METRO EAST EDUCATION DISTRICT

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1 METRO EAST EDUCATION DISTRICT COMMON PAPER GRADE 1 MATHEMATICS P1 SEPTEMBER 018 MARKS: 150 TIME: 3 hours This questio paper cosists of 10 pages ad 1 iformatio sheet.

2 INSTRUCTIONS AND INFORMATION Read the followig istructios carefully before aswerig the questios This questio paper cosists of 10 questios. Aswer ALL the questios. Number the aswers correctly accordig to the umberig system used i this questio paper. Clearly show ALL calculatios, diagrams, graphs, et cetera that you have used i determiig your aswers. Aswers oly will NOT ecessarily be awarded full marks. You may use a approved scietific calculator (o-programmable ad ographical), uless stated otherwise. If ecessary, roud off aswers to TWO decimal places, uless stated otherwise. Diagrams are NOT ecessarily draw to scale. A iformatio sheet with formulae is icluded at the ed of the questio paper. Write eatly ad legibly.

3 3 QUESTION Solve for x i each of the followig: x 10x 3 = 0 (correct to decimal places) (3) 1.1. x(x + ) 9(x + ) = 0 (3) x (3 x 9) = 0 () 1. The roots of the equatio y = ax + bx + c are give as: x = 4 ± 16 4k 1..1 For which values of k will the roots be real? () 1.. Give oe value of k for which the roots will be ratioal, if k Z (Itegers). (1) 1.3 Solve for x ad y simultaeously: x + y = 0 ad x(x y + 1) = 0 (6) 1.4 Give: f(x) = (x 1)(x + ) ad g(x) = x + 3 QUESTION For which value(s) of x will f(x) be o-real? (3) 1.4. Solve for x if f(x) = g (x) (5) [5].1 Prove that i ay arithmetic series of which the first term is a ad where the costat differece is d, the sum of the first terms is give by S a 1d (4). Give the followig sequece: 5 ; 1 ; 3 ; 7 ;. ; Determie the umber of terms i the sequece. (3).. Calculate the sum of the sequece. ()

4 4.3 For a arithmetic series, cosistig of 15 terms, S = Determie:.3.1 The first term of the sequece. ().3. The sum of the last 3 terms. (3).4 A series of bridges, formed by equilateral triagles, are built with steelbars with legth 5 m each. Please ote: the legth of the bridge (drivig surface) is the legth of the bottom side(s) Arragemet 1 Arragemet Arragemet 3 Calculate how may bars are eeded for a bridge with a drivig surface of 75m. (3) [17] QUESTION A quadratic umber patter T = a + b + c has a third term equal to 1, while the first differeces of the quadratic sequece are give by: 1; 8; Write dow the values of the first four terms of the quadratic sequece. () 3.1. Calculate the values of a, b ad c. (3) 3. Cosider the geometric series 4 + p + p 4 + p Calculate the value(s) of p for which the series coverges. () 3.. Calculate the value of p if the sum to ifiity is 3. (3) [10]

5 5 QUESTION 4 I the sketch below the graphs of g(x) = k x+p + 4 ad f(x) = ax² + bx + c are give. The asymptotes of g itersect at B, the turig poit of f. The graphs of f ad g itersect at C, the y itercept of both graphs. The axis of symmetry of g that has a egative gradiet, is the lie h(x) that itersects the graph of f at A ( ; 3) ad B. 4.1 Determie the equatio of h. () 4. Show that the coordiates of B are (1; 4). Clearly show all your calculatios. 4.3 Show that the equatio of f(x) is give by f(x) = x + x + 3. Clearly show all your calculatios. () (4) 4.4 Determie the equatio of g. (3) 4.5 Determie the equatios of the asymptotes of g(x + 1). () 4.6 Determie the value(s) of x for which g (x). f (x) 0. () [15]

6 6 QUESTION 5 The graph of g(x) = log x is give below. A is the poit of itersectio of g with the x-axis. y O 5.1 Write dow the coordiates of A. () 5. Determie the equatio of g 1 i the form y =... () 5.3 Sketch the graph of p(x) = g(x). Clearly idicate all itercept(s) with the axes as well as the coordiates of ay other poit o the graph. () 5.4 Use the graph to determie for which value(s) of x, log ½ x 0 () [8]

7 7 QUESTION 6 Sketch the graph of f(x) = k x+p + q if: The domai is give as: x R ; x 1. The rage is give as: y R ; y k < 0 x itercept ( 1 ; 0) f(0) = 1 [5] QUESTION After how may years will the price of a vehicle, that has a value of R85 000, depreciate to R if the depreciatio rate is 13,4% p.a. o the reduced balace? (4) 7. A ma takes out a mortgage loa to buy a house. He plas to pay back the loa mothly over a period of 0 years. The size of the bod is R ad the iterest rate is calculated at 11% p.a. compouded mothly Calculate his mothly re-paymet that he will have to pay. (4) 7.. After 14 years he wats to settle the mortgage loa. Calculate the outstadig balace. (3) 7..3 The ma ca oly pay R of the settlemet amout after the 14 years. He makes a arragemet with the bak to settle the rest of the loa amout 3 moths later i oe paymet. Calculate the value of the secod settlemet amout if : he does t make ay further paymets o the loa he will pay a oce-off amout three moths later the iterest rate remais the same. (3) [14]

8 8 QUESTION Give: f(x) = 3 x Determie f (x) from first priciples. (5) 8.1. Determie the gradiet of the taget to f where x = 1. () 8. Differetiate with respect to x: 8..1 f(x) = 3x 3 x (3) 8.. xy = (x 1 x² ) (x + 1 x² ) (4) 8.3 Give the fuctios h(x) = k x ad g(x) = 3x Determie the equatio of h (x) i terms of k. () 8.3. Calculate the value of k if g is a taget to h. (5) 8.4 The sketch below shows the graph of p (x) where p(x) = x 3 + bx + 4x + c. A(;0) is a x-itercept of both p(x) ad p (x). C is the other x-itercept of p (x). y 0 A C p x B B Show that the umeric value of b = 9. Clearly show all your calculatios. (3) 8.4. Calculate the coordiates of C. (3) For which value(s) of x will p(x) icrease? (3) Calculate the value(s) of x for which the graph of p is cocave up. () Sketch a possible graph of p(x). Clearly idicate the x-coordiates of the turig poits ad the poit of iflectio. (4) [36]

9 9 QUESTION 9 The parabolic flight of a aeroplae that uses weightlessess, ca be modelled by the quadratic equatio: h(t) = 10t + 300t Where h is the height of the aeroplae, i metres, ad t the time, i secods, whe weightlessess is reached Determie the height whe weightlessess is reached iitially. () 9. Calculate the umber of secods that it takes the aeroplae to reach its maximum height after weightlessess has bee reached. (3) 9.3 Calculate the umber of secods (duratio) of weightlessess if weightlessess is lost at the same height as whe it is reached. () [7]

10 10 QUESTION I a certai survey 3% of the people who were iterviewed, said their favorite icecream flavour is Vailla. What is the probability that the favorite ice-cream flavour of a perso, chose radomly, is NOT Vailla? (3) 10. At a certai political meetig there are 0 Republicas (R); 13 Democrats (D) ad 6 members of the Idepedet party (I). What is the probability that a perso, chose radomly, will be a Democrat or a Republica? (3) 10.3 There are 8 swimmers who will participate i a certai Olympic evet; of the qualifyig participats are from Frace. I how may differet ways ca the swimmers be arraged if the Frech swimmers must swim ext to each other? () 10.4 Cosider the word: CAPETOWN Aswer the followig questios, if repetitio of letters are ot allowed: How may differet possible arragemets ca be compiled from the word CAPETOWN? () Calculate the probability that the word will start with a C ad ed with a N. (3) [13] TOTAL: 150 Copyright reserved

11 Mathematics/P1 INFORMATION SHEET b b 4 ac x a A P( 1 i) A P( 1 i) A P( 1 i) A P( 1 i) S a ( 1 d T a ( 1) d ) 1 T ar ar 1 S F x 1 i 1 i P x1 r 1 1 i i ; r 1 S a 1 r ; 1 r 1 f '( x) lim h 0 f ( x h) h f ( x) x1 x y1 y d ( x x1 ) ( y y1) M ; y mx c y y m x ) 1 ( x1 y y1 m m ta x x 1 x a y b r I ABC: a si A b si B c si C si cos a b c oppervlakt e bc. cos A 1 ABC 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 ( xi x) fx x i 1 ( A) P( A) P(A of B) = P(A) + P(B) P(A e B) S yˆ a bx b x x ( y y) ( x x)

METRO EAST EDUCATION DISTRICT NATIONAL SENIOR CERTIFICATE GRADE 12 MATHEMATICS PAPER 1 SEPTEMBER 2014

METRO EAST EDUCATION DISTRICT NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS PAPER SEPTEMBER 04 MARKS: 50 TIME: 3 hours This paper cosists of 7 pages ad a iformatio sheet. GR Mathematics- P MEED September