NATIONAL
SENIOR CERTIFICATE
GRADE 12/GRAAD 12
MATHEMATICS P1/WISKUNDE V1
NOVEMBER 2015
MEMORANDUM
MARKS: 150
PUNTE: 150
This memorandum consists of 25 pages.
Hierdie memorandum bestaan uit 25 bladsye.
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NSC/NSC – Memorandum
DBE/November 2015
NOTE:
• If a candidate answers a question TWICE, only mark the FIRST attempt.
• Consistent accuracy applies in ALL aspects of the marking memorandum.
LET WEL:
• Indien 'n kandidaat 'n vraag TWEE keer beantwoord, merk slegs die EERSTE poging.
• Volgehoue akkuraatheid is op ALLE aspekte van die memorandum van toepassing.
QUESTION/VRAAG 1
1.1.1
x 2 − 9 x + 20 = 0
(x − 4)(x − 5) = 0
x = 4 or
1.1.2
factors
x =4
x =5
(3)
standard form
x=5
3x 2 + 5 x − 4 = 0
− 5 ± (5) 2 − 4(3)(−4)
x=
2(3)
substitution into
correct formula
− 5 ± 73
6
x = −2,26 or x = 0,59
answers
x=
(4)
OR/OF
5
25 4 25
= +
x2 + x +
3
36 3 36
for adding
2
5
73

x+  =
6
36

x+
1.1.3
2x
x
−5
3
−5
3
5
73
=±
6
6
− 5 ± 73
x=
6
x = −2,26 or
both sides
x =
x = 0,59
25
on
36
− 5 ± 73
6
answers
(4)
= 64
= 32
( )
x = 25
−3
5
x = 2 −3 or
1
or 0,125
8
OR/OF
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dividing both
sides by 2
32 = 2 5 or
64 = 26
raising RHS to
answer
−3
5
(4)
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2x
x
−5
3
−5
3
3
NSC/NSC – Memorandum
DBE/November 2015
= 64
dividing both
sides by 2
raising RHS to
= 32
−3
x = (32 ) 5
 5 32 −3
x = 5 32 −3
x = 2 −3 or
−3
5
1
or 0,125
8
answer
(4)
OR/OF
−3
−3
 −35  5
 2 x  = 64 5




0,659 x = 0,0825
x = 0,125
raising both sides to
−3
5
0,659 and 0,0825
dividing both
sides by 0,659
answer
(4)
OR/OF
x
−5
3
= 32
dividing both
sides by 2
logs on both sides
−5
log x = log 32
3
3
log x =
log 32
−5
log x = −0,903
 log x = −0,903
x = 10 −0,903
= 0,125 or
1.1.4
1
8
answer
2
squaring both
sides
2−x =x−2
2 − x = (x − 2)
2 − x = x 2 − 4x + 4
x 2 − 3x + 2 = 0
(x − 1)(x − 2) = 0
x = 1 or
(4)
factors
x=2
if x = 1, 2 − x = 1 and x − 2 = −1
x = 2 only
 x = 1 or x = 2
 x = 2 only
(4)
OR/OF
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2−x = x−2
2 − x = (x − 2)
squaring both
sides
 2 − x = 1 or 2 − x = 0
2
2 − x = (2 − x )
2 − x = 1 or 2 − x = 0
x = 1 or x = 2
2
 x = 1 or x = 2
if x = 1, 2 − x = 1 and x − 2 = −1
∴ x = 2 only
 x = 2 only
OR/OF
(4)
2−x =x−2
2− x ≥ 0
 x−2≥0
2 − x ≥ 0 and x − 2 ≥ 0
x ≤ 2 and x ≥ 2
 x ≤ 2 and x ≥ 2
∴ x = 2 only
1.1.5
x=2
x + 7x < 0
x ( x + 7) < 0
factors
+
-7
OR/
0 OF
–
-7
+
0
inequality or
interval
− 7 < x < 0 OR/OF x ∈ (− 7; 0)
1.2
(4)
2
The square of any number is always positive or zero
So for the sum of two squares to be zero, both squares must be
zero, i.e.
Die kwadraat van enige getal is altyd positief of nul. Vir die som
van twee kwadrate om nul te wees, moet beide die kwadrate nul
wees, d.i.
(3x − y )2 = 0 and/en (x − 5)2 = 0
3 x − y = 0 and/en
3(5) − y = 0
y = 15
x−5 = 0
x=5
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(3)
 3x − y = 0
 x−5 = 0
x=5
 y = 15
(4)
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1.3
5
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DBE/November 2015
x2 + x = k
x2 + x − k = 0
standard form
∆<0
∆ < 0
b 2 − 4ac < 0
12 − 4(1)(− k ) < 0
1 + 4k < 0
k<
OR/OF
 12 − 4(1)(− k )
−1
4
k <
−1
4
(4)
x2 + x = k
x2 + x +
1
1
=k+
4
4
adds
sides
2
1
1

x+  = k +
2
4

1
for nonreal roots k + < 0
4
−1
k<
4
1
to both
4
2
1
1

x +  = k +
2
4

k +
1
<0
4
k <
−1
4
OR/OF
(4)
Consider the functions y = x 2 + x and y = k
Beskou die funksies y = x 2 + x en y = k
y
y = x2 + x
sketch or
explanation
x
y=k
 −1 −1 
Turning point of/Draaipunt van y = x + x is  ;

 2 4 
x 2 + x = k does not have real roots when the line y = k does not
2
intersect y = x 2 + x .
x 2 + x = k het geen reële wortels as die lyn y = k nie met
y = x 2 + x sny nie.
−1
Therefore k <
4
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1
2
1
y = −
4
x = −
k <
−1
4
(4)
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QUESTION/VRAAG 2
2.1
r=
T2
T1
5
10
1
=
2
=
r =
4
1
T5 = 1,25 
2
5
= or 0,625
8
OR/OF
1
T5 = 10 
2
5
= or 0,625
8
2.2
1
Tn = 10 
2
2.3
For convergence/Om te konvergeer − 1 < r < 1
1
1
Since/Aangesien
r =
and/en − 1 < < 1
2
2
n −1
the sequence converges/die ry konvergeer
2.4
(
n
)
= 20
−
1
= 20 
2
1
20 + 20 
2
n
answer
substitutes a = 10
into GP formula
1
substitutes r =
2
into GP formula
 −1 < r < 1
 show that r =
−1 < r < 1
(2)
(2)
1
is
2
(2)
a
a 1− r
−
1− r
1− r
 1n 
101 − 
2 
10
=
− 
1
1
1−
1−
2
2
 1n 
= 20 − 201 − 
 2 
S∞ - Sn =
1
2
10
1
1−
2
 1n 
101 − 
2 
 
1
1−
2
 1n 
 201 − 
 2 

n
answer
(4)
OR/OF
constructing the
series
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S ∞ - S n = Tn +1 + Tn + 2 + Tn +3 + 
1
= 10 
2
n
 1 1

1 + 2 + 4 + 


n
1  1 
= 10  

 2  1 − 1 
 2
1
= 20 
2
n
DBE/November 2015

1
10 
2

n
 1 1

1 + 2 + 4 + 
1
1
2
answer
1−
(4)
OR/OF
(
a
a 1− rn
−
1− r
1− r
a − a + ar n
=
1− r
n
ar
=
1− r
S∞ - S n =
1
10 
2
=  
1
2
n
1
= 20 
2
n
)
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a − a + ar n

1− r
ar n

1− r
n
1
10 
2
  
1
2
answer
(4)
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QUESTION/VRAAG 3
3.1
 d value
d=8
Tk = a + (k − 1)d
= −3 + (k − 1)(8)
= −3 + 8k − 8
= 8k − 11
3.2
n
∑ (8k − 11)
OR/OF
k =1
3.3
answer
n −1
n −1
k =0
k =0
∑ (8(k + 1) − 11) = ∑ (8k − 3)
n
[2a + (n − 1)d ]
2
n
= [2(− 3) + (n − 1)(8)]
2
n
= [− 6 + 8n − 8]
2
n
= [8n − 14]
2
= n( 4n − 7)
Sn =
(2)
for general term
lower and upper
values in sigma
notation
(2)
formula
substitution

n
[8n − 14]
2
(3)
= 4n 2 − 7 n
OR/OF
n
[2a + (n − 1)d ]
2
n
= [2(− 3) + (n − 1)(8)]
2
n
= [− 6 + 8n − 8]
2
n
= [8n − 14]
2
= 4n 2 − 7 n
Sn =
formula
substitution

n
[8n − 14]
2
(3)
OR/OF
n
[a + l ]
2
n
= [− 3 + 8n − 11]
2
n
= [8n − 14]
2
= 4n 2 − 7 n
Sn =
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formula
substitution

n
[8n − 14]
2
(3)
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OR/OF
S1
–3
S2
2
S3
15
5
S4
36
13
21
8
8
S 2 = −3 + 5 = 2
S 3 = 2 + 13 = 15
S 4 = 15 + 21
calculates S 1 , S 2 ,
S 3 and S 4 ,
S n = an 2 + bn + c
8
2
a=4
S1 = 4 + b + c = −3
a=
a=4
b + c = −7 ...............(1)
S 2 = 16 + 2b + c = 2
2b + c = −14 ..............(2)
b = −7 ................(2) − (1)
Hence S n = 4n − 7n
solves
simultaneously for
b and c.
(3)
3.4.1
Q 6 = −6 − 3 + 5 + 13 + 21 + 29
 answer
3.4.2
Q129 = −6 + S128
c=0
2
= −6 + 4(128) 2 − 7(128)

= 64634
answer
OR/OF
Q1
–6
− 6 + 4(128) 2 − 7(128)
Q2
–9
Q3
–4
–3
(2)
5
8
(3)
Q4
9
13
8
8
2
Qn = an + bn + c
a=4
Q1 = 4 + b + c = −6
Q2 = 16 + 2b + c = −9
a = 4
b + c = −10 ...............(1)
2b + c = −25 ...............(2)
b = −15 ................(2) − (1)
c=5
2
Hence Qn = 4n − 15n + 5
Q129 = 4(129 ) − 15(129 ) + 5

Qn = 4n 2 − 15n + 5
2
= 64 634
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answer
(3)
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QUESTION/VRAAG 4
Given: f ( x) = 2 x +1 − 8
4.1
y=–8
y=–8
4.2
(1)
y
8
6
4
f
2
x
-8
-6
-4
-2
0
2
-2
4
6
8
x-intercept
y-intercept
shape
asymptote
(only if the graph
does not cut the
asymptote)
-4
-6
-8
y = -8
(4)
4.3
g ( x ) = 2 − x +1 − 8
answer
(1)
OR/OF
1
g (x ) =  
2
answer
x −1
−8
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(1)
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QUESTION/VRAAG 5
DBE/November 2015
Given h( x ) = 2 x − 3 for − 2 ≤ x ≤ 4 .
y
h
-2
O
Q
x
4
P
-3
5.1
For x-intercepts, y = 0
2x − 3 = 0
 x = 1,5
y=0
x = 1,5
Q(1,5 ; 0)
5.2
h:
x = −2 : y = 2(−2) − 3 = −7
x = 4:
y = 2(4) − 3 = 5
Domain of h −1 : −7 ≤ x ≤ 5 OR/OF [− 7; 5]
h-1
1,5
-7
x
5
0
 h(–2) = –7
 h(4) = 5
−7 ≤ x ≤ 5
(3)
 y-intercept
on a straight
line
y
5.3
(2)
 line segment
 accurate
endpoints (x
or y or
both)
OR/OF
y
(3)
4
h-1
1,5
x
0
-2
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5.4
h( x ) = 2 x − 3
For the inverse of h,
x = 2y − 3
x+3
y=
2
x+3
h −1 ( x) =
2
−1
h( x ) = h ( x )
x+3
2
4x − 6 = x + 3
2x − 3 =
x=3
12
NSC/NSC – Memorandum
DBE/November 2015
y=
x+3
2
 2x − 3 =
x+3
2
x=3
OR/OF
(3)
h( x ) = 2 x − 3
h and h −1 intersect when y = x
h( x ) = x
2x − 3 = x
x=3
 h( x ) = x
 2x − 3 = x
x=3
OR/OF
h( x ) = 2 x − 3
For the inverse of h,
x = 2y − 3
x+3
y=
2
−1
h ( x) = x
x+3
=x
2
x + 3 = 2x
x=3
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y=

(3)
x+3
2
x+3
=x
2
x=3
(3)
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5.5
OP 2 = (x − 0 ) + ( y − 0 )
2
= x 2 + (2 x − 3)
13
NSC/NSC – Memorandum
DBE/November 2015
 OP 2 = x 2 + y 2
2
2
substitute
y = 2x − 3
= x 2 + 4 x 2 − 12 x + 9
= 5 x 2 − 12 x + 9
For OP to be at its minimum, OP 2 has to be a minimum
Vir OP om minimum te wees, moet OP2 'n minimum wees
d (OP 2 )
b
x=−
=0
OR / OF
dx
2a
− 12
=−
10 x − 12 = 0
2(5)
6
∴x =
5
 5 x 2 − 12 x + 9
x-value
2
Minimum length of OP =
3
9
6
6
5  − 12  + 9 =
or
or 1,34 units
5
5
5
5
answer
OR/OF
For minimum distance OP ⊥ the line
m h = 2 (given)
mOP =
−1
2
∴ OP has equation y =
−1
x = 2x − 3
2
− x = 4x − 6
5x = 6
x P = 1,2
−1
x
2
yP = −
1
(1,2) = −0,6
2
OP =
(1,2 − 0)2 + (− 0,6 − 0)2
= 1,34 or 1,8 units
(5)
 mOP =
−1
2
equation of
OP

−1
x = 2x − 3
2
x-value
answer
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(5)
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OR/OF
For minimum distance OP ⊥ the line
3 
O(0;0 ) P(x; 2 x − 3) Q ; 0 
2 
OP 2 + PQ 2 = OQ 2 (pythag)
(x − 0)2 + (2 x − 3 − 0)2 +  x − 3 
2
3
2
+ (2 x − 3 − 0 ) =  
2
2
2
9
9
x 2 + 4 x 2 − 12 x + 9 + x 2 − 3 x + + 4 x 2 − 12 x + 9 =
4
4
2
10 x − 27 x + 18 = 0
(5 x − 6)(2 x − 3) = 0

6
3
x = or
5
2
6
at P
5
2
OP 2 = x 2 + (2 x − 3)
 OP 2 = x 2 + y 2
substitute
y = 2x − 3

10 x 2 − 27 x + 18
x-value
Hence, x =
2
6  6 
=   +  2  − 3 
5  5 
36 9
=
+
25 25
9
=
5
OP = 1,34
2
OR/OF
For minimum distance OP ⊥ the line
tan Qˆ = 2
Qˆ = 63,43
OP
sin 63,43 =
1,5
OP = 1,34
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answer
(5)
 tan Qˆ = 2
 Qˆ = 63,43
 sin 63,43
OP

1,5
 answer
(5)
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OR/OF
OP =
=
DBE/November 2015

(x − 0)2 + ( y − 0)2
(x − 0)2 + (2 x − 3 − 0)2
2
OP =
substitute
y = 2x − 3
2
= x + 4 x − 12 x + 9
= 5 x 2 − 12 x + 9
By using the chain rule (which is not in the CAPS):
−1
dOP 1 2
= 5 x − 12 x + 9 2 ⋅ (10 x − 12 )
2
dx
−1
1
0 = 5 x 2 − 12 x + 9 2 ⋅ (10 x − 12 )
2
1
0 = (10 x − 12 )
2
0 = 5x − 6
6
x=
5
(
)
(
)
(x − 0)2 + ( y − 0)2
 5 x 2 − 12 x + 9
x-value
2
5
6
OP = 5  − 12  + 9
6
5
= 1,34
OR/OF
For minimum distance OP ⊥ the line
Let the y-intercept be R
OR = 3 units
3
units
OQ =
2
3
RQ =
5 (Pythagoras)
2
1
Area OQR = × base × ⊥height
2
1 3 
1
5 .OP
.OR.OQ = .
2 2 
2
1 3 1 3 
5 .OP
.3.  = .
2 2 2 2 
3
OP =
= 1,34
5
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answer
O
3
1,5
(5)
Q
P
 RQ =
R
3
5
2
 1 . 3 5 .OP
2 2


1 3
.3. 
2 2
 equating
 answer
(5)
Please turn over/Blaai om asseblief
Mathematics/P1/Wiskunde V1
5.6.1
f ′(x ) = 2 x − 3
16
NSC/NSC – Memorandum
DBE/November 2015
 Turning
3
2
3
f ′′( x ) = 2 > 0 or f ′′  > 0
2
Turning point at x =
3
f has a local minimum at x =
2
f het 'n lokale minimum by x =
point at x =
3
2
 f ′′( x ) = 2 > 0
3
2
(2)
OR/OF
h( x) = f ′( x) < 0 for x ∈ (− 2 ; 1,5) ⇒ f is decreasing on the left of Q /
f is dalend links van Q.
′
h( x) = f ( x) > 0 for x ∈ (1,5 ; 4 ) ⇒ f is increasing on the right of Q /
f is stygend regs van Q.
3
∴ f (x) has a local minimum when x = /
2
3
∴ f (x) het ‘n lokael minimum by x =
2
decreasing
left of Q
increasing
right of Q
(2)
OR/OF
f ( x) = x 2 − 3 x + c
f has a minimum value since a > 0
f het 'n minimum waarde omdat a > 0
5.6.2
m = f ′(4) = h(4 ) = 5
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
f ( x) = x 2 − 3 x + c
 explanation
(2)
answer
(1)
[19]
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17
NSC/NSC – Memorandum
DBE/November 2015
QUESTION/VRAAG 6
6.1.1
6.1.2
T (0 ;18)
 (0 ;18)
(1)
y= 0
factors
x = 3
2
− 2 x + 18 = 0
(x − 3)(x + 3) = 0
Q(3 ; 0)
OR/OF
 y= 0
− 2 x 2 + 18 = 0
 x2 = 9
(3)
x=3
x2 = 9
(3)
Q(3 ; 0)
6.1.3
x-coordinate of S is 4,5/x-koördinaat van S is 4,5
By symmetry about the line x = 4,5/Deur simmetrie om die
lyn x = 4,5:
R = (4,5 + 4,5–3 ; 0) = (6 ; 0)
x =6
y=0
(− ∞; ∞ )
6.1.4
For all x ∈ R OR/OF
6.2
If C ( x; y ) is the centre of the hyperbola/As C ( x; y ) die middelpunt is van
die hiperbool
y = x + 6 and x = −2
∴ y = −2 + 6 = 4
answer
y
x = -2
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(2)

asymptote
y=4
asymptote
x = −2
y=4
0
(2)
x
shape
(increasing
hyperbolic
function)
(4)
[12]
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NSC/NSC – Memorandum
DBE/November 2015
QUESTION/VRAAG 7
7.1
R450 000
7.2
answer
A = P(1 − i )
f ( x) = 450000(1 − i ) x
243 736,90 = 450000(1 − i ) 4
i =1− 4
243 736,90
450000
i = 0,1421
The rate of depreciation is 14,21 % p.a.
Die waardeverminderingskoers is 14,21% p.j.
7.3
(1)
n
substitution of
450 000 into
correct formula
substitution of
(4 ; 243 736,90) into
correct formula
making i the
subject
answer
At T :
(4)
A = P(1 + i )
n
g ( x) = 450000(1 + i ) x
a = 450000(1 + 0,081)
4
7.4
= R614490,66
Future Value = R614 490, 66 − R243 736 ,90
= R370 753,76
Let x be the value of monthly payment
n
x (1 + i ) − 1
Fv =
i
 0.062  36 
x 1 +
 − 1
12 


370753,76 =
0.062
12
x = R9397,11
[
]
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i = 0,081 & n = 4
correct
substitution into
formula
answer
(3)
R370 753,76
0,062
12
n = 36
i =
substitution into
correct formula
answer
(5)
[13]
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NSC/NSC – Memorandum
DBE/November 2015
QUESTION/VRAAG 8
8.1
f ( x + h) = ( x + h) 2 − 3( x + h)
= x 2 + 2 xh + h 2 − 3x − 3h
f ( x + h) − f ( x) = x 2 + 2 xh + h 2 − 3x − 3h − ( x 2 − 3x)
= 2 xh + h 2 − 3h
f ( x + h) − f ( x )
f ′( x) = lim
h →0
h
2 xh + h 2 − 3h
= lim
h →0
h
h(2 x + h − 3)
= lim
h →0
h
= lim(2 x + h − 3)
finding f(x+h)
 2 xh + h 2 − 3h
formula
factorisation
h →0
= 2x − 3
answer
(5)
OR/OF
f ( x + h) − f ( x )
h
2
( x + h) − 3( x + h) − ( x 2 − 3 x)
= lim
h →0
h
2
2
x + 2 xh + h − 3 x − 3h − x 2 + 3 x
= lim
h →0
h
2
2 xh + h − 3h
= lim
h →0
h
h(2 x + h − 3)
= lim
h →0
h
= lim(2 x + h − 3)
f ′( x) = lim
h →0
formula
finding f(x+h)
 2 xh + h 2 − 3h
factorisation
h →0
= 2x − 3
8.2.1
answer
(5)
2
1 

y =  x2 − 2 
x 

1
y = x4 − 2 + 4
x
4
= x − 2 + x −4
dy
= 4 x 3 − 4 x −5
dx
OR/OF
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 x4 − 2 +
1
x4
 4x 3
 − 4 x −5
(3)
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NSC/NSC – Memorandum
DBE/November 2015
By using the chain rule (which is not part of CAPS):
(
)
2
y = x 2 − x −2
dy
= 2 x 2 − x − 2 2 x + 2 x −3
dx
= 2 2 x 3 + 2 x −1 − 2 x −1 − 2 x −5
(
(
= 2(2 x
8.2.2
)(
3
− 2x
−5

)
)
(
)(
2 x 2 − x −2 2 x + 2 x −3
)
)
(3)
= 4 x 3 − 4 x −5
 ( x − 1)( x 2 + x + 1) 
Dx 

x −1


2
= Dx x + x + 1
[
]
= 2x + 1
factorisation
 x2 + x +1
 2x + 1
(3)
OR/OF
By using the quotient rule (with is not part of CAPS):
 x 3 − 1
Dx 

 x −1 
(
)
3 x 2 (x − 1) − x 3 − 1
=
(x − 1)2
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
3 x 2 ( x − 1) − x 3 − 1
(x − 1)2
(3)
[11]
(
)
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NSC/NSC – Memorandum
DBE/November 2015
QUESTION/VRAAG 9
9.1
Substitute Q(2; 10) into
h( x) = − x 3 + ax 2 + bx
substitute Q into
h
− 2 3 + a (2 2 ) + b(2) = 10
− 8 + 4a + 2b = 10
finding
derivative
 h′(2)
equating
derivative to 0
2a + b = 9 ..................line 1
h ′( x) = −3 x 2 + 2ax + b
At Q : h ′(2) = 0
− 3(2) 2 + 2a (2) + b = 0
− 12 + 4a + b = 0
4a + b = 12 .............line 2
line 2 − line 1 :
2a = 3
3
a=
2
Substitute in line 1 : b = 6
9.2
f (− 1) = −(− 1) +
3
= −3,5
solving
simultaneously
for a and b
(5)
 f (− 1) = –3,5
3
(− 1)2 + 6(− 1)
2
Average gradient/Gemiddelde gradiënt =
f ( xQ ) − f ( x P )
xQ − x P
10 − (−3,5)
2 − (−1)
= 4,5
Average gradient/ Gemiddelde gradiënt =
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formula
substitution
answer
(4)
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9.3
22
NSC/NSC – Memorandum
DBE/November 2015
h ′( x ) = −3 x 2 + 3 x + 6
 h′( x) = −3x 2 + 3x + 6
 h′′( x) = −6 x + 3
h ′′( x) = −6 x + 3
= −3(2 x − 1)
x
1
1
, h is concave up and for x > , h is concave down
2
2
1
1
Vir x < ,is h konkaaf na bo en vir x > ,is h konkaaf na onder
2
2
For x <
1
/
2
1
∴ konkwiteit verander by x =
2
explanation
using h // ( x )
(3)
∴ concavity changes at x =
9.4
The graph of h has a point of inflection at x =
1
Die grafiek van h het 'n buigpunt by x = .
2
1
/
2
answer
(1)
OR/OF
The graph of h changes from concave up to concave down at
1
1
x = / Die grafiek van h verander by x = van konkaaf op
2
2
na konkaaf af
9.5
Gradient of g is – 12/Gradiënt van g is –12
Gradient of tangent is/Gradiënt van die raaklyn is:
h ′(x ) = −3 x 2 + 3 x + 6
h ′( x) = −12
answer
(1)
 h′(x ) = −3x 2 + 3x + 6
 h′( x) = −12
− 3x 2 + 3x + 6 = −12
3x 2 − 3x + 18 = 0
x2 − x + 6 = 0
( x − 3)( x + 2) = 0
x = − 2 only
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factors
selection of
x-value
(4)
[17]
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NSC/NSC – Memorandum
DBE/November 2015
QUESTION/VRAAG 10
10.1
10.2
h
= tan 60 0
r
h
r=
tan 60 0
h
∴r =
3

h
= tan 600
r
answer
1
Vcone = π r 2 h
3
(2)
formula
2
1  h 
 h
= π 
3  3 
1
= π h3
9
dV 1
= π h2
dh 3
1
dV
2
= π (9)
dh h = 9 3
= 27π or 84,82 cm 3 /cm
Copyright reserved/Kopiereg voorbehou
substitution of
the value of r in
terms of h
simplified
volume answer
derivative
answer
(5)
[7]
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NSC/NSC – Memorandum
DBE/November 2015
QUESTION/VRAAG 11
11.1
11.2.1
P(A) × P(B)
= 0,2 × 0,63
= 0,126
i.e. P(A) × P(B) = P(A and B)
Therefore A and B are independent/Dus is A en B onafhanklik
7 7 = 823 543
11.2.2 7!=5040
11.2.3 There are 3 vowels ⇒ 3 options for first position
There are 4 consonants ⇒ 4 options for last position
The remaining 5 letters can be arranged in 5 × 4 × 3 × 2 × 1 ways
3 × (5 × 4 × 3 × 2 × 1) × 4 = 1440
0,2 × 0,63
P(A) × P(B)=P(A
and B)
conclusion
(3)
77
(2)
7!
(2)
×3
×4
 5 × 4 × 3 × 2 ×1
answer
Daar is 3 klinkers ⇒ 3 opsies vir die eerste posisie
Daar is 4 konsonante ⇒ 4 opsies vir die laaste posisie
Die oorblywende 5 letters kan as volg gerangskik word
5 × 4 × 3 × 2 ×1 ways/maniere
3 × (5 × 4 × 3 × 2 × 1)× 4 = 1440
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(4)
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25
NSC/NSC – Memorandum
11.3
Orange
t
t+2
t
t+2
2
t+2
DBE/November 2015
OO
 t 

t +2
Orange
P(O) = 
2
t+2
t
t+2
Yellow
OY
Orange
YO
Yellow
YY
 2 

t +2
P(Y) = 
Yellow
2
t+2
52
100
 t  t   2  2  52


+

=
 t + 2  t + 2   t + 2  t + 2  100
P(Orange , Orange) + P(Yellow , Yellow) =
P(O,O) =  t 
t +2
 P(Y,Y) =  2 
(
)
(

 t  t   2  2  52
=

+


 t + 2  t + 2   t + 2  t + 2  100
)
3t 2 − 13t + 12 = 0
(3t − 4)(t − 3) = 0
t =3
There are 3 orange balls in the bag/Daar is 3 oranje balle in die
sak
TOTAL/TOTAAL:
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2
t + 2
4
13
t2
+ 2
=
2
t + 4t + 4 t + 4t + 4 25
25 t 2 + 4 = 13 t 2 + 4t + 4
2
 t = 3 (no ca)
(6)
[17]
150 marks