NATIONAL SENIOR CERTIFICATE GRADE 12

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1 NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P FEBRUARY/MARCH 009 MARKS: 50 TIME: 3 hours This questio paper cosists of 0 pages, a iformatio sheet ad diagram sheets. Please tur over

2 Mathematics/P DoE/Feb. March 009 INSTRUCTIONS AND INFORMATION Read the followig istructios carefull before aswerig the questios This questio paper cosists of 4 questios. Aswer ALL the questios. Clearl show ALL calculatios, diagrams, graphs, et cetera that ou have used i determiig our aswers. A approved scietific calculator (o-programmable ad o-graphical) ma be used, uless stated otherwise. If ecessar, aswers should be rouded off to TWO decimal places, uless stated otherwise. Diagrams are NOT ecessaril draw to scale. TWO diagram sheets for aswerig QUESTION 7.4, QUESTION 8. ad QUESTION 4. are icluded at the ed of this questio paper. Write our eamiatio umber o these sheets i the spaces provided ad had them i together with our ANSWER BOOK. Number the aswers correctl accordig to the umberig sstem used i this questio paper. It is i our ow iterest to write legibl ad to preset the work eatl. Please tur over

3 Mathematics/P 3 DoE/Feb. March 009 QUESTION. Solve for : = 4 (4).. 5 ( 3) = (5)..3 > 3 (4). Solve simultaeousl for ad : 3 = + 9 = 7 (7).3 Calculate the value of (3) [3] QUESTION Cosider the series: Epress each of the followig sums as a fractio of the form b a :.. The sum of the first two terms of the series ().. The sum of the first three terms of the series ()..3 The sum of the first four terms of the series (). Make a cojecture about the sum of the first terms of the give series. ().3 Use our cojecture to predict the value of the followig: () [6] Please tur over

4 Mathematics/P 4 DoE/Feb. March 009 QUESTION 3 The followig is a arithmetic sequece: p ; p 3 ; p + 5 ; Calculate the value of p. (3) 3. Write dow the value of: 3.. The first term of the sequece () 3.. The commo differece () 3.3 Eplai wh oe of the umbers i this arithmetic sequece are perfect squares. () [7] QUESTION 4 Cosider the sequece: 6 ; 6 ; ; 6 ; 8 ; Write dow the et term of the sequece, if the sequece behaves cosistetl. () 4. Determie a epressio for the th term, T. (5) 4.3 Show that 6838 is i this sequece. (4) [0] QUESTION 5 A sequece of squares, each havig side, is draw as show below. The first square is shaded, ad the legth of the side of each shaded square is half the legth of the side of the shaded square i the previous diagram. DIAGRAM DIAGRAM DIAGRAM 3 DIAGRAM 4 5. Determie the area of the ushaded regio i DIAGRAM 3. () 5. What is the sum of the areas of the ushaded regios o the first seve squares? (5) [7] Please tur over

5 Mathematics/P 5 DoE/Feb. March 009 QUESTION 6 a Sketched below are the graphs of f ( ) = ( p) + q ad g ( ) = + c. b A(½ ; 0) is a poit o the graph of f. P is the turig poit of f. The asmptotes of g are represeted b the dotted lies. The graph of g passes through the origi. 4 3 g g f A( 5 ; 0) - P 6. Determie the equatio of g. (4) 6. Determie the coordiates of P, the turig poit of f. (4) 6.3 Write dow the equatios of the asmptotes of g( ). () 6.4 Write dow the equatio of h, if h is the image of f reflected i the -ais. () [] Please tur over

6 Mathematics/P 6 DoE/Feb. March 009 QUESTION 7 The graph of h ( ) = a is sketched below. A ; is a poit o the graph of h. h A(- ; ½) O Q 7. Eplai wh the coordiates of Q are (0 ; ). () 7. Calculate the value of a. () 7.3 Write dow the equatio for the iverse fuctio, h, i the form =... () 7.4 Draw a sketch graph, o DIAGRAM SHEET, of h. Idicate o this graph the coordiates of two poits that lie o this graph. (3) 7.5 Read off from our graph the values of for which log >. () 7.6 If g( ) = (00). 3, determie the value of for which h ( ) = g( ). (3) [4] Please tur over

7 Mathematics/P 7 DoE/Feb. March 009 QUESTION 8 Cosider: f ( ) = si 8. Draw a sketch graph of f o DIAGRAM SHEET, for [ 80 ; 360 ]. () 8. Write dow the rage of h ( ) = f ( ). () 8.3 Write dow the period of h ( ) = f. () 8.4 Give a value of θ if f ( + θ ) = cos. () [8] QUESTION 9 9. R 000 was ivested i a fud paig i% iterest compouded mothl. After 8 moths the value of the fud was R 860,00. Calculate i, the iterest rate. (4) 9. O 3 Jauar 008 Farouk baked R00 i a accout that paid 8% iterest per aum, compouded mothl. He cotiued to deposit R00 o the last da of ever moth util 3 December. He was hopig to have eough moe o Jauar 009 to bu a bike for R 300. Determie whether he will be able to do so, or ot. (5) [9] QUESTION 0 Rowa plas to bu a car for R5 000,00. He pas a deposit of 5% ad takes out a bak loa for the balace. The bak charges,5% p.a. compouded mothl. Calculate: 0. The value of the loa borrowed from the bak () 0. The mothl repamet o the car if the loa is repaid over 6 ears (5) [6] QUESTION. Differetiate f b first priciples where f ( ) =. (5). Evaluate:.. D [( 3 3) ] (3).. 3 d 4 if = d 9 (3) [] Please tur over

8 Mathematics/P 8 DoE/Feb. March 009 QUESTION 3 The graph of h ( ) = + a + b is show below. A( ; 3,5 ) ad B( ; 0 ) are the turig poits of h. The graph passes through the origi ad further cuts the -ais at C ad D. B( ; 0) C 0 D A( ; 3,5). Show that a = 3 ad b = 6. (6). Calculate the average gradiet betwee A ad B. ().3 Determie the equatio of the taget to h at =. (5).4 Determie the -value of the poit of iflectio of h. (3).5 Use the graph to determie the values of p for which the equatio p = 0 will have ONE real root. () [8] Please tur over

9 Mathematics/P 9 DoE/Feb. March 009 QUESTION 3 Sketched is the graph of =. A(t ; t ) ad B(3 ; 0) are show. A(t ; t ) O B(3 ; 0) 3. A(t ; t ) is a poit o the curve = ad the poit B(3 ; 0) lies o the -ais. 4 Show that AB = t + t 6t + 9. () 3. Hece, determie the value of t which miimises the distace AB. (5) [7] Please tur over

10 Mathematics/P 0 DoE/Feb. March 009 QUESTION 4 A clothig compa maufactures white shirts ad gre trousers for schools. A miimum of 00 shirts must be maufactured dail. I total, ot more tha 600 pieces of clothig ca be maufactured dail. It takes 50 machie miutes to maufacture a shirt ad 00 machie miutes to maufacture a pair of trousers. There are at most machie miutes available per da. Let the umber of white shirts maufactured i a da be. Let the umber of pairs of gre trousers maufactured i a da be. 4. Write dow the costraits, i terms of ad, to represet the above iformatio. (You ma assume: 0, 0 ) (3) 4. Use the attached graph paper (DIAGRAM SHEET ) to represet the costraits graphicall. (5) 4.3 Clearl idicate the feasible regio b shadig it. () 4.4 If the profit is R30 for a shirt ad R40 for a pair of trousers, write dow the equatio idicatig the profit i terms of ad. () 4.5 Usig a search lie ad our graph, determie the umber of shirts ad pairs of trousers that will ield a maimum dail profit. () [3] TOTAL: 50

11 Mathematics/P DoE/Feb. March 009 b ± = b 4 ac a A = P( + i) A = P( i) INFORMATION SHEET: MATHEMATICS INLIGTINGSBLAD: WISKUNDE A = P( i) A = P( + i) i= i= = ar F = f '( i ) ( r ) a = r [( + i) ] i = lim h 0 f ( + h) f ( ) h i= ; r = i ( + ) i = [ ( + i) ] P = i a = r ( a + ( i ) d ) = ( a + ( ) d ) i= i ar ; < r < d = ( ) ( ) + M + + ; = m + c = m ) ( a) + ( b) = r ( m = m = taθ I ΔABC: si a A b c = = a b c = + bc. cos A area Δ ABC = ab. si C si B si C ( α + β ) = siα.cosβ cosα. si β si( α β ) = siα.cosβ cosα. si β si + cos ( α + β ) = cosα.cosβ siα. si β cos ( α β ) = cosα.cosβ + siα. si β cos α si α cos α = si α si α = siα. cosα cos α ( i ) = σ = i= f ( A) P( A) = P(A or B) = P(A) + P(B) P(A ad B) ˆ = a + b ( S ) b ( ) ( ) = ( )

12 Mathematics/P DoE/Feb. March 009 EXAMINATION NUMBER: DIAGRAM SHEET QUESTION QUESTION

13 Mathematics/P DoE/Feb. March 009 EXAMINATION NUMBER: DIAGRAM SHEET QUESTION

NATIONAL SENIOR CERTIFICATE GRADE 12

NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P ADDITIONAL EXEMPLAR 008 MARKS: 50 TIME: hours This questio paper cosists of 9 pages, diagram sheets ad a -page formula sheet. Mathematics/P DoE/Additioal