13
Copyright Reserved Please Turn Over MATHEMATICS COMMON TEST JUNE 2018 MEMORANDUM NATIONAL SENIOR CERTIFICATE GRADE 12
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MATHEMATICS
COMMON TEST
JUNE 2018
MEMORANDUM
NATIONAL SENIOR CERTIFICATE
GRADE 12
Mathematics ! NSC Memo June Comm Test 20182MARKS: 125
TIME: 2.5 hours
This memorandum consists of 11 pages.
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Mathematics ! NSC Memo June Comm Test 20183QUESTION 1
1.1.1 A✓ 0 A✓ 6 (2)
1.1.2A✓formula
CA✓substitution
CACA✓✓answers
[Penalise for rounding off here only in the entire paper]
(4)
1.1.3A✓Isolating surd term
CA✓ squaring both sides
CA✓standard form
CA✓factors
CA✓answers and rejecting
(5)
1.2A✓y / x the subject
CA✓substitution
CA✓standard form
CA✓factors
CA✓ x – values
CA✓ y – values
(6)
……(1) and ! ….(2)
01 =+− xy 32 =+ xyx
( )( )
221
123
013203233)1(
)2(into)3(ngSubstituti)3......(1:)1(From
2
22
2
−==
−==
=+−
=−−
=−+
=−+
−=
yory
xorx
xxxxxxx
xxx
xy
58,858,0)1(2
)5)(1(4)8()8(2
4
058
2
2
2
−=
−−±−=
−±−=
=−+
or
aacbbx
xx
( )( )
an
xorx
xxxx
xxxxx
xx
/491
0941095410144
1012
1102
2
2
=−=
=−+
=−−
+=+−
+=−
=+−
60 or
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Mathematics ! NSC Memo June Comm Test 20184
QUESTION 2
1.4A✓factorization
A✓reasoning
A✓ conclusion (3)
[20]
The graph y = ! has its x – intercept at x = 0 and the graph y = – 33x2 has its maximum value at x = 0. Therefore the equation has only one solution
( )( )
2
2
2
3
3313033130
033130333.
xxorx
xxxxx
x
x
x
x
−=−
=+−=
=+−
=+−
13 −x
2.1 x 4 8
y
T1 T2 T3
T4
1D 4-x 4
y-8
2D x y-12
x = 4 and y = 16
A✓ x – value
A✓ y – value (2)
2.2 CA✓ a value
CA✓b value
CA✓ c value
CA✓answer
(4) 862
84
603242
2 +−=
=
=++
−==+
==
nnTc
cbabba
aa
n
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Mathematics ! NSC Memo June Comm Test 201852.3
A✓quadratic
CA✓factors
CA✓ answer (3)
[9]
( )( )79820798206478301295662
12964862
2
2
2
≠=∴
=+−
=−−
=−−
=+−=
nornnn
nnnn
nnTn
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Mathematics ! NSC Memo June Comm Test 20186QUESTION 3
3.1
(5)
3.2 A✓correct formula
A✓substitution
CA✓ answer(3)
3.3 A✓condition
A✓substitution of r value
CA✓answer(3)
[11]
1228512)12(3
1)1(
12
12
=−
−=
−
−=
S
rraSn
n
A✓ !
CA✓a – d = 0
A✓
CA✓ a + d = 4
CA✓a – value
)3(27 dada +=+
122 =++++ dadaa
28353
1351
11
−<<−
<+<−
<+
<−
<<−
xx
xr
242:)2()1()2(4
1233122
)1(0627)3(27
.......;)2(;)(;
=⇒=+
→=+
=+
=++++
→=−
+=+
+=+
++
aadada
dadaada
dadadadadadaa
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Mathematics ! NSC Memo June Comm Test 20187
QUESTION 4
4.1(2)
4.2 A✓ y – intercept value
A✓ x – intercept value
(2)
4.3
A✓ shape
CA✓ x – intercept
CA✓ y – intercept
CA✓ asymptotes(4)
[8]
A✓ !
A✓
1−=x
3−=y
31 −=−= yandx
35:intercept
5:intercept
−=−
−=−
xx
yy
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Mathematics ! NSC Memo June Comm Test 20188QUESTION 5
5.1 A✓ axis if symmetry value
CA✓y – value
CA✓answer [coordinate form](3)
5.2CACA✓✓answer (2)
5.3A✓equation equal to 0
CA✓factors
CA✓x – values
CA✓answer(4)
5.4 A✓equating
CA✓standard form
CA✓factors
CA✓x – values
CA✓answer
(5)
5.5 A✓substituting (x – 3 )
CA✓ x – value
CA✓y – value (3)
[17]
( )( )
⎟⎠
⎞⎜⎝
⎛ −
=−=
=−+
=−−
+−=++−
+−=++−
814;
43P
243
02340654
631224
32362
2
2
2
xorx
xxxx
xxx
xxx
816≤y
New turning point
( )24132
69623)3()3(2)(
2
2
2
−+−=
−++−−=
−−+−−=
xxxxx
xxxh
⎟⎠
⎞⎜⎝
⎛816;
413A
( )( )
units213BC
223
0232062062
2
2
=
=−=
=−+
=−−
=++−
xorx
xxxxxx
⎟⎠
⎞⎜⎝
⎛
=+⎟⎠
⎞⎜⎝
⎛+⎟⎠
⎞⎜⎝
⎛−=
816;
41A
8166
41
412
2
y
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Mathematics ! NSC Memo June Comm Test 20189QUESTION 6
6.1A✓ substitution of the point A
CA✓ answer (2)
6.2 CACA✓✓ answer (2)
6.3 AA✓ ✓answer or any other point
that satisfies the equation. (2)
6.4
A✓ x – value
CA✓ answer(2)
6.5
CACA✓✓ answer (2)
[10]
910
913
2log2)(
2
3
1
≤<
==
−=
−≤
−
−
x
x
xxh
xy 3log=
⎟⎠
⎞⎜⎝
⎛=
⎟⎠
⎞⎜⎝
⎛= −
2log
2)(
3
1
x
xhxf
⎟⎠
⎞⎜⎝
⎛− 2;
91
3391)(
22
2
=
=
=
=
−−
−
aa
a
axh x
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Mathematics ! NSC Memo June Comm Test 201810QUESTION 7
7.1 A✓formula
A✓substitution into formula
CA✓answer(3)
7.2.1A✓substitution into formula
CA✓value
CA✓answer (3)
7.2.2 A✓formula
A✓ substitution of F
A✓substitution of i
A✓substitution of n
CA✓answer
(5)
7.2.3 A✓formula
A✓substitution of F and i values
A✓substitution of n value
CA✓ R630,79
(4)
[15]
New monthly deposit = R 13337,73
79,6301212,0
11212,01
8000
]1)1[(
12
R
x
iixFn
v
=
⎥⎥⎦
⎤
⎢⎢⎣
⎡−⎟
⎠
⎞⎜⎝
⎛+
=
−+=
He will pay R1337771,79 – R300 000
=R1037771,79
( )79,77711330825,01000900
)1(5
R
iPA n
=
+=
+=
94,127061212,0
11212,01
79,1037771
]1)1[(
60
R
x
iixFn
v
=
⎥⎥⎦
⎤
⎢⎢⎣
⎡−⎟
⎠
⎞⎜⎝
⎛+
=
−+=
42,452253)0925,01(000500
)1(7
R
iPA n
=
−=
−=
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Mathematics ! NSC Memo June Comm Test 201811QUESTION 8 [Penalise once only in questions 8 and 9 for notational error]
8.1 A✓formula
A✓substitution
CA✓ simplification of numerator
CA✓factors
CA✓answer
(5)
8.2.1
(4)
8.2.2
(4)
[13]
2/
1
2
41)(4
4)(
−
−
+=
−=
−=
xxfxx
xxxf
A✓ A ✓
CA✓answer CA✓answer
34
x2
91 −x
331
/
234
23 4
92
34)(
91
)3()(
−
−
−
−=
+=
+=
xxxg
xx
xxxg
( ) ( )
( )
56
536
536
5355363
)53()(5)(3
)(
lim
lim
lim
lim
lim
0
2
0
222
0
22
0
0
−=
−+=
−+=
+−−−++=
−−+−+=
−+=ʹ
→
→
→
→
→
xhhxhh
hhxhh
xxhxhxhxh
xxhxhxh
xfhxfxf
h
h
h
h
h
A✓x A✓
CA✓answer CA✓answer
14 −− x
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Mathematics ! NSC Memo June Comm Test 201812QUESTION 9
9.1 AA✓✓x – intercepts / point form
A✓y – intercept /point form (3)
9.2
A✓derivative and equal to 0
CA✓standard form
CA✓factors
CA✓x – values
CA✓y – values (5)
9.3
A✓shape
CA✓ x – intercepts
CA✓ y – intercept
CA✓ turning points
(4)
9.4CA✓second derivative
CA✓answer (2)
( )( )
)0;3()32;1(03231
031032
0963)(2793)(
2
2/
23
−−
=−=
=−=
=−+
=−−
=++−=
−++−=
yoryxorx
xxxx
xxxfxxxxf
11
066)(963)(
//
2/
>
=
=+−=
++−=
xx
xxfxxxf
2733
−=
=−=
yxorx
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Mathematics ! NSC Memo June Comm Test 201813
Total Marks : 125
9.5A✓substituting x = 0 into derivative
CA✓gradient value
CA✓equation (3)
9.6(3)
9.7(2)
[22]
032 <<− k
A✓
CA✓substitution of ! into f
)3( +xf
)3( +x 27)3(9)3(3)3()3()(
23 −+++++−=
+=
xxxxfxt
2799)0(
963)(/
2/
−=
=
++−=
xyf
xxxf
CA✓ 32
CA✓0
A✓inequality
−
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