MARKING GUIDELINE
NATIONAL CERTIFICATE
MATHEMATICS N4
6 April 2021
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MARKING GUIDELINE
-2MATHEMATICS N4
QUESTION 1
1.1
Graph√
Axis√
(2)
1.2
Graph√
Axis√
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MARKING GUIDELINE
1.3
-3MATHEMATICS N4
1.3.1
Graph√
Axis√
It is a relation.√
1.3.2
1.4
Continuous
(3)
(1)
r = x2 + y2
= 32 + (− 4)
2
= 25
=5√
tan θ =
y
x
4
3
0
θ = 53 √
θ = 3600 − 530
θ = 307 0 √
θ = tan −1  
(
)
3
∴ 5 307 0 √
= 125Cis9210
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OR
(
125 cos 9210 − i sin 9210
)
OR
125 9210 √√
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MARKING GUIDELINE
1.5
1.6
-4MATHEMATICS N4
7 − j9
1− j
(x − jy )(1 − j ) = 7 − j 9
x − jy − jx + j 2 y = 7 − j √
x − y = 7...........(1) √
− x − y = −1..........(2 ) √
ADD (1) AND (2 )
− 2y = 6
y = −3 √
Substitute y in (1)
x − (− x ) = 7
x = 4√
x − jy =
(5)
j9 − j2
= j +1√
(1)
[20]
QUESTION 2
2.1
cos 2 θ + tan θ . cos θ
2
 39   − 5  − 39 
 +


= 
  − 13  8  √
8




39 5
= − √
64 8
39 − 40
=
√
64
1
=− √
64
2.2
(
)
cot 900 + θ = − tan θ
PROVE: cot 900 + θ
=
(
(
(
)
(4)
)
)
cos 90 + θ
√
sin 900 + θ
0
cos 90 0. cos θ − sin 90 0. sin θ
√
sin 900. cos θ + cos 900. sin θ
0. cos θ − 1. sin θ
=
√
1. cos θ + 0. sin θ
− sin θ
=
cos θ
= − tan θ √
=
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2.3
-5MATHEMATICS N4
sin x + tan x
cos ecx + cot x
sin x
sin x +
cos x √
=
1
cos x
+
sin x sin x
sin x. cos x + sin x
cos x
√
=
1 + cos x
sin x
=
sin x(cos x + 1) sin x
√
.
(1 + cos x )
cos x
sin 2 x
cos x
sin x sin x
=
.
cos x 1
= tan x. sin x √
=
2.4
(4)
sec θ + tan θ
1
=
cos θ
1 − sin θ
1
sin θ
+
LHS = cos θ cos θ √
cos θ
1+ sin θ 1
=
.
cos θ cos θ
1+ sin θ
=
cos 2 θ
1 + sin θ
=
√
1 − sin 2 θ
(1 + sin θ )
=
√
(1 + sin θ )(1 − sin θ )
1
=
√
1 − sin θ
= RHS
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MARKING GUIDELINE
2.5
-6MATHEMATICS N4

α
sec + 50  = − cos ec 200 − α

2

α
sec + 50  = sec 900 − − 200 + α √

2
(
[
)
)]
(
α
+ 50 = 900 + 200 − α √
2
3
α = 1050
2
3 α = 2100 √
α = 700 √
ALTERNATIVE METHOD
α


cos ec 900 − − 50  = cos ec − 200 + α √
2


(
900 −
α
)
− 50 = −200 + α √
2
3
− α = −1050 √
2
− 3α = −2100
α = 700 √
(4)
[20]
QUESTION 3
3.1
y = sec x
y= u
1
dy 1 − 2
= u
du 2
1
√
=
2 u
dy dy du
=
.
dx du dx
1
=
. sec x. tan x √
2 u
sec x. tan x
=
√
2 sec x
3.2
u = sex
du
= sec x. tan x √
dx
(4)
dy
1
3
1
= 2 cos x + sec x. tan x + e 7 x − 2 x. ln 2 + π 3 x . ln π − 0 +
dx
2
4
2x
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3.3
-7MATHEMATICS N4
y = 5x5 − 2 x
f (x + h ) − f (x )
lim
h
h→0
(
)
5(x + h ) − 2(x + h ) − 5 x 5 − 2 x
√
h

 5 5 x 5 −1.h 5(5 − 1)x 5 − 2 .h 2 5(5 − 1)(5 − 2 )x 5 − 3 .h3
+
− 2 x − 2h − 5 x 5 + 2 x 
 x + 1! +
2!
3!
= 5
√
h




5
=
=
5 x 5 + 25 x 4 h + 50 x 3 h 2 + 50 x 2 h 3 + 50 xh 4 − 2 x − 2h − 5 x 5 + 2 x
√
h
 25 x 4 + 50 x 3 h + 50 x 2 h 2 + 50 xh 3 − 2 
= h
√
h


= 25 x 4 − 2 √
3.4
(5)
y = x 3 − 12 x 2 + 16 x − 12
d2y
= 6 x − 24 √
dx 2
6 x − 24 = 0
x = 4√
Substitute x to the corresponding y − value:
y = 43 − 12(4 ) + 16(4 ) − 12
y = −76 √
Points of inflection: (4,−76 ) √
2
(4)
[20]
QUESTION 4
4.1
3x + 2 = ln19
ln 3( x + 3) = ln(ln 19 ) √
1,098 x + 2,197 = 1,079 √
1,098 x = −1,118
x = –1,018√
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4.2
-8MATHEMATICS N4
3x + 4 y = 6
x − 2 y = −3
3 4
= −6 − 4 = −10 √
1 -2
6 4
= −12 + 12 = 0 √
-3 -2
3 6
= −9 − 6 = −15 √
Dy =
1 -3
D
0
x= x =
=0√
D − 10
D y − 15
y=
=
= 1,5 √
D − 10
Dx =
4.3
(5)
3
7 9
− 3 −1 6
4
8 5
4.3.1
4.3.2
−3 6
= −15 − 24 = −39 √√
4 5
(− 1)2+ 2
3 9
= 15 − 36 = −21 √√
4 5
(2 × 2)
4.4
(4)
x + y = 4...........(1) √
4( x − y ) = 15...........(2 ) √
4 x − 4 y = 15
y = 4− x
Substitute y − in (2)
4 x − 4(4 − x ) = 15 √
4 x − 16 + 4 x = −64
x = 6√
y = −2 √
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4.5
-9MATHEMATICS N4
DB  N 
ln 
C R
 N  PC
√
ln  =
 R  DB
PC
N
= e DB √
R
P=
N = Re
PC
DB
√
(3)
[20]
QUESTION 5
5.1
5.1.1
Indicating the area√
Drawing the strip√
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MARKING GUIDELINE
5.1.2
-10MATHEMATICS N4
∫
4, 6
1

 −1 2 2
 x + x + 4 dx √
3

 3
4
 − x3 1 2

=
+ x + 4 x √
3
 9
1
3
 13 1

− (4 ) 1 2
+ (4 ) + 4(4 ) −  − + .12 + 4(1) √
9
3
 9 3

= 14,2 − 4,2 √
= 10 units 2 √
=
5.2
5.3
=
∫
(5)
e −3 x
4.π 3 x
1
5−7 x
− 5π ln x + sec x − cot 3x −
−
+ px + c
3 ln π
3
3 ln e 7 ln 5
π
2
0
(7)
sin 3 xdx
π
 −1
2
=  cos x + c  √
3
0
−1  π 

 1
cos  + c −  − cos 00 + c  √
=
3

 3
2
1
= / 0.333 √
3
5.4
(3)
∫ (− cos ecx. cot x − (coesc x ))dx
2
= − cos ecx + cot x √
− 1 cos x
√
=
+
sin x sin x
cos x − 1
√
=
sin x
(3)
[20]
TOTAL:
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