GRADE 12 MATHEMATICS PAPER 1 MEMORANDUM - NSC PAST PAPERS AND MEMOS JUNE 2016

MATHEMATICS P1
GRADE 12
JUNE 2016
MEMORANDUM
NATIONAL SENIOR CERTIFICATE

NOTE

If a candidate answered a question TWICE, mark the FIRST attempt ONLY.
Consistent accuracy(CA) applies in ALL aspects of the memorandum.
If a candidate crossed out an attempt of a question and did not redo the question, mark the crossed-out attempt.
The mark for substitution is awarded for substitution into the correct formula.

QUESTION 1

1.1.1		✓ factorisation ✓✓ ?-values (3)
1.1.2		✓ standard form ✓ substitution ✓✓ ?-values (4)
1.1.3		✓ critical values with method ✓✓ answer (3)
1.1.4	3^? . 3^?+1 = 27^? 3^2?+1 = 3^3? ∴ 2? + 1 = 3? ? = 1	✓ 3^2?+1 ✓ 3^3? ✓ equating ✓ answer(4)
1.2	3 + ? = 2? ? = 2? − 3 …….. (1) 4?² + ?² = 2?? + 7 4?² + (2? − 3)² = 2?(2? − 3) + 7 4?² + 4?² − 12? + 9 = 4?² − 6? + 7 4?² + 4?² − 12? + 9 − 4?² + 6? − 7 = 0 4?² − 6? + 2 = 0 2?² − 3? + 1 = 0 (2? − 1)(? − 1) = 0 ? = 1 or/of ? = 1 2 ? = −2 or/of ? = −1 If formula is used, award factor’s mark for substitution.	✓ substitution ✓ removing brackets ✓ standard form ✓ factors ✓ ?-values ✓ ?-values (6)
1.3	?(?) = (? − 2)(?² − 6? + 10) Consider the quadratic factor : (?² − 6? + 10) Δ = ?² − 4?? = (−6)² − 4(1)(10) = 36 − 40 = −4 => Δ < 0, therefor NO Solutions ? = 2 is the only solution.	✓ substitution into delta ✓ answer ✓ conclusion (4) [23]

QUESTION 2

2.1.1		✓ first differences ✓ second differences (2)
2.1.2	Next term= 25	✓ answer(1)
2.1.3		✓ ? = 3 2 ✓ ? = − 11 2 ✓ ? = 4 ✓ answer(4)
2.1.4	?₃₀ = 3 (30)² −11(30) + 4 2 2 = 1189	✓ substitution ✓ answer (2)
2.2.1	?₃ − ?₂ = ?₄ − ?₃ ? − 13 = 27 − ? 2? = 40 ? = 20 13 − ? = 7 ? = 6	✓ method ✓ values of a and b (2)
2.2.2	? + (? − 1)? = ?_? 6 + (? − 1)(7) = 230 7? = 231 ? = 33	✓ formula ✓ substitution ✓ answer (3)
2.3	? =1 − ? 5 −1 < 1 − ? < 1 5 −5 < 1 − ? < 5 −6 < −? < 4 −4 < ? < 6	✓? = 1−? 5 ✓ substitution ✓ answer (3)
2.4.1		✓method ✓substitution ✓simplification ✓answer (4)
2.4.2		(2)
2.4.3	?∞ = ? 1 − ?	✓ substitution ✓ answer (2) [25]

QUESTION 3

3.1.1	? = −2 (horizontal asymptote )	✓ answer(1)
3.1.2	? = 1 (vertical asymptote)	✓ answer(1)
3.2	3 − 2 = 0 ? − 1 3 = 2 ? − 1 3 = 2? − 2 5 = 2? ? = 5 2 ? = 3 − 2 0 − 1 ? = −5	✓ y = 0 ✓ x – intercept ✓ y – intercept (3)
3.3		✓ asymptotes ✓ x-and y-intercepts ✓shape (3)
3.4	(4 ; 5)	(2)
		[10]

QUESTION 4

4.1		✓ ? -coordinate ✓y-coordinate ✓coordinates of C (3)
4.2	−?² − 2? + 3 = 0 ?2 + 2? − 3 = 0 (? + 3)(? − 1) = 0 ? + 3 = 0 ?? ? − 1 = 0 ? = −3 ?? ? = 1 ?(−3; 0) ?(1; 0)	✓standard form ✓factors ✓ both values (3)
4.3	m = 2 & c = 6 y = 2? + 6	✓value of m ✓value of c (2)
4.4		✓method ✓substitution ✓answer (3)
4.5	?> 1	✓✓answer (2)
		[13]

QUESTION 5

5.1.1		✓substitution ✓answerd (2)
5.1.2		✓swop ? and ? ✓ answer (2)
5.1.3		✓equating ✓answer (2)
5.1.4	? ∈ R	answer(1)
5.2		✓asymptote ✓negative y-intercept ✓shape (3)
		[10]

QUESTION 6

6.1.1	effective rate = 8,77%p.a	✓ formula ✓ substitution ✓ answer (3)
6.1.2		✓formula ✓substitusie ✓answer(3)
6.2		✓substitution ✓simplification ✓correct use of logs ✓answer (4)
6.3		✓✓ setting up equation ✓ x the subject of the formula ✓ answer (4)

[14]

QUESTION 7

7.1

= 4? - 3

Answer ONLY: 0 marks

✓ substitute (? + ℎ)
✓ formula

Penalise 1 mark for incorrect use
of formula. Must show ?′(?).

✓ simplification
✓ common factor
✓ answer
(5)

7.2

(4)

Penalise 1 mark for incorrect notation

[9]

QUESTION 8

8.1.1	?(?) = ?3 − 4?2 − 11? + 20 ?′(?) = 3?2 − 8? − 11 = 0 (3? − 11)(? + 1) = 0 3? − 11 = 0 ?? ? + 1 = 0	✓ ?′(?) = 0 ✓ factors ✓ x-values ✓ y-values ✓ coordinates (5)
8.1.2	?′′(?) = 6? − 8 = 0 6? = 8 ? =8 or/of 1 1 or/of 1, 33 6 3 OR	✓ ?′′(?) = 0 ✓ answer (2)
8.1.3	? = ?′(2) = 3(2)2 − 8(2) − 11 = −15 ?(2) = (2)³ − 4(2)² − 11(2) + 30 = 0 ? − ?₁ = ?(? − ?1) ? − 0 = −15(? − 2) ? = −15? + 30	✓ ? = −15 ✓ ?(2) = 0 ✓ substitution ✓ answer (4)
8.1.4	36 units downwards 14,8 units upwards	✓ answer ✓ answer (2)
8.2.1	? = −9	✓ answer (1)
8.2.2	? = 1 ?? ? = 5	✓✓ answers (2)
8.2.3	? <1 or ? > 5	✓ ? < 1 ✓ ? > 5 (2)

[18]

QUESTION 9

9.1

? = ?²ℎ
8 = ?²ℎ
∴ ℎ = 8
?²

✓substitution
✓answer (2)

9.2

? = 2?² + 4?ℎ
= 2?² + 4? ( 8)
?²
= 2?² + 32
?

No mark for the answer

✓ 2?²
✓ 4?ℎ
✓ substitution (3)

9.3

?(?) = 2?² + 32?⁻¹
?′(?) = 4? − 32?⁻² = 0
4?³ − 32 = 0
4?³ = 32
?³ = 8
= 2³
∴ ? = 2

✓ ?′(?) = 0
✓ standard form
✓exponential law
✓ value of x
✓ dimensions
(5)

[10]

QUESTION 10

10.1.1	?(? and ?) = ?(?) × ?(?) = 0,4 × 0,5 = 0,2	✓ rule ✓ answer(2)
10.1.2	?(? or ?) = ?(?) + ?(?) − ?(? and?) = 0,4 + 0,5 − 0,2 = 0,7	✓ rule ✓ answer(2)
10.1.3	?(??? ? and ?) = 1 − ?(? or ?) = 1 − 0,7 = 0,3	✓ rule ✓ answer (2)
10.2.1		✓ first branch ✓ second branches ✓ outcomes If probabilities not listed, maximum 1 mark (3)
10.2.2	?(?????? from ??? ?)= 2 5	✓answer (1)
10.2.3	?(????) = 3 + 5 10 18 = 26 45 (0,58)	✓ 3 10 ✓ 5 18 ✓answer (3)
10.3.1	? = 288 − ? ? = 372 − ?	✓ answer ✓ answer (2)
10.3.2	288 − ? + ? + 372 − ? + 56 = 600 −? = −116 ? = 116	✓ equation ✓ answer (2)
10.3.3	No P(A and B) ≠ 0	✓ answer (1) [18]
		TOTAL: 150

Last modified on Tuesday, 15 June 2021 08:06