NATIONAL SENIOR CERTIFICATE GRADE/GRAAD 11

Transcription

1 NATIONAL SENIOR CERTIFICATE GRADE/GRAAD MATHEMATICS P/WISKUNDE V NOVEMBER 04 MEMORANDUM MARKS/PUNTE: 50 This memorandum consists of 6 pages. Hierdie memorandum bestaan uit 6 bladsye.

2 Mathematics/P/Wiskunde/V DBE/November 04 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, mark the crossed out version. Consistent accuracy applies in ALL aspects of the marking memorandum. Assuming answers/values in order to solve a problem is NOT acceptable. LET WEL: Indien 'n kandidaat 'n vraag twee keer beantwoord, merk slegs die eerste poging. Indien 'n kandidaat 'n antwoord doodgetrek het, maar nie oorgedoen het nie, merk die doodgetrekte antwoord. Volgehoue akkuraatheid geld in ALLE aspekte van die memorandum. Aanname van antwoorde/waardes om 'n probleem op te los, is ONaanvaarbaar. QUESTION/VRAAG.. IQR (A) Data of Supermarket A is skewed to the left/data van Supermark A is skeef na links. OR Negatively skewed/negatief skeef.. Range/Omvang (B) Supermarket A Supermarket A received 5 or more deliveries on more than 7 days whilst Supermarket B received 5 or more deliveries on less than 7 days/ Supermark A het op meer as 7 dae 5 of meer aflewerings ontvang terwyl Supermark B op minder as 7 dae soveel aflewerings ontvang het.. x 4,5 x + 9 4,5 4 x x 50 x () comment/kommentaar correct choice/ regte keuse reason/rede x x () () () [0]

3 Mathematics/P/Wiskunde/V DBE/November 04 QUESTION/VRAAG. 8 days/dae answ/antw (). days No. of days/ getal dae ,86% percentage/ persentasie Accept/Aanvaar,5 days which is/dae, wat gelyk is aan 44,64% OR Accept/Aanvaar days which is/dae, wat gelyk is aan 46,4%..4 Frequency Temperature, T, in degrees Celsius Frequency 9 T < T < 6 T < T < T < T < FREQUENCY POLYGON Average daily temperature (in degrees Celsius) and/en 6 9 and/en 5 4 and/en () anchored at / geanker by (8 ; 0) and/en ( ; 0) points at midpoints/ punte by middelpunte straight lines joining pts/ reguitlyne verbind punte all points plotted/ alle punte geplot (4) [0]

4 Mathematics/P/Wiskunde/V 4 DBE/November 04 QUESTION/VRAAG m AC m m (parallel lines/parallelle lyne) RNS AC 7 y x a ( a 9) NB + (7 9) (5 5 8a a 8a 40 0 ( a 0)( a + ) 0 a 0 a 7 tanα mac α 80 74,05 05,95 5) substitution into gradient formula/ subst in gradiëntformule 7 () gradients equal/ gradiënte gelyk equation/vgl () subst into distance formula/subst in afstandformule st form/st vorm factors/faktore answ/antw (4) 7 tanα α 05,95 tan β m BC β 56, θ 05,95 56, 49, 64 tan β β 56, θ 49,64 (5) []

5 Mathematics/P/Wiskunde/V 5 DBE/November 04 QUESTION/VRAAG 4 4. Perimeter/Omtrek ABCD 4 AD (sides of rhombus all equal/sye van ruit almal gelyk) AD ( + 8) + (9 6) 0,40 Perimeter/Omtrek ,6 4. m BD m AC mbd (diagonals of a rhombus/hoeklyne van ruit) m AC y y m( x x) y mx + c y 6 ( x + 8) 6 ( 8) + c y x + c 0 0 y x + y x + 4. At/By K: x x + 9x x + 0 0x 0 x y () K( ; ) 0 subst into distance form/ subst in afstandformule 0, ,6 m BD m AC subst into correct form/subt in korrekte formule answer/antw equate equations/ stel vgls gelyk x y (4) y x x + (x) 0 0x 0 x y () K( ; ) x y +0 ( y + 0) y 0 9 y + 0 y 0 0y 0 y x + 0 x K( ; ) subst of/subst van y x into x + y 0 x y subst of/subst van x y +0 into x y 0 y x

6 Mathematics/P/Wiskunde/V 6 DBE/November xb + x + B xb B( ; ) yb + 9 y B y B x value/waarde y value/waarde () 4.5 x B y B (by translation/deur translasie) B( ; ) m m m AB m AD AB AD 9 7 ABCD is not a square/ is nie 'n vierkant ( B ÂD 90 ) C(0 ; 0) BD ( ( )) + (9 ( )) 60 BD 4 0,65 AC ( 8 0) + (6 0) 60 AC 6 0 8,97 ABCD is not a square/is nie 'n vierkant (BD AC) x value/waarde y value/waarde () subst into gradient formula/subst in gradiëntformule gradient AB gradient AD /R (5) C(0 ; 0) subst into distance formula/subst in afstandformule 4 0 OR, OR 8,97 /R (5) [7]

7 Mathematics/P/Wiskunde/V 7 DBE/November 04 QUESTION/VRAAG cos 0 cos p sin 9 sin 67 cos p sin(60 x).tan( x) cos(80 + x).(sin A + cos ( sin x)( tan x) ( cos x)() sin x ( sin x) cos x cos x sin x cos x tan x cos x + ( + sin x) LHS ( + sin x).cos x A) cos x + + sin x + sin x ( + sin x).cos x + + sin x ( + sin x).cos x ( + sin x) ( + sin x).cos x cos x RHS 5.. Undefined if/ongedefinieerd as: sin x or cos x 0 x 90 ; sin x 4cos x tan x 4 x 75,96 + k. 80 x 7,98 + k.90 ; k Z reduction/herlei answer/antw () reduction/herlei co-ratio/ko-verh answ/antw ito/v p sin x tan x cos x sin x cos x tan x (6) numerator/teller denominator/ noemer multiplication/ vermenigvuldiging identity/identiteit fact/faktor numerator/teller tan x 4 75,96 7,98 k.90 k Z (5) () (5)

8 Mathematics/P/Wiskunde/V 8 DBE/November x + y r x + ( ) x x ± x (since P lies in the st quadrant/aangesien P in die ste kwadrant lê) sin PÔT PÔT 60 PÔT + α 90 α b sin( 0 ) 0 b 0 sin( 0 ) b 0 a cos( 0 ) 0 a 0 cos( 0 ) a 0 7, Q(0 ; 0) Q(7, ; 0) OQ 400 a² + b² 400 PQ² ² + 0² PQ² 404 ( a ) + ( b ) 404 a a + + b b a + 4 b 404 a b a b ( b ) + b 400 4b 400 b 00 b 0 (b < 0) a ( 0) a 0 Q( 0 ; 0) 404 subst x correct ratio/ korrekte verh 60 answer/antw correct ratio/ korrekte verh b 0 sin( 0 ) b 0 () correct ratio/ korrekte verh a 0 OR7, (5) subst into distance formula/subst in afstandformule subst into distance formula/subst in afstandformule a b b 0 0 a OR7, (5) []

9 Mathematics/P/Wiskunde/V 9 DBE/November 04 QUESTION/VRAAG 6 6. a b p x ( 90 ; 0 ) 90 < x < 0 between 90 and 0 /tussen 90 en 0 6. f ( x) cos (x) cos4x period/periode h ( x) cosx Minimum value/waarde move 45 to the left and then reflect about the x-axis/ skuif 45 na links en reflekteer dan om die x-as a b p 45 extreme values/ uiterste waardes correct notation/ korrekte notasie () extreme values/ uiterste waardes correct notation/ korrekte notasie () extreme values/ uiterste waardes correct notation/ korrekte notasie () cos 4x 90 () 45 left/links reflection x-axis/ refleksie om x-as () The graph of g must be moved 5 to the right/ Die grafiek van g moet 5 na regs beweeg. 5 right/regs () []

10 Mathematics/P/Wiskunde/V 0 DBE/November 04 QUESTION/VRAAG A ĈB AB 6 sin 0 sin0 AB,9m 7.. Area ABC:.AB.BC.sin B,9 6 sin ,5m Volume of pyramid area of base height 6,5 8 6,4m 7. radius of base/radius van basis: r tan 6 h r tan 6,45m slant height/skuinshoogte: S h cos6 S,47m cos6 SA π(tan 6 ) + π(tan 6 ) 7,9 m cos 6 Surface area of cone area of base + area of curved surface buite-opp van keël opp van basis + opp van geboë opp πr + πrs π (,45) + π (,45)(,47) 7,86m 0 subst into sine rule/ subst in sin-reël AB,9 subst into correct form/subst in korrekte formule 6,5 () correct formula/ korrekte formule subst answer/antw r tan6 h r tan 6, 45 l h cos6 S,47 cos6 subst into correct form/subst in korrekte formule answer/antw subst into correct form/subst in korrekte formule answer/antw (6) [4]

11 Mathematics/P/Wiskunde/V DBE/November 04 QUESTION/VRAAG 8 8. M 7 T O S P 8.. Mˆ + Mˆ 90 ( in semi circle/ in halfsirkel or/of diameter Mˆ 5 subtends right /midlyn onderspan regte / or/of ) /R () Ô 74 ( at centre/midpt at circum/by omtrek) /R 8.. Ô 06 ( s on a str line/ e op reguitlyn) Mˆ 5 ( at centre/midpt at circum/by omtrek) Ô Mˆ ( at centre/midpt at circum/by omtrek) Ô 06 /R () () Ô 74 ( at centre/midpt at circum/by omtrek) Ô 06 ( s on a str line/ e op reguitlyn) /R ()

12 Mathematics/P/Wiskunde/V DBE/November K O 06 T M Ô Tˆ 7 L ( s round a pt or s in a rev) ( e om 'n pt of e omw) Ô ( at centre/midpt at circum/by omtrek) 8.. KO OL (radii equal/radiusse gelyk) KM ML (Tans from common/same pt/rklyne van dies pt) KOLM is a kite (adj sides of quad are /aangr sye v vh ) 8.. O Kˆ M 90 (tan/rkl radius or/of tan/rkl diam/midlyn) O Lˆ M 90 (tan/rkl radius or/of tan/rkl diam/midlyn) O Kˆ M + OLˆ M 80 OKML cyc quad/kdvh (opp s quad supp or converse opp s of cyclic quad)/ (tos e vierh supp of omgek tos e van kdvh) 8..4 Mˆ + Ô 80 (opp s of cyclic quad/tos e van kdvh) Mˆ 74 R /R /R /R R R () [5]

13 Mathematics/P/Wiskunde/V DBE/November 04 QUESTION/VRAAG 9 K L 0 4 M N P 9. NKˆ M Kˆ 0 (equal chords; equal s) (gelyke koorde; gelyke e) R 9. Alternate s are equal/verwiss e gelyk R 9. NM LM (radii) NM KL (given/gegee) KL LM 9.4. MKˆ L Kˆ 0 ( s/e opp equal sides/to gelyke sye) KLˆ M Lˆ 40 ( s sum in / e som in ) KNˆ M Nˆ (opp s of cyclic quad/ tos e van kdvh) 9.4. KMˆ N Mˆ 80 ( ) 0 ( s sum in / e som in ) LMˆ N Mˆ + Mˆ LPˆN Pˆ + Pˆ 70 ( at centre at circumference) ( by midpt by omtrek) /R R R () () () (4) []

14 Mathematics/P/Wiskunde/V 4 DBE/November 04 QUESTION/VRAAG 0 0. Construction: Draw diameter AD and join DB. Konstruksie: Trek middellyn AD en verbind DB D construction/ konstruksie C B O P Proof/Bewys: A T B ÂT + DÂB 90 (tangent/rklyn radius) D Bˆ C + CBˆ A 90 ( in semi circle/halfsirkel) D ÂB + ADˆ B 90 ( s/e of/van ) B ÂT ADˆ B B ĈA ADˆ B ( s in same segment/ e in dies segment) B ÂT BĈA Construct diameter AD and join DC. Konstrueer middellyn AD en verbind DC. D R R /R construction/ konstruksie (6) C B O Proof/Bewys: B ÂT + DÂB 90 (tangent/rklyn radius) D ĈB + BĈA 90 ( in semi circle/halfsirkel) BĈA 90 DĈB DÂB 90 BÂT D ĈB DÂB ( s in same segment/ e in dies segment) B ÂT BĈA P A T R R /R (6)

15 Mathematics/P/Wiskunde/V 5 DBE/November 04 Construction: Draw radii OA and OB. Konstruksie: Trek radiusses OA en OB. construction/ konstruksie C B O P A T Proof/Bewys: O ÂB + BÂT 90 (tangent/rklyn radius) OÂB 90 BÂT OBˆ A 90 BÂT ( s opp sides/ e to sye) AÔB 80 (90 BÂT) ( s/e of/van ) A ÔB BÂT A ÔB Ĉ ( at centre at circumference/ B ÂT BĈA by midpt by omtrek) R /R (6) 0. P S Q R T

16 Mathematics/P/Wiskunde/V 6 DBE/November (a) Tan chord theorem/rklyn-koordstelling R 0..(b) s in same segment/ e in dieselfde segment R 0.. Rˆ Pˆ + Tˆ (ext of / buite v ) Pˆ Qˆ (from/vanaf 0..(b)) Qˆ Tˆ (from/vanaf 0..(a)) Qˆ + Qˆ Pˆ + Tˆ Qˆ + Qˆ Rˆ 0.. PQ PR (sides opp s/sye to e) PQR isosceles triangle/gelykbenige driehoek Rˆ Qˆ ( s in same segment/ e in dies segment) Tˆ Qˆ (from/vanaf 0..(a)) Rˆ Tˆ PR is a tangent to circle RST at R (converse tan chord th) PR is 'n rklyn aan sirkel RST by R (omgekeerde rkl-kdst) /R /R R () () (4) Pˆ 80 (Qˆ + Qˆ + Rˆ ) ( s/e of/van ) Rˆ Qˆ ( s in same segment/ e in dies segment) /R Qˆ Tˆ (from/vanaf 0..(a)) R Rˆ Tˆ TOTAL/TOTAAL: 50