NATIONAL SENIOR CERTIFICATE GRADE...

Copyright reserved Please turn over

MARKS: 150

This memorandum consists of 19 pages.

NATIONAL

SENIOR CERTIFICATE

GRADE 12

MATHEMATICS P1

SEPTEMBER 2015

MEMORANDUM

Mathematics P1 2 NW/September 2015

NSC – Memorandum


NOTE:

If a candidate answered a question TWICE, only mark the FIRST attempt.

If a candidate crossed out an attempt of a question and did not redo the question, mark the

crossed-out question.

Consistent accuracy applies throughout in ALL aspects of the marking memorandum.

QUESTION 1

1.1.1

2

2

( 1)( 8) 10

8 8 10 0

7 18 0

( 9)( 2) 0

9 0 or 2 0

9 2

x x

x x x

x x

x x

x x

x x

OR

2

2

2

2

( 1)( 8) 10

8 8 10 0

7 18 0

4

2

7 7 4(1)( 18)

2(1)

7 121

2

9 or 2

x x

x x x

x x

b b acx

a

x

x

x x

simplification

standard form

factors

both answers

(4)

simplification

standard form

substitution into the

correct formula

both answers

(4)

1.1.2

2

2

2

2

44 11 0

4 4 11 0

4 11 4 0

4

2

11 11 4(4)(4)

2(4)

11 57

8

0,43 or 2,32

xx

x x

x x

b b acx

a

x x

standard form


correct formula

x = − 0,43

x = − 2,32 (4)


NSC – Memorandum


1.1.3 2

2

2

6 3

0 3 6

0 2

0 ( 2)

0 or 2; R

x x

x x

x x

x x

x x x

0 2

OR 2

2

2

6 3

0 3 6

0 2

0 ( 2)

x x

x x

x x

x x

( ; 0) or (2 ; )x x

OR 2

2

2

6 3

0 3 6

0 2

0 ( 2)

x x

x x

x x

x x

( ; 0) (2 ; )x

standard form

factors

method

both x-values

notation

(5)

standard form

factors

method

both x-values

notation

(5)

standard form

factors

method

both x-values

notation

(5)

1.2 2 2

2 2

2 2 2

2

2

3 2 and 4 2 7

2 3

(2 3) 4 2(2 3) 7

4 12 9 4 4 6 7

4 6 2 0

2 3 1 0

(2 1)( 1) 0

2 1 or 1

1

2

12 3 or 2(1) 3

2

2 1

x y x y xy

x y

y y y y

y y y y y

y y

y y

y y

y y

y

x x

OR

x subject

substitution

standard form

factors

y-values

x-values

(7)


NSC – Memorandum


2 2

2

2

2 2 2

2

3 2 and 4 2 7

3

2

3 34 2 7

2 2

9 6 3 7

3 2 0

( 2)( 1) 0

2 or 1

3 2 3 1or

2 2

11

2

x y x y xy

xy

x xx x

x x x x x

x x

x x

x x

y y

y subject

substitution

standard form

factors

x-values

y-values

(7)

1.3

( )

( ) ( ) 0

only if is an odd number

m m

m m

x y

f x x y

f y y y

m

substitution

answer (2)

[22]

QUESTION 2

2.1 6 ; x ; 26 ; 45 ; y

x − 6 26 − x 19 y − 45

(26 – x) − (x – 6) 19 – (26 – x) (y – 45) − 19

32 – 2x − 7 + x y – 64

32 – 2x = − 7 + x − 7 + x = y – 64

39 = 3x − 7 +13 = y – 64

13 = x 70 = y

x − 6

26 − x

equations

x = 13

y = 70 (6)

2.2.1

34

11 4 3 ... 220

( 1)

220 11 ( 1)(7)

231 ( 1)7

33 1

34

2 ( 1)2

342( 11) (34 1)(7)

2

17 22 231

3553

n

n

T a n d

n

n

n

n

nS a n d

S


correct formula

n = 34


correct formula

answer

(4)

or


NSC – Memorandum


OR

34

11 4 3 ... 220

( 1)

220 11 ( 1)(7)

231 ( 1)7

33 1

34

2

3411 220

2

3553

n

n

T a n d

n

n

n

n

nS a l

S


correct formula

n = 34


correct formula

answer

(4)

2.2.2

34

1

( 1)

11 ( 1)7

11 7 7

7 18

(7 18) 3553

n

n

T a n d

n

n

n

n

7n – 18

n = 1 to 34

34

1

(7 18)n

n

(3)

2.3 0,89x; 0,892x; 0,893x; …

𝑇𝑛 = 𝑎𝑟𝑛−1

= 0,89x(0,89)n-1

= x(0,89)n

0,2x = x(0,89)n

0,2 = (0,89)n

𝑛 = 𝑙𝑜𝑔0,890,2

n = 13,8

∴ 13 copies can be made

sequence

10,89 (0,89)n

nT x

0,2x = x(0,89)n

answer

If x is omitted, penalise

by 1 mark

(4)

2.4 2

1

21

85

14

5

12

5

Tr

T

k

k

k

value of r

0,2

13,8


NSC – Memorandum


Converge if :

1 1

11 2 1

5

5 51

2 2

7 3

2 2

3 7

2 2

r

k

k

k

k

substitution

simplification

answer

(4)

2.5 5

75

80

74

79

80 80 79

75 74

75 74

74

74

3 2

3 2

3 2

(3 2) (3 2)

3 3

3 (3 1)

2.3

n

nS

S

S

T S S

OR

80 80 79

75 74

75 74

74

74

(3 2) (3 2)

3 3

3 (3 1)

2.3

T S S

75

80 3 2S

74

79 3 2S

74

80 2.3T

(3)

75

80 3 2S

74

79 3 2S

74

80 2.3T

(3)

[24]


NSC – Memorandum


QUESTION 3

3.1 x = − 3

y = 2

x = − 3

y = 2 (2)

3.2

23

0 21 3

22

4

4( ) 2

3

ay q

x p

a

x

a

a

a

f xx

substitution of p and q

substitution of (− 1; 0)

a = − 4

(3)

3.3 ( 3) 2

3 2

1

y x

x

x

OR

2 ( 3)

1

1

y x c

c

c

y x

substitution

answer

(2)

substitution

answer

(2)

3.4 R; 2x x Rx

2x

(2)

3.5 4( ) 2

3

42

( 3)

42

3

k xx

x

x

answer

(2)

3.6 3 1 or 0x x

OR

( 3; 1] or [0; )x x

3 1x

0x

(2)

( 3; 1]x

[0; )x

(2)


NSC – Memorandum


3.7 2

2

2

( ) 2

2 ( 3) 2

4 (9)

4

9

4( ) 2

9

g x bx

b

b

b

g x x

c = − 2

substitution (− 3 ; 2)

4

9b

(3)

3.8

2

1 2

2

4:

9

4:

9

9

4

9; 0

4

3; 0

2

h y x

h x y

y x

y x x

y x x

24( )

9h x x

swop x and y

answer with restriction

(3)

3.9

0x

h y = x

h - 1

OR

0x h

y = x

h - 1

form of h

form of h – 1

(must fit form of h)

(2)

form of h

form of h – 1

(must fit form of h)

(2)

[21]


NSC – Memorandum


QUESTION 4

4.1

2

0 :

(0 1) 9

1 9

8

(0; 8)

x

y

C

x-coordinate of C

y-coordinate of C

(2)

4.2

1

1

8 (Horisontal asymptote)

.2 8

Turning point: ( 1;9)

9 .2 8

12

2

( ) 2.2 8

2 8

x

x

x

q

y a

D

a

a

a

g x

q = 8

substitution of

D( − 1 ; 9)

a = 2

(3)

4.3 8y OR (8; )y

notation

answer

(2)

4.4 D'(–1 ; 7)

answer

(1)

4.5 Reflection about the x-axis, and a translation of 1 unit

left and 18 units up.

OR

Reflection about the line y = 9 and a translation of 1 unit

left.

Reflection x-axis

1 unit left

18 units up

(3)

Reflection y = 9

1 unit left

(3)

4.6 1

3

logy x

OR

3logy x

OR

3

1logy

x

answer (1)

answer (1)

answer (1)


NSC – Memorandum


4.7

9 5

4

1is a decreasing function

3

the bigger the value the smaller the value

maximum value of = 9

1minimum value :

3

1

3

1

81

x

y

x y

f

y

OR

2

2

( ( ) 5)

( ) 5

( 1) 9 5

( 1) 4

4

3

3

3

3

minimum 3

1

81

f x

f x

x

x

y

substitution of 9

answer 4

1 1or

3 81

Accept 0,01

(2)

substitution of f(x)

answer 4

1 1or

3 81

Accept 0,01

(2)

[14]

QUESTION 5

5.1

12 4

1 1

0,131 1

12 4

1,136475928...

1 1,010718046...12

0,0107180464...12

0,1286165568...

12,86% p.a. compounded monthly

m

nomeff

m

m

m

m

m

ii

m

i

i

i

i

r

12

112

mi

40,13

14

1 1,010718046...12

mi

(3)


NSC – Memorandum


5.2

0,12861655681

12

[1 (1 ) ]

0,12861655689 000 1 1

12600 000

0,1286165568

12

0,12861655680,7145364267 1 1

12

0,12861655681 1 0,7145364267

12

0,2854635733

log 0,2854

n

v

n

n

n

x iP

i

n

635733

117,5911312

117,59

117 payments of 9 000

n

R


correct formula

simplify RH

correct log

1,010718...log 0,28546

log 0,28546...or =

log1,010718...

n

n

n = 117,59

117 payments

(5)

5.3

117

117

1

0,1286165568600 000 1

12

2 088 640.059

1 1

0,12861655689 000 1 1

12

0,1286165568

12

2 083 364,827

Outstanding balance after 117 payments:

2 088 640,059 2 083 364,827

5 27

n

n

v

A P i

R

x iF

i

R

R R

R

5,232

0,1286165568Final payment 5 275,232 1

12

5 331,77R

OR


correct formula


correct formula

subtract: A − F

interest of 1 month

answer

(5)


NSC – Memorandum


118

118

1

0,1286165568600 000 1

12

2 111 026, 2

1 1

0,12861655689 000 1 1

12

0,1286165568

12

2 114 694, 428

Outstanding balance after 118 payments

2 111 026, 2 2 114 694, 428

3 668, 22

n

n

v

A P i

R

x iF

i

R

R R

R

8

Final payment 9 000 3 668, 228

5 331,77

R R

R

OR

0,5911

1 11

0,12861659 000 1 1

12 0,12861651

0,1286165568 12

12

5 331,77

n

v

x iP i

i

R


correct formula


correct formula

subtract: A − F

subtract from 9 000

answer

(5)

substitution of i and n

into the correct formula

one month interest

answer

(5)

5.4 117(R 9 000) 5 331,77

1 058 331,77

R

R

117(9 000)

1 058 331,77R

(2)

[15]


NSC – Memorandum


QUESTION 6

6.1 2

2

2 2

2 2

( ) 3

( ) 3( ) ( )

3 3 ( 2 )

3 3 2

f x x x

f x h x h x h

x h x xh h

x h x xh h

0

2 2 2

0

2 2 2

0

2

0

0

0

( ) ( )'( ) lim

(3 3 2 ) (3 )lim

3 3 2 3lim

3 2lim

(3 2 )lim

lim 3 2

3 2

h

h

h

h

h

h

f x h f xf x

h

x h x xh h x x

h

x h x xh h x x

h

h xh h

h

h x h

h

x h

x

subst (x + h) into f(x)

expansion

simplification

common factor

answer

(5)

6.2 2

1

1 2

3

2 2

3 1

5 2

3

5 2

3

5 4

xy

x x

x x

dy x x

dx

1

1 23

5 2

x xy

23

5

x

3

2

4

x

(4)

[9]


NSC – Memorandum


QUESTION 7

7.1.1 3 2

2

2

( ) 4 11 30

'( ) 3 8 11

0 3 8 11

0 (3 11)( 1)

3 11 or 1

11

3

400

27

14,81 36

11; 14,81 ( 1; 36)

3

f x x x x

f x x x

x x

x x

x x

x

y

y

B A

OR 3 2

2

2

2

( ) 4 11 30

'( ) 3 8 11

0 3 8 11

8 8 4(3)( 11)

2(3)

11or 1

3

400

27

14,81 36

11; 14,81 ( 1; 36)

3

f x x x x

f x x x

x x

x

x x

y

y

B A

2'( ) 3 8 11f x x x

'( ) 0f x

factors

both x-values

y-values

(6)

2'( ) 3 8 11f x x x

'( ) 0f x

subst into formula

both x-values

y-values

(6)

7.1.2 ''( ) 6 8

0 6 8

6 8

8 4

6 3

f x x

x

x

x

0 = 6x − 8

answer

(2)

7.1.3 3 2

2

1 1

(2) (2) 4(2) 11(2) 30

0

'(2) 3(2) 8(2) 11

15

( )

0 15( 2)

15 30

f

f

y y m x x

y x

y x

OR

f (2) = 0

f ' (2) = − 15

substitution

answer

(4)


NSC – Memorandum


3 2

2

(2) (2) 4(2) 11(2) 30

0

'(2) 3(2) 8(2) 11

15

0 15(2)

30

15 30

f

f

y mx c

c

c

y x

f (2) = 0

f ' (2) = − 15

substitution

answer

(4)

7.1.4 ( )

36 or 14,81

f x k

k k

OR

40036 or

27k k

36k

14,81k

(2)

36k

400

27k

(2)

7.2.1 '(0) 12g answer (1)

7.2.2 1symmetry at

2

1

x

x

x = 1

(1)

7.2.3

y

g

x

-2 ½ 3

turning point at x = − 2

and at x = 3

point of inflection at

x = ½

shape

(3)


NSC – Memorandum


QUESTION 8

8.1 2( ) 2 12 32

(0) 2(0) 12(0) 32

32

h t t t

h

m

answer

(1)

8.2 '( ) 4 12

0 4 12

4 12

3 sec

h t t

t

t

t

OR

2

12

4

3 sec

bt

a

'( ) 4 12h t t

'( ) 0h t

answer

(3)

correct formula


correct formula

answer

(3)

8.3 2

2

2

( ) 2 12 32

0 2 12 32

0 6 16

0 ( 8)( 2)

8 2 . .

8

'(8) 4(8) 12

32 12

20 /

20 / downwards

h t t t

t t

t t

t t

t or t n a

t

h

m s

m s

h(t) = 0

factors

t = 8

'(8) 4(8) 12h

–20m/s or

20 m/s downwards

(5)

8.4 2''(t) 4 /h m s

OR 2''( ) 4 / downwardsh t m s

answer (1)

answer (1)

[10]


NSC – Memorandum


QUESTION 9

9.1.1 ( , )

18 20

120 120

19

60

0,32

P boy tennis or squash

18 20

120 120

19

or 0,3260

(2)

9.1.2 41( , )

120

0,34

P learner tennis

41or 0,34

120

(1)

9.1.3 50 5( ) 0,42

120 12P girl

50 5or or 0,42

120 12

(1)

9.2 32 4( ) 0,27

120 15

70 44 77( ). ( ) . 0,21

120 120 360

0,27 0,21

Not independent

P boy and golf

P boy P golf

32 4

or or 0,27120 15

70 44

.120 120

77

or 0,21360

not independent

(4)

[8]

QUESTION 10

10.1.1 1014! 8,72 10 1014! or 8,72 10

(1)

10.1.2 4! 5! 3! 4! 2! 829 440 4! 5! 3! 4! 2!

829 440

(2)

10.2.1 PROBABILITY

11!9 979 200

2! 2!

11!

2! 2!

answer

(3)

10.2.2 10!907 200

2! 2!

10!

2! 2!

907 200

(2)

[8]

TOTAL: 150


NSC – Memorandum


COGNITIVE GRID

Question Levels: Exam/Mod Levels: Teacher Topics

L1 L2 L3 L4 L1 L2 L3 L4 Alg Patt Func Fin Diff Prob

1.1.1 4 4

1.1.2 4 4

1.1.3 5 5

1.2 7 7

1.3 2 2

2.1 6 6

2.2.1 4 4

2.2.2 3 3

2.3 4 4

2.4 4 4

2.5 3 3

3.1 2 2

3.2 3 3

3.3 2 2

3.4 2 2

3.5 2 2

3.6 2 2

3.7 3 3

3.8 3 3

3.9 2 2

4.1 2 2

4.2 3 3

4.3 2 2

4.4 1 1

4.5 3 3

4.6 1 1

4.7 2 2

5.1 3 3

5.2 5 5

5.3 5 5

5.4 2 2

6.1 5 5

6.2 4 4

7.1.1 6 6

7.1.2 2 2

7.1.3 4 4

7.1.4 2 2

7.2.1 1 1

7.2.2 1 1

7.2.3 3 3

8.1 1 1

8.2 3 3

8.3 5 5

8.4 1 1


NSC – Memorandum

Copyright reserved

Question Levels: Exam/Mod Levels: Teacher Topics

L1 L2 L3 L4 L1 L2 L3 L4 Alg Patt Func Fin Diff Prob

9.1.1 2 2

9.1.2 1 1

9.1.3 1 1

9.2 4 4

10.1.1 1 1

10.1.2 2 2

10.2.1 3 3

10.2.2 2 2

Policy:

Marks 30 52,5 45 22,5 30 52,5 45 22,5 25 25 35 15 35 15

% 20 35 30 15 20 35 30 15 17 17 23 10 23 10

Actual:

Marks 31 54 45 20 31 54 45 20 22 24 35 15 38 16

% 21 36 30 13 21 36 30 13 15 16 23 10 25 11

NATIONAL SENIOR CERTIFICATE GRADE...

Documents

Transcript of NATIONAL SENIOR CERTIFICATE GRADE...