10
Copyright reserved Please turn over NATIONAL CERTIFICATE APRIL EXAMINATION MATHEMATICS N5 4 APRIL 2016 This marking guideline consists of 10 pages. MARKING GUIDELINE
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NATIONAL CERTIFICATE
APRIL EXAMINATION
MATHEMATICS N5
4 APRIL 2016
This marking guideline consists of 10 pages.
MARKING GUIDELINE
MARKING GUIDELINE -2- T970(E)(A4)T MATHEMATICS N5
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INSTRUCTIONS AND INFORMATION 1. 2. 3. 4. 5. 6. 7.
Half marks are not allocated, unless indicated otherwise. Where formula is required, using the wrong formula is a principle error and NO marks are allocated. Students should show ALL formulae and intermediate steps and simplify where possible. ALL final answers must be rounded off to THREE decimal places (unless indicated otherwise). Questions may be answered in any order, but subsections of questions must be kept together. If subsections are separated, the student can be penalised by ONE mark. Where a student copied wrong from the question paper, and the standard of the question is still the same, the student will be penalised by ONE mark. If the copying error simplifies the question and makes it easier, the student forfeits the marks. Questions must be answered in blue or black ink. Answers in PENCIL will not be marked as it is regarded as rough work.
MARKING GUIDELINE -3- T970(E)(A4)T MATHEMATICS N5
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xxx
hx
hx
hxxx
hxhx
h
h
h
sin)1(sin)0(cos
sinh.sin1cosh.coslim
cossinh.sincosh.coslim
cos)cos(lim
0
0
0
-=-=
--
=
--=
-+
®
®
®
xxxxxx 2.)4(4.)4sin( 2).4sin().4(cos4 3242223 ------=
¥=
=01
010
sincoslim
)00(
cossin1lim
2
2
=-
=
--
=
-=
®
®
xx
xx
x
x
p
p
3 )(
96
543
27)3()3(3
- at xcontinuousisxfx
f
=
=--
=---
=-
QUESTION 1 1.1 1.1.1
ü ü
(2)
1.1.2
ü ü ü
(3) 1.2
ü ü
(2)
[7] QUESTION 2 2.1 WENK:
ü ü ü ü
(4) 2.2 2.2.1
ü ü ü ü
(4)
xxeex
x
x 2lim
0 -®
)tan(seclim2
xxx
-®p
1sinhlim; 01coshlim00
==-
®® hh hh
4224 )4cos()4(cos -+-= xxy
MARKING GUIDELINE -4- T970(E)(A4)T MATHEMATICS N5
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.)4cosln2(
4cos4cot.4cos8.1tan34cosln2.
121.1sec.1tan6
2
22
xecxec
xxecxxecx
xx ---
-
---
=
(
2ln2.312
2ln2.3112
33
33
atdxdy
xyx
dxdyxyx
x
x
=--
=--
364,0
12ln8.3)5(12
=
---=
dxdydxdy
364,5364,0:364,5
)1(364,05
-==-
+=-+=
xyequationc
ccmxy
1)4.2(4.24ln4.6233
3
-=
xx
x
dxdy
)ln(
ln.1ln.ln .
xexex
dxdy
xexe
dxdyy
xeyxy
xxe
xx
x
e
x
x
+=
+=
=
=
ü ü ü ü
2.2.2 ü
(5)
2.2.3
üü
(2) 2.3
üü ü
(4) 2.4 2.4.1
ü ü : 1,-5) ü
(3)
2.4.2 ü ü
(2) [24]
)4.2sec( 3xarcy =
MARKING GUIDELINE -5- T970(E)(A4)T MATHEMATICS N5
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303,0 303,8291,1or 291,1
05345)(
2
3
=-=-==
=-
--=
yyxx
xxxxf
571,2129,07,2
129,087,16183,2
)7,2(')7,2(
=-=
-=-=-=
a
ffe
QUESTION 3 3.1 3.1.1
mark for both x-values mark for both y-values
(3) 3.1.2 Use values on the table:
X 0 1 2 3 4 marks for any 4 correct values Y -4 -8 -6 8 40 y-value changes from negative to positive indicating a root
(4) 3.1.3 40
8 shape of the graph y-intercept -2 -1 0 1 2 3 4 -4 -6 -8
(2) 3.1.4
(4)
40 ££ x
MARKING GUIDELINE -6- T970(E)(A4)T MATHEMATICS N5
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dtdy
dtdx
dtdP
== ifonly 0
122,0
3.7100)3(2
.1002
1002
2
2
1
-=
-=
-=
+= -
dtdPdtdP
dtdx
xdtdx
dtdP
xxP
=
--=
--=
v
ttt
v
tttts
23
21
25
23
21
25
29
23
33)(
=
---
=
a
ttt
a 21
21
23 4
15
4
9
4
3
3.2
3.2.1
(5) 3.2.2
Therefore: x=y Meaning the figure is a square
(2) 3.3
3.3.1 -23,987
(4) 3.3.2
-7,159
(3) [27]
yxxyxyA
=
==
-15050
MARKING GUIDELINE -7- T970(E)(A4)T MATHEMATICS N5
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cee xx
++ -
5)( 5
cx
cu
duu
xdu
ux
xdudx
xu
++-
+-
-
-
-=
+=
ò
ò-
cos12
2
sin.sinsin
cos1
21
21
cxxx
xdxxx
xxgxfxxgxxf
+-
-
====
ò)ln(sectan.
tantan.
tan)( 1)('sec)(' )( 2
cxx
dxxx
++=
+ò
44sin
88sin
)4cos()8cos(
cx
dxx
+=
+-
ò
3tan
32432
1
2
QUESTION 4 4.1
(2) 4.2 4.2.1
(3) 4.2.2
(3) 4.2.3
(2) 4.2.4
(3)
xxycos1
sin+
=
xxy 2sec.=
xxy 2cos.6cos=
2432x
y+
=
MARKING GUIDELINE -8- T970(E)(A4)T MATHEMATICS N5
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cxx
dxxx
Ax
Bx
BxxAxxB
xA
xxx
++--=
++
-
=-=
=-=
++=-
++=
+-
ò
)2ln(29ln
21
)2(29
21
21:0
29:2
)2(12)2)((
1
3
3
cxxx
dxx
x
+-++=
-++= ò
)5ln(2552
5255
2
4.3
(5) 4.4
(4) [22]
xxxy
22
2
3
+-
=
MARKING GUIDELINE -9- T970(E)(A4)T MATHEMATICS N5
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566,4(
]2
41624
sin8[
)4(16
40
221
4
0
2
==
+--+=
---
-
òxxxxx
dxxx
02,67)]0()333,21[(
]32
8163
16[
)4(16
40
323
4
0
22
2
=-=
-+--=
---ò
p
p
p
xxxxx
dxxx
s
s
es
se
st
st
5
)10(5
]1[5
][5
0
0
-=
-=
=
--=
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QUESTION 5 5.1
5.1.1
4,566-0)
(3) 5.1.2
(4) 5.2
.
(4) [11]
MARKING GUIDELINE -10- T970(E)(A4)T MATHEMATICS N5
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cxxyy
dxxdyyec
xdxydy
dxxec
dyy
+-=--
-=-
=
=
42sin
21cot
)2cos21
21()1(cos
sincotcos1
tan1
2
22
22
dd
cc
dcxxxxy
cxxxdxdy
=+-=
=-+=-
++++-
=
+++-
=
946,6946,42
725,6725,33
2123
24
43
61
234
23
p
p
QUESTION 6 6.1
(4) 6.2
(5) [9] TOTAL: 100
xecy
dxdy
2
2
costan
=
