NATIONAL SENIOR CERTIFICATE GRADE 12

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1 NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 04 MARKS: 50 TIME: 3 hours This questio paper cosists of 8 pages ad iformatio sheet. Please tur over

2 Mathematics/P DBE/04 NSC Grade Eemplar INSTRUCTIONS AND INFORMATION Read the followig istructios carefully before aswerig the questios This questio paper cosists of 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/P 3 DBE/04 NSC Grade Eemplar QUESTION. Solve for : = 0 () = 0 ; 0. (Leave your aswer correct to TWO decimal places.) (4) 3..3 = 4 ()..4 3 ( 5) < 0 (). Solve for ad y simultaeously: y = 6 ad y = (6).3 Simplify, without the use of a calculator: (3).4 Give: f ( ) = 3( ) + 5 ad g( ) = 3 QUESTION.4. Is it possible for f () = g ()? Give a reaso for your aswer. ().4. Determie the value(s) of k for which f () = g () + k has TWO uequal real roots. () [3]. Give the arithmetic series: Determie the umber of terms i this series. (3).. Calculate the sum of this series. ()..3 Calculate the sum of all the whole umbers up to ad icludig 300 that are NOT divisible by 6. (4). The first three terms of a ifiite geometric sequece are 6, 8 ad 4 respectively... Determie the th term of the sequece. ().. Determie all possible values of for which the sum of the first terms of this sequece is greater tha 3. (3)..3 Calculate the sum to ifiity of this sequece. () [6] Please tur over

4 Mathematics/P 4 DBE/04 NSC Grade Eemplar QUESTION 3 3. A quadratic umber patter T = a + b + c has a first term equal to. The geeral term of the first differeces is give by Determie the value of a. () 3.. Determie the formula for T. (4) 3. Give the series: ( ) + (5 6) + (9 0) + (3 4) (8 8) Write the series i sigma otatio. (It is ot ecessary to calculate the value of the series.) (4) [0] QUESTION 4 4. Give: f ( ) = Calculate the coordiates of the y-itercept of f. () 4.. Calculate the coordiates of the -itercept of f. () 4..3 Sketch the graph of f i your ANSWER BOOK, showig clearly the asymptotes ad the itercepts with the aes. (3) 4..4 Oe of the aes of symmetry of f is a decreasig fuctio. Write dow the equatio of this ais of symmetry. () 4. The graph of a icreasig epoetial fuctio with equatio f ( ) a b + q the followig properties: Rage: y > 3 The poits (0 ; ) ad (; ) lie o the graph of f. 4.. Determie the equatio that defies f. (4) 4.. Describe the trasformatio from f ( ) to ( ) =. + =. has h () [5] Please tur over

5 Mathematics/P 5 DBE/04 NSC Grade Eemplar QUESTION 5 The sketch below shows the graphs of f ( ) = ad g ( ) = a + q. The agle of icliatio of graph g is 35 i the directio of the positive -ais. P is the poit of itersectio of f ad g such that g is a taget to the graph of f at P. P y g 0 <35º f 5. Calculate the coordiates of the turig poit of the graph of f. (3) 5. Calculate the coordiates of P, the poit of cotact betwee f ad g. (4) 5.3 Hece or otherwise, determie the equatio of g. () 5.4 Determie the values of d for which the lie k ( ) = + d will ot itersect the graph of f. () [0] Please tur over

6 Mathematics/P 6 DBE/04 NSC Grade Eemplar QUESTION 6 The graph of g is defied by the equatio g ( ) = a. The poit (8 ; 4) lies o g. 6. Calculate the value of a. () 6. If g ( ) > 0, for what values of will g be defied? () 6.3 Determie the rage of g. () 6.4 Write dow the equatio of g, the iverse of g, i the form y =... () 6.5 If h( ) = 4 is draw, determie ALGEBRAICALLY the poit(s) of itersectio of h ad g. (4) 6.6 Hece, or otherwise, determie the values of for which ( ) h( ) QUESTION 7 g >. () [] Siphokazi bought a house. She paid a deposit of R0 000, which is equivalet to % of the sellig price of the house. She obtaied a loa from the bak to pay the balace of the sellig price. The bak charges her iterest of 9% per aum, compouded mothly. 7. Determie the sellig price of the house. () 7. The period of the loa is 0 years ad she starts repayig the loa oe moth after it was grated. Calculate her mothly istalmet. (4) 7.3 How much iterest will she pay over the period of 0 years? Roud your aswer correct to the earest rad. () 7.4 Calculate the balace of her loa immediately after her 85 th istalmet. (3) 7.5 She eperieced fiacial difficulties after the 85 th istalmet ad did ot pay ay istalmets for 4 moths (that is moths 86 to 89). Calculate how much Siphokazi owes o her bod at the ed of the 89 th moth. () 7.6 She decides to icrease her paymets to R8 500 per moth from the ed of the 90 th moth. How may moths will it take to repay her bod after the ew paymet of R8 500 per moth? (4) [6] QUESTION 8 8. Determie f () from first priciples if f ( ) = 3. (5) 8. Determie dy if y = 4 d 5. () [7] Please tur over

7 Mathematics/P 7 DBE/04 NSC Grade Eemplar QUESTION 9 3 Give: f ( ) = Use the fact that f () = 0 to write dow a factor of f(). () 9. Calculate the coordiates of the -itercepts of f. (4) 9.3 Calculate the coordiates of the statioary poits of f. (5) 9.4 Sketch the curve of f i your ANSWER BOOK. Show all itercepts with the aes ad turig poits clearly. (3) 9.5 For which value(s) of will f ( ) < 0? () [5] QUESTION 0 Two cyclists start to cycle at the same time. Oe starts at poit B ad is headig due orth to poit A, whilst the other starts at poit D ad is headig due west to poit B. The cyclist startig from B cycles at 30 km/h while the cyclist startig from D cycles at 40 km/h. The distace betwee B ad D is 00 km. After time t (measured i hours), they reach poits F ad C respectively. A F B C D 0. Determie the distace betwee F ad C i terms of t. (4) 0. After how log will the two cyclists be closest to each other? (4) 0.3 What will the distace betwee the cyclists be at the time determied i QUESTION 0.? () [0] Please tur over

8 Mathematics/P 8 DBE/04 NSC Grade Eemplar QUESTION. Evets A ad B are mutually eclusive. It is give that: P(B) = P(A) P(A or B) = 0,57 Calculate P(B). (3). 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. (4).. What is the probability that a yellow ball will be chose from Bag A? ()..3 What is the probability that a pik ball will be chose? (3) [] QUESTION Cosider the word M A T H S.. How may differet 5-letter arragemets ca be made usig all the above letters? (). Determie the probability that the letters S ad T will always be the first two letters of the arragemets i QUESTION.. (3) [5] TOTAL: 50

9 Mathematics/P NSC Grade Eemplar DBE/04 INFORMATION SHEET b ± b 4 ac = a A = P( + i) A = P( i) A = P( i) A = P( + i) T a + ( ) d = S = [ a + ( d] ) a( r ) T = ar S = F = f '( ) [( + i) ] i = lim h 0 f ( + h) f ( ) h r ; r [ ( + i) ] P = i ( ) ( ) + y + y d = + y y M ; y = m + c y y = m ) ( a) + ( y b) = r I ABC: si a A area ABC ( b c = = a = b + c bc. cos A si B si C = ab. si C S a = ; < r < r y y m = m = taθ ( α + β ) = siα.cos β cosα. si β si( α β ) = siα.cos β cosα. si β si + cos ( α + β ) = cosα.cos β siα. si β cos ( α β ) = cosα.cos β + siα. si β cos α si α cos α = si α si α = siα. cosα cos α ( i ) = σ = i= f ( A) P( A) = P(A or B) = P(A) + P(B) P(A ad B) y ˆ = a + b ( S ) b ( ) ( ) ( y y) =