GRADE 12 JUNE 2016 MATHEMATICS P1

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1 NATIONAL SENIOR CERTIFICATE GRADE 1 JUNE 016 MATHEMATICS P1 MARKS: 150 TIME: 3 hours *MATHE1* This questio paper cosists of 14 pages, icludig a iformatio sheet.

2 MATHEMATICS P1 (EC/JUNE 016) INSTRUCTIONS AND INFORMATION Read the followig istructios carefully before aswerig the questios. 1. This questio paper cosists of 10 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.

3 (EC/JUNE 016) MATHEMATICS P1 3 QUESTION Solve for x, i each of the followig: x 7x = 0 (3) x + 4 x + 11 = 0 ; x 0 (correct to TWO decimal places) (4) (x 1)(x 3) > 0 (3) x. 3 x+1 = 7 x (4) 1. Solve simultaeously for x ad y i the followig equatios: 3 + y = x ad 4x + y = xy + 7 (6) 1.3 Give: f(x) = x 3 4x x + 0 = (x + )(x 6x + 10) Prove that f(x) has oly oe real root. (3) [3]

4 4 MATHEMATICS P1 (EC/JUNE 016) QUESTION.1 Give: 0 ; -1 ; 1 ; 6 ; 14 ; Show that this sequece has a costat secod differece. ().1. Write dow the ext term of the sequece. (1).1.3 Determie a expressio for the th term of the sequece. (4).1.4 Calculate the 30 th term. (). I the arithmetic series: a b Prove that a = 6 ad b = 0. ().. Determie which term of the series will be equal to 30. (3).3 For which value(s) of k will the series: ( 1 k k ) + (1 5 5 ) + ( 1 k 3 5 ) +... coverge? (3).4 Give: Determie the sum of the first 40 terms of the series, to the earest iteger. (4).4. Write the series: i the form T k k= where T k = ar k 1 ad a ad r are ratioal umbers. ().4.3 Determie S of the series i QUESTION.4.. () [5]

5 (EC/JUNE 016) MATHEMATICS P1 5 QUESTION 3 Give: f(x) = 3 x Write dow the equatio of the: horizotal asymptote of f. (1) 3.1. vertical asymptote of f. (1) 3. Determie the x- ad y-itercepts of f. (3) 3.3 Sketch the graph of f, showig clearly the asymptotes ad the itercepts with the axes. (3) 3.4 If aother fuctio g is defied as g(x) = f (x 3) + 7, determie the coordiates of the poit of itersectio of the asymptotes of g. () [10]

6 6 MATHEMATICS P1 (EC/JUNE 016) QUESTION 4 The fuctios f(x) = x x + 3 ad g(x) = mx + c are draw below, with g passig through E, C ad A. A ad B are the x-itercepts of f, ad CD is the axis of symmetry of f. E is the y-itercept of g. y E g C f A D O B x 4.1 Determie the coordiates of C, the turig poit of the graph of f. (3) 4. Determie the coordiates of A ad B. (3) 4.3 Determie the values of m ad c. () 4.4 Calculate the legth of CE. (leave your aswer i surd form) (3) 4.5 Determie the values of x, for which f(x). g(x) < 0. () [13]

7 (EC/JUNE 016) MATHEMATICS P1 7 QUESTION The graph of f(x) = a x, where a > 0 ad a 1, passes through the poit (3 ; 7 8 ). y.(3 ; 7 ) 8 O x Use the sketch ad the give iformatio to aswer the followig questios Determie the value of a. () 5.1. Write dow the equatio of f 1 i the form y =... () Determie the value(s) of x for which f 1 (x) = 1. () If h(x) = f(x 5), write dow the domai of h. (1) 5. Draw a clear sketch graph of the fuctio g defied by the equatio g(x) = a. b x + q, where a < 0 ; b > 1 ad q < 0. (a, b ad q are real umbers). Idicate all the itercepts with the axes ad the asymptotes. (3) [10]

8 8 MATHEMATICS P1 (EC/JUNE 016) QUESTION Jerry receives R1 000 to ivest for a period of 5 years. He is offered a iterest rate of 8,5% p.a. compouded quarterly Determie the effective iterest rate. (3) 6.1. What is the amout that Jerry will receive at the ed of the 5 years? (3) 6. A compay bought office furiture that cost R After how may years will the furiture depreciate to a value of R41 611,57 accordig to the reducig-balace method, if the rate of depreciatio is 1,4% p.a.? (4) 6.3 Adrew plas to save R0 000 for a deposit o a ew car. He decided to use a part of his aual bous to pay three eve aual deposits ito a savigs accout at the begiig of every year. Calculate how much moey he must deposit to save up R0 000 after three years. Iterest o the savigs accout is 8% p.a. compouded quarterly. (4) [14]

9 (EC/JUNE 016) MATHEMATICS P1 9 QUESTION Determie the derivative of f(x) = x 3x from first priciples. (5) 7. Determie dy dx if y = x 3x 5x (4) [9]

10 10 MATHEMATICS P1 (EC/JUNE 016) QUESTION The graph of f(x) = x 3 4x 11x + 30 is draw below. A ad B are turig poits of f. A y f O x B Determie the coordiates of A ad B. (5) 8.1. Determie the x-coordiate of the poit of iflectio of f. () Determie the equatio of the taget to f at x =, i the form y = mx + c. (4) Explai how the graph of f ca be shifted for it to have two equal roots. ()

11 (EC/JUNE 016) MATHEMATICS P The diagram below shows the graph of f (x), the derivative of f(x) = ax 3 + bx + cx + d. The graph of f (x) itersects the x-axis at 1 ad 5. A(4 ; 9) is a poit o the graph of f (x). y y = f (x) O 1 5. A(4 ; -9) x 8..1 Write dow the gradiet of the taget to f at x = 4. (1) 8.. Determie the x-coordiates of the turig poits of f. () 8..3 For which value(s) of x is f strictly icreasig? () [18]

12 1 MATHEMATICS P1 (EC/JUNE 016) QUESTION 9 A solid square right prism is made of 8 m 3 melted metal. The legth of the sides of the base are x metres ad the height is h metres. The block will be coated with oe layer of pait. h x x 9.1 Express h i terms of x. () 9. Show that the surface area of the block is give by: A(x) = x + 3 x (3) 9.3 Calculate the dimesios of the block that will esure that a miimum quatity of pait will be used. (5) [10]

13 (EC/JUNE 016) MATHEMATICS P1 13 QUESTION The evets A ad B are idepedet. P(A) = 0,4 ad P(B) = 0,5. Determie: P(A ad B) () P(A or B) () P(ot A ad ot B) () 10. Two idetical bags are filled with balls. Bag A cotais 3 pik ad yellow balls. Bag B cotais 5 pik ad 4 yellow balls. It is equally likely that Bag A or Bag B is chose. Each ball has a equal chace of beig chose from the bag. A bag is chose at radom ad a ball is the chose at radom from the bag Represet the iformatio by meas of a tree diagram. Clearly idicate the probability associated with each brach of the tree diagram ad write dow all the outcomes. (3) 10.. What is the probability that a yellow ball will be chose from Bag A? (1) What is the probability that a pik ball is chose? (3) 10.3 Eastside High School offers oly two sportig activities, amely rugby (R) ad hockey (H). The followig iformatio is give ad partly represeted i the diagram. There are 600 learers i the school. 37 learers play hockey. 88 learers play rugby. 56 of the learers play NO sport. The umber of learers that play both hockey ad rugby is x. R a x b S = 600 H Write dow the values of a ad b i terms of x. () Calculate the value of x. () Are the evets playig rugby ad playig hockey mutually exclusive? Justify your aswer. (1) [18]

15 14 MATHEMATICS P1 (EC/JUNE 016) 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 f ' ( S x 1 i 1 i x) lim h 0 h f ( x h) a r 1 r 1 f ( x) ; r 1 S = a ; 1 r 1 1 r P x[1 (1 i) ] i d ( x x1 ) ( y y1 ) M x1 x y1 y ; 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 b c 1 a b c bc. cos A area ABC ab. si C si A si B si C si si.cos cos.si cos 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 i1 fx x ad B) y a bx b x x y y x x ( A) P( A) P (A or B) = P (A) + P (B) P (A S

GRADE 12 JUNE 2017 MATHEMATICS P1

GRADE 12 JUNE 2017 MATHEMATICS P1 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