NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P FEBRUARY/MARCH 03 MARKS: 50 TIME: 3 hours This questio paper cosists of pages, 3 diagram sheets ad iformatio sheet. Please tur over
Mathematics/P DBE/Feb. Mar. 03 INSTRUCTIONS AND INFORMATION Read the followig istructios carefully before aswerig the questios... 3. 4. 5. 6. 7. 8. 9. 0.. 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. 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. Diagram sheets for QUESTION., QUESTION 3. ad QUESTION 0.3 are attached at the ed of this questio paper. Write your cetre umber ad eamiatio umber o these sheets i the spaces provided ad isert the sheets iside the back cover of your ANSWER BOOK. A iformatio sheet, with formulae, is icluded at the ed of this questio paper. Number the aswers correctly accordig to the umberig system used i this questio paper. Write eatly ad legibly. Please tur over
Mathematics/P 3 DBE/Feb. Mar. 03 QUESTION The table below gives the average rad/dollar echage rate ad the average mothly oil price for the year 00. Echage rate i R/$ Oil price i $ Ja. Feb. Mar. Apr. May Ju. Jul. Aug. Sep. Oct. Nov. Dec. 7,5 7,7 7, 7,4 7,7 7,7 7,6 7,3 7, 7,0 6,9 6,8 69,9 68,0 7,9 70,3 66,3 67, 67,9 68,3 7,3 73,6 76,0 8,0. Draw a scatter plot o DIAGRAM SHEET to represet the echage rate (i R/$) versus the oil price (i $). (3). Describe the relatioship betwee the echage rate (i R/$) ad the oil price (i $). ().3 Determie the mea oil price. ().4 Determie the stadard deviatio of the oil price. ().5 Geerally there is cocer from the public whe the oil price is higher tha two stadard deviatios from the mea. I which moth(s) would there have bee cocers from the public? () [] QUESTION The bo-ad-whisker diagrams below represet Vuyai ad Peter's scores for their Schoolbased Assessmet Tasks i a certai subject throughout the year. Vuyai's scores Peter's scores. Give the rage of Peter's scores. (). Give the miimum of Vuyai's scores. ().3 Commet o who you thik had a more cosistet performace throughout the year. Motivate your aswer by referrig to values i the bo-ad-whisker diagrams. () [5] Please tur over
Mathematics/P 4 DBE/Feb. Mar. 03 QUESTION 3 The average percetage of 50 learers for all their subjects is summarised i the cumulative frequecy table below. PERCENTAGE INTERVAL CUMULATIVE FREQUENCY 0 5 0 30 50 40 70 50 88 60 0 70 35 80 4 90 47 00 50 3. Draw the ogive (cumulative frequecy graph) represetig the above data o DIAGRAM SHEET. (4) 3. Use the ogive to approimate the followig: 3.. The umber of learers who scored less tha 85% () 3.. The iterquartile rage (Show ALL calculatios.) (3) [9] Please tur over
Mathematics/P 5 DBE/Feb. Mar. 03 QUESTION 4 I the diagram below, trapezium ABCD with AD // BC is draw. The coordiates of the vertices are A( ; 7); B(p ; q); C( ; 8) ad D( 4 ; 3). BC itersects the -ais at F. DĈB = α. y A( ; 7) E B(p ; q) O F D( 4 ; 3) α C( ; 8) 4. Calculate the gradiet of AD. () 4. Determie the equatio of BC i the form y = m + c. (3) 4.3 Determie the coordiates of poit F. () 4.4 AB / CD is a parallelogram with B / o BC. Determie the coordiates of B /, usig a trasformatio ( ; y) ( + a ; y + b) that seds A to B /. () 4.5 Show that α = 48, 37. (4) 4.6 Calculate the area of DCF. (6) [9] Please tur over
Mathematics/P 6 DBE/Feb. Mar. 03 QUESTION 5 Circles C ad C i the figure below have the same cetre M. P is a poit o C. PM itersects C at D. The taget DB to C itersects C at B. The equatio of circle C is give by + + y + 6y + = 0 ad the equatio of lie PM is y =. y C A y = E C M D P B 5. Determie the followig: 5.. The coordiates of cetre M (3) 5.. The radius of circle C () 5. Determie the coordiates of D, the poit where lie PM ad circle C itersect. (5) 5.3 If it is give that DB = 4, determie MB, the radius of circle C. (3) 5.4 Write dow the equatio of C i the form 5.5 Is the poit F ( 5 ; 0) ( = r a) + ( y b). () iside circle C? Support your aswer with calculatios. (4) [8] Please tur over
Mathematics/P 7 DBE/Feb. Mar. 03 QUESTION 6 6. Write dow the coordiates of the image of poit A( 5 ; 3) after it has udergoe the followig trasformatios: 6.. Traslatio of 3 uits dowwards ad 4 uits to the right () 6.. Reflectio about the -ais () 6. Cosider the followig diagram: 4 3 y K / M / K P M 0-6 -5-4 -3 - - 3 4 5 6 7 8 9 - N - N / -3-4 -5 6.. I the above diagram, triagle KMN is elarged by a certai factor to / / / form triagle K M N. Determie the factor of elargemet. () 6.. Give the geeral rule for the trasformatio i QUESTION 6... () 6..3 Use the aswer to QUESTION 6.. to determie the image P / of P(3 ; ). () 6..4 M is the reflectio of K about the lie with equatio = a. Determie the value of the costat a. () 6..5 KMN is rotated 80 o about the origi to form coordiates of K //. K // M // N //. Give the () 6..6 KMN is traslated 3 uits to the right ad uit upwards to obtai K /// M /// N /// / /// K K. Write dow the ratio of after the traslatio. / /// K M (3) [7] Please tur over
Mathematics/P 8 DBE/Feb. Mar. 03 QUESTION 7 I the diagram below, poit K(a ; b) is rotated clockwise through a agle of 90 about the origi to K / ad the rotated clockwise through a agle θ to K //. y K / K(a ; b) θ O K // 7. Write dow the coordiates of poit K / i terms of a ad b. () 7. Write dow the coordiates of K // i terms of a, b, siθ ad cosθ. Simplify if ecessary. () 7.3 T( 4 ; ) is rotated clockwise through a agle of (90 + θ ) about the origi to obtai image T /. Determie, i the simplest form, the coordiates of T / i terms of θ. () 7.4 Hece, or otherwise, calculate the size of θ if it is give that T / ( 3 + ; 3 ) ad 90 < θ < 80. (5) [] Please tur over
Mathematics/P 9 DBE/Feb. Mar. 03 QUESTION 8 8. Simplify as far as possible: si θ + 3 cos θ () 8. Simplify WITHOUT the use of a calculator: 8.3 4 cos si + cos Prove that = + si si 4 si50 3ta 5 (4) (4) 8.4 Prove that for ay agle θ, 3 cos3θ = 4cos θ 3cosθ. (Hit: 3 θ = θ + θ ) (4) 8.5 If = cos 0, use QUESTION 8.4 to show that 8 3 6 = 0. () [6] QUESTION 9 9. Simplify to ONE trigoometric fuctio WITHOUT usig a calculator: cos60 ta 00 si( 0 ) (6) 9. Cosider cos( + 45 )cos( 45 ). 9.. Show that cos( + 45 )cos( 45 ) = cos. (4) 9.. Hece, determie a value of i the iterval 0 80 for which cos( + 45 )cos( 45 ) is a miimum. (3) [3] Please tur over
Mathematics/P 0 DBE/Feb. Mar. 03 QUESTION 0 The graph of f() = si for 80 90 is show i the sketch below. y f 0,5 80 o 90 o 0 90 o 0,5 0. Write dow the rage of f. () 3 0. Determie the period of f. () 0.3 Draw the graph of g() = cos( 30 ) for 80 90 o the system of aes o DIAGRAM SHEET 3. Clearly label ALL -itercepts ad turig poits. (4) 0.4 Hece, or otherwise, determie the values of i the iterval 80 90 for which f ( ). g( ) < 0. (4) 0.5 Describe the trasformatio that graph f has to udergo to form y = si ( + 60 ). () 0.6 Determie the geeral solutio of si = cos( 30 ). (6) [0] Please tur over
Mathematics/P DBE/Feb. Mar. 0 QUESTION I the diagram below, ABC is a right-agled triagle. KC is the bisector of AC = r uits ad B CK ˆ =. C r A ĈB. B K A. Write dow AB i terms of ad r. (). Give the size of AKˆ C i terms of. ().3 If it is give that AK =, calculate the value of. AB 3 (8) [] TOTAL: 50
Mathematics/P DBE/Feb. Mar. 03 CENTRE NUMBER: EXAMINATION NUMBER: DIAGRAM SHEET QUESTION. Scatter plot of echage rate versus oil price 8 8 80 79 78 77 76 Oil price (i $) 75 74 73 7 7 70 69 68 67 66 65 6.7 6.8 6.9 7 7. 7. 7.3 7.4 7.5 7.6 7.7 7.8 Echage rate (i R/$)
Mathematics/P DBE/Feb. Mar. 03 CENTRE NUMBER: EXAMINATION NUMBER: DIAGRAM SHEET QUESTION 3. 60 Cumulative Frequecy Graph 50 40 30 0 0 Cumulative frequecy 00 90 80 70 60 50 40 30 0 0 0 0 0 0 30 40 50 60 70 80 90 00 Percetage iterval
Mathematics/P DBE/Feb. Mar. 03 CENTRE NUMBER: EXAMINATION NUMBER: DIAGRAM SHEET 3 QUESTION 0.3 y f 0,5 80 o 90 o 0 90 o 0,5
Mathematics/P DBE/Feb. Mar. 03 INFORMATION SHEET: MATHEMATICS b ± b 4 ac = a A = P( + i) A = P( i) A = P( i) A = P( + i) i= = i= ( + ) i = a( r ) T = ar S = F = f '( ) [( + i) ] i lim = h 0 f ( + h) f ( ) h r = S = ( a + ( d ) T a + ( ) d ; r [ ( + i) ] P = i ( ) ( ) + y + y d = + y y M ; ( y = m + c y y = m ) ( a) + ( y b) = r I ABC: si area ABC a A = S ) a = ; < r < r y y m = m = taθ b c = = a = b + c bc. cos A si B si C ab. si C ( 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 ( ; 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) =