SENIOR CERTIFICATE EXAMINATIONS

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1 SENIOR CERTIFICATE EXAMINATIONS MATHEMATICS P1 016 MARKS: 150 TIME: 3 hours This questio paper cosists of 9 pages ad 1 iformatio sheet. Please tur over

2 Mathematics/P1 DBE/016 INSTRUCTIONS AND INFORMATION Read the followig istructios carefully before aswerig the questios This questio paper cosists of 11 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 ot ecessarily be awarded full marks. You may use a approved scietific calculator (o-programmable ad o-graphical), 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. Please tur over

3 Mathematics/P1 3 DBE/016 QUESTION Solve for x: x 5 0 (3) 1.1. x 5x 0 (correct to TWO decimal places) (3) ( x )( x 4) 0 (3) x x 4 (5) 1. Solve for x ad y: x y y 1 0 ad x 3x 4 y (6) 1.3 Give: f x x 1 QUESTION Write dow the domai of f. (1) 1.3. Solve for x if x x 1.1 Give the arithmetic series: a 13 b 7... f. (5) [6].1.1 Show that a = 6 ad b = 0 ().1. Calculate the sum of the first 0 terms of the series. (3).1.3 Write the series i QUESTION.1. i sigma otatio. () 3. Give the geometric series: ( x ) ( x 4) ( x x 4x 8) Determie the values of x for which the series coverges. (4).. If x = 3, calculate the sum to ifiity of the give series. (3) [14] Please tur over

4 Mathematics/P1 4 DBE/016 QUESTION 3 The first four terms of a quadratic umber patter are 1 ; ; 9 ; Determie the geeral term of the quadratic umber patter. (4) 3. Calculate the value of the 48 th term of the quadratic umber patter. () 3.3 Show that the sum of the first differeces of this quadratic umber patter ca be give by (3) S 3.4 If the sum of the first 69 first differeces i QUESTION 3.3 equals (that is, S ), which term of the quadratic umber patter has a value of 9 590? () [11] QUESTION 4 The sketch below shows the graphs of f ( x) x x 3 ad g( x) mx q. Graph f has x-itercepts at A ad B(1 ; 0) ad a turig poit at C. The straight lie g, passig through A ad C, cuts the y-axis at E. y g E C f A O B x 4.1 Write dow the coordiates of the y-itercept of f. (1) 4. Show that the coordiates of C are 1; 4. (3) 4.3 Write dow the coordiates of A. (1) 4.4 Calculate the legth of CE. (6) 4.5 Determie the value of k if h( x) x k is a taget to the graph of f. (5) 4.6 Determie the equatio of 4.7 For which value(s) of x is gx g 1 x? 1 g, the iverse of g, i the form y =... () (3) [1] Please tur over

5 Mathematics/P1 5 DBE/016 QUESTION 5 3 The sketch below shows the graphs of f ( x) q ad g( x) x r x p g itersects the vertical asymptote of f at A. B is the commo y-itercept of f ad g. y = is the commo horizotal asymptote of f ad g. y g B f A f O x 5.1 Write dow the value of r. (1) 5. Determie the value of p. (4) 5.3 Determie the coordiates of A. (3) 5.4 For which value(s) of x is x g( x ) 0? f () 5.5 If h ( x) f ( x ), write dow the equatio of h. () [1] Please tur over

6 Mathematics/P1 6 DBE/016 QUESTION How log would the price of a asset take to reduce by a third of its origial value if it depreciates o a reducig balace at a rate of 4,7% p.a.? (4) 6. Lebogo bought a tractor for Rx o 1 April 016. QUESTION 7 She will trade i this tractor whe she replaces it with a similar oe i 5 years' time o 1 April 01. The tractor depreciates by 0% p.a. accordig to the reducig-balace method. The price of a similar tractor icreases by 18% aually. Lebogo calculated that if she deposited R8 000 per moth ito a sikig fud, which paid iterest at 10% p.a. compouded mothly, she would have eough moey to cover the replacemet cost of the tractor. She made the first deposit i this fud o 30 April 016 ad will cotiue to do so at the ed of every moth util 31 March Determie, i terms of x, what the book value of the curret tractor will be o 1 April 01 (that is, 5 years after it was bought). Give your aswer correct to FIVE decimal places. () 6.. Determie, i terms of x, what the price of a similar ew tractor will be o 1 April 01. Give your aswer correct to FIVE decimal places. () 6..3 Calculate the amout accumulated i the sikig fud o 1 April 01. (4) 6..4 Calculate the value of x, the price of the curret tractor. Roud off your aswer to the earest thousad. (4) [16] 7.1 Determie f (x) from first priciples if f ( x) 3x 5 (5) 7. Determie dy if: dx 7..1 y x (3) 3 x y x x (4) [1] Please tur over

7 Mathematics/P1 7 DBE/016 QUESTION 8 Sketched below are the graphs of f ( x) ( x ) ( x k) ad g( x) mx 1 A ad D are the x-itercepts of f. B is the commo y-itercept of f ad g. C ad D are turig poits of f. The straight lie g passes through A. C y g B A O D f x 8.1 Write dow the y-coordiate of B. (1) 8. Calculate the x-coordiate of A. (3) 8.3 If k = 3, calculate the coordiates of C. (6) 8.4 For which values of x will f be cocave dow? (3) [13] Please tur over

8 Mathematics/P1 8 DBE/016 QUESTION 9 A 340 ml ca with height h cm ad radius r cm is show below. r 1 ml = 1 cm 3 h 9.1 Determie the height of the ca i terms of the radius r. (3) 9. Calculate the legth of the radius of the ca, i cm, if the surface area is to be a miimum. (6) [9] QUESTION A touramet orgaiser coducted a survey amog 150 members at a local sports club to fid out whether they play teis or ot. The results are show i the table below. PLAYING TENNIS NOT PLAYING TENNIS Male Female What is the probability that a member selected at radom is: (a) Female () (b) Female ad plays teis (1) Is playig teis idepedet of geder? Motivate your aswer with the ecessary calculatios. (3) Please tur over

9 Mathematics/P1 9 DBE/ The probability of evets A ad B occurrig are deoted by P(A) ad P(B) respectively. For ay two evets A ad B it is give that: P( B) = 0,8 P(B) = 3P(A) P(A or B) = 0,96 Are evets A ad B mutually exclusive? Justify your aswer. (4) [10] QUESTION 11 Five boys ad four girls go to the movies. They are all seated ext to each other i the same row Oe boy ad girl are a couple ad wat to sit ext to each other at ay ed of the row of frieds. I how may differet ways ca the etire group be seated? (3) 11. If all the frieds are seated radomly, calculate the probability that all the girls are seated ext to each other. (3) [6] TOTAL: 150

10 Mathematics/P1 DBE/016 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 f '( x 1 i 1 i x) lim h 0 f ( x h) f ( x) h P x1 r 1 1 i i ; r 1 x1 x y1 y d ( x x1 ) ( y y1) M ; y mx c y y m x ) x a y b r a b c I ABC: si A si B si C si cos a b c bc. cos A 1 area ΔABC ab.sic 1 ( x1 1 S a ; 1 r 1 1 r y y1 m m ta x x 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 x i 1 (A) P(A) P(A or B) = P(A) + P(B) P(A ad B) yˆ a bx S b x x) x ( y y) ( x x)

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