GRADE 12 NATIONAL SENIOR CERTIFICATE MATHEMATICS P1 PREPARATORY EXAMINATION 2008

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1 GRADE NATIONAL SENIOR CERTIFICATE MATHEMATICS P PREPARATORY EXAMINATION 008 MARKS: 50 TIME: 3 hours This questio paper cosists of 9 pages, a formula sheet ad diagram sheet.

2 Mathematics/P DoE/Preparatory Eamiatio 008 INSTRUCTIONS AND INFORMATION Read the followig istructios carefully before aswerig the questios This questio paper cosists of questios. Aswer ALL the questios. Clearly show ALL calculatios, diagrams, graphs, et cetera 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. Graph paper is used i QUESTION oly. Write your ame i the space provided ad had it i together with the ANSWER BOOK. Diagrams are NOT ecessarily draw to scale. Number the aswers correctly accordig to the umberig system used i this questio paper. Write eatly ad legibly.

3 Mathematics/P 3 DoE/Preparatory Eamiatio 008 QUESTION Solve for :. = 0 (3) 8. + = 8 (4) +.3 Solve for ad y simultaeously: y = ad y + y + = 7 (7).4 ( )( + 4) 6 (5) [9] QUESTION. Determie the time, take i years, for a sum of moey to double if the iterest rate is,64% p.a., compouded half-yearly. (4). Dudu wats to buy a house for R ,00. She has a deposit of R50 000,00 ad takes out a loa for the balace at a rate of 8% p.a. compouded mothly. QUESTION 3.. How much moey must Dudu borrow from the bak? ().. Calculate the mothly paymet if she wishes to settle the loa i 5 years. (4)..3 Dudu wo a lottery ad wishes to settle the loa after the 50 th paymet. What is the outstadig balace? (4) [3] Cosider the followig sequece of umbers: ; ; ; 5; ; 8; ; ; 3. What is the 0th term of the above sequece? () 3. Calculate the sum of the first 50 terms of the sequece. (4) [6]

4 Mathematics/P 4 DoE/Preparatory Eamiatio 008 QUESTION 4 Cosider the followig sequece: 3; 6; ; 8; 7; 4. Determie the 6th ad 7th terms of the give sequece, if the sequece behaves cosistetly. () 4. Determie a formula for the geeral term, p, of the sequece. (4) 4.3 Use your formula to calculate p if the th p term i the sequece is 67. (4) [0] QUESTION 5 5. Kopao wats to buy soccer boots costig R800, but he oly has R90,00. Kopao's ucle Stephe challeges him to do well i his homework for a reward. Ucle Stephe agrees to reward him with 50c o the first day he does well i his homework, R o the secod day, R o the third day, ad so o for 0 days. 5.. Determie the total amout ucle Stephe gives Kopao for 0 days of homework well doe. (5) 5.. Is it worth Kopao's time to accept his ucle's challege? Substatiate your aswer. () Cosider the geometric sequece: 8( ) ; 4( ) ; ( ) ; 5.. Determie the value of for which the sequece coverges. (3) 5.. Determie the sum to ifiity of the series if =,5. (3) [3]

5 Mathematics/P 5 DoE/Preparatory Eamiatio 008 QUESTION 6 The diagram below shows the graphs of f ( ) = a ad g( ) =. The poit M( ; -) is the poit of itersectio of f ad g. y g 0 g f M( ; ) 6. Determie the value of a. () 6. If g() is traslated to give h ( ) = +, write dow the asymptotes of h (). () 6.3 Sketch the graph of h ( ) = +. (3) [6]

6 Mathematics/P 6 DoE/Preparatory Eamiatio 008 QUESTION 7 7. The diagram below shows the graphs of f 4 a ( ) = ad g ( ) =. The poit P( ; 4) is the poit of itersectio of f ad g. y f P( ; 4) g (0 ; ) Write dow the equatio of f i the form y =. () 7.. Is f a fuctio? Substatiate your aswer. () 7..3 Determie the equatio of h(), the resultat fuctio whe f () is reflected about the y-ais. () 7. Determie the value of a i g (). () 7.3 Determie the equatio of m(), the resultat fuctio whe g () is shifted horizotally uits to the right ad vertically uit dow. () 7.4 Calculate the itercepts of m() with the aes. (3) [3]

7 Mathematics/P 7 DoE/Preparatory Eamiatio 008 QUESTION 8 I the diagram below f ( ) = ta( 45 ) ad g() = si. The poit O is the origi. y f f g f 8. Write dow the amplitude of g (). () 8. Write dow the domai of g (). () 8.3 Write dow the ew equatio of g if the traslatio of the graph is 60 horizotally to the right. () 8.4 Determie the value of for which g ( ) f ( ) =. () [7]

8 Mathematics/P 8 DoE/Preparatory Eamiatio 008 QUESTION 9 9. Give: f ( ) = 9.. Determie f '( ) from first priciples. (5) 9.. Determie the average gradiet of f () betwee = ad = 3. (3) dy 9. Determie if: d y = ( + )( ) (4) y = (4) [6] QUESTION 0 3 Give: f ( ) = Determie all itercepts of f(). (4) 0. Determie the coordiates of the turig poits of f(). (5) 0.3 Draw a sketch of f(). (4) 0.4 What is the equatio of the taget to f () whe =? (3) 0.5 What are the coordiates of the turig poit f ( )? () 0.6 Determie the -value of the poit of iflectio. (4) [] QUESTION A + B 3 - C. Show that the area of the triagle i the figure above is give by: 5 A ( ) = + 4 (). Calculate the value of for which the area will be maimum. (3).3 Hece, calculate the maimum area of the give triagle. () [7]

9 Mathematics/P 9 DoE/Preparatory Eamiatio 008 QUESTION Tsholaag is the maager of a small busiess that maufactures hadmade sadals. Two types of pairs of sadals are maufactured: Elegace sadals ad Classic sadals. The compay ca maufacture betwee 40 ad 50 pairs of Elegace sadals per moth. The compay ca maufacture betwee 0 ad 0 pairs of Classic sadals per moth. All together o more tha 00 pairs of sadals ca be maufactured per moth. The profit o a pair of Elegace sadals is R60 ad R00 o a pair of Classic sadals. ELEGANCE SANDALS () CLASSIC SANDALS (y) Let the umber of pairs of Elegace sadals be ad the umber of pairs of Classic sadals y.. Write dow the costraits of the above sceario. (6). Sketch the costrait i QUESTION. o the grid o the DIAGRAM SHEET. Clearly idicate the feasible regio. (6).3 Usig a search lie ad your graph, determie how may pairs of each type of sadal should be sold to geerate maimum profit. (3).4 Write dow the objective fuctio for maimisig profit i the form P =. ().5 Determie the maimum mothly profit. () [8] TOTAL: 50

10 Mathematics/P 0 DoE/Preparatory Eamiatio 008 FORMULA SHEET: MATHEMATICS FORMULEBLAD: WISKUNDE b ± b 4 ac = A = P( + i) a A = P( i) F = A = P( i) [( + i) ] i A = P( + i) [ ( + i) ] P = i = ( a + i ) d ) = ( a + ( ) d ) i= i= i a( r ) ( ar = i= r ; r i= ( + ) i = i= a r i ar ; < r < = f '( ) = lim h 0 f ( + h) h f ( ) ( ) ( ) + y + y d = + y y M ; y = m + c y y = m ) ( m = y y area m = taθ ( a) + ( y b) = r I ABC: ABC = ab. sic cos α si α cos α = si α cos α a A si b c = = a = b + c bc. cos A si B sic Δ si ( α + β ) = siα.cos β + cosα. si β si α = siα. cosα f = ( A) P( A) = ( S ) ( S ) cos si ( α β) = si α.cosβ cosα. siβ ( α + β ) = cosα.cos β siα. si β ( α β ) = cosα.cos β siα. si β cos + ( ) i= σ = ( A) P( A) = P(A or/of B) = P(A) + P(B) P(A ad/e B)

11 Mathematics/P DoE/Preparatory Eamiatio 008 NAME: DIAGRAM SHEET QUESTION Classic sadals (y) Elegace Sadals ()

NATIONAL SENIOR CERTIFICATE GRADE 12

NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P FEBRUARY/MARCH 04 MARKS: 50 TIME: 3 hours This questio paper cosists of 9 pages, diagram sheet ad iformatio sheet. Please tur over Mathematics/P DBE/Feb.