NATIONAL SENIOR CERTIFICATE GRADE 12

Transcription

1 NTIONL SENIOR ERTIFITE GRDE MTHEMTIS P EXEMPLR 04 MRKS: 50 TIME: 3 hours This questio paper cosists of pages, 3 diagram sheets ad iformatio sheet. Please tur over

2 Mathematics/P DE/04 NS Grade Eemplar INSTRUTIONS ND INFORMTION Read the followig istructios carefully before aswerig the questios This questio paper cosists of 0 questios. swer LL the questios. learly show LL calculatios, diagrams, graphs, et cetera which you have used i determiig your aswers. swers oly will NOT ecessarily be awarded full marks. You may use a approved scietific calculator (o-programmable ad o-graphical), uless stated otherwise. If ecessary, roud off aswers to TWO decimal places, uless stated otherwise. THREE diagram sheets for QUESTION., QUESTION 8., QUESTION 9, QUESTION 0., ad QUESTION 0. 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 them iside the back cover of your NSWER OOK. 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 DE/04 NS Grade Eemplar QUESTION Twelve athletes traied to ru the 00 m sprit evet at the local athletics club trials. Some of them took their traiig more seriously tha others. The followig table ad scatter plot shows the umber of days that a athlete traied ad the time take to ru the evet. The time take, i secods, is rouded to oe decimal place. Number of days of traiig Time take (i secods) ,9 3, 7,0,3 8, 6,5 4,3,7 0,,7 7, 4,3 Scatter plot 0 8 Time take (i secods) Number of days of traiig. Discuss the tred of the data collected. (). Idetify ay outlier(s) i the data. ().3 alculate the equatio of the least squares regressio lie. (4).4 Predict the time take to ru the 00 m sprit for a athlete traiig for 45 days. ().5 alculate the correlatio coefficiet. ().6 ommet o the stregth of the relatioship betwee the variables. () [] Please tur over

4 Mathematics/P 4 DE/04 NS Grade Eemplar QUESTION The table below shows the amout of time (i hours) that learers aged betwee 4 ad 8 spet watchig televisio durig 3 weeks of the holiday. Time (hours) umulative frequecy 0 t < t < t < t < t < t < 0 7. Draw a ogive (cumulative frequecy curve) o DIGRM SHEET to represet the above data. (3). Write dow the modal class of the data. ().3 Use the ogive (cumulative frequecy curve) to estimate the umber of learers who watched televisio more tha 80% of the time. ().4 Estimate the mea time (i hours) that learers spet watchig televisio durig 3 weeks of the holiday. (4) [0] Please tur over

5 Mathematics/P 5 DE/04 NS Grade Eemplar QUESTION 3 I the diagram below, M, T( ; 5), N( ; y) ad P(7 ; 3) are vertices of trapezium MTNP havig TN MP. Q( ; ) is the midpoit of MP. PK is a vertical lie ad S Pˆ K = θ. The equatio of NP is y = + 7. y N( ; y) T( ; 5) θ P(7 ; 3) S 0 Q( ; ) K M 3. Write dow the coordiates of K. () 3. Determie the coordiates of M. () 3.3 Determie the gradiet of PM. () 3.4 alculate the size of θ. (3) 3.5 Hece, or otherwise, determie the legth of PS. (3) 3.6 Determie the coordiates of N. (5) 3.7 If (a ; 5) lies i the artesia plae: 3.7. Write dow the equatio of the straight lie represetig the possible positios of. () 3.7. Hece, or otherwise, calculate the value(s) of a for which T ÂQ = 45. (5) [] Please tur over

6 Mathematics/P 6 DE/04 NS Grade Eemplar QUESTION 4 I the diagram below, the equatio of the circle havig cetre M is ( + ) + (y + ) = 9. R is a poit o chord such that MR bisects. T is a taget to the circle havig cetre N(3 ; ) at poit T(4 ; ). y N(3 ; ) M T(4 ; ) R 4. Write dow the coordiates of M. () 4. Determie the equatio of T i the form y = m + c. (5) If it is further give that MR = uits, calculate the legth of. Leave your aswer i simplest surd form. (4) 4.4 alculate the legth of MN. () 4.5 other circle havig cetre N touches the circle havig cetre M at poit K. Determie the equatio of the ew circle. Write your aswer i the form + y + + Dy + E = 0. (3) [5] Please tur over

7 Mathematics/P 7 DE/04 NS Grade Eemplar QUESTION 5 5. Give that si α = 5 4 ad 90 < α < 70. WITHOUT usig a calculator, determie the value of each of the followig i its simplest form: 5.. si ( α) () 5.. cos α () 5..3 si (α 45 ) (3) 8si(80 )cos( 360 ) 5. osider the idetity: = 4 ta si si (90 + ) 5.. Prove the idetity. (6) 5.. For which value(s) of i the iterval 0 < < 80 will the idetity be udefied? () 5.3 Determie the geeral solutio of cos θ + 4si θ 5si θ 4 = 0. (7) [] Please tur over

8 Mathematics/P 8 DE/04 NS Grade Eemplar QUESTION 6 I the diagram below, the graphs of f () = ta b ad g() = cos ( 30 ) are draw o the same system of aes for The poit P(90 ; ) lies o f. Use the diagram to aswer the followig questios. y f P(90 ; ) 80 g Determie the value of b. () 6. Write dow the coordiates of, a turig poit of g. () 6.3 Write dow the equatio of the asymptote(s) of y = ta b( + 0 ) for [ 80 ; 80 ]. () 6.4 Determie the rage of h if h() = g() +. () [6] Please tur over

9 Mathematics/P 9 DE/04 NS Grade Eemplar QUESTION 7 7. Prove that i ay acute-agled, si si =. (5) a b 7. The framework for a costructio cosists of a cyclic quadrilateral PQRS i the horizotal plae ad a vertical post TP as show i the figure. From Q the agle of elevatio of T is y. PQ = PS = k uits, TP = 3 uits ad S Rˆ Q =. T 3 P y Q k S R 7.. Show, givig reasos, that P ŜQ =. () 7.. Prove that SQ = k cos. (4) 7..3 Hece, prove that SQ = 6cos ta y. () [3] Please tur over

10 Mathematics/P 0 DE/04 NS Grade Eemplar Give reasos for your statemets i QUESTIONS 8, 9 ad 0. QUESTION 8 8. omplete the followig statemet: The agle betwee the taget ad the chord at the poit of cotact is equal to... () 8. I the diagram,,,, D ad E are poits o the circumferece of the circle such that E. E ad D produced meet i F. GH is a taget to the circle at. ˆ = 68 ad Fˆ = 0. G F E 3 D 0 H Determie the size of each of the followig: Ê () ˆ () 3 Dˆ () Ê () 8..5 Ĉ () [9] Please tur over

11 Mathematics/P DE/04 NS Grade Eemplar QUESTION 9 I the diagram, M is the cetre of the circle ad diameter is produced to. ME is draw perpedicular to such that DE is a taget to the circle at D. ME ad chord D itersect at F. M =. M 3 F D E 9. If Dˆ =, write dow, with reasos, TWO other agles each equal to. 4 (3) 9. Prove that M is a taget at M to the circle passig through M, E ad D. (4) 9.3 Prove that FMD is a cyclic quadrilateral. (3) 9.4 Prove that D = 5. (3) 9.5 Prove that D DFM. (4) 9.6 Hece, determie the value of DM. () FM [9] Please tur over

12 Mathematics/P DE/04 NS Grade Eemplar QUESTION 0 0. I the diagram, poits D ad E lie o sides ad respectively of such that DE. Use Euclidea Geometry methods to prove the theorem which states that D E =. D E D E (6) 0. I the diagram, DE is a triagle havig ED ad E GF. It is also give that : E = : 3, = 3 uits, EF = 6 uits, FD = 3 uits ad G = uits. 3 G E 6 F 3 D alculate, givig reasos: 0.. The legth of D (3) 0.. The value of (4) 0..3 The legth of (5) 0..4 The value of area Δ area ΔGFD (5) [3] TOTL: 50

13 Mathematics/P NS Grade Eemplar DE/04 NME: GRDE/LSS: DIGRM SHEET QUESTION. Ogive (umulative Frequecy urve) umulative Frequecy Time (i hours)

14 Mathematics/P NS Grade Eemplar DE/04 NME: GRDE/LSS: DIGRM SHEET QUESTION 8. G F E 3 D 0 H QUESTION 9 M 3 F D E

15 Mathematics/P NS Grade Eemplar DE/04 NME: GRDE/LSS: DIGRM SHEET 3 QUESTION 0. D E QUESTION 0. 3 G E 6 F 3 D

16 Mathematics/P NS Grade Eemplar DE/04 INFORMTION SHEET: MTHEMTIS b ± b 4 ac = a = P( + i) = P( i) = P( i) = P( + i) T a + ( ) d = S = [ a + ( d] ) T = ar a( r ) S = F = f '( ) [( + i) ] i = lim h 0 f ( + h) f ( ) h r ; r [ ( + i) ] P = i ( ) ( ) + y + y d = + y y M ; ( y = m + c y y = m ) ( a) + ( y b) = r I : m y S y a = ; < r < r m = ta = θ a b c = = a b c = + bc. cos area Δ = ab.si si si si ( α + β ) = siα.cos β cosα. si β si( α β ) = siα.cos β cosα. si β ( α + β ) = cosα.cos β siα. si β cos ( α β ) = cosα.cos β + siα. si β si + cos cos α si α cos α = si α si α = siα. cosα cos α ( i ) = σ = i= f ( ) P( ) = P( or ) = P() + P() P( ad ) y ˆ = a + b ( S ) b ( ) ( y y) ( ) =