# NATIONAL SENIOR CERTIFICATE GRADE 12

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1 NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P NOVEMBER 0 MARKS: 50 TIME: 3 hours This questio paper cosists of 8 pages, diagram sheet ad iformatio sheet. Please tur over

2 Mathematics/P DBE/November 0 INSTRUCTIONS AND INFORMATION Read the followig istructios carefully before aswerig the questios This questio paper cosists of questios. Aswer ALL the questios. 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 ographical), uless stated otherwise. If ecessary, roud off aswers to TWO decimal places, uless stated otherwise. Diagrams are NOT ecessarily draw to scale. ONE diagram sheet for aswerig QUESTION. is attached at the ed of this questio paper. Write your cetre umber ad eamiatio umber o this sheet i the spaces provided ad isert the page iside the back cover of your ANSWER BOOK. A iformatio sheet, with formulae, is icluded at the ed of the questio paper. Number the aswers correctly accordig to the umberig system used i this questio paper. Write eatly ad legibly. Please tur over

3 Mathematics/P 3 DBE/November 0 QUESTION. Solve for :.. ( + ) = 6 (3) = 8 (4) (4). Cosider the equatio: + 5y + 6y = 0.. Calculate the values of the ratio y. (3) QUESTION.. Hece, calculate the values of ad y if + y = 8. (5) [9]. Give the sequece: 4 ; ; 3 Determie the value(s) of if the sequece is:.. Arithmetic ().. Geometric (3). Determie the value of P if P = 3 3 k = k 5 (4).3 Prove that for ay arithmetic sequece of which the first term is a ad the costat differece is d, the sum to terms ca be epressed as S = ( a + ( ) d ). (4) [3] QUESTION 3 The followig sequece is a combiatio of a arithmetic ad a geometric sequece: 3 ; 3 ; 9 ; 6 ; 5 ; ; 3. Write dow the et TWO terms. () 3. Calculate T5 T5. (5) 3.3 Prove that ALL the terms of this ifiite sequece will be divisible by 3. () [9] Please tur over

4 Mathematics/P 4 DBE/November 0 QUESTION 4 A quadratic patter has a secod term equal to, a third term equal to 6 ad a fifth term equal to Calculate the secod differece of this quadratic patter. (5) 4. Hece, or otherwise, calculate the first term of the patter. () [7] QUESTION Cosider the fuctio: f ( ) = Calculate the coordiates of the y-itercept of f. () 5.. Calculate the coordiates of the -itercept of f. (3) 5..3 Sketch the graph of f i your ANSWER BOOK, showig clearly the asymptotes ad the itercepts with the aes. (4) 5..4 For which values of is f ( ) > 0? () 5..5 Calculate the average gradiet of f betwee = ad = 0. (4) 5. Draw a sketch graph of y = a + b + c, where a < 0, b < 0, c < 0 ad a + b + c = 0 has oly ONE solutio. (4) [9] Please tur over

5 Mathematics/P 5 DBE/November 0 QUESTION 6 The graphs of ( ) = f 8 ad g ( ) = a + b + c are sketched below. B ad C(0 ; 4,5) are the y-itercepts of the graphs of f ad g respectively. The two graphs itersect at A, which is the turig poit of the graph of g ad the -itercept of the graphs of f ad g. y f C(0 ; 4,5) g O A B 6. Determie the coordiates of A ad B. (4) 6. Write dow a equatio of the asymptote of the graph of f. () 6.3 Determie a equatio of h if h ( ) = f () + 8. () 6.4 Determie a equatio of h i the form y =... () 6.5 Write dow a equatio of p, if p is the reflectio of h about the -ais. () Calculate g ( k) g( k). Show ALL your workig. (4) [4] k= 0 5 k= 4 Please tur over

6 Mathematics/P 6 DBE/November 0 QUESTION 7 7. How may years will it take for a article to depreciate to half its value accordig to the reducig-balace method at 7% per aum? (4) 7. Two frieds each receive a amout of R6 000 to ivest for a period of 5 years. They ivest the moey as follows: Radesh: 8,5% per aum simple iterest. At the ed of the 5 years, Radesh will receive a bous of eactly 5% of the pricipal amout. Thadi: 8% per aum compouded quarterly. Who will have the bigger ivestmet after 5 years? Justify your aswer with appropriate calculatios. (6) 7.3 Nicky opeed a savigs accout with a sigle deposit of R 000 o April 0. She the makes 8 mothly deposits of R700 at the ed of every moth. Her first paymet is made o 30 April 0 ad her last paymet o 30 September 0. The accout ears iterest at 5% per aum compouded mothly. QUESTION 8 Determie the amout that should be i her savigs accout immediately after her last deposit is made (that is o 30 September 0). (6) [6] 8. Determie f () from first priciples if f ( ) = 4. (5) 8. Evaluate: 8.. dy 3 if y = (3) d 8.. f () if ( ) = (7 + ) f (4) [] Please tur over

7 Mathematics/P 7 DBE/November 0 QUESTION 9 3 The fuctio f ( ) = + a + b + c is sketched below. The turig poits of the graph of f are T( ; 9) ad S(5 ; 8). y S(5 ; 8) f O T( ; 9) 9. Show that a =, b = 60 ad c = 43. (7) 9. Determie a equatio of the taget to the graph of f at =. (5) 9.3 Determie the -value at which the graph of f has a poit of iflectio. () [4] QUESTION 0 The graph of y = f (), where f is a cubic fuctio, is sketched below. y 4 0 y = f / () Use the graph to aswer the followig questios: 0. For which values of is the graph of y = f () decreasig? () 0. At which value of does the graph of f have a local miimum? Give reasos for your aswer. (3) [4] Please tur over

8 Mathematics/P 8 DBE/November 0 QUESTION Water is flowig ito a tak at a rate of 5 litres per miute. At the same time water flows out of the tak at a rate of k litres per miute. The volume (i litres) of water i the tak at time t (i miutes) is give by the formula V ( t) = 00 4t.. What is the iitial volume of the water i the tak? (). Write dow TWO differet epressios for the rate of chage of the volume of water i the tak. (3).3 Determie the value of k (that is, the rate at which water flows out of the tak). () [6] QUESTION A school is plaig a trip for 500 learers. The compay that will be providig the trasport has two types of buses, red buses ad blue buses, available. Each red bus has 50 seats ad each blue bus has 5 seats. The compay has at most 5 bus drivers available. There are at most 8 blue buses available. Let the umber of red buses hired by the school be ad the umber of blue buses hired by the school be y.. Write dow ALL the costraits, i terms of ad y, to represet the above iformatio. (6). Represet the costraits graphically o the attached DIAGRAM SHEET. Clearly idicate the feasible regio. (4).3 The cost of hirig a red bus is R600 for the day ad the cost of hirig a blue bus is R300 for the day. Write dow the total trasport cost. ().4.4. Determie ALL possible values of ad y so that the cost will be a miimum. (3).4. Calculate the miimum cost of hirig the buses. ().5 If eactly bus drivers are to be used, determie the umber of each type of bus which the school will ow eed to still esure miimum cost. () [7] TOTAL: 50

9 Mathematics/P DBE/November 0 CENTRE NUMBER: EXAMINATION NUMBER: DIAGRAM SHEET QUESTION. y

10 Mathematics/P DBE/November 0 INFORMATION SHEET: MATHEMATICS b ± b 4 ac = a A = P( + i) A = P( i) A = P( i) A = P( + i) i= = i= ( + ) i = T = ar a( r ) S = F = f [( + i) ] i f ( + h) f ( ) '( ) = lim h 0 h r T a + ( ) d = S = ( a + ( d ) ; r [ ( + i) ] P = i ( ) ( ) + y + y d = + y y M ; y = m + c y y = m ) ( a) + ( y b) = r I ΔABC: si a A area Δ 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.cosβ cosα. si β si( α β ) = siα.cosβ 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 ( ; y) ( cosθ + y siθ ; y cosθ siθ ) ( ; y) ( cosθ y siθ ; y cosθ + siθ ) ( 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) =

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