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NATIONAL CERTIFICATE

NOVEMBER EXAMINATION

MATHEMATICS N2

24 NOVEMBER 2016

This marking guideline consists of 11 pages.

MARKING GUIDELINE

MARKING GUIDELINE -2- T860(E)(N24)T MATHEMATICS N2

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INSTRUCTIONS AND INFORMATION 1. 2. 3. 4. 5. 6. 7. 8.

ü equals 1 mark. Half marks are not allocated, unless indicated otherwise (√). When a formula is required and the wrong formula used, it 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 decimals (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 wrongly 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 are NOT marked as it is regarded as rough work.

MARKING GUIDELINE -3- T860(E)(N24)T MATHEMATICS N2

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)1)(1(

)1)(1(88)1(6)1(2

)1)(1(1

)1(86

)1(41

+-=

+--+++

=

+--

-+

-

xxx

xxxx

xxxx

QUESTION 1 1.1

(8) 1.2 1.2.1

(4) 1.2 1.2.2

(4) [16]

2

2

3 2 2

2

2

4 ( 2)( 2)8 14 4 2(4 1)( 2)

2 4 8 ( 2) 4( 2) ( 2)( 4) =( 2)( 2)( 2) =( 2) ( 2)

( 2) 2(4

x x xx x x xx x x x x x

x xx x xx x

HCF xLCM

- = - +

- - = + -

+ - - = + - +

= + -+ + -

+ -

= -

= 21)( 2)( 2)

x x x+ - +

2

1 6 1 4 4 8( 1) ( 1)x x x

+ -- - -

( )

2

1 6 14 4 8( 1) ( 1)

1 3 14( 1) 4( 1) ( 1)( 1)4 1

4( 1) ( 1)( 1)1 11 ( 1)( 1)1 1

( 1)( 1)1 1

( 1)( 1)

( 1)( 1)

x x x

x x x x

x x x

x x xxx xxx x

xx x

+ -- - -

= + -- - - +

= -- - +

= -- - +

+ -=

- ++ -

=- +

=- +

( )

2

2 2

2

2

9 8 12 8 124 9 4 9 4 12

3 ( 3) 4 9 8 124 9 8 12 4( 3)

34

x x xx x x

x x x xx x x

x

- + +÷ ´

- - -

- + - += ´ ´

- + -

+=

ü

ü

ü

ü

ü ü

ü ü

ü

ü

ü

ü

ü

ü

ü

ü

ü

ü

ü

ü

MARKING GUIDELINE -4- T860(E)(N24)T MATHEMATICS N2

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QUESTION 2 2.1

(4) 2.2 ….eq(1)

…..eq(2)

From eq(1) and eq(2)

(4)

2.3

(3)

2

2

2

2

(5 6) 2(3 )5 6 6 25 8 6 0

5 b 8 6

42

( 8) ( 8) 4(5)( 6) 8 1842(5) 10

8 184 8 184 10 10

2,156 0,556

x x xx x xx xa c

b b acxa

x

x or x

x x

- = +

- = +

- - == = - = -

- ± -=

- - ± - - - ±= =

+ -= =

= = -

1y x= -

44xy = - +

1 44

4 4 16 5 20 4

(1) : 4 1 3

xx

x xxx

From eq yy

- = - +

\ - = - +\ =\ =

= -\ =

3 5

5 3

5 3

5 3

53

Ax y

Ay x

Axy xy x

Axxy

Ax

= -

\ = -

-\ =

\ =-

=-

ü

ü

ü ü

ü

ü

ü

ü

ü

ü

ü

MARKING GUIDELINE -5- T860(E)(N24)T MATHEMATICS N2

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2.4

(4) [15]

QUESTION 3 3.1 3.1.1

(3) 3.1.2

(3) 3.2 3.2.1

(3)

Let the number be 10x+y2 ....(1)

10 10 88 ......(2)11 11 88

8 .......(3)From (1) and (3) : 2 8 2

x yy x x yx y

x yy y

y

\ = ++ + + =

\ + =\ + =

+ + =\ 6

3 From (1): 3 2 5 Number is

yxx

=\ =

= +\ =

\ 10(5)+3=53

1 2

13 927

x x

x

- +

+

´

1 2 4

3 3

3 3

3 3

3 33

331

x x

x

x

x

- +

+

+

+

´=

=

=

1 2

13 927

x x

x

- +

+

´

( )( )

21 2

13

1 2 4

3 3

1 2 4 3 3

0

3 3

3

3 33

331

xx

x

x x

x

x x x

+-

+

- +

+

- + + - -

´=

´=

=

==

3 8 4

6 6 12

2

64

22

x x

xx

=

=

3 8 4

3 6 12

2 4

2

64

2

22

x x

x

xx

=

=

1 3

1 3

2 2 3

16 4 016 44 42 2 32 1

12

x

x

x

xx

x

+

+

+

- =

\ =

\ =\ + =\ =

\ =

ü

ü

ü

ü

ü

ü

ü

ü

ü

ü

ü

ü

ü

ü

ü

ü

ü

ü

ü

MARKING GUIDELINE -6- T860(E)(N24)T MATHEMATICS N2

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smVV

DnV

/876,30)1,35)(28,0(

===pp

3.2.2

(5) 3.3

can only be awarded with a mark if the root is excluded. (5)

[19] QUESTION 4 4.1 4.1.1

ü (1)

4.1.2

ü

diameter in meters

(3) 4.2

Radius =21 mm Diameter=42 mm

(4) 4.3

= -0,411

(1)

2

2

4 1

4 1 0

2

2

2

3 1

3 34 1 04 1

14

1 1 or 2 2

x

x

xx

x

x x

-

-

=

\ =

\ - =

\ =

\ =

\ = = -

3)2(log 22 =- xx

2 3

2

2 22 8 0

( 4)( 2) 04 or 2( / )

x xx xx xx x n a

- =

- - =- + == = -

2x = -

sr /1,35602106

=

2

2

2

D44 4

4(42)(10) 4(10)

1680 400

128035,777

x hh

x Dh x

x mm

= +

= -

= -

= -

==

0,5 cos ln sin 270 tan4

oe p p- æ ö+ -ç ÷è ø

ü

ü

ü ü

ü

ü ü

ü

ü ü

ü

ü

ü

ü

ü

ü

ü

MARKING GUIDELINE -7- T860(E)(N24)T MATHEMATICS N2

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4.4

4.4.1

(4) 4.4.2

Follow up on wrong height of QUESTION 4.3.1

(4) 4.5

(4)

[21]

2

2

2 22π(28) +2π(28)h = 7891

4926,017+175,929h = 7891175,929h = 2964,983

2964,983h = 175,929

h = 16,853cm

A r rhp p= +

\\

\

2

2

3

(28) (16,853)41509,08462

41,509

V r hVV cm

l

p

p

=

=

==

3

3

3

3

3 3

4Volume of sphere=34 (60)3

904778,684Volume of small cubes= 20 8000

904778number of smaller cubes

r

mml

mm

p

p=

=

= =

=,684

8000 113 small cubes=

3

3

3

3

3 3

4Volume of sphere=34 (6)3

904,779Volume of small cubes= 2 8

904,779number of smaller cubes8

r

cml

cm

p

p=

=

= =

=

113 small cubes=

ü

ü ü

ü

ü

ü

ü

ü

ü

ü

ü

ü

ü

ü

ü

ü

MARKING GUIDELINE -8- T860(E)(N24)T MATHEMATICS N2

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QUESTION 5 5.1

5.1.1

(3)

5.1.2

(2)

2 2 2

2

(65,8) (49,7)6799,73

82,460 m

DBDBDB

= +

==

o

65,8sin10439,272 ( 39,249 )

In ACD

y

y or °

D

=

\ =

ü

ü ü

ü

ü

MARKING GUIDELINE -9- T860(E)(N24)T MATHEMATICS N2

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5.1.3

Or

(4)

5.2

5.2.1

(2) 5.2.2

ü

(2) [13]

2 2 2

2

65,8 1046486,3680,538

80,538 49,730,838

65,8tan39,272

65,8tan39,27280.472( 80,538)30,772( 30,838)

ACACAC

BCBC

OR

AC

AC

AC orBC or

o

o

+ =

==

= -=

=

=

==

2 2 2

2 2

104 65,8 (49,7 )

104 65,8 49,780,538 49,7

30,84

let BC xx

xx

BC cm

=

= + +

- = +- =

\ =

2 2 2

2 2

2

sin1

In Δ ABC OA =OB BA = 1

1

mm

m

OA m

q = =

-

-

= -

2tan

1mm

q =-

( )2 2

22 2

2 2

sin cos

1

11

m m

m m

q q+

= + -

= + -=

ü

ü ü

ü

ü

ü

ü

ü

ü

ü

ü

ü

ü

ü

MARKING GUIDELINE -10- T860(E)(N24)T MATHEMATICS N2

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QUESTION 6 6.1 6.1.1

H Starting point

( ;2) TP(maxvalue) ( ;3)

Shape Half mark

Total =2½

G Starting point ( ; 2)

TP(min value)( ;-2)

Shape Half mark

Total =2½

(5)

6.1.2 (0° ; 2)

ü

(1)

o0 o90

o0 o180

√

ü

ü

ü

ü

ü

√

MARKING GUIDELINE -11- T860(E)(N24)T MATHEMATICS N2

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6.2.3 (0 ; 7)

(2) [16]

TOTAL: 100

6.2 (6.2.1 & 6.2.2)

Marking instructions-mark critical points only under the condition that the shape of the graphs are correct. Critical

points values Values Marks

Parabola üü Roots -2,646 2,646 2 ü y-intercept (0;7) 1 üü TP (0;7) 2 Total 5

ü

Root(x-intercept)

(7;0)

1

ü Y-intercept (0;7) 1 Total 2 üH(x)=7 Y-intercept (0;7) 1 Total 1

(8)

( ) 7g x x= -

ü ü

ü

ü

ü

ü

ü ü

ü

ü

ü