NATIONAL SENIOR CERTIFICATE GRADE 12

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1 NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 008 MARKS: 50 TIME: 3 hours This questio paper cosists of pages, diagram sheets ad a formula sheet. Please tur over

2 Mathematics/P DoE/Eemplar 008 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 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. Diagrams are NOT ecessarily draw to scale. It is i your ow iterest to write legibly ad to preset the work eatly. TWO diagram sheets for aswerig QUESTION 3.., QUESTION 9.3, QUESTION 0. ad QUESTION. are attached at the ed of this questio paper. Write your ame/eamiatio umber i the spaces provided ad had them i together with your ANSWER BOOK. Please tur over

3 Mathematics/P 3 DoE/Eemplar 008 QUESTION ABCD is a quadrilateral with vertices A( ; 5), B( 3 ; 0), C( 4 ; 3) ad D( ; ). B( 3 ; 0) y A( ; 5) C( 4 ; 3) M θ D( ; ). Calculate the legth of AC. (Leave the aswer i simplest surd form.) (). Determie the coordiates of M, the midpoit of AC. ().3 Show that BD ad AC bisect each other at right agles at M. (5).4 Calculate the area of ABC. (4).5 Determie the equatio of DC. (3).6 Determie θ, the agle of icliatio of DC. ().7 Calculate the size of A Dˆ C. (4) [] Please tur over

4 Mathematics/P 4 DoE/Eemplar 008 QUESTION A(8 ; 5) ad B( ; 7) are two poits i a Cartesia plae. BA produced itersects the -ais at D. AD is a diameter of the circle cetred at C. y B( ; 7) A(8 ; 5) C D O...3 Show that the equatio of the lie through A ad B ca be give as y + = 0. Determie the coordiates of D. Determie the coordiates of C. (3) () ().4 Determie the equatio of the circle. (5).5 Determie the equatio of the taget passig through A(8 ; 5). (3) /.6 Determie A, the image of A reflected about the straight lie through C, perpedicular to the -ais. (3) [8] Please tur over

5 Mathematics/P 5 DoE/Eemplar 008 QUESTION 3 3. The poit P( ; 3 ) lies i a Cartesia plae. Determie the coordiates of the image of P if: 3.. P is reflected across the -ais () 3.. P has bee rotated about the origi through 90 i a aticlockwise directio () 3. A trasformatio T of the Cartesia plae is described as follows: A poit is first rotated about the origi through 80 i the aticlockwise directio. Thereafter it is elarged through the origi by a factor of. I the diagram below quadrilateral ABCD is give with A( ; ), B( ; 3), C( ; 4) ad D(3 ; ). 4 y B C A D Use the grid o the attached DIAGRAM SHEET to sketch ad label PQRS, the image of ABCD uder the trasformatio T. (4) 3.. Write dow the image of ( ; y) i terms of ad y. (4) 3..3 Write dow the ratio of area ABCD : area PQRS. () Please tur over

6 Mathematics/P 6 DoE/Eemplar 008 / 3.3 Show that the coordiates of P, the image of P( ; y) rotated about the origi through a agle of 30º i the aticlockwise directio, is give by 3 y 3 ; y + P / ( / ; y / ) 30º P( ; y) O α (8) 3.4 / / K ad L are the images of K(4 ; 3) ad L(3 ; 6) uder a rotatio of 30, i the aticlockwise directio, about the origi. / / Usig the results i QUESTION 3.3, determie the coordiates of K ad L. (4) [5] QUESTION 4 4. Simplify, without the use of a calculator: si 40.ta( 35 ) cos 30.si 40 (6) 4. Simplify: si (90 ) ta(80 + ).cos(540 + ) si( ) + (8) cos(90 + ) 4.3 Show, without the use of a calculator: ( 3 ) si5 = (5) Show that cos + 3cos = cos + 3cos θ θ θ θ. () 4.4. Hece determie the geeral solutio for: cos θ + 3cosθ = 0 (4) [5] Please tur over

7 Mathematics/P 7 DoE/Eemplar 008 QUESTION 5 The followig questio was give to cadidates to aswer: If cos = t, determie, without the use of a calculator, the value of si66º i terms of t. Joyce gave the followig solutio: si66º = si( 45 + ) lie = si 45 + si lie = + t lie 3 = + t lie 4 Ufortuately, Joyce's aswer is icorrect. º t t 5. Eplai why Joyce's aswer is icorrect. () 5. Give a correct solutio to Joyce's problem. (4) [5] QUESTION 6 OCB is a semicircle with cetre D ad radius a. O ĈB = 90 ad D Bˆ C = θ y C θ O a D a B 6. Show that BC = a.cos θ () 6. Show that the area OCB = a.si θ (3) 6.3 Determie the coordiates of C such that the area of OCB is a maimum. () [5] Please tur over

8 Mathematics/P 8 DoE/Eemplar 008 QUESTION 7 CD is a vertical mast. The poits B, C ad E are i the same horizotal plae. BD ad ED are cables joiig the top of the mast to pegs o the groud. DE = 8, m ad BC = 0,7 m. The agle of elevatio of D from B is 43,6º. C Bˆ E = 63 ; B Dˆ E = 35,7. D B 63º C E Give your aswers correct to ONE decimal place i each of the followig questios: 7. Calculate the legth of BD. (3) 7. Show that the legth of BE rouds to 7,4 m. (4) 7.3 Calculate the area of BEC. () [9] Please tur over

9 Mathematics/P 9 DoE/Eemplar 008 QUESTION 8 Sketched below are the graphs of the fuctios f ( ) = cos ad g ( ) = si( 30 ) for [ 80 ; 80 ]. The curves itersect at poits A ad B. y f B g A Determie the coordiates of the poits A ad B. (7) 8. For which values of is f ( ) > g( )? () [9] Please tur over

10 Mathematics/P 0 DoE/Eemplar 008 QUESTION 9 Geoff, a driver of a courier motorcycle, recorded the distaces he travelled (i kilometres) o 5 trips. The data is give below: What is the media for the above-metioed data? () 9. Write dow the upper ad lower quartiles. () 9.3 Draw a bo ad whisker diagram for the data of Geoff's travels, i the space provided o the attached diagram sheet. (5) 9.4 Aother driver, Thabo, i the same compay had also travelled ad recorded (i kilometres) the distace he travelled o 5 trips. The five umber summary of his data is ( ; ; 5 ; 3 ; 34). The bo ad whisker diagram is show below Carefully aalyse the bo ad whisker diagrams, o the diagram sheet, of Geoff ad Thabo's travels ad commet o the differeces or similarities, if ay, betwee the distaces covered by each o the 5 trips. () [] Please tur over

11 Mathematics/P DoE/Eemplar 008 QUESTION 0 A compay that rets out cars calculated the average cost per kilometre of maitaiig a ew car for differet distaces covered durig the first year. The data gathered is give i the table below. Aual mileage (i thousads of kilometres) Cost per kilometre (i rad) ,50,46,4,37,3,5 0. Draw a scatter diagram to represet the above data o DIAGRAM SHEET. (3) 0. Decide which of the followig graphs fit the above-metioed data best: straight lie, parabola or epoetial. () 0.3 Estimate, by usig a suitable graph, the average cost per kilometre of operatig a ew car if it is drive kilometres durig the first year. () [6] QUESTION. Fifty shoppers were asked what percetage of their icome they sped o groceries. Si aswered that they sped betwee 0% ad 9%, iclusive. The full set of resposes is give i the table below. PERCENTAGE FREQUENCY (f) By usig the table o DIAGRAM SHEET, calculate the mea percetage of family icome allocated to groceries. (5). The marks of 8 learers i a test for which the maimum mark is 0, were: 7, 4, 9, 4, 9, 5, 4, 6. Calculate the stadard deviatio of this data. (7) [] Please tur over

12 Mathematics/P DoE/Eemplar 008 QUESTION Your school makes use of two suppliers of light bulbs. Both compaies claim that o average their light bulbs last 000 hours. The graphs below show the distributio of time take i hours before a light bulb from each supplier burt out. The horizotal ad vertical scales of the two graphs are the same. Supplier A: sd 000 hours sd Supplier B: sd 000 hours sd. Compare the graphs above. Which compay supplies bulbs that has a higher deviatio from the mea? (). The clerk asks for your assistace i selectig the supplier from which he will purchase 00 light bulbs for the school. Which supplier would you select? Eplai the reaso for your choice. () [3] TOTAL: 50

13 Mathematics/P DoE/Eemplar 008 NAME/EXAMINATION NUMBER: DIAGRAM SHEET QUESTION y B C A D QUESTION Geoff's travels Thabo's travels

14 Mathematics/P DoE/Eemplar 008 NAME/EXAMINATION NUMBER: DIAGRAM SHEET QUESTION y QUESTION. PERCENTAGES FREQUENCY ( f ) Mea =

15 Mathematics/P 3 DoE/Eemplar 008 b ± = b 4 ac a FORMULA SHEET: MATHEMATICS FORMULEBLAD: WISKUNDE A = P( + i) A = P( i) A = P( i) F = i= = [( + i) ] i A = P( + i) [ ( + i) ] P = i i= ( + ) i = ( a + ( i ) d ) = ( a + ( ) d ) i= i= f ar i ( r ) a = r f ( + h) f ( ) '( ) = lim h 0 h ; r = i a r i ar = ; < < r d = ( ) ( ) + y y M + y + y ; y = m + c y y = m ) y y m = m = taθ ( a) + ( y b) = r ( I ABC: si a A b c = = a b c = + bc. cos A area ABC = ab. si C si B si C si ( α + β) = si α.cosβ + cosα. siβ si ( α β) = si α.cosβ cosα. siβ 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 i = = σ = f ( 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 MATHEMATICS P FEBRUARY/MARCH 009 MARKS: 50 TIME: 3 hours This questio paper cosists of 0 pages, a iformatio sheet ad 3 diagram sheets. Please tur over Mathematics/P DoE/Feb.