t log e p log e w ds (3)
3.4 Alex paid a deposit of R3x for a computer. He paid the rest in 9 monthly
instalments. He paid a total of R33x . 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
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(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 k 7 . (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 y x 2 and
ˆ where B and C
y 3 x 3 respectively, intersect at A. Determine the size of BAC
are the intercepts on the x-axis as shown.
(5)
FIGURE B [21]
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(16030143) -6- T950(E)(A1)T
QUESTION 5
5.1 Draw the graph defined by the equation: 3x 2 3 y 2 27 (2)
5.2 Given : y x3 6 x 2 9 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)
dy 1
5.3 Determine if y 2 x . Leave the answers with positive indices and in
dx x
surd form. (4)
[14]
QUESTION 6
6.1 Prove the following trigonometric identity:
sin 2 A tan 2 A cos 2 A sec 2 A (4)
6.2 Calculate the value(s) of which will satisfy the equation if 0o 270o :
sin 1 cos 2 (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 25o
and the angle of depression of the foot of the tower is 20o . 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)
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(16030143) -7- T950(E)(A1)T
FIGURE C
6.4 Consider FIGURE D below. The sketch represents the graph of f ( x) a sin px
where 0 x . Determine the values of a and p .
(2)
FIGURE D [17]
TOTAL: 100
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(16030143) -1- T950(E)(A1)T
FORMULA SHEET
Any applicable formula may also be used.
1. Factors 2. Logarithms
a3 - b3 = (a - b)(a2 + ab + b2) log ab log a log b
a
a3 + b3 = (a + b)(a2 - ab + b2) log log a log b
b
logc a
3. Quadratic formula logb a
logc b
b b 2 4ac
x log a m m log a
2a
1
4. Parabola logb a
loga b
y ax 2 bx c loga a 1 ∴ 1n e 1
4ac b 2 aloga t t ∴ e1n m m
y
4a
b
x
2a
5. Circle 6. Straight line
x2 y2 r 2 y y1 m(x x1 )
Perpendicular: m1 m2 1
x2
D h
4h
Parallel lines: m1 m2
x 4Dh 4h 2
Distance: D (x 2 x1 )2 (y 2 y1 )2
x x 2 y1 y 2
Midpoint: P 1 ;
2 2
Angle of inclination: θ tan -1m
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(16030143) -2- T950(E)(A1)T
7. Differentiation
dy lim f x h f x
dx h → 0 h
d n
dx
x nx n - 1
Max/Min
For turning points: f ' x 0
8. Trigonometry
y 1
sinθ
r cosecθ
x 1
cosθ
r secθ
y 1
tanθ
x cotθ
sin 2 θ cos 2 θ 1
1 tan 2 θ sec 2 θ
1 cot 2 θ cosec2 θ
sinθ
tanθ
cosθ
cosθ
cotθ
sinθ
sinA sinB sinC
a b c
a 2 b 2 c 2 2bc cosA
Area of ∆ABC = ½ ac sin B
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MARKING GUIDELINE
NATIONAL CERTIFICATE
APRIL EXAMINATION
MATHEMATICS N3
1 APRIL 2016
This marking guideline consists of 10 pages.
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MARKING GUIDELINE -2- T950(E)(A1)T
MATHEMATICS N3
QUESTION 1
1.1 1.1.1 x (3x 2) y (3 y 2)
3x 2 2 x 3 y 2 2 y
3( x 2 y 2 ) 2( x y )
3( x y )( x y ) 2( x y )
( x y )[3( x y ) 2]
( x y )[3x 3 y 2] (4)
1.1.2 4n 4 p 3n 2 p 1
(n 2 p 1)(4n 2 p 1) (2)
1.2 f ( x) 2 x 3 x 2 5 x 2
f (1) 2(1)3 (1) 2 5(1) 2
=2 1 5 2
=0
x-1 is a factor of f ( x)
2 x 2 3x 2
x-1 2 x3 x 2 5 x 2
2 x3 2 x 2
3x 2 5 x
3x 2 3 x
- 2x 2
-2 x 2
. .
f ( x) ( x 1)(2 x 2 3 x 2)
= ( x 1)(2 x 1)( x 2)
(5)
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MARKING GUIDELINE -3- T950(E)(A1)T
MATHEMATICS N3
1.3
x 1 2 x 1 2 x 2 7 x 17
x 1 x 3 ( x 1)( x 3)
( x 1)(x 3) (2 x 1)(x 1) 2 x 2 7 x 17
=
( x 1)(x 3)
x 2 4 x 3 2 x 2 x 1 2 x 2 7 x 17
=
( x 1)(x 3)
5( x 2 2 x 3)
( x 1)(x 3)
(6)
[17]
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MARKING GUIDELINE -4- T950(E)(A1)T
MATHEMATICS N3
QUESTION 2
2.1 1
x
2 x
3
3x 2
2x 1 3
1
3x 2
2
2x
2x 1 1
1
3
2 2
2x 3x
2x 1
6 x2 (4)
2.2 log 0,5 128
log 2 128
1
log 2
2
log 2 27
log 2 21
7 log 2 2
1log 2 2
7 (4)
2.3 2.3.1 16 32 x 3x 2
2
16 32 x 3x 2
16 32 x 32 x 4.3x 4
12 4.3x
3x 3
x 1
TEST : If x 1 then LHS=RHS=5 (4)
2.3.2 (log x 2) log( x 2) 0
log x 2 0 or log( x 2) 0
log x 2 x 2=100
x 102 x 2 1
x 100 x3 (4)
[16]
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MARKING GUIDELINE -5- T950(E)(A1)T
MATHEMATICS N3
QUESTION 3
3.1
(4)
3.2
xb
D2
xb
(4)
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MARKING GUIDELINE -6- T950(E)(A1)T
MATHEMATICS N3
3.3
tw
loge ds
p
tw
e ds
p
p ds
w e
t (3)
3.4 Deposit is R3x
Total price is R33x
Total amount for 9 instalments= R30x
30 x 10 x
Each monthly instalment = R R
9 3 (4)
[15]
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MARKING GUIDELINE -7- T950(E)(A1)T
MATHEMATICS N3
QUESTION 4
4.1 4.1.1
x2 x1 y2 y1
2 2
AB
= 4 2 5 1
2 2
= 4 16
= 20
=2 5 (2)
4.1.2 BC=2 5
0 4 k 5 20
2 2
16 k 5
2
= 20
k 5 = 4
2
k 5 = 2
k 5 2 or k 5 2
k 7 k 3 (4)
4.1.3 5 1
M AB 2
42
75 1
M CB
04 2
M AB M CB 1
therefore CB AB (3)
4.1.4 1
Area of ΔABC = × base × height
2
1
=AB BC
2
1
= 20 20
2
=10 units 2
(3)
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MARKING GUIDELINE -8- T950(E)(A1)T
MATHEMATICS N3
4.2
x x y y
coordinates of M 1 2 ; 1 2
2 2
2 4 1 7
= ;
2 2
=(1;3)
Equation x =1
(4)
4.3 M AB 1 M AC 3
tan 1 tan =3
=45o =71,565o
ˆ = = 71,565o 45o = 26,565o
BAC (5)
[21]
QUESTION 5
5.1 3x 2 3 y 2 27
Y
3
-3 0 3 X
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MARKING GUIDELINE -9- T950(E)(A1)T
MATHEMATICS N3
5.2.1 y x3 6 x 2 9 x
dy
3x 2 12 x 9
dx
Let y 0
3 x 2 12 x 9 0
3( x 2 4 x 3) 0
( x 1)( x 3) 0
x 1 or x 3
f (1) (1)3 6(1) 2 9(1) 4
f (3) (3)3 6(3) 2 9(3) 0
(5)
Turningpoints (3;0) and ( 1; 4)
5.2.2
(3)
5.3 1
y 2 x
x
1
1
y x 2x 2
1
dy 2
x 2(0,5)x 2
dx
dy 1 1
2
dx x x (4)
[14]
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MARKING GUIDELINE -10- T950(E)(A1)T
MATHEMATICS N3
QUESTION 6
6.1
(4)
6.2 sin 1 cos 2
sin sin 2
sin 2 sin 0
sin (sin 1) 0
sin 0 or sin 1
0 o or 90o
or 180o (5)
6.3 6.3.1
In ABC :
AB 30
o
sin 70 sin 45o
30sin 70o
AB
sin 45o (3)
39,868m
6.3.2 In ABD :
AD
cos 25 o
AB
AD AB cos 25o
39,868 cos 25o
(3)
36,133m
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MARKING GUIDELINE -11- T950(E)(A1)T
MATHEMATICS N3
6.4 a2
p2 (2)
[17]
TOTAL: 100
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1
T950(E)(A2)T
APRIL EXAMINATION
NATIONAL CERTIFICATE+
MATHEMATICS N3
(16030143)
2 April 2015 (Y-Paper)
13:00–16:00
This question paper consists of 7 pages and a formula sheet of 2 pages.
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(16030143) -2- T950(E)(A2)T
DEPARTMENT OF HIGHER EDUCATION AND TRAINING
REPUBLIC OF SOUTH AFRICA
NATIONAL CERTIFICATE
MATHEMATICS N3
TIME: 3 HOURS
MARKS: 100
NOTE: Diagrams are NOT drawn to scale.
INSTRUCTIONS AND INFORMATION
1. Answer ALL the questions.
2. Read ALL the questions carefully.
3. Number the answers according to the numbering system used in this question paper.
4. Show ALL the calculations and intermediary steps.
5. Questions may be answered in any order but subsections of questions must NOT be
separated.
6. ALL final answers must be accurately approximated to THREE decimal places.
7. ALL graph work must be done in the ANSWER BOOK. Graph paper is NOT
supplied.
8. A formula sheet is attached to this question paper. The list is NOT necessarily
complete. Any other applicable formula may be used.
9. Write neatly and legibly.
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(16030143) -3- T950(E)(A2)T
QUESTION 1
1.1 Simplify the following expression without using a calculator.
1
1
12
1
x x
2 2
x x 2
(4)
1.2 Determine the value of p if x 2 is a factor of the following function:
f ( x) 2 x 3 px 2 4 x 5 (3)
1.3 Factorise as far as possible in prime factors:
1.3.1 x2 y 2 2 x 2 y (3)
1.3.2 2
2x2 3 (4)
x2
1.4 Simplify:
p 2 121 p2 2 p 22
2 (6)
p 4 2 p 22 p 2 p 11( p 2)
2
[20]
QUESTION 2
2.1 Solve for x :
2.1.1 27 x 9 x 2 1 (3)
2.1.2 2x 2x2 12 (4)
2.1.3 log x log 3 log(3 x 1) log 4 (4)
2.2 The area of a rectangular swimming pool, 10 m long and 3 m wide, doubles if its
length and width are increased by the same amount. Calculate the dimensions. (4)
2.3 Make m the subject of the following formula:
2mgh
W
mr 2 I (4)
2.4 Make T the subject of the following formula:
900
t 0,1e T
(4)
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(16030143) -4- T950(E)(A2)T
2.5 Determine x by completing the square:
3 x 10 x 2
(4)
[27]
QUESTION 3
3.1 In FIGURE 1 below L(-5;-2), M(-1;-6) and K(5;4) are vertices of triangle KLM in a
Cartesian plane.
FIGURE 1
3.1.1 Calculate the coordinates of the midpoint of MK. (2)
3.1.2 Determine the gradient of LM. (2)
3.1.3 Determine the length of LM. Leave the answer in surd form. (2)
3.1.4 Determine the equation of the line parallel to LM passing through the
point K. Leave your answer in general form. (3)
3.2 Determine the equation of the straight line that passes through the points A(2;4) and
B(-1;-3). Leave your answer in gradient-intercept form. (3)
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(16030143) -5- T950(E)(A2)T
1
3.4 Determine the equation of the line AB that is perpendicular to the line f ( x) x in
3
FIGURE 2 below. AB cuts the x-axis at x 6 . Leave the answer in surd form and in
gradient-intercept form.
(3)
FIGURE 2
[18]
QUESTION 4
dy
4.1 Determine by using the rules of differentiation. Leave the answers with positive
dx
indices and in surd form.
2
4.1.1 y 2 x (3)
x
x5 x3
4.1.2 y (3)
5x2
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(16030143) -6- T950(E)(A2)T
4.2 Consider FIGURE 3 below. Determine the x-coordinates of the turning points P and
Q of f ( x) x3 9 x .
(3)
FIGURE 3 [9]
QUESTION 5
5.1 The design of a bridge in FIGURE 4 shows an arch represented by a parabola, which
is defined by the equation y 2 x 0, 25 x 2 .
ww.dragoart.com
FIGURE 4
5.1.1 Draw the graph of the parabola. (3)
5.1.2 Use the graph to determine the height of the bridge (in meters) if the line
y = 0 represents the water level of the river. (1)
5.1.3
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(1)
(16030143) -7- T950(E)(A2)T
5.2 Draw the graphs of the following trigonometric functions on the same system of axes
for 0 x 180 :
y 3 sin 2 x and y 2 cos 3 x
ALL values at the point of intersection with the system of axes and the co-ordinates
of the turning points must be shown. (6)
[11]
QUESTION 6
6.1 Simplify the following without using a calculator:
tan(180o A) 1 sin 2 A
cos 2 (180o A) sin 2 (360o A) (4)
6.2 Make use of basic trigonometric identities to prove that:
1 cos sin
2 cos ec
sin 1 cos (4)
6.3 Calculate the values of that will satisfy the following equation for 0 360 :
sin 2 cos (3)
6.4 Consider FIGURE 5. From the top of a cliff 60 m above sea level the angles of
depression of two ships on the same vertical plane as the observation point are 20°
and 25° respectively. Calculate the distance between the two ships.
FIGURE 5 (4)
[15]
TOTAL: 100
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(16030143) -8-
-1- T950(E)(A2)T
MATHEMATICS N3
FORMULA SHEET
Any applicable formula may also be used.
1. Factors/Faktore 2. Logarithms/Logaritmes
a 3 b 3 a b a 2 ab b 2 log ab log a log b
a 3 b 3 a b a 2 ab b 2
a
log log a log b
b
3. Quadratic formula/ log c a
logb a
Kwadratiese formule log c b
b b 2 4ac
x
2a log a m m log a
1
4. Parabola/Parabool logb a
log a b
5. y ax bx c
2
log a a 1 ln e 1
4ac b 2
y
4a
a loga t t e ln m m
b
x
2a
5. Circle/Sirkel 6. Straight Line/Reguitlyn
x2 y2 r 2 y y1 mx x1
Perpendicular:
x2 m1 m2 1
D h Loodreg:
4h
Parallel Lines:
x 4 Dh 4h 2 Ewewydige lyne: m1 m2
Distance:
Afstand: D x 2 x1 2 y 2 y1 2
Midpoint:
x x 2 y1 y 2
Middelpunt: P 1 ;
2 2
Angle of inclination:
Hellingshoek: tan 1 m
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(16030143) -2--9- T950(E)(A2)T
7. Differentiation/Differensiasie
dy lim f x h f x
dx h 0 h
d n
dx
x nx n 1
Max/ Min
Maks/ Min
For turning points:
Vir draaipunte: f 1 x 0
8. Trigonometry/Trigonometrie
y 1
sin
r cos ec
x 1
cos
r sec
y 1
tan
x cot
sin 2 cos 2 1
1 tan 2 sec 2
1 cot 2 cos ec 2
sin
tan
cos
cos
cot
sin
sin A sin B sin C
a b c
a 2 b 2 c 2 2bc cos A
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MARKING GUIDELINE
NATIONAL CERTIFICATE
APRIL EXAMINATION
MATHEMATICS N3
2 APRIL 2015
This marking guideline consists of 9 pages.
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MARKING GUIDELINE -2- T930(E)(A2)T
MATHEMATICS N3
QUESTION 1
1.1 1
12
11 1
1
12
1
x x x x 2
2 2
x x2 2
x x 2
1 1 1
1 1
1
x x x2 x 2 x2 x 2
x x 1
x 1 x 1 2x 2
x x 2 2 x
x 1 x 1 x x
x
2 2 x (4)
x x
1.2 x 2
f ( x) 2 x3 px 2 4 x 5
f (2) 2(2)3 p(2)2 4(2) 5
16 4 p 8 5
4 p 13
4 p 13 0
13
p
4
(3)
1.3 1.3.1 x2 y 2 2 x 2 y
( x y )( x y ) 2( x y )
( x y )( x y 2)
(3)
1.3.2 2
2x2 3
x2
2 x 4 3x 2 2
x2
(2 x 2 1)( x 2 2)
x2 (4)
1.4 p 2 121 p2 2 p 22
2
p 4 2 p 22 p 2 p 11( p 2)
2
p 11 ( p 11) p 2 p(p 2) 11( p 2)
p 2 ( p 2) 2( p 11) 2( p 11)
p 11 ( p 11) (p 2) (p 2)(p 11)
p 2 ( p 2) 2( p 11) 2( p 11)
p 11
(6)
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MARKING GUIDELINE -3- T930(E)(A2)T
MATHEMATICS N3
QUESTION 2
2.1 2.1.1 33x 3 2 x4 30
3x 2x 4 0
5x 4
4
x (3)
5
2.1.2 2 x 2 x 2 12
x x2
2 2 2 2 12
x x
2 2 2 2 .2 12
x
2 2 1 2 12
x
2 2 22
x4
(4)
2.1.3 log x log 3 log(3 x 1) log 4
log x log 3 log(3 x 1) log 4
3x
log log 4
(3x 1)
3x 12 x 4
4
x
9 (4)
2.2 (10 x)(3 x) 30 2
30 13x x 2 60
x 2 13x 30 0
x 15 or x 2
New length = 12 m
New width = 5 m
(4)
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MARKING GUIDELINE -4- T930(E)(A2)T
MATHEMATICS N3
2.3 2mgh
w2
mr 2 I
w2 mr 2 w2 I 2mgh
w2 mr 2 2mgh w2 I
m( w2 r 2 2 gh) w2 I
w2 I w2 I
m or m
w2 r 2 2 gh 2 gh w2 r 2 (4)
2.4 900
t 0,1e T
900
t
e T
0,1
900
t
ln ln e T
0,1
t 900
ln
0,1 T
900 900
T
ln
t ln t ln 0,1
0,1 (4)
2.5 3
2
3
2
x 2 3 x 10
2 2
2
3 9
x 10
2 4
3 9
x 10
2 4
x
3 49
2 4
3 7
x
2 2
x 2 or x 5 (4)
[27]
QUESTION 3
3.1 3.1.1 midpoint of MK
1 5 6 4
;
2 2
(2)
= (2;-1)
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MARKING GUIDELINE -5- T930(E)(A2)T
MATHEMATICS N3
3.1.2 62
m
1 5
4
=
4
m 1 (2)
3.1.3 LM (1 5)2 (6 2)2
= 16 16
= 2 16 (2)
=4 2
3.1.4 y y1 m( x x1 )
y 4 1( x 5)
y 4 x 5
y x90
(3)
3.2 43
m
2 1
7
3
y y1 m( x x1 )
7
y 4 ( x 2)
3
7 14
y4 x
3 3
7 2
y x or y 2,333x 0,667 (3)
3 3
3.3 m tan
7
tan 1
3
(3)
= 66,8
3.4 m2 3
y y1 m( x x1 )
y 0 3( x 6)
y 3x 6 3
(3)
[18]
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MARKING GUIDELINE -6- T930(E)(A2)T
MATHEMATICS N3
QUESTION 4
4.1 4.1.1 2
y 2 x
x
1
y 2 x 1 2 x 2
dy 1
2 x 2 x 2
dx
2 1
= 2
x x
(3)
4.1.2 x5 x3
y
5x2
1 1
y x3 x
5 5
dy 3 2 1
x
dx 5 5 (3)
4.2 y x 3 9x
dy
3x 2 9
dx
3( x 2 3) 0
x2 3
x 3
x-coordinate of P 3 (3)
[9]
x coordinate of Q= 3
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MARKING GUIDELINE -7- T930(E)(A2)T
MATHEMATICS N3
QUESTION 5
5.1 5.1.1
(3)
5.1.2 4 meters (1)
5.1.3 8 meters (1)
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MARKING GUIDELINE -8- T930(E)(A2)T
MATHEMATICS N3
5.2
y 3sin 2 x y 2 cos 3 x
x-intercepts: 0o ,90o ,180o x-intercept : 30o ,90o ,150o
y-intercept:0o y-intercept : 2
maxpoint:(45o ,3) maxpoint:(0o , 2), (120o , 2) (6)
minpoint : (135 , 3),
o
minpoint : (60 , 2), (180 , 2)
o o [11]
QUESTION 6
6.1 tan(180o A) 1 sin 2 A
cos 2 (180o A) sin 2 (180o A)
tan A cos 2 A
cos A sin A
2 2
tan A cos A
cos 2 A sin 2 A
sin A (4)
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MARKING GUIDELINE -9- T930(E)(A2)T
MATHEMATICS N3
6.2 (1 cos )(1 cos ) (sin )(sin )
sin (1 cos )
1 2 cos cos2 sin 2
=
sin (1 cos )
2(1 cos )
=
sin (1 cos )
2
=
sin
= 2 cos ec
LHS RHS (4)
6.3 sin 2 cos
cos cos
tan 2
tan 1 2
reference α 63, 4
116,6
or (3)
296,6
6.4 60
In ABD tan 20o
BD
BD 164,849 m
60
In ACD tan 25o
CD
CD 128, 67 m
BC=BD CD=36,179 m (4)
[15]
TOTAL: 100
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16030143 -2- T930(E)(N18)T
T940(E)(N18)T
NOVEMBER EXAMINATION
NATIONAL CERTIFICATE
MATHEMATICS N3
(16030143)
18 November 2014 (Y-Paper)
13:00–16:00
This question paper consists of 6 pages and 1 formula sheet of 2 pages.
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(16030143) -2- T940(E)(N18)T
DEPARTMENT OF HIGHER EDUCATION AND TRAINING
REPUBLIC OF SOUTH AFRICA
NATIONAL CERTIFICATE
MATHEMATICS N3
TIME: 3 HOURS
MARKS: 100
________________________________________________________________________
INSTRUCTIONS AND INFORMATION
1. Answer ALL the questions.
2. Show ALL the calculations and intermediary steps.
3. Questions may be answered in any order but subsections of questions must NOT be
separated.
4. ALL final answers must be accurately approximated to THREE decimal places.
5. All graph work must be done in the answer book. Graph paper is NOT supplied.
6. Note that diagrams are NOT drawn to scale.
7. A formula sheet is attached to this question paper. The list is NOT necessarily
complete. Any other applicable formula may be used.
8. Write neatly and legibly.
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(16030143) -3- T940(E)(N18)T
QUESTION 1
1.1 Simplify the following WITHOUT using a calculator:
1.1.1 log x a (log a b) (2)
1.1.2 x y
(3)
x x
1.1.3 2 18 3 32
(3)
2 8
1.2 Make use of the factor theorem to prove that x 7 is a factor of the function
f ( x) x 3 39x 70 (3)
1.3 Factorise the following two expressions as far as possible in prime factors:
1.3.1 p 2 4m 2 4m 1 (3)
1.3.2 12a 2 27b 2 (2)
1.4 Simplify the following expression:
x 2 2x x 3 x 3
3 (4)
x 4 x x 4x 4 x 2
2 2
[20]
QUESTION 2
2.1 Solve for x in the following equations :
2.1.1 x5 x 3 (3)
2.1.2 log5 x log5 x3 log5 x5 18 (3)
2.1.3 x
x 1 x2 1
3 3 11
3 (4)
2.2 Make r the subject of the following formula:
P 0, 4 e k r (4)
2.3 Make x the subject of the formula:
x 1
H (4)
x2
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(16030143) -4- T940(E)(N18)T
2.4 Solve for x by completing the square:
x 2 12x 10 (4)
2.5 Calculate the coordinates of the points of intersection of the graphs defined by the
following two equations:
x
2 y 17
5
y
x7 (5)
4
2.6 The length of the rectangular field is 6 m longer than 3 times its breadth. What are
the measurements of the rectangle if the perimeter is 188 m ? (3)
[30]
QUESTION 3
3.1 Consider FIGURE 1. The parabola f in the sketch cuts the x-axis at E and P and the
y-axis at M. The straight line g cuts the x- and y-axis at E and M respectively. The
functions f and g are defined by f ( x) x 2 4 x 5 and g ( x) mx c .
FIGURE 1
Determine the following:
3.1.1 The co-ordinates of M, E and P which are the intercepts of the graphs
with the system of axes (3)
3.1.2 The co-ordinates of the turning point N of the graph of f (2)
3.1.3 The values of m and c in g ( x) mx c (2)
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3.1.4 The length of EM in simplified surd form (2)
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(16030143) -5- T940(E)(N18)T
3.2 Determine the equation of a straight line that passes through the point 1; 2 and
which is perpendicular to the line y 4 x 2 .Write your answer in general form.
(3)
3.3 Determine the value of p if:
3.3.1 p is the x -coordinate of the midpoint of the line segment joining (3, 1)
and (1, 3)
3.3.2 p is the angle of inclination of the line y 3x 1
x2 y 2
3.3.3 p is the radius of the circle defined by 81
9 9
3.3.4 p is the gradient of the line which is parallel to x 2 y 0
3.3.5 p is the y-intercept of the ellipse defined by x 2 9 y 2 81
(5 x 2) (10)
[22]
QUESTION 4
4.1 f ( x h) f ( x )
Determine lim for the following function:
h 0 h
f ( x) 4 x 2 (4)
4.2 dy
Determine by making use of the rules of differentiation. Write your answer with
dx
positive exponents and in surd form when applicable.
4.2.1 1
y 4 4 x (4)
x4
4.2.2 x3 x
y (3)
x 1
[11]
QUESTION 5
5.1 Solve the following equation if 0o 360o :
3cot 2 9,5 (3)
5.2 Use trigonometric identities to prove that tan 2 sin 2 tan 2 .sin 2 (4)
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(16030143) -6- T940(E)(N18)T
5.3 Consider FIGURE 2. A window washer on a ladder looks at a nearby building 100 m
away, noting that the angle of elevation of the top of the building is 18, 7 o and the
angle of depression of the foot of the building is 6,5o . How tall is the nearby building?
FIGURE 2 (4)
5.4 Consider FIGURE 3. Calculate the length of AB by using the sine rule.
FIGURE 3 (2)
5.5 Draw the graphs which are represented by the following trigonometric equations on
the same system of axes for 0 x 180 :
f ( x) cos x
g ( x) sin( x 30o ) (4)
[17]
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100
16030143 -1- T930(E)(N18)T
FORMULA SHEET
Any other applicable formula may also be used.
1. Factors/Faktore 2. Logarithms/Logaritmes
a 3 b 3 a b a 2 ab b 2 log ab log a log b
a 3 b 3 a b a 2 ab b 2
a
log log a log b
b
3. Quadratic formula/ log c a
Kwadratiese formule logb a
log c b
b b 2 4ac log a m m log a
x
2a
1
logb a
4. Parabola/Parabool log a b
5. y ax bx c
2
log a a 1 ln e 1
4ac b 2
y
4a a loga t t e ln m m
b
x
2a
5. Circle/Sirkel 6. Straight line/Reguitlyn
x2 y2 r 2 y y1 mx x1
Perpendicular:
x2 m1 m2 1
D h Loodreg:
4h
Parallel lines:
x 4 Dh 4h 2 Ewewydige lyne: m1 m2
Distance:
Afstand: D x 2 x1 2 y 2 y1 2
Midpoint:
x x 2 y1 y 2
Middelpunt: P 1 ;
2 2
Angle of inclination:
Hellingshoek: tan 1 m
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(16030143) -2- T930(E)(N18)T
7. Differentiation/Differensiasie
dy lim f x h f x
dx h 0 h
d n
dx
x nx n 1
Max/Min
Maks/Min
For turning points:
Vir draaipunte: f 1 x 0
8. Trigonometry/Trigonometrie
y 1
sin
r cos ec
x 1
cos
r sec
y 1
tan
x cot
sin 2 cos 2 1
1 tan 2 sec 2
1 cot 2 cos ec 2
sin
tan
cos
cos
cot
sin
sin A sin B sin C
a b c
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a 2 b 2 c 2 2bc cos A
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MARKING GUIDELINE
NATIONAL CERTIFICATE
NOVEMBER EXAMINATION
MATHEMATICS N3
18 NOVEMBER 2014
This marking guideline consists of 9 pages.
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MARKING GUIDELINE -2- T940(E)(N18)T
MATHEMATICS N3
QUESTION 1
1.1 1.1.1 log x a (log a b) or log x a
log x b apply law
log x a simplify and
log a log b
write it as ONE
log x log a log x b log
log b
log x
log x b (2)
1.1.2 x y
x x
common
x . x xy denominator
x x factorisation
x xy x (1 y ) divide and
rationalized
x x x x denominator
(1 y )
x
x (1 y )
x (3)
1.1.3 2 3 2 3 4 2
prime factors in
22 2 roots
2 6 12 factorisation
2 1 2 value
(3)
2
1.2 Let x 7
f (7) 7 3 39(7) 70 value of x
substitution
343 273 70
=0 deduction
x 7 is a factor of f(x) (3)
1.3 1.3.1 p 2 4m 2 4m 1
factorise
p 2 (2m 1)(2m 1) factorise
p 2 (2m 1) 2 simplify
[ p 2m 1][ p 2m 1] (3)
1.3.2
3 4a 2 9b 2 factorise
factorise
32a 3b2a 3b
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(2)
MARKING GUIDELINE -3- T940(E)(N18)T
MATHEMATICS N3
1.4 x2 2x x3 x3 division sign
3 2 factorise first
x 4 x x 4x 4 x 2
2
term
x( x 2) x 2 ( x 1) 4( x 1) x 3 grouping
2
x 4 x 3 x2 answer
x( x 2) ( x 4)( x 1) x 3
2
x2 4 x 3 x2
x( x 1) (4)
[20]
QUESTION 2
2.1 2.1.1 x5 x 3
square both sides
x 5 x 3
2
factorise
x 5 x2 6x 9 solve
(penalise with ½
x2 5x 4 0 mark if -4 is not
( x 1)( x 4) 0 shown as invalid)
x 1 or x 4 (3)
2.1.2 log 5 x log 5 x 3 log 5 x 5 18 write it as one log
by applying law
log 5 x 9 18
9 log 5 x 18 applying
log 5 x 2 definition of log
x 52
value
x 25 (3)
2.1.3 x
x 1 1
x2
3 3 11 applying
3 exponent law
x 1 x2
3 3 3x 11 factorising
3x 3 9 1 11 use exponent rule
to find value
3 1x
value
3x 30 (4)
x0
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MARKING GUIDELINE -4- T940(E)(N18)T
MATHEMATICS N3
2.2 P 0, 4 e k r manipulate
using logs
P
e k r simplify
0, 4 make r the subject
P
ln ln e k r
0, 4
P
ln kr
0, 4
P
ln
r
0, 4
k
ln P ln 0, 4 ln 0, 4 ln P
(4)
k k
2.3 x 1
H square both sides
x2
cross multiply
x 1 take out t as
H2
x2 common factor
H x 2 x 1
2
simplify
H 2 x 2H 2 x 1
H 2 x x 2 H 2 1
x( H 2 1) 2 H 2 1
(2 H 2 1) (4)
x
H 2 1
2.4 x 2 12 x 10 0 write LHS as
2 2 perfect square
12 12 x subject of the
x 12 x 10
2
2 2 formula
both values
x 6 46
2
correct rounded
x 6 46 off-penalize only
with one here
x 6 46
x 12, 782 or x 0, 782
(4)
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MARKING GUIDELINE -5- T940(E)(N18)T
MATHEMATICS N3
2.5 x
2 y 17 ...(1) simplify
5
y substitutions
x7 ....(2) simplifying
4 value of x
x 10 y 85 x 85 10 y value of y
Replace x by 85 -10 y in equation 2:
y 4 x 28
y 4(85 10 y ) 28
y 340 40 y 28
39 y 312
y 8
Replace y by 8 in equation 2 :
8
x7
4
x5
(5)
2.6 Let breath be equal to x Any steps
Then length 3x 6 answer
Perimeter 2 l b
188 2[ x 3 x 6]
8 x 12 188
x 22m
3x 6 72m (3)
[30]
QUESTION 3
3.1 3.1.1 let y 0 M
let x 0 E
0 x2 4 x 5
y 005 P
x 5
y5
or x 1
M (0;5) (3)
E (5;0) and P (1;0)
3.1.2 4 x-coordinate
x
2 1 y-coordinate
x 2
y 2 4 2 5
2
y 9
(2)
N 2;9
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MARKING GUIDELINE -6- T940(E)(N18)T
MATHEMATICS N3
3.1.3 50 y mx c m
m
05 5 10 c c (2)
m 1 c 5
3.1.4 EM ( x 2 x1 ) 2 ( y 2 y1 ) 2
answer in
(5 0) 2 (0 5) 2 surd form
25 25
(2)
50 = 5 2
3.2 1
m1 4 m2 gradient
4
y y1 m( x x1 ) substitution
simplify
y 2 1 ( x 1)
4
1
y21 x
4 4
1 7
y x 0 (3)
4 4
4y x 7 0
3.3.1 x1 x2 value of p
p
2
3 1
2 (2)
2
3.3.2 tan p 3 value of p
(2)
p 60o
3.3.3 x2 y 2 value of p
81
9 9
x 2 y 2 729
r p 27 (2)
3.3.4 x 2y 0 value of p
1
y x
2
(2)
1
p
2
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MARKING GUIDELINE -7- T940(E)(N18)T
MATHEMATICS N3
3.3.5 or y-intercept value of p
x 2 9 y 2 81 (2)
9 y 2 81
x2 y 2
1 y2 9
92 32
y 3
p 3 [22]
p 3
QUESTION 4
4.1 f ( x h) f(x)
lim multiplying of
h 0 h
4 x 2 8 xh 4h 2 4 x 2
2(x+h)² and
lim substituting
h 0 h factorising
4h(2 x h) simplifying
lim answer
h 0 h
4(2 x 0)
(4)
8 x
4.2.1 1
y 4
4 4 x rewriting by
x
1 applying
y x 4 4 x 4 exponential rules
1
1
applying
dy 1
4 x 5 4 x 4 differentiation
dx 4 rule
5
3
writing it as
4 x x 4
positive exponent
4 1 and in surd form (4)
5
x 4
x3
4.2.2 x( x 2 1) factorise
y differentiate
x 1
x( x 1)( x 1)
y
x 1
yx x
2
dy (3)
2x 1
dx [11]
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MARKING GUIDELINE -8- T940(E)(N18)T
MATHEMATICS N3
QUESTION 5
5.1 3cot 2 9,5
cot 2,5 manipulate for
two answers
tan 0, 4 (3)
21,801 o
or 201,801 o
5.2 tan 2 sin 2 tan 2 .sin 2
applying quotient
LHS tan 2 sin 2
identity
sin 2 finding die LCD
sin 2
cos
2 factorise
sin 2 sin 2 cos 2 proof
cos 2
sin 2 (1 cos 2 )
cos 2
tan 2 .sin 2
(4)
RHS
5.3 BC determining
tan18, 7 o BC
100
BC 33,848 determining CD
adding BC and
CD
tan 6,5o CD to determine
100 the height
CD 11,394
Height BC CD
(4)
45, 242m
5.4 17 sin 72o replacing
AB
sin 50o answer
21,106 m (2)
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MARKING GUIDELINE -9- T940(E)(N18)T
MATHEMATICS N3
5.5
x
f ( x ) cos
2
y-intercept=1
x-
intercept=90°
Or any other
point on the
graph
g ( x) sin( x 300 )
y-intercept=
- 0,5
x-
intercept=30°
Or any other
point on the
(4)
graph
[17]
TOTAL: 100
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T930(E)(J28)T
AUGUST EXAMINATION
NATIONAL CERTIFICATE
MATHEMATICS N3
(16030143)
28 July 2014 (Y-Paper)
13:00–16:00
Diagrams are NOT drawn to scale.
This question paper consists of 7 pages, and 1 formula sheet of 2 pages.
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(16030143) -2- T930(E)(J28)T
DEPARTMENT OF HIGHER EDUCATION AND TRAINING
REPUBLIC OF SOUTH AFRICA
NATIONAL CERTIFICATE
MATHEMATICS N3
TIME: 3 HOURS
MARKS: 100
INSTRUCTIONS AND INFORMATION
1. Answer ALL the questions.
2. Show ALL the calculations and intermediary steps.
3. Questions may be answered in any order but subsections of questions must NOT be
separated.
4. ALL final answers must be accurately approximated to THREE decimal places.
5. All graph work must be done in the ANSWER BOOK. Graph paper is NOT supplied.
6. A formula sheet is attached to this question paper. The list is NOT necessarily
complete. Any other applicable formula may be used.
7. Write neatly and legibly.
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(16030143) -3- T930(E)(J28)T
QUESTION 1
1.1 Simplify the following WITHOUT using a calculator: (5)
a 1 a 1
2 8
3
4a 1
1.2 Prove that log c a logb c log a b 1
(3)
1.3 Determine the factors of the following function if x 1 is one of the factors.
f ( x) x 3 3 x 2 x 3 (4)
1.4 Factorise the following expressions as far as possible in prime factors
1.4.1 x 2 ( x 1) (1 x) (3)
1.4.2 a 2 2a 3 ab b (3)
1.5 Simplify the following:
a 2 b 2 a 2 2ab b 2 a b
2a b 4a 2 b 2 ab (4)
[22]
QUESTION 2
2.1 Solve for ' a ' by completing the square:
4a 2 12a 7 0 (5)
2.2 Solve for ' x ' :
2.2.1 x (4)
x2 x 2 1
2 2 42
2
2.2.2 2x 4 4 x
2
x 1 x 1 x 1 (4)
2.3 Make ‘ n ‘ the subject of the formula :
P =100e 0,3n (4)
2.4 Make ‘ r ’ the subject of the formula:
a r 2 rs (2)
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(16030143) -4- T930(E)(J28)T
2.5 Calculate the coordinates of the points of intersection of the graphs defined by the
following two equations:
y 2 x 2 8 x 10 (4)
y 9 x
2.6 The product of two consecutive odd numbers is 143. Calculate the numbers. (3)
[26]
QUESTION 3
3.1 ABC has vertices A(2;3) , B ( 2; 1) and C (4;1) .
3.1.1 Draw ABC on a set of axes. (1)
3.1.2 Determine the gradient of AB . (1)
3.1.3 Determine the angle of inclination of the line AB . (1)
3.1.4 Show that AB is perpendicular to AC . (2)
3.1.5 Determine the equation of the line through C and parallel to AB . (3)
3.1.6 Find D and E , the midpoints of AC and BC respectively. (4)
3.1.7 Show that AB 2 DE (5)
3. 2 Choose a description from COLUMN B that matches an item in COLUMN A. Write
only the letter (A-G) next to the question number(3.2.1-3.2.4) in the ANSWER
BOOK.
COLUMN A COLUMN B
3.2.1
A x 2 y 2 10
x y
B 100
1 10 (2)
x2 y2
C 100
1 10
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(16030143) -5- T930(E)(J28)T
D x y2
3.2.2
(2)
E y 1 x2
x2 y 2
F 1
1 1
3.2.3 x y (2)
G 1
1 1
H y x2
x2 y 2
I 1
1 100
(2)
3.2.4
[25]
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(16030143) -6- T930(E)(J28)T
QUESTION 4
4.1 dy
Determine by making use of the rules of differentiation. Write your answer with
dx
positive exponents and in surd form when applicable.
4.1.1 2
y2 x
x (4)
4.1.2 x8 1
y
x4 1 (2)
4.2 Determine the gradient of the tangent to the following curve at the point where x 2 :
y ( x 2 3)( x 1) (3)
[9]
QUESTION 5
5.1 Make use of basic trigonometric identities to prove that
1 sin cos
2 sec
cos 1 sin (5)
5.2 ˆ 250 , length AB 80 and length AC 46 .
Given: ABC with BAC
Use the cosine rule to determine the length of BC .
(3)
5.3 Solve the following equation if 0 360 :
2 sec 4 0 (3)
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(16030143) -7- T930(E)(J28)T
5.4 A vertical antenna AC is mounted on top of a pole CD . The technician measures the
angles of elevation of two points on the antenna at a distance of 40 m from D , the base
ˆ and ABD
of the pole . If the angles, CBD ˆ , are 20° and 24° respectively, find the length (4)
of the antenna.
5.5 The graphs represent the functions f ( x) 2 sin 2 x and g ( x) cos x . A , B and P are
turning points. Determine the coordinates of A , B and P .
(3)
[18]
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(16030143) -8-
-1- T930(E)(J28)T
MATHEMATICS N3
FORMULA SHEET
Any applicable formula may also be used.
1. Factors/Faktore 2. Logarithms/ Logaritmes
a3 - b3 = (a - b)(a2 + ab + b2) log ab log a log b
a
a3 + b3 = (a + b)(a2 - ab + b2) log log a log b
b
logc a
3. Quadratic formula/ logb a
logc b
Kwadratiese formule
b b 2 4ac
x log a m m log a
2a
1
4. Parabola/ Parabool logb a
loga b
y ax 2 bx c loga a 1 ∴ 1n e 1
4ac b 2 aloga t t ∴ e1n m m
y
4a
b
x
2a
5. Circle/ Sirkel 6. Straight line/ Reguitlyn
x2 y2 r 2 y y1 m(x x1 )
Perpendicular:
x2 Loodreg: m1 m2 1
D h
4h
Parallel lines:
x 4Dh 4h 2 Ewewydige lyne: m1 m2
Distance:
Afstand: D (x 2 x1 )2 (y 2 y1 )2
Midpoint:
x x 2 y1 y 2
Middelpunt: P 1 ;
2 2
Angle of inclination:
Hellingshoek: θ tan -1m
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(16030143) -9-
-2- T930(E)(J28)T
7. Differentiation/ Differensiasie
dy lim f x h f x
dx h → 0 h
d n
dx
x nx n - 1
Max/Min
Maks/Min
For turning points:
Vir draaipunte: f ' x 0
8. Trigonometry/ Trigonometrie
y 1
sinθ
r cosecθ
x 1
cosθ
r secθ
y 1
tanθ
x cotθ
sin 2 θ cos 2 θ 1
1 tan 2 θ sec 2 θ
1 cot 2 θ cosec2 θ
sinθ
tanθ
cosθ
cosθ
cotθ
sinθ
sinA sinB sinC
a b c
a 2 b2 c 2 2bc cosA
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MARKING GUIDELINE
NATIONAL CERTIFICATE
AUGUST EXAMINATION
MATHEMATICS N3
28 JULY 2014
This marking guideline consists of 9 pages.
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MARKING GUIDELINE -2- T1020(E)(J28)T
MATHEMATICS N3
QUESTION 1
MARKING
INSTRUCTION &NOTES
1.1 2a 1 3 8a 1 writing 8 as power with
base 2
4a 1 writing 4 as power with
3a 3
base 2
2 2 2 2
a 1 3 3
simplifying root
22 a 22 applying multiplication
rule of powers by adding
2a 1 a 1 2 a 2 24 16 exponents
simplifying (5)
1.2 LHS log c a log b c log a b RHS 1 applying log rule where
bases changed
log a log c log b
simplifying
log c log b log a
1 RHS
(3)
1.3 x2 4 x 3 for correct quotient
x 1 x 3x 2 x 3
3
two factors-
(x+3)(x+1)
x 3 -1x 2 no credit to (x-1) it
4 x2 x was given
4 x2 - 4 x
3x 3
3x 3
0 0
Factors : (x-1)(x+3)(x+1) (4)
1.4 1.4.1 x 2 ( x -1) (1- x) taking out -1 as common
factor
x 2 ( x -1) - ( x -1) factorise
( x -1)( x 2 -1) applying factorisation of
( x -1)( x -1)( x 1) difference between two
squares (3)
1.4.2 a 2 2a 3 ab b factorising of quadratic
trinomial
(a 2 2a 3) ab b taking out of common
(a 3)(a 1) b(a 1) factor
(a 1)[a 3 b] taking out of common
factor (3)
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MARKING GUIDELINE -3- T1020(E)(J28)T
MATHEMATICS N3
1.5 a 2 b 2 a 2 2ab b 2 a b change of division sign to
multiplication sign
2a b 4a 2 b 2 ab
(x2) factorisation using
difference between two
(a b)(a b) (2a b)(2a b) a b squares
2a b (a b)(a b) a b factorisation of trinomial
2a b simplifying
(4)
[22]
QUESTION 2
2.1 4a 2 12a 7 0 taking out 4 as common
factor
4a 2 12a 7
adding the
7 2
a 2 3a coeffcient of a
4 both sides
2
2
3 7 9 writing lhs as a square and
(a 3a )
2
✓
2 4 4 simplification of RHS
2 get rid of squares by drawing
3 16
a 4 ✓ square roots both sides-
2 4 Mark will not be awarded if
3 is left out.
a 2 ✓
2 (x2) two answers
3
a 2
2
7 1
a or a
2 2
(5)
2.2.1 1
x simplifying of negative
x2 x2
2 2 42 power
2 taking out common factor
2 x.22 2 x.22 2 x 42 simplification of power
2 x 22 22 1 42 solving by equating like
powers
2 5, 25 42
x
(4)
2 2
x 3
x3
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MARKING GUIDELINE -4- T1020(E)(J28)T
MATHEMATICS N3
2.2.2 2x 4 4 x correct LCD(denominator)
2 rewrite fractions to
x 1 x 1 x 1
2 x( x 1) 4( x 1) 4 x LCM(numerator)
simplification
x 1 ( x 1) x 1 ( x 1) (x2)-solution
2 x2 2 x 4 x 4 4 x Check the roots
2 x2 x 0
x(2 x 1) 0
1
x 0 or x
2
(4 )
2.3 P =100e 0,3n manipulation
using logarithms to manipulate
P
e 0,3n simplify
100
P
ln e
0,3 n
ln
100
ln P ln100 0,3n ln e
ln P ln100
n or
0,3
ln100 ln P
n or
0,3
100
ln
P n
0,3 (4)
2.4 a r 2 rs using correct values in the
place of a,b,c in the quadratic
r 2 rs a 0 formula
s 2 s 2 4 a manipulation and simplification
r
2
(2)
2.5 y 2 x 2 8 x 10 finding the x-coordinates
finding the y-coordinates of
y 9 x
points of intersection
2 x 2 8 x 10 9 x
2 x 2 x 10 0
(2 x 5)( x 2) 0
5
x or x 2
2
(4)
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MARKING GUIDELINE -5- T1020(E)(J28)T
MATHEMATICS N3
5 1
( ; 22 ) and 2; 18
2 2
2.6 x ( x 2) 143 any steps to show how the
x 2 x 143 0
2 student determined the answers
answers
( x 13)( x 11) 0 Students could determine answers
also using inspection.
x= -13 x=11
Two numbers: 11 and 13
Or -11 and -13 (3)
[26]
QUESTION 3
3.1 3.1.1 sketch of
triangle
(1)
3.1.2 YA YB substituting values and
M AB
XA XB simplifying
3 (1)
2 (2)
4 (1)
1
4
3.1.3 M AB 1 substituting value and
tan 1
1 simplifying
450
(1)
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MARKING GUIDELINE -6- T1020(E)(J28)T
MATHEMATICS N3
3.1.4 YA YC finding the gradient of
M AC line AC
X A XC
multiplication of gradients
3 (1)
to show that lines are
2 (4) perpendicular
2
1
2
M AB M AC (1) (1) 1
Therefore the two lines are perpendicular. (2)
3.1.5 Parallel lines have equal gradients. using identity on parallel
The gradient of the line through C (4;1) lines to find gradient of line
and parallel to AB with M AB 1 AB
finding y-intercept by
y yC m( x xC ) substituting
y 1 (1)(x 4) equation
y x 4 1
y x 3
(3)
3.1.6 x x y y substituting
D PAC 1 2 ; 1 2 simplify
2 2
substituting
2 4 3 1 simplify
;
2 2
D (3; 2)
x x y y
E PBC 1 2 ; 1 2
2 2
2 4 1 1
;
2 2
E (1;0)
(4)
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MARKING GUIDELINE -7- T1020(E)(J28)T
MATHEMATICS N3
3.1.7 DAB ( x A xB ) 2 ( y A yB ) 2 using the distance formula
and substitute
(2 (2)) 2 (3 (1)) 2 simplify
using the distance formula
16 16 and substitute
32 simplify
rewriting surds to prove
DDE ( x A xB ) 2 ( y A yB ) 2 that AB=2DE
(3 1) 2 (2 0) 2
44
8
AB 32 4 8 2 8
AB 2 DE (5)
3.2 3.2.1 G Choosing the correct
equation to describe the
graph (2)
3.2.2 E Choosing the correct (2)
equation to describe the
graph
3.2.3 D Choosing the correct
equation to describe the
graph (2)
3.2.4 I Choosing the correct
equation to describe the
graph (2)
[25]
QUESTION 4
4.1 4.1.1 2 rewriting square root
y2 x
x rewriting a fraction
1 differentiate function
2 x 2 x 1
2 √ √ differentiate function
rewriting function
dy
1
rewriting function
x 2 2 x 2
dx
1 2
2
x x (4)
√ √
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MARKING GUIDELINE -8- T1020(E)(J28)T
MATHEMATICS N3
4.1.2 x8 1 (2x)simplify fraction
y differentiation
x4 1
( x 4 1)( x 4 1)
y
x4 1
y x4 1
dy
4 x3
dx
(2)
4.2 y ( x 2 3)( x 1) multiplication
differentiation
y x3 x 2 3x 3 substitution
f '( x) 3x 2 2 x 3
f '(2) 3(2) 2 2(2) 3
3(2) 2 4 3
13 (3)
[9]
QUESTION 5
5.1 1 sin cos LCD
2 sec writing fractions to
cos 1 sin
LCD
LHS (1 sin )(1 sin ) cos cos RHS 2sec applying square identity
cos (1 sin ) cos 2 x 1 sin 2 x
simplifying
1 2sin sin 2 (1 sin 2 ) indicating that lhs = rhs
cos(1 sin )
2 2sin
cos (1 sin )
2(1 sin )
cos (1 sin )
2
cos
2sec
LHS RHS
(5)
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MARKING GUIDELINE -9- T1020(E)(J28)T
MATHEMATICS N3
5.2 a 2 462 802 2 46 80 Cos 25 substituting using cosine
rule
a 2 2116 6400 2 46 80 0,906 simplifying
a 2 1845,575 simplifying
a 42,96 units
(3)
5.3 2Sec 4 0 manipulating
sec 2 finding solutions
reference angle 60
120 and 240
(3)
5.4 CD trigonometric equation
tan 200 length CD
40
CD 14,558 length AD
length AC
AD
tan 240
40
AD 17,809
AD CD AC (lenght of antenna)
17,809 -14,558 AC
AC 3, 250m (4)
5.5 A (450 ; 2) (x2)
(x2)
B (2250 ; 2) (x2)
(3)
P (180 ; 1)
0
[18]
TOTAL: 100
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T930(E)(M31)T
APRIL EXAMINATION
NATIONAL CERTIFICATE
MATHEMATICS N3
(16030143)
31 March 2014 (Y-Paper)
13:00–16:00
This question paper consists of 6 pages and 2 formula sheets.
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(16030143) -2- T930(E)(M31)T
DEPARTMENT OF HIGHER EDUCATION AND TRAINING
REPUBLIC OF SOUTH AFRICA
NATIONAL CERTIFICATE
MATHEMATICS N3
TIME: 3 HOURS
MARKS: 100
NOTE: Diagrams are NOT drawn to scale.
INSTRUCTIONS AND INFORMATION
1. Answer ALL the questions.
2. Read ALL the questions carefully.
3. Number the answers according to the numbering system used in this question paper.
4. ALL final answers must be accurately approximated to THREE decimal places.
5. All graph work must be done in the ANSWER BOOK. Graph paper is NOT supplied.
6. Two FORMULA SHEETS are attached to this question paper. The list is NOT
necessarily complete. Any other applicable formula may also be used.
7. Write neatly and legibly.
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(16030143) -3- T930(E)(M31)T
QUESTION 1
1.1 Simplify the following WITHOUT using a calculator.
1.1.1 1
x
9 2 3x 1
3x 2 (3)
1.1.2 log 27 log 8 log 125
log 6 log 5
(5)
1.2 Factorise as far as possible in prime factors:
1.2.1 2( x 2)2 9( x 2) 5 (3)
1.2.2 (m n)36 x2 49ny 2 49my 2 (3)
1.3 Given that x 1 is a factor of f ( x) .
Determine the other factors if f ( x) x3 13x 12 (3)
1.4 Simplify:
1.4.1 1 1
2
2
x y
1 1
y x (3)
1.4.2 4x 2x 5x2
3x 1 1 3x 9 x 2 1 (5)
[25]
QUESTION 2
2.1 Solve for x :
2.1.1 3x 10 x 2 (3)
3 x3
2.1.2 x5
x x (3)
2.2 Make ' x ' the subject of the formula by completing the square:
x 2 8 x 5a 0 (3)
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(16030143) -4- T930(E)(M31)T
2.3 Make 't' the subject of the formula:
V1 V0 e at (4)
2.4 Thomas is 4 times as old as John. In 12 years' time Thomas will be twice as old as
John. What are their present ages? (3)
2.5 Determine the points of intersection of the graphs represented by the following
equations algebraically:
x 2 y 2 25
yx5 (5)
[21]
QUESTION 3
3.1 Complete the following sentences by filling in the missing values, expressions or
equations. Write only the missing answer next to the question number (3.1.1–3.1.3) in
the ANSWER BOOK.
P(-1;-1) and Q(-3;3) are two points. The coordinates of the midpoint of
3.1.1
PQ are … (2)
3.1.2 The acute angle of inclination formed between the x-axis and y x 3 is
equal to … (2)
3.1.3 The equation of the line passing through the points (2,0) and (2, 2) is … (1)
3.2 In the diagram below, P(-3;2) is a point on the circle with the centre at the origin. AB
is a tangent to the circle at point P.
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(16030143) -5- T930(E)(M31)T
3.2.1 Calculate the length of OP. (2)
3.2.2 Determine the equation of the circle. (2)
3.2.3 Determine the equation of the tangent AB. (4)
3.3 Determine the equation of the straight line that passes through point O (0;0) and
which is parallel to the line 18 x 3 y 9 0 . (3)
[16]
QUESTION 4
4.1 Consider the function y 1 4 x2
dy
4.1.1 Determine the derivative, , from first principles.
dx (4)
4.1.2 Determine the gradient of the function at the point where x =1. (1)
dy
4.2 Determine if
dx
x2 2 x
y x
x2
(Leave the answer with positive exponents and in surd form) (4)
4.3 Sketch the graphs defined by the following equations in the ANSWER BOOK. Each
graph should be on its own system of axes. Calculations need NOT be shown. ALL
values at the points of intersection with axes and the coordinates of the turning points
(where applicable) must be shown.
x2 y2
4.3.1 1
256 144 (2)
4.3.2 y 3x 2 2 x 3 (4)
4.3.3 x y
1
4 3 (2)
[17]
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(16030143) -6- T930(E)(M31)T
QUESTION 5
5.1 Calculate the exact value WITHOUT using a calculator:
sin 2 120 sec2 150 sin150
cosec 30 cos 60 (3)
5.2 Calculate the value(s) of which will satisfy the equation if 00 3600 :
2sin cos sin 0
(5)
5.3 From the top of a tower, 100 m high, the angles of depression of two objects on the
ground, due west of the tower, are 41,45 and 21,68 respectively.
Calculate the distance between the TWO objects. (5)
5.4 Given : ABC with, A = 600, AB 2 and AC 8 .
Calculate:
5.4.1 The value of 'x' (3)
5.4.2 The area of the triangle. (2)
5.5 Sketch the graph of 3 y sin 3 for 0 1200 (3)
[21]
TOTAL: 100
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(16030143) -7- -1- T930(E)(M31)T
FORMULA SHEET
Any applicable formula may also be used.
1. Factors/Faktore 2. Logarithms/Logaritmes
a3 - b3 = (a - b)(a2 + ab + b2) log ab log a log b
a
a3 + b3 = (a + b)(a2 - ab + b2) log log a log b
b
log c a
3. Quadratic formula/ log b a
log c b
Kwadratiese formule
b b 2 4ac
x log a m m log a
2a
1
4. Parabola/ Parabool log b a
log a b
y ax 2 bx c loga a 1 ∴ 1n e 1
4ac b 2 aloga t t ∴ e1n m m
y
4a
b
x
2a
5. Circle/ Sirkel 6. Straight line/ Reguitlyn
x2 y2 r 2 y y1 m(x x1 )
Perpendicular:
x2
D h Loodreg: m1 m 2 1
4h
Parallel lines:
x 4Dh 4h 2 Ewewydige lyne: m1 m 2
Distance:
Afstand: D (x2 x1 )2 (y2 y1 )2
Midpoint:
x x 2 y1 y 2
Middelpunt: P 1 ;
2 2
Angle of inclination:
Hellingshoek: θ tan -1 m
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(16030143) -2-
-8- T930(E)(M31)T
7. Differentiation/Differensiasie
dy lim f x h f x
dx h → 0 h
d n
dx
x nx n - 1
Max/Min
Maks/Min
For turning points:
Vir draaipunte: f ' x 0
8. Trigonometry/ Trigonometrie
y 1
sinθ
r cosecθ
x 1
cosθ
r secθ
y 1
tanθ
x cotθ
sin 2 θ cos 2 θ 1
1 tan 2 θ sec 2 θ
1 cot 2 θ cosec 2 θ
sinθ
tanθ
cosθ
cosθ
cotθ
sinθ
sinA sinB sinC
a b c
a 2 b 2 c 2 2bc cosA
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MARKING GUIDELINE
NATIONAL CERTIFICATE
APRIL EXAMINATION
MATHEMATICS N3
31 MARCH 2014
This marking guideline consists of 12 pages.
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MARKING GUIDELINE -2- T930(E)(M31)T
MATHEMATICS N3
QUESTION 1
1.1 1.1.1 1
x
9 2 3x 1
3x 2
x
3 3 2 2 x 1
x2
3
3 3x.31x
3x.32
1
3x 1
x
3
3 .9
2
2 1 2
3
9 3 9 27
(3)
1.1.2 log 27 log 8 log 125 log (3)3 log (2)3 log (5)3
log 6 log 5 log 6 log 5
27 8 (3)3 (2)3
log log
125 (5)3
6
log 6
5 log
5
216
log (3)3 (2)3
125 log
(5)3
6
log 6
5 log
3
5
6
log 6
3
5 log
5
6
log 6
5 log
5
3 6
log 3 6
2 5
3 log
2 5 3
6 2
log 6 2
5 log
5 (5)
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MARKING GUIDELINE -3- T930(E)(M31)T
MATHEMATICS N3
1.2 1.2.1 2( x 2) 2 9( x 2) 5 or 2( x 2) 2 9( x 2) 5
let x 2 k 2( x 2 4 x 4) 9 x 18 5
2k 2 9k 5 2 x 2 8 x 8 9 x 18 5
(2k 1)(k 5) 2 x 2 x 15
Factors of 2( x 2) 2 9( x 2) 5 (2 x 5)( x 3)
[2( x 2) 1][( x 2) 5]
(2 x 5)( x 3)
(3)
1.2.2 (m n)36 x 2 49ny 2 49my 2
(m n)36 x 2 49 y 2 m n
(m n) 36 x 49 y
2 2
(m n) 6 x 7 y 6 x 7 y
(3)
1.3
x 2 x 12
x 1 x 3
13x 12
x3 + x 2
- x 2 -13x
- x2 - x
-12 x - 12
-12x - 12
f ( x) ( x 1)( x 2 x 12) ( x 1)( x 4)( x 3)
(3)
1.4 1.4.1 1 1 1 1
2 2
x y y x
y 2 x2 x y
2 2
x y xy
( y x)( y x) xy
2 2
x y ( x y)
yx
xy (3)
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MARKING GUIDELINE -4- T930(E)(M31)T
MATHEMATICS N3
1.4.2 4x 2x 5x2
2
3x 1 1 3x 9 x 1
4x 2x 5x2
3x 1 3 x 1 (3 x 1)(3 x 1)
4 x(3x 1) 2 x(3x 1) 5 x 2
(3x 1)(3x 1)
12 x 2 4 x 6 x 2 2 x 5 x 2
(3x 1)(3x 1)
11x 2 6 x
(3x 1)(3x 1) (5)
[25]
QUESTION 2
2.1 2.1.1 3x 10 x 2
3x 10 x 2
2
3x 10 x 2 4 x 4
x2 x 6 0
x 3 x 2 0
x 3 0 or x - 2 0
x 3 or x=2
x -3 (n.a) extraneous solution
x=2 is the only solution
any clear rejection of the root is acceptable. (3)
2.1.2 3 x3
x5
x x
3 x 5x x 3
2
x2 4 x 0
x( x 4) 0
x0 or x+4=0
x0 x -4 (3)
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MARKING GUIDELINE -5- T930(E)(M31)T
MATHEMATICS N3
2.2 x 2 8 x 5a 0
x 2 8 x 5a
x 2 8 x 16 5a 16
x 4 5a 16
2
x 4 5a 16
x 4 5a 16 (3)
2.3 V1 V0e at
V
1 e at
V0
V
ln 1 ln e at
V0
V
ln 1 at
V0
V
ln 1
t 0
V
a
V
ln 0
ln V0 ln V1
or t 1 or
V
a a (4)
2.4
AGE NOW AGE IN 12 YEARS
John x (x+12)
Thomas 4x (4x+12)
age of Thomas=2 age of John
4 x 12 2( x 12)
4 x 12 2 x 24
2 x 12
x6
present ages = 6 & 24 years (3)
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MARKING GUIDELINE -6- T930(E)(M31)T
MATHEMATICS N3
2.5 x 2 y 2 25.......(1)
y x 5.............(2)
replace y by x 5 in (1)
x 2 ( x 5) 2 25
x 2 x 2 10 x 25 25
2 x 2 10 x 0
2 x( x 5) 0
x 0 or x 5
If x 0 then y 5
or if x -5 then y 0
(5)
[21]
QUESTION 3
3.1 3.1.1
(-2;1) (2)
3.1.2
45° (2)
3.1.3 x2 (1)
3.2 3.2.1 OP (3) 2 (2) 2
= 9 4 13 units (2)
3.2.2 x2 y 2 r 2
x 2 y 2 ( 13) 2
x 2 y 2 13 (2)
3.2.3 mOP mAB 1 OP AB
2
mAB 1
3
3
mAB
2
y y1 m( x x1 )
3
y2 ( x 3)
2
3 9
y2 x
2 2
3 13
y x
2 2 (4)
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MARKING GUIDELINE -7- T930(E)(M31)T
MATHEMATICS N3
3.3
18 x 3 y 9 0
y 6x 3
Required line : y 6 x
(3)
[16]
QUESTION 4
4.1 4.1.1 y 1 4 x2
f ( x h) f ( x )
lim
h 0 h
1 4( x h) 2 (1 4 x 2 )
lim
h 0 h
1 4( x 2 xh h 2 ) 1 4 x 2
2
lim
h 0 h
1 4 x 8 xh 4h 2 1 4 x 2
2
lim
h 0 h
8 xh 4h 2
lim
h 0 h
h(8 x 4h)
lim
h 0 h
8 x (4)
4.1.2 dy
if x 1 then -8(1) -8
dx (1)
4.2 x2 2 x
f ( x) x
x2
1
f ( x) x 1 2 x 1
2
1 1
f '( x) x 2 2 x 2
2
1 2
f '( x) 2
2 x x (4)
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MARKING GUIDELINE -8- T930(E)(M31)T
MATHEMATICS N3
4.3
4.3.1 y
12
-16 16 x
-12
MARK ALLOCATION MARKS
x intercepts at (16;0) and (-16;0) 1
y intercepts at (0;12) and (0;-12) 1 (2)
4.3.2 y 3x 2 2 x3
dy
6 x 6 x2
dx
0 6 x(1 x)
x 0 or 1 x 0
x 0 or x 1
TP:(0;0) or (1;1)
x-intercepts 3 x 2 2 x3 0
x 2 (3 2 x) 0
x 0 or x 0 or 3 - 2 x 0
3
x0 x
2
y-intercept y 0
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MARKING GUIDELINE -9- T930(E)(M31)T
MATHEMATICS N3
ALLOCATION OF MARKS MARKS
1 mark for turning point(1;1) 1
1 mark for turning point(0;0)/ 1
y-intercept at 0/x-intercept at 0
3 1
x-intercepts at and 0
2
Shape 1
(4)
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MARKING GUIDELINE -10- T930(E)(M31)T
MATHEMATICS N3
4.3.3
y
3
-4 0 x
ALLOCATION OF MARKS MARKS
1 mark for x-intercept (-4;0) 1
1 mark for y-intercept at (0;3) 1 (2)
[17]
QUESTION 5
5.1 sin 2 120 sec 2 150 sin150
cosec 30 cos 60
2
3 2 1
2
2
3 2
2 1
1 2
1
1
2 3
2 1 2
1 2
(3)
5.2 2sin cos sin 0
sin (2 cos 1) 0
sin 0 or 2 cos 1 0
00 or 1800 or 3600 600 or 3000
(5)
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MARKING GUIDELINE -11- T930(E)(M31)T
MATHEMATICS N3
5.3 x y
Tan 68,32 Tan 48,55
100 100
x 251,544 m y 113,228 m
Distance between objects = 251,544 m – 113,228 m
= 138,316 m
100 100
or tan 41, 45 y
x tan 21, 65
x
100 251,929 m
tan 41, 45 Distance y x
113, 228 m 138, 701m (5)
5.4 5.4.1
x ( 8 ) 2 ( 2 ) 2 2. 8. 2 Cos 60
8 2 8Cos 60
= 6 OR 2,449 (3)
5.4.2 1
Area bcSinA
2
1
8 2 Sin60
2
1,732 m 2 (2)
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MARKING GUIDELINE -12- T930(E)(M31)T
MATHEMATICS N3
5.5 3 y sin 3
sin 3
y
3
ALLOCATION OF MARKS MARKS
1 mark for x-intercept at 00 ,600 or1200 1
1 mark for y-intercept at 00 1
Height 1 (3)
[21]
TOTAL: 100
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