NATIONAL SENIOR CERTIFICATE EXAMINATION MATHEMATICS P1 SEPTEMBER 2016 GRADE 12

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1 NATIONAL SENIOR CERTIFICATE EXAMINATION MATHEMATICS P SEPTEMBER 06 GRADE MARKS: 50 TIME: 3 HOURS This questio paper cosists of 9 pages ad iformatio sheet, Please tur over

2 Mathematics/P September 06 INSTRUCTIONS TO CANDIDATES Read the followig istructios carefully before aswerig the questios This questio paper cosists of 9 questios. Aswer ALL the questios. Show clearly ALL the calculatios, diagrams, graphs, etcetera, which you have used i determiig the aswers. A approved scietific calculator (o-programmable ad o-graphical) may be used, uless stated otherwise. If ecessary, aswers should be rouded off to TWO decimal places, uless stated otherwise. Number the aswers correctly accordig to the umberig system used i this questio paper. Diagrams ad graphs are ot ecessarily draw to scale. Write eatly ad legibly. A formula sheet is attached. Please tur over

3 Mathematics/P 3 September 06 QUESTION. Solve for correct to ONE decimal places if ecessary:.. 3 (3) (3) (4) (4). Cosider y ad y 3.. Show that y 3 (3).. Solve for ad y simultaeously (5).3 For which values of p will the equatio p 49 p 0 has equal roots? (4) [6] Please tur over

4 Mathematics/P 4 September 06 QUESTION. The first four terms of a quadratic sequece are: ; 5; 3; 3... Calculate the geeral term of the sequece. (4).. Which term has a value of 643? (3). k ; 3k ; 5k 5 are the first three terms of a arithmetic sequece.. Calculate the value of k. ().. Write dow this sequece i sigma otatio for the first 0 terms. (3) Give the geometric series: For what value(s) of will the series coverge? (3) 8.3. Calculate the values of if S. (4) 3.4 The first, third ad thirteeth terms of arithmetic sequece are the first 3terms of a geometric sequece. If the first term of both sequeces is, determie:.4. the first three terms of the geometric sequece if r > (6).4. the sum of 7 terms of the geometric sequece if the sequece is ; 5 ; 5. () [7] Please tur over

5 Mathematics/P 5 September 06 QUESTION 3 Sketched below are the graphs of: g 8 f 6 ad h() = A ad B are the - ad y - itercepts of h respectively, C ( 6 ; 0) ad E are the poits of itersectio of f ad g. y c C( 6 ; 0) D f h A O B E g 3. Calculate the coordiates of A, B ad E. (4) 3. Show that the value of c 6 () 3.3 Write dow the values of for which f 0 g () 3.4 Determie the equatio of the symmetry ais of h if the gradiet is egative. () 3.5 Show that the legth of BE = 5 () 3.6 Write dow the rage of s, if s() = h() +. () 3.7 If t is a taget to f ad parallel to g, determie the equatio of the taget, t i the form y = m + c. (5) [9] Please tur over

6 Mathematics/P 6 September 06 QUESTION 4 Give: g ad f 4 6 log 4. Determie the equatio of g, the iverse of g, i the form g = () 4. Write f i the form f a p q (4) 4.3 The graph of f is shifted uit to the left to form p. Write dow the equatio of the ew graph i the form p() =.. () 4.4 Write dow the values of for which log 0 () 4.5 If h is the reflectio of f 3 g about the -ais, for which value of will h (3) [3] QUESTION 5 5. Read the advertisemet carefully. The questio below refers to the advertisemet. BANK WITH THE REFORMED BANK 8,75% iterest compouded Mothly How may moths will it take ay ivestmet to double if the moey is ivested i the accout as advertised above? (4) Please tur over

7 Mathematics/P 7 September Joh, the ower of a mie, has just bought a ew ecavator for R He has decided to replace the ecavator i 5 years time, whe its trade-i value will be R The replacemet cost of the ecavator is epected to icrease by 7% per aum, compouded semi-aually. 5.. Joh wats to pay cash for the ew ecavator after tradig i his curret ecavator i 5 years time. How much will he eed to pay? (3) 5.. Oe moth after purchasig the preset ecavator, Joh deposited rad ito a accout that pays iterest at a rate of 6% per aum, compouded mothly. He cotiues to deposit the same amout at the ed of each moth for a total of 60 moths. After 5 years he has the eact amout that is eeded to purchase the ew ecavator after tradig i his curret oe. Calculate the value of. (4) 5..3 Joh s brother, Micheal, wats to ivest i the mie 3 years after Joh has bought the mie. They agree that Micheal pays the rest of the mothly istalmets i a lump sum. What is the amout that Micheal has to pay, i order to still buy the ew ecavator i 5 years time? (4) [5] QUESTION 6 6. Give: f 3 Determie f () from FIRST PRINCIPLES. (5) 6. Determie: 6.. dy if y 3 d (3) 6.. dy if y = d (4) [] Please tur over

8 Mathematics/P 8 September 06 QUESTION 7 3 Sketched below is the graph of 8 g. 3 ad f ( ) c A ad D are the -itercepts of g. OB is 6 uits. A ad C are the turig poits of g. y A B O D f g C C 7. Write dow the coordiates of A ad D. () 7. Determie the coordiates of C. (4) 7.3 Write dow the equatio of f. () 7.4 Calculate the -value of the poit of iflectio of g. () 7.5 Use the graph to determie the value(s) of t for which g () = t will have oly oe real root. () / 7.6 For which value(s) of is. g 0 f? () [3] Please tur over

9 Mathematics/P 9 September 06 QUESTION 8 I the diagram ABC is a equilateral triagle whose sides have a legth of 3 uits. DEFG is a rectagle iside the triagle as idicated, D lies o AB, G lies o AC ad EF lies o BC. BE = FC = A D G B E F C 8. Show that DE = ta 60⁰ (3) 8. Prove that the area (A) of the rectagle is give by A = 3 (3 ) () 8.3 Calculate the maimum area that the rectagle ca be. (5) [0] QUESTION If P(A) = ad P(B) =, calculate P(A or B) if : A ad B are mutually eclusive evets. (3) 9.. A ad B are idepedet evets. (3) 9. The probability of gettig the first aswer i a quiz correct, is 0,6. If the first aswer is correct, the probability of gettig the secod aswer correct rises to 0,7. However, if the first aswer is icorrect, the probability of gettig the secod aswer correct, is oly 0,4. Determie the probability of gettig the secod aswer correct. (4) 9.3 Three married couples Mr ad Mrs Small, Mr ad Mrs Beaver ad Mr ad Mrs Khoza are to be seated o a bech How may differet ways ca the people be arraged? () 9.3. Calculate the probability that Mr ad Mrs Khoza are seated et to each other o the bech. (3) [5] TOTAL: 50

10 Mathematics/P September 06 FORMULA SHEET b b 4ac a A P i A P i A P i A P i a d S a T S r ; r a r a r S ; r P i i d f F lim h 0 T ar i i f h f h d y y y y M ; y m c y y m m y m ta y a y b r I a b c ABC : si A si B a b c bc. cos A sic Area of si cos ABC ab. sic si.cos cos si si si.cos cos si cos.cos si. si cos cos.cos si. si cos A si A cosa si A sia si A. cos A cos A A P A PA or B PA PB PA ad B S b y y y a b t