national senior certificate examination ......mathematics paper 2 grade 11 june 2015 5 3.4 prove...
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MATHEMATICS PAPER 2 GRADE 11 JUNE 2015
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NATIONAL SENIOR CERTIFICATE EXAMINATION
MATHEMATICS
JUNE EXAMINATION
2015
GRADE 11
PAPER 2
MARKS: 100
TIME : 2 HOURS
This paper consists of 7 pages including diagram sheet
MATHEMATICS PAPER 2 GRADE 11 JUNE 2015
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INSTRUCTIONS AND INFORMATION Read the following instructions carefully before answering the questions.
1 This paper consists of 5 questions.
2 Answer ALL the questions
3 Clearly show ALL calculations, diagrams, graphs et cetera which you have used in
determining the answers.
4 Answers only will not necessarily be awarded full marks.
5 You may use an approved scientific calculator (non-programmable and non-graphical),
unless stated otherwise.
6 If necessary, round off answers to TWO decimal places, unless stated otherwise.
7 Diagrams are NOT necessarily drawn to scale.
8 Number the answers correctly according to the numbering system used in this question paper.
9 Write neatly and legibly.
10 One diagram sheet for QUESTION 5.1 is attached at the end of this question paper. Please
write your NAME in the diagram sheet.
MATHEMATICS PAPER 2 GRADE 11 JUNE 2015
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QUESTION 1 In the diagram below, ABCD is a quadrilateral with vertices A (2; 6), B (4; 0); C (7; 1) and
D(8; 8) in a Cartesian plane. E and F are on AD and CD respectively. AD // BC.
1.1 Determine
1.1.1 the coordinates of F, the mid-point of DC. (2)
1.1.2 gradient of BC (2)
1.1.3 the value of E, the angle of inclination of BC (3)
1.1.4 the equation of AD in the form y = …… (4)
1.1.5 the equation of CE if CE A AD (4)
1.2 Calculate the length of AD and AB (4)
1.3 Show that AD is perpendicular to AB (4)
1.4 Calculate the area of 'ABD (3) [26]
x
y
A (2; 6)
B (4; 0)
D (8; 8)
C(7; 1)
0
F
E
E
MATHEMATICS PAPER 2 GRADE 11 JUNE 2015
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QUESTION 2
In the diagram, ABCD is a quadrilateral with A(4; –1) and B(6; 9), The diagonals of ABCD intersect at M(0; 3), The gradient of AB = 5 and E = 45q 2.1 What type of a quadrilateral is ABCD? Give a reason for your answer. (2) 2.2 Determine the coordinates of C (2) 2.3 Determine the value of T (2) 2.4 Hence or otherwise, determine the value of D (3) [9] QUESTION 3
3.1 Reduce the following to one trigonometric ratio:
TTT 222 sintantan � (3)
3.2 If 5
62sin A and > @360;90q�A
With the aid of a diagram, calculate the value of AA cos.tan15 (5)
3.3 If cos 20q = t, express the following in terms of t:
3.3.1 tan 20q (3)
3.3.2 sin 340q (2)
3.3.3 2cos 110q (2)
x T
D
B(6; 9)
M(0; 3)
C
A(4; –1) 0
y
E
D
MATHEMATICS PAPER 2 GRADE 11 JUNE 2015
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3.4 Prove that
TT
TTcos
1sin1
costan �
� (5)
3.5 Simplify:
»¼º
«¬ª
��
� 1sin1
1sin1cos2
xxx (6)
[26]
QUESTION 4 4.1 Simplify the expressions fully
)90sin()180cos()360tan()180sin()cos()180sin(xxx
xxx�q�qq��
�q���q (7)
4.2 Without the use of a calculator, calculate
qqq135tan.170sin
80cos.225sin.180cos (7)
4.3 Determine the general solution of 0sin3cos3 � TT (5)
[19] QUESTION 5 Consider the functions defined by:
]180;90[tan21)(2cos)( qq�� xforxxgandxxf
5.1 Sketch the graphs of f and g on the same system of axes. (6)
5.2 Determine
5.2.1 the period of f (1)
5.2.2 the amplitude of f (1)
5.2.3 the range of f (2)
5.3 For which values of x will:
5.3.1 f(x) t 0 (4)
5.3.2 g(x) > f(x) (2)
MATHEMATICS PAPER 2 GRADE 11 JUNE 2015
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5.4 If the graph of f is shifted 2 units downwards and 45q to the left, write down
the equation of f in the form h(x) = ….. (2)
5.5 If g(x) is reflected about the y-axis, write down the image of g(x) in the form y = …. (2)
[20] TOTAL: 100
MATHEMATICS PAPER 2 GRADE 11 JUNE 2015
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DIAGRAM SHEET FOR QUESTION 5 NAME:………………………………….
NATIONAL SENIOR CERTIFICATE EXAMINATION NASIONALE SENIOR SERTIFIKAAT EKSAMEN
MATHEMATICS / WISKUNDE
JUNE EXAMINATION / JUNIE EKSAMEN
2015
GRADE/ GRAAD11
PAPER/VRAESTEL 2
MARKING MEMORANDUM
MARKS/PUNTE: 100 TIME/ TYD: 2 hours/uur
This memorandum consists of 7 pages / Hierdie memorandum bestaan uit 7 bladsye
Mathematics P1 / Wiskunde V1NSC Grade /Graad 12 memorandumMDoE/June/Junie 2015
2
QUESTION/ VRAAG 1�1.1.1
� �5,4;5,7/29;
215
218;
278);(
FOFOR
yxF
¸¹·
¨©§
¸¹·
¨©§ ��
9 2
15 x
9 29 y
(2)�1.1.2
31
4701
��
BCm
9 substitution/ vervang �9 answer/ antw. (2)�
1.1.3 Etan BCm
31tan ? E
¸¹·
¨©§ �
31tan 1E
q 44,18E
9 Etan BCm
9
31tan E
9 q 44,18E (3)�
1.1.4 BCADmAD //,31
cmxy � � � )6;2(/2
316 Aasforc�
316 c
316/
316
31
�
�
xyofor
xy
931 ADm
9substitution/vervang �9value of c/ waarde van c 9correct equation/ Korrekte vergelyking (4)�
1.1.5 31 ADm
,/ CEADmaarBut A 3� ? CEm cmxy �
� � )1;7(,731 Cforc�� 22 ?c
223 �� xy
9 3� CEm �9substitution/ vervang �9value of c/ waarde van c �9correct equation/ Korrekte vergelyking (4)�
Mathematics P1 / Wiskunde V1NSC Grade /Graad 12 memorandumMDoE/June/Junie 2015
3
1.2
9substitution for AD/ Vervang vir AD �9answer /antw. ���9substitution for AB Vervang vir AB �9answer Antw. (4)
1.3
�9substitution into gradient formula Vervang in gradient formule �9gradient of/van AB �9method/ metode �9 1� u ADAB mm (4) �
1.4 �9substitution into area formula/ vervang in oppervlak formule ��9simplification/ vereenvoudig �9answer/antw. (3)
QUESTION/ VRAAG 2�2.1 Parallelogram
Both pairs of opposite sides are parallel/ albei pare teenoorst. sye //
99 answer and reason Antw. en rede (2)�
2.2
4420
2
� ?� u
�
C
C
ACm
xx
xxx
9 substitution into/ vervang in
2AC
m
xxx � ��
And/ en 2
ACm
yyy � �
� � � �
units
AD
102
40
364
2868 22
�
���
ABAD
mm
m
m
ADAB
AD
AB
A?�
�u u?
� �
��
1
331
31
32
64206
enhedevierkanteeunitssquare
hbABDoppervlofArea
eenhedeunitsABenandAD
/202104
10210221
21/
/102/102
u
uu
uu '?
� � � �
units
AB
102
40
3640624 22
�
���
Mathematics P1 / Wiskunde V1NSC Grade /Graad 12 memorandumMDoE/June/Junie 2015
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2AC
M
yyy �
)1(23 �� u Cy
)7;4(7
�? ?
Cyc
�9x = –4 and/ en y = 7 (2)�
2.3
RTT
69,785tan
5
ABm
9 tan T = 5 �9answer/ antw. (2)�
2.4 E + D = T (ext. angle of Δ) / (buite hoek van Δ) ?D = 78,69q – 45q = 33,69q OR/ OF 45q + 101,31 + D = 180q (sum of angles in a ')/ ( hoeke van Δ)
?D = 33,69q
9E + D = T 9reason/ rede 9answer/ antw. OR/ of 945q+101,31+D=180q 9reason/ rede 9answer/ antw. (3)
QUESTION/VRAAG 3�3.1 TTT 222 sintantan �
)sin1(tan 22 TT � TT 22 cos.tan
TTT 2
2
2
cos.cossin
T2sin
�9 factorization/ faktore 9 T2cos 9
TT
2
2
cossin
(3)�3.2
� �
11
2425562
2
222
222
� ?r
�
�
�
xxxx
ryx
¸¹·
¨©§�¸̧¹
·¨̈©
§�
51
16215costan15 AA
66
�9 sketch in correct quadrant/ skets in korrekte kwadrant �9 value of x/ waarde van x �9
162tan
� A
951cos � A
9 answer/ antw. (5)
3.3.1 22 1 ty � 21 ty �
tt2120tan � q
�9 sketch/ skets 9 value of y/ waarde van y 9 answer / antw. OR/ of
5630
y
0 x
5
–1
A 62
y
x
1
20q t
y
Mathematics P1 / Wiskunde V1NSC Grade /Graad 12 memorandumMDoE/June/Junie 2015
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OR/ of
qq q
20cos20sin20tan
sin220q = 1 – cos220q ? 2120sin t� q
tt 2120tan � q?
��9 identity for tan 20q Identiteit vir tan20˚ 9 identity for sin2 20q Identiteit vir sin220˚ 9 answer/ antw. (3)�
3.3.2 )20360sin(380sin q�q q q� 20sin
21 t�� 9 q� 20sin 9 21 t�� (2)�
3.3.3 2cos 110q = 2cos (90q+20q) = –2sin 20q = –2t
9–sin 20q 9–2t (2) �
3.4 TTT
sin1costan/�
� LKLHS
TT
TT
sin1cos
cossin
��
)sin1(cos
cos)sin1(sin 2
TTTTT
���
)sin1(cos
cossinsin 22
TTTTT
���
)sin1(cos
1sinTT
T��
Tcos
1
RKRHS /
9TT
cossin
�9 TTT 2cos)sin1(sin �� 9Denominator/noemer, (cosT (1+sinT )) 9 1cossin 22 � TT 9simplification/ vereenvoudig (5)�
3.5 »¼º
«¬ª
��
� 1sin1
1sin1cos2
xxx
»¼
º«¬
ª�����
)1)(sin1(sin1sin1sincos2
xxxxx
»¼º
«¬ª
�
1sinsin2cos 2
2
xxx
»¼º
«¬ª�
x
xx 22
cossin2cos
xsin2�
9Numerator/ teller �9Denominator/ noemer �9 2sin x �9 1sin2 �x �9 x2cos� 9answer/ antw. (6)
QUESTION/ VRAAG 4�
Mathematics P1 / Wiskunde V1NSC Grade /Graad 12 memorandumMDoE/June/Junie 2015
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4.1
9 sin x 9 cos x 9 –tan x 9 –cos x 9 cos x �9
xx
cossin
9 answer / antw. (7)�
4.2
9 –sin 45q �9 sin 10q 9 –tan 45q 9 –1 9
22
9 1 9 answer/ antw. (7)�
4.3 0sin3cos3 � TT
TT
TT
cossin3
coscos3
Ttan33
33tan T
q � 30ref Zkk �q�q ,.18030T OR Zkk �q�q ,180210T
�9division by cos T / deling deur cosT�9
33tan T
9 kq�q 18030T 9 k.180210 q�q T 9 Zk� (5)�
OUESTION/VRAAG 5�
.22
122
45tan45sin.1
45tan10sin10sin.45sin.180cos
)45180tan()10180sin()1090cos()45180sin(.180cos
�
�
�
��
����
q
q
qqq
qqqq
qqqqq
.1sinsin
coscos.sin
sincos.cos.tan
cos.sin)90sin()180cos()360tan(
)cos()180sin(
��
������q
qqq
xx
xxx
xxxx
xxxxx
xx
Mathematics P1 / Wiskunde V1NSC Grade /Graad 12 memorandumMDoE/June/Junie 2015
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5.1
9 shape of/ vorm van f 9 x-intercepts/ afsnitte of f 9 turning points of/ draaipunte van f �9 shape of/ vorm van g 9 x-intercepts/ afsnitte of g 9 asymptote of/ asimptote van f (6)
5.2.1 Period = 180q 9answer/ antw. (1)�5.2.2 Amplitude = 1 9answer/ antw. (1)�5.2.3 y � [–1; 1] or
–1 d y d 1 9critical values/ kritiese punte 9notation/ notasie (2)�
5.3.1 –45q d x d 45q or/ of x t 135q 99–45q d x d 45q�9or/ of�9 x t 135q (4)�
5.3.2 45q d x < 90q 9critical values/ kritiese waardes 9notation/ notasie (2)�
5.4 h(x) = cos 2(x + 45q) – 2 h(x) = cos(2x + 90q) – 2
9cos 2(x + 45q) 9– 2 (2)�
5.5 )tan(21 xy �
xy tan21�
9 )tan(21 xy � �
9 xy tan21�
(2)�
f
g