NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 007 MARKS: 50 TIME: 3 hours Ths questo paper cossts of 9 pages, a sheet of graph paper ad a -page formula sheet.
Mathematcs/P INSTRUCTIONS AND INFORMATION Read the followg structos carefully before aswerg the questos:.. 3. 4. 5. 6. 7. 8. Ths questo paper cossts of questos. Aswer ALL the questos. Clearly show ALL calculatos, dagrams, graphs, et cetera, whch you have used determg the aswers. A approved scetfc calculator (o-programmable ad o-graphcal) may be used, uless stated otherwse. If ecessary, aswers should be rouded off to TWO decmal places, uless stated otherwse. Number the aswers correctly accordg to the umberg system used ths questo paper. Dagrams are NOT ecessarly draw to scale. It s your ow terest to wrte legbly ad to preset the work eatly. A sheet of graph paper for aswerg QUESTION. ad QUESTION.3 s cluded at the ed of ths questo paper. Wrte your ame/eamato umber o ths sheet the space provded ad had t together wth your ANSWER BOOK.
Mathematcs/P 3 QUESTION. Solve for :.. 9 4 0.. 3 (Show your aswer correct to ONE decmal place.) (5)..3 (6). Solve smultaeously for ad y the followg system of equatos: y y 5 0 (7).3 If A 4, determe the values of for whch:.3..3. QUESTION A s udefed A s o-real () () [5] Smplfy each of the followg:. 3 4 6 6 6. 8 98.3 Show that y ca be wrtte as y. (4) [0]
Mathematcs/P 4 QUESTION 3 A researcher s vestgatg the umber of trees a forest over a perod of years. After vestgatg umerous data, the followg data model emerged: YEAR NUMBER OF TREES IN HUNDREDS 3 3 9 4 7 3. 3. 3.3 How may trees, hudreds, are there the SIXTH year f ths patter s cotued? Determe a algebrac epresso that descrbes the umber of trees the th year the forest. Do you thk ths model, whch determes the umber of trees the forest, wll cotue deftely? Gve a reaso for your aswer. () () () [6] QUESTION 4 Two learers, Aa ad Dpka, dd a vestgato o umber patters. They maaged to acheve the same results for t, but otced that the patters they had arrved at, looked dfferet. Aa's Table Dpka's Table t Patter t Patter 3 3 3 + 8 4 8 + 3 5 3 5 3 5 3 3 + 3 4 4 4 6 4 4 4 + 4 4. 4. 4.3 Determe a algebrac formula for Aa's patter. If Dpka's patter formula ca be wrtte as t = +, state whether the formula for Dpka's patter ad the formula for Aa's patter are equvalet. Motvate your aswer. Determe the value of for whch the learers would have obtaed a result of t = 43. () () (5) [9]
Mathematcs/P 5 QUESTION 5 Busswe s usg regular heagoal (s-sded) carpet tles to make floor rugs. She has arraged the tles the followg patters to make rugs of dfferet szes: RUGS: Patter Patter 3 Patter 4 Patter 5 (7 tles) (9 tles) (37 tles) (6 tles) 5. 5. 5.3 How may tles wll Busswe eed to make Patter 6 ths sequece? Make a cojecture that descrbes the relatoshp betwee the patter umber ad the umber of tles eeded for the patter. Use varables to wrte a algebrac statemet to geeralse the relatoshp betwee the patter umber ad the umber of tles. () () (4) [8] QUESTION 6 6. A computer s purchased for R6 000. It deprecates at 5% per aum. 6.. 6.. Determe the book value of the computer after 3 years f deprecato s calculated accordg to the straght-le method. Fd the rate, accordg to the reducg-balace method, that would yeld the same book value as QUESTION 6.. after 3 years. (5) 6. Peter vests R 500,00 for 5 years at % per aum compouded mothly for the frst years ad 4% per aum compouded sem-aually for the et 3 years. How much wll Peter receve total after 5 years? (4)
Mathematcs/P 6 6.3 Thada vests R0 000. He s quoted a omal terest rate of 7,% per aum compouded mothly. 6.3. 6.3. 6.3.3 QUESTION 7 Calculate the effectve rate per aum correct to THREE decmal places. Use the effectve rate to calculate the value of Thada's vestmet f he vested the moey for 3 years. Suppose Thada vests hs moey for a total perod of 4 years, but after 8 moths makes a wthdrawal of R0 000, how much wll he receve at the ed of the 4 years? (4) (5) [4] The graphs of f () = ( + 3) ad g() = +, are represeted below: L y f M O g P 7. 7. 7.3 7.4 7.5 7.6 Determe the values of for whch f () = 0. Calculate the coordates of P, the turg pot of f. Determe the average gradet of the curve f betwee = 5 ad = 3. Hece state what you ca deduce about the fucto, f, betwee = 5 ad = 3. Determe the values of for whch f () > 0. Gve the coordates of the turg pot of f ( ). () (5) () () 7.7 L s a pot o the straght le ad M s a pot o the parabola such that LM s perpedcular to the -as. Show that the epresso for LM ca be wrtte as: 7 8 LM (5) 4 6 7.8 Hece state the mamum legth of LM ad the correspodg -value at whch t occurs. () [4]
Mathematcs/P 7 QUESTION 8 The followg graph s a graph of f () = a.b (a 0). P( ; 44) s a pot o f. y y P (; 44) f 0 8. 8.. If 3 b, calculate the value of a. 4 8.. Hece wrte dow the equato of f. () 8. Determe, correct to TWO decmal places, the value of f. () 8.3 Descrbe the trasformato of the curve of f to h f h() = f ( ) () [8] QUESTION 9 h. 4 Gve: 9. 9. 9.3 Determe the equatos of the asymptotes of h. Determe the coordates of the tercepts of h wth the - ad y-aes. Sketch the graph of h showg clearly the asymptotes ad ALL tercepts wth the aes. () (6) (4) []
Mathematcs/P 8 QUESTION 0 The dagram below shows the graphs of f () = s a ad g() = cos( + b ). y (-360; 0,7) f (360; 0,7) 0 g 0. 0. 0.3 What s the perod of f? Determe the values of a ad b. State the rage of h f h() = g(). () () () [5]
Mathematcs/P 9 QUESTION A bcycle maufacturer makes two dfferet models of bcycles, amely mouta bkes ad speed bkes. The bcycle maufacturer works uder the followg costrats: No more tha 5 mouta bcycles ca be assembled daly. No more tha 3 speed bcycles ca be assembled daly. It takes oe ma to assemble a mouta bcycle, two me to assemble a speed bcycle ad there are 8 me workg at the bcycle maufacturer. Let represet the umber of mouta bcycles ad let y represet the umber of speed bcycles....3.4.5 Determe algebracally the costrats that apply to ths problem. Represet the costrats graphcally o the graph paper provded. By meas of shadg, clearly dcate the feasble rego o the graph. The proft o a mouta bcycle s R00 ad the proft o a speed bcycle s R600. Wrte dow a epresso to represet the proft o the bcycles. Determe the umber of each model bcycle that would mamse the proft to the maufacturer. TOTAL: (6) (4) () (4) [9] 50
Mathematcs/P NAME/EXAMINATION NUMBER: GRAPH PAPER QUESTION. ad.3
Mathematcs/P INFORMATION SHEET : MATHEMATICS INLIGTINGSBLAD : WISKUNDE b b 4 ac a A P( ) A P( ) A P( ) a ( ) d a ( ) d ar a r r F f ; r f ( h) f ( ) '( ) lm h 0 h A P( ) ( ) a r ar ; r [ ( ) ] P d ) ( y ( y ) M y ; y ( y m c y y m ) y y m m ta a y b r I ABC: s cos a s A b s B c s C a b c bc. cos A area ABC ab. s C s.cos cos. s s s.cos cos. s cos.cos s. s cos cos.cos s. s cos s cos s cos s s. cos
Mathematcs/P f var var d s. s A A P ) ( ) ( P(A or B) = P(A) + P(B) P(A ad B)