NATIONAL SENIOR CERTIFICATE GRADE 12

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1 NATIONAL SENIOR CERTIFICATE GRADE 1 MATHEMATICS P1 FEBRUARY/MARCH 011 MARKS: 150 TIME: 3 hours This questio paper cosists of 8 pages, 3 diagram sheets ad 1 iformatio sheet. Please tur over

2 Mathematics/P1 DBE/Feb. Mar. 011 INSTRUCTIONS AND INFORMATION Read the followig istructios carefully before aswerig the questios This questio paper cosists of 1 questios. Aswer ALL the questios. Clearly show ALL calculatios, diagrams, graphs, et cetera that you have used i determiig your aswers. Aswers oly will ot ecessarily be awarded full marks. You may use a approved scietific calculator (o-programmable ad ographical), uless stated otherwise. If ecessary, roud aswers off to TWO decimal places, uless stated otherwise. Diagrams are NOT ecessarily draw to scale. THREE diagram sheets for aswerig QUESTION 5.3, QUESTION 10.4 ad QUESTION 1. are attached at the ed of this questio paper. Write your cetre umber ad eamiatio umber o these sheets i the spaces provided ad isert them iside the back cover of your ANSWER BOOK. A iformatio sheet, with formulae, is icluded at the ed of the questio paper. Number the aswers correctly accordig to the umberig system used i this questio paper. Write legibly ad preset your work eatly. Please tur over

3 Mathematics/P1 3 DBE/Feb. Mar. 011 QUESTION Solve for, correct to TWO decimal places, where ecessary: ( 1) 1 (3) (4) (4) 1. Solve for ad y simultaeously: y 7 y 1 y (7) 1.3 Simplify completely, without the use of a calculator: (3) [1] QUESTION The sequece 3 ; 9 ; 17 ; 7 ; is a quadratic sequece..1 Write dow the et term. (1). Determie a epressio for the th term of the sequece. (4).3 What is the value of the first term of the sequece that is greater tha 69? (4) [9] QUESTION The first two terms of a ifiite geometric sequece are 8 ad 8. Prove, without the use of a calculator, that the sum of the series to ifiity is (4) 3. The followig geometric series is give: = to 0 terms Write the series i sigma otatio. () 3.. Calculate the value of. (3) [9] Please tur over

4 Mathematics/P1 4 DBE/Feb. Mar. 011 QUESTION The sum to terms of a sequece of umbers is give as: Calculate the sum to 3 terms of the sequece. () 4.1. Hece calculate the 3 rd term of the sequece. (3) 4. The first two terms of a geometric sequece ad a arithmetic sequece are the same. The first term is 1. The sum of the first three terms of the geometric sequece is 3 more tha the sum of the first three terms of the arithmetic sequece. QUESTION 5 Determie TWO possible values for the commo ratio, r, of the geometric sequece. (6) [11] 3 Cosider the fuctio f ( ) Write dow the equatios of the asymptotes of f. () 5. Calculate the itercepts of the graph of f with the aes. (3) 5.3 Sketch the graph of f o DIAGRAM SHEET 1. (3) 5.4 Write dow the rage of y = f (). (1) Describe, i words, the trasformatio of f to g if g ( ). () 1 [11] S Please tur over

5 Mathematics/P1 5 DBE/Feb. Mar. 011 QUESTION 6 A parabola f itersects the -ais at B ad C ad the y-ais at E. The ais of symmetry of the 7 parabola has equatio 3. The lie through E ad C has equatio g ( ). y B O 3 C g E f 6.1 Show that the coordiates of C are (7 ; 0). (1) 6. Calculate the -coordiate of B. (1) 6.3 Determie the equatio of f i the form y a( p) q. (6) 6.4 Write dow the equatio of the graph of h, the reflectio of f i the -ais. (1) 6.5 Write dow the maimum value of t() if t() = 1 f(). () 6.6 Solve for if f ( ) 0. (4) [15] Please tur over

6 Mathematics/P1 6 DBE/Feb. Mar. 011 QUESTION 7 Cosider the fuctio 1 f ( ) Is f a icreasig or decreasig fuctio? Give a reaso for your aswer. () 7. Determie f 1 ( ) i the form y = () 7.3 Write dow the equatio of the asymptote of f() 5. (1) 7.4 Describe the trasformatio from f to g if g( ) log 3. () [7] QUESTION R1 430,77 was ivested i a fud payig i% p.a. compouded mothly. After 18 moths the fud had a value of R1 711,41. Calculate i. (4) 8. A father decided to buy a house for his family for R He agreed to pay mothly istalmets of R o a loa which icurred iterest at a rate of 14% p.a. compouded mothly. The first paymet was made at the ed of the first moth. QUESTION Show that the loa would be paid off i 34 moths. (4) 8.. Suppose the father ecoutered uepected epeses ad was uable to pay ay istalmets at the ed of the 10 th, 11 st, 1 d ad 13 rd moths. At the ed of the 14 th moth he icreased his paymet so as to still pay off the loa i 34 moths by 111 equal mothly paymets. Calculate the value of this ew istalmet. (7) [15] 9.1 Use the defiitio to differetiate Calculate D f ( ) 1 3. (Use first priciples.) (4) (3) dy 9.3 Determie d if y 1. (3) [10] Please tur over

7 Mathematics/P1 7 DBE/Feb. Mar. 011 QUESTION 10 Give: g ( ) ( 6)( 3)( ) 10.1 Calculate the y-itercept of g. (1) 10. Write dow the -itercepts of g. () 10.3 Determie the turig poits of g. (6) 10.4 Sketch the graph of g o DIAGRAM SHEET. (4) 10.5 / For which values of is g().g () 0? (3) [16] QUESTION 11 A farmer has a piece of lad i the shape of a right-agled triagle OMN, as show i the figure below. He allocates a rectagular piece of lad PTOR to his daughter, givig her the freedom to choose P aywhere alog the boudary MN. Let OM = a, ON = b ad P( ; y) be ay poit o MN. N (0 ; b) y T P( ; y) O R M (a ; 0) 11.1 Determie a equatio of MN i terms of a ad b. () 11. Prove that the daughter's lad will have a maimum area if she chooses P at the midpoit of MN. (6) [8] Please tur over

8 Mathematics/P1 8 DBE/Feb. Mar. 011 QUESTION 1 While preparig for the 010 Soccer World Cup, a group of ivestors decided to build a guesthouse with sigle ad double bedrooms to hire out to visitors. They came up with the followig costraits for the guesthouse: There must be at least oe sigle bedroom. They ited to build at least 10 bedrooms altogether, but ot more tha 15. Furthermore, the umber of double bedrooms must be at least twice the umber of sigle bedrooms. There should ot be more tha 1 double bedrooms. Let the umber of sigle bedrooms be ad the umber of double bedrooms be y. 1.1 Write dow the costraits as a system of iequalities. (6) 1. Represet the system of costraits o the graph paper provided o DIAGRAM SHEET 3. Idicate the feasible regio by meas of shadig. (7) 1.3 Accordig to these costraits, could the guesthouse have 5 sigle bedrooms ad 8 double bedrooms? Motivate your aswer. () 1.4 The retal for a sigle bedroom is R600 per ight ad R900 per ight for a double bedroom. How may rooms of each type of bedroom should the cotractors build so that the guesthouse produces the largest icome per ight? Use a search lie to determie your aswer ad assume that all bedrooms i the guesthouse are fully occupied. (3) [18] TOTAL: 150

9 Mathematics/P1 DBE/Feb. Mar. 011 CENTRE NUMBER: EXAMINATION NUMBER: DIAGRAM SHEET 1 QUESTION y O

10 Mathematics/P1 DBE/Feb. Mar. 011 CENTRE NUMBER: EXAMINATION NUMBER: DIAGRAM SHEET QUESTION 10.4 y O

11 Mathematics/P1 DBE/Feb. Mar. 011 CENTRE NUMBER: EXAMINATION NUMBER: DIAGRAM SHEET 3 QUESTION 1. y

12 Mathematics/P1 DBE/Feb. Mar. 011 INFORMATION SHEET: MATHEMATICS b b 4 ac a A P( 1 i) A P( 1 i) A P( 1 i) A P( 1 i) i1 1 i1 ( 1) i 1 T ar ar 1 S F f '( 1 i 1 i ) lim h 0 f ( h) f ( ) h r 1 T a ( 1) d S a ( 1 d ; r 1 [1 (1 i) ] P i ( ) ( ) 1 y1 y d 1 y y1 M ; S ) a ; 1 r 1 1 r y m c y y m ) 1 ( 1 y y1 m m ta 1 a y b r I ABC: si cos si a A area ABC b c a b c bc. cos A si B si C 1 ab. si C si.cos cos. si si si.cos cos. si cos.cos si. si cos cos.cos si. si cos si cos 1 si si si. cos cos 1 ( ; y) ( cos ysi ; ycos si ) ( ; y) ( cos ysi ; ycos si ) i i1 f ( A) P( A) P(A or B) = P(A) + P(B) P(A ad B) yˆ a b S b ( y y) ( )

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.