NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P FEBRUARY/MARCH 00 MARKS: 00 TIME: hours This questio paper cosists of 0 pages, a iformatio sheet ad diagram sheets.
Mathematics/P DoE/Feb. March 00 INSTRUCTIONS AND INFORMATION Read the followig istructios carefully before aswerig the questios.... 4. 5. 6. 7. 8 This questio paper cosists of questios. Aswer ALL the questios. Clearly show ALL calculatios, diagrams, graphs, et cetera, 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. Diagrams are NOT ecessarily draw to scale. THREE diagram sheets for aswerig QUESTION 7., QUESTION 7., QUESTION 8, QUESTION 9., QUESTION 0 ad QUESTION are attached at the ed of this questio paper. Write your cetre umber ad examiatio umber o these sheets i the spaces provided ad had them i together with your ANSWER BOOK. Number the aswers correctly accordig to the umberig system used i this questio paper. It is i your ow iterest to write legibly ad to preset the work eatly.
Mathematics/P DoE/Feb. March 00 QUESTION Cosider the followig sequece: ; ; ; 7 ; 5 ;. Write dow the ext TWO terms i the sequece. (). Write dow a recursive formula for the sequece. () [5] QUESTION The graphs below represet the aual profits of a compay. Aswer the questios that follow with referece to the graphs. Aual Profit (i millios of rads) 00 80 60 40 0 00 80 60 40 0 0 Aual Profit (i millios of rads) 000 900 800 700 600 500 400 00 00 00 0 004 005 006 007 008 004 005 006 007 008 Year Year Graph Graph Aual Profit (i millios of rads) 00 80 60 40 0 00 80 60 40 0 0 008 007 006 005 004 Year Graph. Do the graphs display the same iformatio? Justify your aswer. (). Explai the impressio created by graphs ad whe compared to graph. (). Which graph would you recommed for the maagig director to use i the Aual Report for the compay? Explai your aswer. () [6]
Mathematics/P 4 DoE/Feb. March 00 QUESTION The time, i miutes, that each learer spet i the library o oe afteroo was recorded. It was observed that the times followed a ormal distributio. The ormal curve for these times are draw as show below. ±4% ±4% ±% ±4% ±4% ±% 47 m 6 Time (i miutes). What was the average time that learers spet i the library? (). Determie the value of m ad. (). Of all the learers who spet time i the library, 0 stayed loger tha 47 miutes. How may learers were at the library that afteroo? ().4 The school maagemet would like to employ a library assistat to work i the library i the afteroos. Before employig this perso, the maagemet observed the times spet by learers i the library i the afteroos over the period of oe moth. They foud the distributio of the time spet to be the same as the distributio show above. For how log should the school employ the assistat every afteroo to keep the cost to a miimum? Justify your aswer. () [7] QUESTION 4 P(A) = 0, ad P(B) = 0,5. Calculate P(A or B) if: 4. A ad B are mutually exclusive evets () 4. A ad B are idepedet evets () [5]
Mathematics/P 5 DoE/Feb. March 00 QUESTION 5 At a school for boys there are 40 learers i Grade. The followig iformatio was gathered about participatio i school sport. boys play rugby (R) 58 boys play basketball (B) 96 boys play cricket (C) 6 boys play all three sports boys play rugby ad basketball 6 boys play cricket ad basketball 6 boys do ot play ay of these sports Let the umber of learers who play rugby ad cricket oly be x. 5. Draw a Ve diagram to represet the above iformatio. (4) 5. Determie the umber of boys who play rugby ad cricket. () 5. Determie the probability that a learer i Grade selected at radom: (Leave your aswer correct to THREE decimal places.) QUESTION 6 5.. oly plays basketball. () 5.. does ot play cricket. () 5.. participates i at least two of these sports. () [] A South Africa bad is plaig a cocert tour with performaces i Durba, East Lodo, Port Elizabeth, Cape Tow, Bloemfotei, Johaesburg ad Polokwae. I how may differet ways ca they arrage their itierary if: 6. There are o restrictios () 6. The first performace must be i Cape Tow ad the last performace must be i Polokwae () 6. The performaces i the four coastal cities (the cities close to the sea or ocea) must be grouped together? (4) [9]
Mathematics/P 6 DoE/Feb. March 00 QUESTION 7 The term latitude refers to how far a place is from the equator. Latitude i the Norther Hemisphere rage from 0 at the equator to 90 N at the orth pole. Below are the latitudes of several cities i the Norther Hemisphere together with the mea maximum temperature for April i degrees Celsius. City Norther Latitude Mea maximum temperature for April Lagos, Nigeria 6 Lodo, Eglad 5 Calcutta, Idia 6 Rome, Italy 4 0 Moscow, Russia 56 8 Cairo, Egypt 0 8 Sa Jua, Puerto Rico 8 9 Copehage, Demark 56 0 Tokyo, Japa 5 7 7. Draw a scatter plot for the above iformatio o DIAGRAM SHEET. () 7. Determie the equatio of the least squares regressio lie. (4) 7. Draw the least squares regressio lie o your scatter plot, o DIAGRAM SHEET. () 7.4 What iformatio does the y-itercept of this lie represet? () 7.5 The city of Madrid has a latitude of 40 N. Determie the mea maximum temperature for April for this city. () 7.6 Calculate the correlatio coefficiet of the data. () 7.7 Explai the correlatio betwee latitude ad the mea maximum temperature for April. () [5]
Mathematics/P 7 DoE/Feb. March 00 QUESTION 8 8. Complete the statemet: The sum of the agles aroud a poit is... () 8. I the figure below, O is the cetre of the circle. K, L, M ad N are poits o the circumferece of the circle such that LM = MN. LÔN = 00. L M K 00 O N Determie, with reasos, the values of the followig: 8.. L Mˆ N () 8.. M Kˆ L () [7]
Mathematics/P 8 DoE/Feb. March 00 QUESTION 9 9. Complete the followig statemet: The agle betwee the taget ad the chord () 9. I the diagram below, two circles have a commo taget TAB. PT is a taget to the smaller circle. PAQ, QRT ad NAR are straight lies. Let Qˆ = x. T R P 4 5 6 A N x B Q 9.. Name, with reasos, THREE other agles equal to x. (5) 9.. Prove that APTR is a cyclic quadrilateral. (5) []
Mathematics/P 9 DoE/Feb. March 00 QUESTION 0 Two circles touch each other at poit A. The smaller circle passes through O, the cetre of the larger circle. Poit E is o the circumferece of the smaller circle. A, D, B ad C are poits o the circumferece of the larger circle. OE CA. C F A O E 4 5 D B 0. Prove, with reasos, that AE = BE. () 0. Prove that ΔAED ΔCEB. () 0. Hece, or otherwise, show that AE = DE.CE. () 0.4 If AE.EB = EF.EC, show that E is the midpoit of DF. () [0]
Mathematics/P 0 DoE/Feb. March 00 QUESTION ΔABC is a right-agled triagle with Bˆ = 90. D is a poit o AC such that BD AC ad E is a poit o AB such that DE AB. E ad D are joied. AD : DC = :. AD = 5 cm. B x C E D A. Prove that ΔBDA ΔCDB. (). Calculate BD (Leave your aswer i surd form). (). Calculate AE (Leave your aswer i surd form). (6) [] TOTAL: 00
Mathematics/P DoE/Feb. March 00 b ± x = b 4 ac a A = P( + i) A = P( i) INLIGTINGSBLAD: WISKUNDE INFORMATION SHEET: MATHEMATICS A = P( i) A = P( + i) i= = S = ( a + ( d ) ) i= ( + ) i = T = ar a( r ) S = F = f x [( + i) ] i f ( x + h) f ( x) '( x) = lim h 0 h r ; r x[ ( + i) ] P = i T = a + ( ) d d = ( x ) ( ) x + y y M x + x y + y ; y = mx + c y y = m x ) ( x a) + ( y b) = r ( x S a = ; < r < r y y m = m = taθ x x I ΔABC: a b c = = a b c = + bc. cos A area Δ ABC = ab. si C si A si B si C si ( α + β ) = siα.cos β + cosα. si β si( α β ) = siα.cos β cosα. si β cos α + β = cosα.cos β siα. si cos ( α β ) = cosα.cos β + siα. si β ( ) β cos α si α cos α = si α si α = siα. cosα cos α ( x ; y) ( x cosθ + y siθ ; y cosθ xsiθ ) ( x ; y) ( x cosθ y siθ ; y cosθ + xsiθ ) ( ) = xi x fx i= x σ = ( A) P( A) = P(A or B) = P(A) + P(B) P(A ad B) S y ˆ = a + bx ( ) b ( x x) ( y y) = ( x x)
Mathematics/P DoE/Feb. March 00 CENTRE NUMBER: EXAMINATION NUMBER: DIAGRAM SHEET QUESTION 7. AND 7.
Mathematics/P DoE/Feb. March 00 CENTRE NUMBER: EXAMINATION NUMBER: DIAGRAM SHEET QUESTION 8 M K 00 O N QUESTION 9. T R P 4 5 6 A N x B Q
Mathematics/P DoE/Feb. March 00 CENTRE NUMBER: EXAMINATION NUMBER: DIAGRAM SHEET QUESTION 0 C F A O E 4 5 D B QUESTION B x C E D A