SENIOR CERTIFICATE EXAMNATIONS SENIORSERTIFIKAAT-EKSAMEN

Size: px

Start display at page:

Download "SENIOR CERTIFICATE EXAMNATIONS SENIORSERTIFIKAAT-EKSAMEN"

Alice Dorsey
5 years ago
Views:

1 SENIOR CERTIFICATE EXAMNATIONS SENIORSERTIFIKAAT-EKSAMEN MATHEMATICS P/WISKUNDE V 07 MARKING GUIDELINES/NASIENRIGLYNE MARKS: 50 PUNTE: 50 Thee marking guideline conit of page. Hierdie naienriglyne betaan uit bladye..

2 Mathematic/P/Wikunde/V DBE/07 NOTE: If a candidate anwer a quetion TWICE, only mark the FIRST attempt. If a candidate ha croed out an attempt of a quetion and not redone the quetion, mark the croed out verion. Conitent accuracy applie in ALL apect of the marking memorandum. Stop marking at the econd calculation error. Auming anwer/value in order to olve a problem i NOT acceptable. Geometry: S a mark for a correct tatement (a tatement mark i independent of a reaon) R a mark for a correct reaon (a reaon mark may only be awarded if the tatement i correct) S/R award a mark if tatement and reaon are both correct NOTA: A 'n kandidaat 'n vraag TWEEKEER beantwoord, merk leg die EERSTE poging. A 'n kandidaat 'n antwoord van 'n vraag doodtrek en nie oordoen nie, merk die doodgetrekte poging. Volgehoue akkuraatheid word in ALLE apekte van die memorandum toegepa. Hou op naien by die tweede berekeningfout. Aanvaar van antwoorde/waarde om 'n probleem op te lo, word NIE toegelaat nie. Euklidiee Meetkunde: S n punt vir n korrekte bewering ( n beweringpunt i onafhanklik van die rede) R n punt vir n korrekte rede ( n punt kan leg vir n rede toegeken word, indien die bewering korrek i S/R n punt word toegeken indien beide die bewering en rede korrek i

3 Mathematic/P/Wikunde/V DBE/07 QUESTION/VRAAG/VRAAG TIME TAKEN (IN HOURS) COST (IN THOUSANDS OF RANDS) a 4, , 8 b,..., y 4,8 +, x. Cot 5,974 5,97 thouand rand (calculator) R5 970 a 4, 8 b, equation () 5,97 anwer (in Rand) () y 4,8 +,(6) ubtitution y 5,9 Cot R5 90 anwer (in Rand) (). r 0,949 0,95 anwer ().4 x 0 x 0 y 4,8 OR (4,80647) R4 80 OR R4806,47 anwer () [8]

4 Mathematic/P/Wikunde/V 4 DBE/07 QUESTION/VRAAG. modal cla: 80 < x 00 correct cla (). Commiion earned (in thouand of Rand) Frequency Cumulative Frequency 0 < x < x Cumulative Frequency 60 < x < x < x OGIVE ; ; 5 () grounded/geanker upper limit/ boonte limiet cum frequency / Kum frekwenie hape/vorm Commiion earned (in thouand of Rand).4 No. of alemen awarded bonue: alemen.5 (0 7) + (50 6) + (70 8) + (90 0) + (0 4) Etimated mean ,86 thouand rand or R68 857,4 R or 69 thouand rand accept (5 7) accept (8 0) top line uing midpt & freq 40 () anwer (nearet) () []

5 Mathematic/P/Wikunde/V 5 DBE/07 QUESTION/VRAAG A ( 4 ; ) y B ( ; 4) α F G O H E x C (4 ; ) θ. ( 5).. D (0 ; 5) mcd 4 0 ubtitution of C & D anwer () 5 ubtitution of A & mad 0 ( 4) D m AD mcd mad product AD DC () AB ( + 4) + (4 ) 50 5 correct ubtitution length of AB BC (4 ) + ( 4) 5 correct ubtitution AB BC length of BC ABC i an icocele triangle/' n gelykbenige driehoek

Mathematic/P/Wikunde/V 6 DBE/07.4.5 m CD m BF 4 () + c 5 c 5 y x + tanα α 6.

equation tanα α 6. 57 θ 6,57 tanα tan θ θ 6,57 () () ().

6 Mathematic/P/Wikunde/V 6 DBE/ m CD m BF 4 () + c 5 c 5 y x + tanα α 6.57 α 90 + θ θ 6,57 [ext ] tanα OR m AD tan θ θ 6,57 [BF DC] m BF ubtitution of B( ; 4) equation tanα α θ 6,57 tanα tan θ θ 6,57 () () ().6 Inclination of DE i β: tan β β 6,57 O Dˆ E 6, 4 θ 90-6,4 6,57 x + y r + ( ) 5 x + y 5 β 6, 57 O Dˆ E 6, 4 θ 6,57 r 5 () equation () [7]

7 Mathematic/P/Wikunde/V 7 DBE/07 QUESTION/VRAAG 4 y S P ( ; 4) R N x M ( ; ) 4. + ( ) 4 + ( ) N ; N ; i the centre of the circle ( ) 4. r ( ( ) ) + ( 4 ) r radiu ( x + ) + ( y ) OR/OR r ( ( ) ) + ( ) r radiu ( x + ) + ( y ) ubtitution M & P x-value of N y-value of N () ubtitution N & P r LHS of eq RHS of eq ubtitution N & M r LHS of eq RHS of eq

8 Mathematic/P/Wikunde/V 8 DBE/07 4. m m m NM NM MR y y y + ( ( x x 4 y m MR ( ) ( ) x x + ) ) OR 4.4 Symmetry of a kite: S( ; 4) [radiu tangent/ raakklyn] m PM P ŜM 90 [ in emi circle] PS SM S( ; 4) ( + ) ( NS) ( radiu) + (y ) 4 ( ) ( ) (y ) 9 y ± y 4 OR y S( ; 4) SR RM...Tangent from common pt/ rklyne v die punt 4.5 ( ) ( ) ( x + ) y + ( y 4) R 7 ; 8y + 6 y ( x + ) y y x 4 5 x + 4y ( y + ) or x 4 x 7 correct ubtitution m m NM MR ubtitution of m MR &( ; ) equation (5) x-value of S y-value of S () x-value of S y-value of S () x-value of S y-value of S () equating length implification y-value of R x-value of R

9 Mathematic/P/Wikunde/V 9 DBE/07 R( x;) x 4 5 x 5 x 5 R ; [RN i a horizontal line] y R horizontal line OR R lie on y equating x-value of R (x < 4,6) 4 m NS + m RS y 4 ( x + ) y x + 6 x 4 x + 6 x 7 5 y ( ) R ; 4.6 RS ( + 7,5) + (4 ) RM ( + 7,5) + ( ) RS 5,4 area of RSNM area of RSN ( ) 9 9,5 quare unit y x + 6 equating x-value of R (x < 4,6) y-value of R RS OR RM method and anwer method MS 6 RN 6,5 anwer

10 Mathematic/P/Wikunde/V 0 DBE/07 area RSNM (MS RN) (area of a kite/ opp v vlieër) (6)(6,5) 9 OR 9,5 quare unit RS OR RM RS RS ( + 7,5) or5,4 + (4 ) area of RSN ( ) RM ( + 7,5) + ( ) method anwer 9 9,75 quare unit 4 area of RSNM area of RSN method 9 9,5 quare unit MS 6 h & 5 SM 6 area of RSNM Area of SMN + Area of RSM anwer (6)() + (6)(5 ) []

11 Mathematic/P/Wikunde/V DBE/07 QUESTION/VRAAG in A tan A co A p p identity value of tan A () p tan A p (p ; p) x y value of tan A 5.. in A + co A ( p) + p 4 p + p 5 p ( p ) + p implification of LHS p 5 anwer p () 5 5. in x 5in x + 0 ( in x )(in x ) 0 factor or formula in x or in x (no olution) both equation no olution/geen opl ref 0 x 0 + k.60 or x 50 + k.60 ; k Z 0 + k k.60 ; k Z (6) 5.. in( x + 00 ) in x co 00 + in 00 expanion/uitbreiding () 5.. in( x + 00 ) co( x 50 ) in x co 00 + in 00 ( co50 + in xin50 ) nd expanion/ in x co60 in 60 ( co x co0 + in xin 0 ) de uitbreiding reduction/redukie in x co60 in 60 + co0 in xin 0 in x + in x pecial angle value/ peiale hoekwaarde 0 anwer (5) ()

12 Mathematic/P/Wikunde/V DBE/07 in( x + 00 ) co( x 50 ) in x co 00 + in 00 ( co50 + in xin50 ) in x co60 in 60 ( co x co0 + in xin 0 ) in x co60 in 60 + co0 in xin 0 in x in 0 in 60 + in 60 in x in tan x + Conider: in x + in x tan x + in x in x + + LHS in x in x. in x + + in x + in x + in x + RHS in x in x + + LHS in x in x. in x + + in x + in x + in x + RHS 5.5. ( ) ( ) + k in x + + k in x + in x + co + k + in x k in x x nd expanion/ de uitbreiding reduction/redukie co-ratio / ko-verh anwer (5) identity of tan x in x + in x + in x + implify identity of tan x in x + co x in x + (5) implify multiplication (5) quare both ide in x + in x ()

13 Mathematic/P/Wikunde/V DBE/ From 5.5. in x + + in x max value: in x + + Maximum value of + in x + maximum value of in x + max of in x anwer () max of in x anwer () (in x + ) in + in x max value (in x + ) max valuein x + x + in x + co + x max of in x anwer () [7]

14 Mathematic/P/Wikunde/V 4 DBE/07 QUESTION/VRAAG 6 y C f h 80 o 90 o 0 o 90 o x B D A 6. Period 80 anwer in x + in x in x co in 45.in x anwer co in 45. in x () () in x co( x 45 ) x [ 75 ;65 ] co( x 45 ) anwer [6]

15 Mathematic/P/Wikunde/V 5 DBE/07 QUESTION/VRAAG 7 A B H K D C 8 cm A 60 B 8 cm E cm Figure/Figuur (i) F G 6 cm D E C cm Figure / Figuur (ii) F L G 7. KC 6 cm anwer 7. Let P be the point of interection of KL and CB () KP KC KP in 60 6 in 60 KP or 5,0 KL 8 + or,0 cm K C P B F L G trig ratio length of KP anwer () 7. DK 6 + DK 80 or 6 5 or,4 cm in KDˆ L in DLˆK KL DK in KDˆ L KL in DLˆ K DK 8 + or 6 5,0,4 or 0,98 DK 6 5 ue of ine rule in KDˆ L in DLˆ K anwer KL DK [8]

16 Mathematic/P/Wikunde/V 6 DBE/07 QUESTION/VRAAG 8 L M K 55 O N 00 S 8. Lˆ 00 [ ext cyclicquad int opp / buite kdvh to ] S R () 8. Nˆ 80 [ on traight line] Lˆ 00 [opp of cyclic quad] Nˆ 80 [ on traight line/ e op reguitlyn ] Ô 60 [ at centre at circumference/midpt omtrek ] S R () S S R () reflex KÔM 00 [ at centre at circumference/midpt omtrek ] Ô 60 [ around a pt/ e om n pt] S R S 8. Mˆ Mˆ 60 ( ) [ um of quad/ om e v vierhoek] 45 S S () () [7]

17 Mathematic/P/Wikunde/V 7 DBE/07 QUESTION/VRAAG 9 A G B F O C D 9.. in emi-circle/ in halfirkel anwer 9.. Opp of quad 80 /Teenoort e v vierhoek80 anwer 9.. OF AC [line from centre biect chord/lyn v midpt halv kd] AC GO [co-interior/ko-binne 80 correp/ooreenkomtige ] 9.. Ĝ Â [correp/ ooreenk ; AC GO] Â Bˆ [ in ame egment/ e in die egment] Ĝ Bˆ Ĝ Bˆ [ext cyclic quad/ buite koordevh] but ABF CBF [,, ] Bˆ Bˆ Ĝ Bˆ 9. OF : FB : DB r DO 5k and DF 8k 8 DF r r r 5 5 DG DO r [line ide of Δ/ lyn y v ] DA DF 8 r 5 DG 5 DA 8 R R S S () () () () []

18 Mathematic/P/Wikunde/V 8 DBE/07 QUESTION/VRAAG 0 B T 4 x R S A P 0. Tangent-chord theorem R 0.. Â Ŝ Ŝ + Â Bˆ Â + Bˆ AB ST Bˆ + Â + Bˆ [ opp ide/ eteenoor ye] [ext cyclic quad/ buite koordevh] [correp/ ooreenk ] R () (5) RTˆ S BÂS [ext cyclicquad/ buite koordevh] BÂS ABˆ T [ opp ide/ eteenoor ye] RTˆ S AB ST ABˆ T [correp/ ooreenk ] R (5)

19 Mathematic/P/Wikunde/V 9 DBE/ Bˆ x x + Tˆ 4 Bˆ + Bˆ Tˆ 4 Bˆ Bˆ Â Tˆ 4 Â Tˆ 4 [tan chord theorem/ raakl koordt] [correp/ ooreenk ; AB // ST] [tan chord theorem/ raakl koordt] Â [proven/ bewy in 0..] RTAP i a cyclicquadrilateral [line ubtend ] I 'n koordevierhoek [lyn onderpan e] R S R (5) () []

20 Mathematic/P/Wikunde/V 0 DBE/07 QUESTION/VRAAG. A D G H B C E F. Contr: On ide AB and AC of ABC, mark point G and H repectively uch that AG DE and AH DF. Draw GH/Merk punt G en H op y AB en AC van ABC onderkeidelik af odanig dat AG DE en AH DF. Trek GH. contruction/ kontrukie Proof/Bewy: AGH ΔDEF [,, ] S/R AĜH Ê Bˆ GH BC AG AH AB AC DE DF AB AC [Bˆ Ê, given/ gegee] [correp/ ooreenk ] [line ide of [contr/ kontrukie] / lyn ye v ] S S /R (6)

21 Mathematic/P/Wikunde/V DBE/07. B A P S T C R D..(a) AP PC [diag But TP PS [given/ gegee] AP TP PC PS AT SC m biect each other/ hoekl m halveer mekaar] S S OR S )..(b) In PSR Pˆ Pˆ Bˆ Ŝ and PBA : [vertically opp [ PSR PBA [,, ] In PSR Pˆ Pˆ Bˆ Â Ŝ Rˆ and PBA : [ PSR PBA [,, ] [um / regoort e] in ame egment/ e in die egment] [vertically opp in ame egment/ e in die egment] / om e ] / regoort e] S R S R R S R S R S (5) (5)

22 Mathematic/P/Wikunde/V DBE/07..(a)..(b) PR PS PA PB PR TR PS PA AD PB PR TR TP PA AD PD ΔRPT ΔAPD [ ] PR [given PA [PS TP;PB PD] [ide of TR ] AD Δ in prop/ ye v indie verhouding] Tˆ Dˆ [ ] ATRD i a cyclicquad [convere: ext of cyclicquad/ Omgekeerde buite v koordevh] S (all ratio) S R S R () () [8] TOTAL/TOTAAL: 50

A/CA Cum frequency values/ Kum.frek. waardes

A/CA Cum frequency values/ Kum.frek. waardes SEPTEMBER P/ V 06 MARKING/ NASIEN MEMORANDUM NOTE: If a candidate answers a question TWICE, only mark the FIRST attempt. If a candidate has crossed out an attempt of a question and not redone the question,