KwaZulu-Natal Department of Education REPUBLIC OF SOUTH AFRICA MATHEMATICS SEPTEMBER PREPARATORY 2018 MEMORANDUM NATIONAL SENIOR CERTIFICATE GRADE 12

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1 Education KwaZulu-Natal Department of Education REPUBLIC OF SOUTH AFRICA MATHEMATICS SEPTEMBER PREPARATY 08 MEMANDUM NATIONAL SENI CERTIFICATE GRADE MARKS: 50 TIME: hours This memorandum consists of 4 pages. Copyright Reserved Please Turn Over

2 Mathematics September Preparatory 08 QUESTION.. 0 or 5 A 0 A 5 () b b 4ac a A formula 0,9 4()( ) () or,7 A substitution in correct formula CACA answers (penalize mark if rounding off is incorrect-once for whole paper) or n / a A squaring both sides CA standard form CA factors CA answers and rejecting n / s or 0 4 log 4,6 A writing 9 as prime base CA factors CA 0 and 4 A use of logs CA,6 or log 4 (5) log 0 4,6 A writing 9 as prime base A dividing by : 0 CA and 4 A use of logs CA,6 or log 4 (5)

3 Mathematics September Preparatory 08. y and 4 y 4 y y 4 y y y 4 y y 4 8 y y 4 0 or or y 0 0 for all R A substitution CA simplification CA standard form CA factors CA both y values CA both values (6) AA 0 CA 4 0 CA answer If graphical Solution is used: Marks for sketches (AA) Marks for solution (CACA) [5] QUESTION. 7 ; 99 AA answers () D D 4 4 a 4 a a b 0 b 4 a b c 9 c T n n 4n A a value CA b value CA c value CA answer

4 Mathematics 4 September Preparatory 08 T n ( n ( n ) T ( n ) d d ( n ( n ) 9 ( n )(0) 9 0n 0 n 6n 4 n 4n A formula CA substitution into correct formula CA simplifying CA answer. T n n ( n n ( n 4n n n ) is even for all n N n ) is odd for all N CA rewriting n th term A reasoning A reasoning () For the first difference T n 4n 6 (n ) An even number of the first difference is always added to first term of the quadratic sequence to get an odd number. This process continues to produce all odd numbers of the sequence. CA for nth term of first difference A reasoning A reasoning () NB. If a candidate presents the following argument: The first term (9) is odd. To get net term, an even number is always added. This will give an odd number all the time. Award / [9]

5 Mathematics 5 September Preparatory 08 QUESTION. t ; t ; 9 t t t 9 t ( t) A equating differences t t 9 t t t 9 t 9 6t t 9 t t 8t 0 t t 8 0 t 0 or n / a t 8 CA standard form of equation CA factors CA answers with rejection. Pattern is ; 8 ; 5 ; ; ;... 4 terms are positive. ; 8 T n ; 5 n n i. e. n 4 4 terms are positive. Answer only full marks AA all 5 terms listed CA answer CA n th term > 0 CA n 4 CA conclusion () () [7] QUESTION r ( ) A answer () 4.. r 4 A condition CA substitution of common ratio CA answer ()

6 Mathematics 6 September Preparatory ; p ; p ;... and ; p ; p ;... T0 9 p A 9p A S p p T 0 9 p p 6 p 9 p 9 p p p p 0 n / a 9 p 6 p 0 or 0 p p CA equating CA standard form CA p values and rejecting (5) [9] QUESTION f ( ) 5 A writing numerator as + 5 () 5. and y A A y () 5. y intercept : 0 ; intercept: ; 0 A y intercept A intercept () 5.4 y c c c 5 f ( ) y c 5 (co-ordinate form not needed) CA substitution of the point ( ; ) CA answer CA y = + (m must be ) CA answer () () [7]

7 Mathematics 7 September Preparatory 08 QUESTION 6 6. f ( ) log log p p p p A substitution of the point ( ; - ) A answer () 6. B (; 0) A answer () 6. At A the co-ordinate is the same as the ais of symmetry value of the graph of g. y log A ; CA value CA substitution CA answer () 6.4 y a( 0)( ) a 0 a a 4 4 y 4( ) y 4 b 4 4 CA substitution of intercepts and TP CA a value (a < 0) CA substitution into equation CA b value

8 Mathematics 8 September Preparatory 08 y a( p) q y a B( ; 0) : 0 a a 4 a 4 y b 4 b 4 a 4 4 b b Substituting 4 4 B(;0) : 0 a b () A ; : a b () 4 () : 4 a b () a b into () CA substitution of pt. B and TP. CA a value CA equation of parabola CA b value CA subst. B( ; 0) to form eq. () CA subst. CA b value CA a value A ; to form eq. () 6.5 y or y AA answer () 6.6 ( 0 ; ] AA answer (penalize for incorrect notation) () 6.7 CACA answer (penalize for incorrect notation) () [6]

9 Mathematics 9 September Preparatory 08 QUESTION 7 7. y dy 6 6 d m 6( ) 6( ) 4 4( ) c c 9 y mc c y A derivative CA substitution of = into derivative and equating to gradient CA substituting m = 4 and given point CA c value CA answer (5) CA equating or CA standard form CA factors CA values CA choosing answer (5) [0]

10 Mathematics 0 September Preparatory 08 QUESTION A P( i) ( i) ( i) i i i 0, n 8 Therefore the interest rate is,95 % p.a. n [ ( i) ] Pv i 0, ,45 R0 74,6 0,45 n log n 9,98569 n 9,98569 n 0 payments ,6 0% of R869,5 n [ ( i) ] Pv i 0,45 869, ,45 n 0,45 0, , n A substitution into correct formula A i value A answer () A formula A substitution of P value A substitution of i and n values CA answer N.B. Substituting i and n in the future value formula Award /4 CA = 869,5 CA substitution into correct formula CA use of logs CA answer

11 Mathematics September Preparatory Balance on loan 0,45 869,5 P v 0,45 R94,5 0, ,45 Final Instalment = 94,5 08, CA n value CA substitution into present value formula CA R 94, 5 CA R08, Balance on loan = A - F 0, R94,5 9 0,45 869,5 0,45 9 CA n value CA substitution into formulae CA R 94, 5 0,45 Final Instalment = 94,5 08, CA R08, [5] QUESTION 9 (penalize mark once for incorrect notation in this question) 9. f h f f ( ) lim h0 lim h0 lim h0 5 5 h ( h) 5 h 0h 5h h 0 5h lim h0 h 0 h h 5 h A formula A substitution CA simplification of numerator CA factorization CA answer (5)

12 Mathematics September Preparatory A rewriting in eponential form g( ) / g ( ) 4 / g D D CA derivative CA substituting 4 into derivative CA answer A cubing the binomial CACACA each answer [] QUESTION 0 0. h( ) c d / h ( ) c / h () () 7 9 c 0 c 8 h( ) 8 d h 8 d d 0 d / h ( ) c 8 d () c 0 c / h ( ) ( )( ) h( ) 8 d h d 0 / h ( ) ( ) ( ) c 0 6 c 0 c 8 8 A derivative A subst. or into derivative and equating to 0 A simplifying A subst. 4 into h and equating to 0 A simplifying A derivative A derivative using stationary values A simplifying A equating coefficients of polynomials to get c value A substituting = 4 onto equation to get d value (5) (5)

13 Mathematics September Preparatory h( ) 8 h( ) ( ) A ;54 ( ) 8( ) 54 A subst. = into h A y value () 0. A CA answer () h( ) 8 / h ( ) 8 // h ( ) 6 0 A second derivative equal to 0 CA value () 0.4 CA answer () ; 54 A = CA y=54 answer () k 54 CACA answer () [4]

14 Mathematics 4 September Preparatory 08 QUESTION.. D 8 t t D() 8 () () 7 7, ,6 8 0 Average Rate of change 0,7 0 7 D 8 t t. 9 7 / D ( t) t t 9 9 / D (6) (6) (6) 9 9 m/ h The water level is decreasing at m/h. A subst. t = A 7,6 CA subst. into average rate of change CA answer A A derivative CA subst. t = 6 CA m/ h [8]

15 Mathematics 5 September Preparatory 08 QUESTION. a = 0 ; b = 60 ; c = 40 ; d = 0 A a value and A b value A c value and A d value P(Male) CA P(Male) P(liking sport) 50 P(Male and liking sport) P(Male) P(liking sport) P(Male liking sport) P(Male) P(liking sport) The events are independent. 8 5 CA P(Male and liking sport) P(Male) P(liking sport) CA CA conclusion P(Female) P(liking sport) 50 P(Femaleand liking sport) P(Female) P(liking sport) P(Female liking sport) P(Female) P(liking sport) The events are independent CA P(Female) 50 CA P(Female and liking sport) 5 P(Female) P(liking sport) CA CA conclusion [8]

16 Mathematics 6 September Preparatory 08 QUESTION. There are 9 letters: Es, Ds and Ns. The number of different words are 9!!!! A 9! A!!! 50 A answer. If we take one of the letters for the first letter, there are seven letters remaining, of which there are Es and Ds. () Hence the number of words. 8!!! 60. If both Ns are used for the first and last, there are 7 letters remaining of which there are Es and Ds. Hence the number of word.7!.!! 40 A 8! A!! A answer A 7! A!! A answer () () [9] Total Marks : 50