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