NATIONAL SENIOR CERTIFICATE GRADE 12

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1 NATIONAL SNIOR CRTIFICAT GRA MATHMATICS P3 XMPLAR 008 MARKS: 00 TIM: hours This questio paper cosists of 9 pages, diagram sheets ad a -page formula sheet.

2 Mathematics/P3 o/xemplar 008 INSTRUCTIONS AN INFORMATION Read the followig istructios carefully before aswerig the questios This questio paper cosists of 9 questios. Aswer ALL the questios. Show ALL calculatios, diagrams, graphs, et cetera clearly, which you have used i determiig the aswers. A approved scietific calculator (o-programmable ad o-graphical) may be used, uless stated otherwise. If ecessary, aswers should be rouded off to TWO decimal places, uless stated otherwise. Number the aswers correctly accordig to the umberig system used i this questio paper. iagrams are NOT ecessarily draw to scale. It is i your ow iterest to write legibly ad to preset the work eatly.

3 Mathematics/P3 3 o/xemplar 008 QUSTION A sequece is defied as follows: T = 5 ad T k+ = T k + 3 for k.. Write dow the FIRST FIV terms of the sequece. (3). Give a explicit formula for the sequece. () [5] QUSTION The followig extract was take from a televisio ews report durig May 007. Traset bosses claim that a chage i drivig attitudes is ecessary to combat accidets at railways crossigs. Sixty oe people were killed over the last three years i collisios ivolvig trais ad motor vehicles. Twety farm workers lost their lives i oe accidet last year. A trai travellig at reasoable speed requires kilometres to come to a stop. I a survey carried out at the same site where the farm workers were killed, the reporter oted that i a two-hour period oe i 0 vehicles stopped before crossig the railway lie, therefore cofirmig the claim made by the Traset bosses.. Calculate the average umber of people killed aually i collisios ivolvig trais ad motor vehicles over the last three years. (). How does the accidet, ivolvig the farm workers, ifluece the average? ().3 Is the ews reporter s observatio valid? xplai. () [5]

4 Mathematics/P3 4 o/xemplar 008 QUSTION 3 Mary is employed as a huma resources officer at a compay. She collected data o the hourly earigs of all o-salaried employees at her compay ad orgaised the data ito the followig table: Hourly earigs (Rads) Number of wome Number of me 9,70 < 9, ,90 <0, ,0 <0, ,30 <0, ,50 <0, Calculate the mea for the hourly earigs of the me. (4) 3. Calculate the stadard deviatio for the hourly earigs of the me. (5) 3.3 As part of her aalysis Mary also calculated the statistics for wome. They are: Mea: R0,5 Stadard deviatio: R0,5 Mary reached the coclusio that there was little differece i the hourly earigs betwee me ad wome. o you agree with Mary? xplai. () 3.4 raw a ogive (cumulative frequecy curve) for the hourly earigs for the me. (4) [5]

5 Mathematics/P3 5 o/xemplar 008 QUSTION 4 4. A cadidate writes a Accoutig examiatio ad a Mathematics examiatio. The cadidate believes that he has a 40% chace of passig the Mathematics examiatio, a 60% chace of passig the Accoutig examiatio ad a 30% chace of passig both of them. What is the probability that the cadidate passes Mathematics or Accoutig? (3) 4. A box of 40 calculators is set to a store. The ower of the store is ot aware that 5 of the calculators are defective. Two calculators are selected at radom from the box, the first oe ot beig replaced before the secod oe is selected. 4.. What is the probability that the first oe chose is ot defective? () 4.. What is the probability that of the two calculators selected oe calculator is defective ad the other is ot? (4) 4..3 What is the probability that both the calculators selected are defective? (3) 4.3 Four Mathematics books ad three Sciece books were left o the table. You eed to place these books o a shelf If you decide to place ay book i ay positio, i how may differet ways ca you arrage the books o the shelf? (3) 4.3. If two particular books must be placed ext to each other, i how may differet ways ca you arrage the books o the shelf? () If all the Mathematics books must be placed ext to each other ad all the Sciece books must be placed ext to each other, i how may differet ways ca you arrage the books o the shelf? (3) [0]

6 Mathematics/P3 6 o/xemplar 008 QUSTION 5 The ower of Star Travel ad Tours compiled the followig data which illustrates the relatioship betwee the aual profit of the busiess ad its aual advertisig expediture (both i thousads of rads) Aual advertisig expediture (x) Aual profit (y) raw a scatter plot for the data. (3) 5. Calculate the equatio of the least squares lie for the data. (4) 5.3 raw the least squares lie for the data. () 5.4 Predict the aual profit if the aual advertisig expediture is R () 5.5 Calculate the correlatio coefficiet. (3) 5.6 What coclusio ca you reach about the stregth of the relatioship betwee the aual profit ad the aual advertisig expediture? () [5] QUSTION 6 6. I the adjacet figure, = 60. ABC is a straight lie. ^ 6.. Calculate the value of x. (3) 6.. Prove AC is a taget to the circle. () 60º 3x x x A B C

7 Mathematics/P3 7 o/xemplar The diagram below shows a clock face. is joied to the 4 ad to the 7 with straight lies AB ad C respectively. O is the cetre of the clock face. A 0 O C B 6.. If AO ad O are joied, show that AO = 60º. 6.. If CO ad BO are joied calculate acute COB Hece, or otherwise, calculate. () () (3) [] QUSTION 7 I the figure below F is draw with = 3t uits; F = (4t + ) uits ad F = (3t ) uits with t > 0.. 3t 3t 4t + F 7. Which side of F is the logest side? Motivate your aswer. () 7. Now calculate the values of t for which F will be right agled. (4) [6]

8 Mathematics/P3 8 o/xemplar 008 QUSTION 8 I the sketch below, A ad B are tagets to the circle at A ad B. AF=FB. AB produced cuts the lie through, which is parallel to FB, at C. AF produced meets C at ad A = x. 3 C F 3 B A x 8. Fid, with reasos, 5 agles each equal to x. (5) 8. Prove that AB is a cyclic quadrilateral. () 8.3 Prove A B = 3 A (3) 8.4 Prove A = BC. (3) [3]

9 Mathematics/P 9 o/november 007 QUSTION 9 I the figure alogside, MR bisects LK // MR // NP. K L R N with M o LN. LRP ad NRK are straight lies ad P R 4 3 Prove that: L M N 9. LR RN LM = (4) MN 9. KLNP is cyclic () 9.3 KLP MRN (3) [9] TOTAL : 00

10 Mathematics/P o/november 007 NAM/XAMINATION NUMBR: IAGRAM SHT QUSTION QUSTION

11 Mathematics/P3 o/xemplar 008 NAM/XAMINATION NUMBR: IAGRAM SHT QUSTION 6 QUSTION 7 0 A O 9 3 3t 3t 8 7 C B 4t + F QUSTION 8 3 C F 3 B A QUSTION 9 K P R 4 3 L M N

12 Mathematics/P3 o/xemplar 008 INFORMATION SHT: MATHMATICS INLIGTINGSBLA: WISKUN b ± x = b 4 ac a A = P( + i) A = P( i) A = P( i) A = P( + i) i= = i= i = ( + ) ( a + i ) d ) = ( i= i= ar F = f i x ( a + ( ) d ) ( r ) a = r [( + i) ] i f ( x + h) f ( x) '( x) = lim h 0 h ; r = i a r i ar = ; < r < x[ ( + i) ] P = i d = ( x ) ( ) x + y y M x + x y + y ; y = mx + c y y = m x ) y y m = m = taθ x x ( x a) + ( y b) = r ( x I ABC: si a A b c = = a b c = + bc. cos A area ABC = ab. si C si B si C ( α + β) = α β + α β ( ) si si.cos cos.si cos si α β = si α.cos β cos α.si β ( 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

13 Mathematics/P3 3 o/xemplar 008 ( xi x) = i= = fx x ( A) P( A) = P(A or B) = P(A) + P(B) P(A ad B) ( S)

NATIONAL SENIOR CERTIFICATE GRADE 12

NATIONAL SENIOR CERTIFICATE GRADE 12 NATIONAL SNIOR CRTIFICAT GRA MATHMATICS P3 XMPLAR 008 MARKS: 00 TIM: hours This questio paper cosists of 9 pages, diagram sheets ad a formula sheet. Please tur over Mathematics/P3 o/xemplar 008 INSTRUCTIONS