Models of Health Behavior

Topics: Power of two, X&Y, 2 Pages: 47 (5969 words) Published: August 25, 2013
NATIONAL SENIOR CERTIFICATE

MATHEMATICS P1 NOVEMBER 2011 MEMORANDUM

MARKS: 150

This memorandum consists of 28 pages.

Mathematics/P1

2 NSC – Memorandum

DBE/November 2011

NOTE: • If a candidate answers a question TWICE, only mark the FIRST attempt. • If a candidate has crossed out an attempt of a question and not redone the question, mark the crossed out version. • Consistent Accuracy applies in all aspects of the marking memorandum. QUESTION 1 1.1.1 x( x + 1) = 6 x2 + x = 6 x2 + x − 6 = 0 (x + 3)(x − 2) = 0 x = −3 or 2 OR Note: Answers by inspection: award 3/3 marks Note: Answer only of x = 2 : award 1/3 marks Note: If candidate converts equation to linear: award 0/3 marks

standard form factors answers (3) standard form substitution into correct formula answers (3) standard form

x2 + x − 6 = 0
x= − b ± b 2 − 4ac 2a

− 1 ± 12 − 4(1)(− 6 ) = 2(1) x = −3 or 2

1.1.2
2

3x − 4 x = 8
2

3x − 4 x − 8 = 0
x= = = − b ± b 2 − 4ac 2a − (−4) ±

(− 4)2 − 4(3)(− 8) 2(3)
Note: Penalise 1 mark for inaccurate rounding off to ANY number of decimal places if candidate gives decimal answers.

Note: If candidate uses incorrect formula: maximum 1/4 marks (for standard form)

substitution into correct formula
Note: If an error in subs and 4 ± − 80 gets: and 6 states “no solution”: maximum 3/4 marks

4 ± 16 + 96 6 4 ± 112 = 6 2±2 7 = 3 = 2,43 or − 1,10

112

If doesn’t conclude with “no solution”: maximum 2/4 marks

4 ± 112 or 6 decimal answer (4)

Mathematics/PI

3 NSC – Memorandum

DBE/November 2011

OR 3x 2 − 4 x = 8

3x 2 − 4 x − 8 = 0
x= = − b ± b − 4ac 2a − (−4) ±
2

(− 4)2 − 4(3)(− 8) 2(3)

Note: Penalise 1 mark for inaccurate rounding off to ANY number of decimal places if candidate gives decimal answers

= 2,43 or − 1,10

1.1.3

4 x 2 + 1 ≥ 5x

(4 x − 1)(x − 1) ≥ 0
+ 0 1 4
x≤
OR

4 x 2 − 5x + 1 ≥ 0

factors

0 1

+ 1 4
1⎤ ⎛ OR ⎜ − ∞; ⎥ ∪ [1; ∞ ) 4⎦ ⎝

1

both critical 1 values of and 1 4 or OR ∪ answer (4)

1 or x ≥ 1 4

1 4

1

x

OR

1 4

1

x

Note: If candidate gives either of these correct graphical solutions but writes down the incorrect intervals or uses AND: max 3/4 marks

NOTES:

If a candidate gives an answer of 1 ≤ x ≤ If a candidate gives an answer of

1 then max 3/4 marks. 4

1 ≤ x ≤ 1 then max 2/4 marks. 4 1 If a candidate gives an answer of x ≤ and x ≥ 1 then max 3/4 marks. 4 If the candidate leaves out the equality of the notation then penalty of 1 mark. 1 If a candidate gives an answer of x ≤ ; x ≥ 1 then max 3/4 marks. 4 1 If candidate gives x ≥ and/or x ≥ 1 , BREAKDOWN: max 2/4 marks. 4 If candidate gives : 0 − 0 + + award 3/4 marks 1 1 4 Copyright reserved Please turn over

Mathematics/PI

4 NSC – Memorandum

DBE/November 2011

1.2.1

x 2 + 5 xy + 6 y 2 = 0 (x + 3 y )(x + 2 y ) = 0 x + 3y = 0 x = −3 y OR x = −3 y OR

x + 2y = 0 x = −2 y x = −2 y

Note: If a candidate gives x x − = 3 or − = 2 y y award 2/3 marks

factors

Let k =

x y 2 x + 5 xy + 6 y 2 = 0
2

⎛x⎞ ⎛x⎞ ⎜ ⎟ + 5⎜ ⎟ + 6 = 0 ⎜ y⎟ ⎜ y⎟ ⎝ ⎠ ⎝ ⎠ 2 k + 5k + 6 = 0 (k + 3)(k + 2) = 0 k = – 3 or k = – 2 x x = −3 or = −2 y y OR

factors

x 2 + 5 xy + 6 y 2 = 0
x= x= − 5 y ± (5 y ) 2 − 4(1)(6 y 2 ) 2(1) − 5y ± y2

2 − 5y ± y x= 2 x = −3 y x = −2 y or x x = −3 = −2 y y

substitutes correctly into correct formula

OR

x 2 + 5 xy + 6 y 2 = 0
⎛5 x 2 + 5 xy + ⎜ ⎝2 ⎛5 ⎞ y ⎟ = −6 y 2 + ⎜ ⎝2 ⎠ 2 2

⎞ y⎟ ⎠

2

5 ⎞ 1 2 ⎛ ⎜ x + y⎟ = y 2 ⎠ 4 ⎝ 5 1 x+ y=± y 2 2 5 1 x=− y± y 2 2

completing the square

Mathematics/PI

x = −3 y x = −3 y

x = −2 y

5 NSC – Memorandum

DBE/November 2011

or x = −2 y

OR

Let k = x = ky

x y

x 2 + 5 xy + 6 y 2 = 0...