NATIONAL SENIOR CERTIFICATE EXAMINATION MATHEMATICS JUNE EXAMINATION 2015...
Transcript of NATIONAL SENIOR CERTIFICATE EXAMINATION MATHEMATICS JUNE EXAMINATION 2015...
NATIONAL SENIOR CERTIFICATE EXAMINATION
MATHEMATICS
JUNE EXAMINATION
2015
GRADE 11
PAPER 1
MARKS : 100
TIME : 2 hours
This question paper consists of 5 pages
Mathematics P1 2 MDoE/June Exam 2015
NSC
INSTRUCTIONS AND INFORMATION
Read the following instructions carefully before answering the questions.
1. This question paper consists of 6 questions.
2. Answer ALL the questions.
3. Clearly show ALL calculations, diagrams, graphs, etcetera which you have used in determining your answers.
4. Answer 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.
Mathematics P1 3 MDoE/June Exam 2015
NSC
QUESTION 1
1.1 Solve for x
1.1.1 0)3)(5( �� xx (2)
1.1.2 232 �
xx (correct to 2 decimal places) (4)
1.1.3 0)1( 2 !�x (2)
1.1.4 27333 �� xxx (3)
1.1.5 42 � �� xx (6)
1.2 Given: 424 �� bba
1.2.1 Show that: ab 24 � (2)
1.2.2 Hence, solve for a and b simultaneously if it is further given that
432 2 � � aba (5)
1.3 If 51 �
xx , without solving for x, determine the value of:
1.3.1 22 1
xx � (3)
1.3.2 33 1
xx � (4)
[31]
QUESTION 2
2.1 Calculate 32
3 2.8.16 without the use of calculator. (4)
2.2 Simplify :
2.2.1 � � nn
nnn
6.33.1854
21
11
�
��� (7)
2.2.2 4
66
325018
xxx � (3)
[14]
Mathematics P1 4 MDoE/June Exam 2015
NSC
QUESTION 3
3.1 Consider the expression 65352
2
2
����
xxxx and determine the value(s) of x for
which the expression is:
3.1.1 equal to 0 (4)
3.1.2 undefined (3)
3.2 Determine the value(s) of p for which the equation ,11 2 �� xxp
where 0zp
has no real roots. (6)
3.3 The roots of a quadratic equation are given by 7
8204 rx �r� , where
^ 3̀;2;1;0;1;2;3 ����r . Write down two values of r for which the roots will be
rational. (2)
[15]
QUESTION 4
4.1 Consider the following number pattern:
1 ; -3 ; -9 ; -17 ; ..........
4.1.1 Determine the general term for the above number pattern: (4)
4.1.2 Determine the value of the 30th term. (2)
4.1.3 Which term in the pattern will equal to -2547? (4)
4.2 The second difference of the quadratic number pattern
8 ; y ; 20 ; …… is 2. Determine the value of y. (3)
[13]
QUESTION 5
Given: 12
4)( ��
x
xh
5.1 State the equations of the asymptotes of h . (2)
5.2 Write down the equation of the line of symmetry of h which has a negative gradient. (2)
5.3 State the domain of h . (1)
5.4 Determine the x - and y -intercept of ).(xh (4)
Mathematics P1 5 MDoE/June Exam 2015
NSC
5.5 Sketch h , on the set of axes, indicating the intercept with the axes and the
asymptotes. (3)
5.6 Determine the average gradient of h between 3 x and 1 x (3)
[15]
QUESTION 6
6.1 The graphs of 2)( bxxf and xaxg )( are drawn below.
K(2;16) is a common point of the two graphs.
6.1.1 Determine the equations of f and g (4)
6.1.2 For which values of x is f decreasing? (1)
6.1.3 For which value of x is 1)()( � xfxg ? (1)
6.1.4 Write down the equation of the axis of symmetry of f . (1)
6.1.5 Write down the equation of g, if g is the graph formed by translating
the graph of f 3 units to the right and 3 units up. (2)
6.2 Draw the sketch graphs of cbxaxxf �� 2)( in the following case:
,0�a 0!b , 0�c , 0�' (3)
[12]
TOTAL: 100
K(2;16) ●
f
g
NATIONAL SENIOR CERTIFICATE
MATHEMATICS/ WISKUNDE
JUNE EXAMINATION
JUNIE EKSAMEN
2015
GRADE/ GRAAD 11
PAPER / VRAESTEL 1
MEMORANDUM
MARKS/ PUNTE: 100
TIME/ TYD : 2 H
This memorandum consists of 8 pages/ Hierdie memorandum bestaan uit 8 bladsye.
Mathematics/Wiskunde/Grade 11 P1 Memo 2 MDoE/June/ Junie Exam 2015 NSC
x Mark according to the method used by candidate. / Merk volgens leerder se metode. x Consistence accuracy to be used ./ Volg antwoorde op met konsekwente akkuraatheid.
QUESTION/ VRAAG 1 1.1.1 0)3)(5( �� xx 5� x or/ of 3 x
9 5� x 9 3 x (2)
1.1.2 232 �
xx
6126
)3(2)2)(3(4)6()6(
0263623
2
2
2
r
��r��
��
�
x
x
xxxx
58,1 x or/ of x = 0,42
9 standard form/standaardvorm 9 substitution/ substitusie 9 58,1 x 9 42,0 x
Penalise 1 mark for wrong rounding off Penaliseer met 1 punt vir afronding wat verkeerd is. (4)
1.1.3 0)1( 2 !�x 0)1)(1( !�� xx critical value/ kritiese waarde: 1 x 1 Rx� , 1zx
9 Critical value/ kritiese waarde 9 Rx� , 1zx
(2)
1.1.4 27333 �� xxx
23393
27)3(3
2
x
x
x
x
9 )3(3x 9 233 x 9 2 x (3)
1.1.5 42 � �� xx 42 � � xx 22 )4()2( � � xx 1682 2 �� � xxx 01492 �� xx � �� � 027 �� xx 2/7 xoforx N/A / nvt 7 ? x
9 42 � � xx 9 squaring both sides/ kwadreer beide kante 9 standard form/standaardvorm 9 both factors/ albei faktore 9 2/7 xoforx 9 choosing/ kies 7 x (6)
Mathematics/Wiskunde/Grade 11 P1 Memo 3 MDoE/June/ Junie Exam 2015 NSC
1.2.1 424 �� bba 4)(2 22 �� bba 422 � � bba ab 24 �
9 )(22 ba� 9 422 � � bba (2)
1.2.2 ab 24 � ………….(1) 432 2 � � aba …. ………(2) Subst./ vervang (1) into/in (2) : 4)24(32 2 � �� aaa 46122 22 � �� aaa 04128 2 �� aa 0132 2 �� aa � �� � 0112 �� aa
121
aora
23 bb
9 substitution/ vervang 9 standard form / standaardvorm 9 factors/ faktore 9 values of a/ waardes van a 9 values of b / waardes van b (5)
1.3.1 51 �
xx
271
2512
51
22
22
22
�
��
¸¹·
¨©§ �
xx
xx
xx
9squaring both sides/ kwadreer beide 9simplification/ vereenvoudig 9answer/ antw.
(3)
1.3.2 33 1
xx �
140
)127)(5(
1112
2
�
¸¹·
¨©§ ��¸¹·
¨©§ �
xx
xx
99factors/ faktore 9 substitution/ vervang 9 answer/ antw. (4)
[31] QUESTION / VRAAG2
2.1 32
3 2.8.16
� �
3222
2.2.2
2.2.2
5
323
34
32
334
32
331
4
��
9 � � 331
4 2.2
9 34
2 9addition of exponents/ tel eksponente op 9 answer/ antw. (4)
Mathematics/Wiskunde/Grade 11 P1 Memo 4 MDoE/June/ Junie Exam 2015 NSC
2.2.1 � � nn
nnn
6.33.1854
21
11
�
���
= � � � �� � � �nn
nnn
2.3.33.2.32.3
21
1123
�
���
= nnn
nnnnn
2.3.33.2.32.3
22
11223
�
����
= nnn
nnnnn
2.32.32.3
22
11223
��
�����
= nn
nnnn
2.32.32.3
23
1133
�
���
=� �
nn
nn
2.3.32.312.3
23
113
�
���
=
91
21.
311�
=
91
611�
=19
65u
= 645
=2
15
9 rewriting the powers with prime bases Herskryf met priembasisse
9multiplying the exponents Vermenigvuldig eksponente
9 adding exponents of like terms optel van eksponente met selfde basisse 9 taking out a common factor Haal gemeenskaplike factor uit
9
91
21.
311�
9 19
65u
9 645
(7)
2.2.2 4
66
325018
xxx �
2
2422
242523
21622529
2
3
2
33
4
66
xxx
xxx
xxx
�
�
�
u
u�u
94
66
21622529
xxx
u
u�u
9 322 x�
9answer/ antw. (3) [14]
Mathematics/Wiskunde/Grade 11 P1 Memo 5 MDoE/June/ Junie Exam 2015 NSC
QUESTION / VRAAG 3 3.1.1
� �� �� �
5/70570352
)0(352
0352
2
22
2
2
� �� ��
��
��
xoforxxx
xxxx
xx
9 03522 �� xx 9standard form/ standaardvorm 9factors/ faktore 9x values/ x-waardes (4)
3.1.2 0652 �� xx
� �� �
2/3023
� � ��
xoforxxx
9 0652 �� xx 9factors/ faktore
9x values / x-waardes (3)
3.2
� � � �
pppppppp
acbppxpxppxpx
4344
)1(4
401
1
2
22
2
2
2
2
��
��
���
� '
���
��
for non-real roots/ nie-reële wortels 0�'
0)43(043
0432
2
!�!�
���
pppp
pp
critical values :
0 p or 34
p
0�p or /of 34
!p
9standard form/ standaardvorm
9substitution/ vervang 9 0�' 9factorisation/ faktore
9 0�p ; 34
!p
9or / of (6)
3.3 -2 and /en 2 9 - 2 9 2 (2)
[15]
3
4
0
Mathematics/Wiskunde/Grade 11 P1 Memo 6 MDoE/June/ Junie Exam 2015 NSC
QUESTION/ VRAAG 4 4.1.1 1 ; -3 ; -9 ; -17 1st diff -4 -6 -8 eerste verskil 2nd diff -2 -2 tweede verskil
33
11
4)1(343
122
2 ���
��
� � ��
� �� �
nnTc
cbab
bba
aa
n
9value of a/ waarde van a 9value of b/ waarde van b 9value of c/ waarde van c 9 32 ��� nnTn (4)
4.1.2 � �927
33030 230
� ��� T
9substitution/vervang 9answer/ antw (2)
4.1.3
� �� � 05150
0255025473
2
2
�� ��
� ���
nnnn
nn
50 n or 51� n (N/A) ? 50 n
9Tn= -2547 9factors/ faktore
9values of n/ waardes van n
9choosing correct n value/ kies die korrekte waarde vir n
(4) 4. 2 8 ; y ; 20 y – 8 20 – y 2 2)8()20( ��� yy 2228 � y 13 y
9 1st differences/ 1 ste verskille 9 2)8()20( ��� yy 9answer/ antw. (3)
[13]
Mathematics/Wiskunde/Grade 11 P1 Memo 7 MDoE/June/ Junie Exam 2015 NSC
QUESTION / VRAAG 5
5.1 21
xy
9 1 y
9 2 x (2) 5.2
3321
��
�� ��
xyc
ccxy
9Substitution/ vervang 9Equation/ vergelyking
(2)
5.3 domain/ defversameling: RxH , 2zx
9domain/ defversameling: RxH , 2zx (1)
5.4 x-int/ afsnit; let/ laat 0 y
� �0;22
24
12
4
012
4
��
��
� �
��
xx
x
x
y-int/afsnit; let/laat 0 x
� �1;0
1
120
4
��
��
y
y
9y = 0 9 simplification/ vereenvoudig 9 2� x 9 1� y
(4)
5.5
9shape/ vorm 9intercecpts with the axes/ afsnitte met asse 9asymptotes/ asimptote (3)
Mathematics/Wiskunde/Grade 11 P1 Memo 8 MDoE/June/ Junie Exam 2015 NSC
5.6
� �
� �� � � � � � 4
235
1313/
3121
41
5123
43
��
��
� ��
��
hhgemAverage
h
h
9h(3)= 5 9h(1)= -3
9average/ gemiddelde gradiënt = 4 (3) [15]
QUESTION / VRAAG 6 6.1.1
� �
2
2
2
4)(4
216
)(
xxfb
b
bxxf
x
x
xga
aa
axg
4)(4
416
)(
22
2
9substitution/ vervang 9equation/ vergelyking 9 substitution/ vervang
9 equation / vergelyking (4)
6.1.2 0�x �x R OR/OF � �0;f� 9 answer / antw. (1) 6.1.3 0 x 9 answer / antw. (1) 6.1.4 0 x 9 answer / antw. (1)
6.1.5 � � 334 2 �� xy 99answer / antw. (2)
6.2
9 shape/ vorm
9 turning point in the 4th quadrant & graph not cutting x axis dp in die 4de kwadrant en grafiek sny nie die x-as nie 9 y intercept below x- axis y-as afsnit onder die x-as
(3)
[12] TOTAL/ TOTAAL
100
x
y
c
0