NATIONAL SENIOR CERTIFICATE GRADE 1 JUNE 017 MATHEMATICS P1 MARKS: 150 TIME: 3 hours *MATHE1* This questio paper cosists of 11 pages, icludig a iformatio sheet.
MATHEMATICS P1 (EC/JUNE 017) INSTRUCTIONS AND INFORMATION Read the followig istructios carefully before aswerig the questios. 1. This questio paper cosists of TEN questios. Aswer ALL the questios.. Clearly show ALL calculatios, diagrams, graphs, et cetera that you have used i determiig your aswers. 3. Aswers oly will ot ecessarily be awarded full marks. 4. You may use a approved scietific calculator (o-programmable ad o- graphical), uless stated otherwise. 5. If ecessary, roud off aswers to TWO decimal places, uless stated otherwise. 6. Diagrams are NOT ecessarily draw to scale. 7. A iformatio sheet, with formulae, is icluded at the ed of the questio paper. 8. Number the aswers correctly accordig to the umberig system used i this questio paper. 9. Write eatly ad legibly.
(EC/JUNE 017) MATHEMATICS P1 3 QUESTION 1 1.1 Solve for x, i each of the followig: 1.1.1 x x 30 = 0 (3) 1.1. 3x + x 1 = 0 (correct to TWO decimal places) (3) 1.1.3 x (x + 4) (4) 1.1.4 3x 5 x = (5) 1. Solve simultaeously for x ad y i the followig equatios: y x 6 = 0 ad (x 3) + (y 3) = 18 (5) 1.3 Solve for x if: 1 x 1 7 1 5 x ; x 0 (5) [5]
4 MATHEMATICS P1 (EC/JUNE 017) QUESTION.1 A petago umber ca be represeted by dots that are arraged i the shape of a petago, as show below. The first four petagoal umbers are give. T 1 1 T 5 T 1 3 T 4.1.1 Determie the ext two petago umbers. ().1. Determie a expressio for the th term of the sequece. (4).1.3 Determie which term i the petago umber patter will be equal to 343. (4). If ; m ; 3 are the first three terms of a arithmetic sequece:..1 Determie the value of m ().. Determie T 51 (leave your aswer i surd form) (3).3 How may terms betwee 50 ad 500 are divisible by 7? (3).4 Give the geometric sequece: ; ; ;... 3 9.4.1 Determie the geeral term for the sequece i the form a.b (3).4. Is the geometric sequece coverget? Give a reaso for your aswer. ().4.3 Solve for p, if 3 p S S 4 (5).5 Expad ad evaluate: 6 k k 1 1 1 () [30]
(EC/JUNE 017) MATHEMATICS P1 5 QUESTION 3 3.1 Give a fuctio, f y + 4 = (x 5) 3.1.1 Write dow the equatio of the axis of symmetry of f. (1) 3.1. Determie the x-itercepts of f. (3) 3.1.3 Sketch the graph of f, clearly showig the itercepts with the axes ad the turig poit. (4) 3.1.4 Write dow the rage of f. (1) 3.1.5 is trasformed to, where the x-itercepts of g(x) is the same as that of f (x) ad the turig poit of g ( x ) is (5 ; 4 ). Describe the trasformatio ad write dow the equatio of g(x). () f ( x) g ( x) 3. Give: f ( x ) x 3 a d g ( x ) kx 1, determie the value(s) of k if g a taget to the graph of f. (5) [16] QUESTION 4 Give the graph of f, a hyperbola of the form y a x p q, aswer the questios that follow. 4.1 Write dow the values of p ad q. () 4. Determie the value of a, ad write dow the equatio of f i the form y = (3) 4.3 The axes of symmetry of f are y = x + 3 ad y = x + 1. The graph of f is trasformed to g such that the axes of symmetry of g are give by y = x 3 ad y = x + 1. Describe the trasformatio. Show all calculatios to support your aswer. (5) [10]
6 MATHEMATICS P1 (EC/JUNE 017) QUESTION 5 The graph of f defied by f(x) = a x + 1, where a > 0 ad a 1, passes through the poits ( ; p) ad (1 ; 5 ), is draw i the sketch below with the graph of g. 6 Use the sketch ad the give iformatio to aswer the followig questios. 5.1 Determie the value of a. () 5. Fid the value of p. () 5.3 Write dow a equatio for g, the reflectio of f i the y-axis. (1) 5.4 If h(x) = g(x) 1, write dow the equatio of h 1 i the form y = () 5.5 Calculate the average gradiet of the curve of f betwee x = ad poit A. (3) [10]
(EC/JUNE 017) MATHEMATICS P1 7 QUESTION 6 6.1 Calculate the effective iterest rate per aum if the omial iterest rate is 15% compouded mothly. (3) 6. Neymar applied for a loa of R75 000 from XYZ Bak at 1% simple iterest per aum for 8 years. 6..1 Calculate Neymar s mothly istalmet. (3) 6.. The bak wats to chage the iterest to a compoud iterest per aum without affectig Neymar s mothly paymet. Calculate the compoud iterest rate that the bak would charge correct to 3 decimal places. (4) 6.3 R60 000 is ivested i a accout which offers iterest at 7% p.a. compouded quarterly for the first 18 moths. Thereafter the iterest rate chages to 5% p.a. compouded mothly. Three years after the iitial ivestmet, R5 000 is withdraw from the accout. How much will be i the accout at the ed of 5 years? (4) [14]
8 MATHEMATICS P1 (EC/JUNE 017) QUESTION 7 7.1 Determie the derivative of f(x) = x from first priciples. (4) 7. Determie: 7..1 dy dx if y = 6x + 4x x (3) 7.. D t 1 3t 6 t (3) [10] QUESTION 8 The graph of f(x) = x 3 + bx + cx 4 is draw below. A ad B are turig poits of f. The x-itercepts ad the y-itercept are clearly idicated. 8.1 Show that the values of b = 6 ad c = 9. (3) 8. Determie the coordiates of B. (4) 8.3 For which values of x is the graph of f icreasig? () 8.4 Show that the poit of iflectio lies o the straight lie g that passes through C ad D. (4) [13]
(EC/JUNE 017) MATHEMATICS P1 9 QUESTION 9 A rectagle with legth x ad width y, is to be iscribed i a isosceles triagle of height 8 cm ad base 10 cm, as show. (Hit: A P Q a d A B C are sim ilar. ) 9.1 Express y i terms of x. () 9. Hece, show that the area of the rectagle ca be express as: A 8 x 8 x 10 () 9.3 Determie the dimesios, i.e. the legth ad the width of the rectagle for it to be a maximum. (4) [8]
10 MATHEMATICS P1 (EC/JUNE 017) QUESTION 10 10.1 Let A ad B be two evets i a sample space. Suppose that the P(A) = 0,4 ; P(B) = k ad P(A or B) = 0,7. Determie: 10.1.1 P (A o r B ) (1) 10.1. The value of k, for which A ad B are mutually exclusive evets. () 10.1.3 The value of k, for which A ad B are idepedet evets. (4) 10. There are 4 bags of marbles for sale i a shop. I te of the bags there are 7 gree marbles ad three yellow marbles. The other bags each have x gree marbles ad 9 yellow marbles. A bag is chose at radom ad a marble is the chose at radom from the bag. The tree diagram below illustrates the process ad the outcomes. 10..1 Determie the values of m ad. () 10.. Determie the value of x. (3) 10..3 What is the probability that a marble chose is gree? () [14] TOTAL: 150
(EC/JUNE 017) MATHEMATICS P1 11 INFORMATION SHEET: MATHEMATICS b b 4 ac x a A P ( 1 i ) A P ( 1 i ) A P ( 1 i ) A P ( 1 i ) T = a + ( 1)d S = (a + ( 1)d) T = ar 1 F x i 1 i 1 S a r 1 r 1 ; r 1 P S = a 1 r ; x[1 (1 i) ] i 1 r 1 f ' ( x ) lim h 0 f ( x h ) h f ( x ) d ( x x ) ( y y ) M x x y y 1 1 ; 1 1 y mx c y y m x ) ( x 1 1 m y y 1 m ta x x 1 x a y b I ABC: r si a A b c 1 a b c bc. cos A area ABC ab. si C si B si C si si. co s co s. si si si. co s co s. si cos cos cos. cos si. si cos cos. cos si. si cos si 1 si si si. cos cos 1 x x x x i i 1 P ( A ) ( A ) P (A or B) = P (A) + P (B) P (A ad B) S y a b x b x x y y x x