NATIONAL SENIOR CERTIFICATE/NASIONALE SENIOR SERTIFIKAAT GRADE/GRAAD 1 JUNE/JUNIE 018 TECHNICAL MATHEMATICS P/TEGNIESE WISKUNDE V MARKING GUIDELINE/NASIENRIGLYN MARKS/PUNTE: 150 This marking guideline consists of 19 pages./hierdie nasienriglyn bestaan uit 19 bladsye.
TECHNICAL MATHEMATICS P/TEGNIESE WISKUNDE V (EC/JUNE 018) NOTE: Continuous accuracy (CA) applies in ALL aspects of the marking guideline. After two mistakes, do not apply CA marking. Assuming values/answers in order to solve a problem is unacceptable. LET WEL: Volgehoue akkuraatheid (CA) is deurgaans in ALLE aspekte van die nasienriglyn van toepassing. Na twee foute word CA nie toegepas nie. Aanvaarding van waardes/antwoorde om ʼn problem op te los, is onaanvaarbaar. Symbol/Simbool Explanation/Verduideling M Method/Metode MA Method with accuracy/metode met akkuraatheid A Accuracy/Akkuraat CA Consistent accuracy/deurlopende akkuraatheid S Simplification or Statement Vereenvoudiging of bewering R Reason/Rede SR Statement and correct reason/bewering en korrekte rede
(EC/JUNIE 018) TECHNICAL MATHEMATICS P/TEGNIESE WISKUNDE V 3 QUESTION/VRAAG 1 1.1 x x y y BC 10 1 1 8 0 0 6 100 MA formula/formule substitution/vervanging CA Answer/antwoord (3) 1. x x y y M ; 4 0 4 6 M ; M ; 1 1 1 A x-coordinate/koördinaat A y-coordinate/koördinaat () 1.3 y y1 mbc x x 1 0 6 8 0 6 8 3 4 0,75 MA formula and substitution/formule en vervanging CA Simplification/ vereenvoudiging ()
4 TECHNICAL MATHEMATICS P/TEGNIESE WISKUNDE V (EC/JUNE 018) 1.4 y y1 m x x1 y 0, 75 x y 0, 75x0,5 M Correct formula/korrekte formule A Correct substitution pt D or E/Korrekte vervanging pt D of E CA Simplification/vereenvoudiging 1.5 inclination of DE = inclination of BC ( lines; same gradient) m DE m BC tan incl DE 0,75 incl DE 36,87 CA Standard form/ standaardvorm (4) m m M DE BC A tan incl DE 0,75 CA incl DE 36,87 m AE 4 6 5 4 0 5 tan incl AE incl AE 180 68,198... 111,8 AED ˆ 111,8 36,87 74,93 5 MA mae CA incl AE 111,8 CA AED ˆ 74,93 (6) [17]
(EC/JUNIE 018) TECHNICAL MATHEMATICS P/TEGNIESE WISKUNDE V 5 QUESTION/VRAAG.1.1 r x y 5 1 169 Equation of circle/vergelyking van sirkel: x y 169 M Correct formula/ korrekte formule A Correct substitution/korrekte vervanging CA Value of r /waarde van r (3)
6 TECHNICAL MATHEMATICS P/TEGNIESE WISKUNDE V (EC/JUNE 018).1. 0 1 1 MA Gradient of m radius 0 5 5 radius/gradiënt van 5 radius mtangent 0,4 1 CA Gradient of Equation of tangent:/vergelyking van raaklyn tangent/gradiënt van y y m x x raaklyn 1 1 y1 0, 4 x 5 y 0, 4x14,1 A Subst. P into equation/vervanging van P in vergelyking CA Equation of line/vergelyking van lyn OR/OF m radius = 0+1 = 1 0+5 5 m tangent/raaklyn = = 5 1 y y m x x 1 1 5 y1 x 5 1 5 5 y1 x 1 1 5 169 y x 1 1 OR/OF x x y y r 1 1 5x1y 169 5 169 y x 1 1 CA Gradient of tangent/ Gradiënt van raaklyn MA Gradient of radius / Gradiënt van radius CA Gradient of tangent/raaklyn A Subst. P into equation/vervanging van P in vergelyking M Correct formula / Korrekte formule A Correct substitution of P/ Korrekte vervanging van P A Correct substitution of grad./ Korrekte vervanging van grad. CA Equation of line/vergelyking van lyn (4)
(EC/JUNIE 018) TECHNICAL MATHEMATICS P/TEGNIESE WISKUNDE V 7. 4x 9y 36 x y 9 4 x y 3 1 1 M Rewrite the equation/ Herskryf die vergelyking, RHS = 1 M Rewrite into/herskryf as a & b CA x-intercepts/afsnitte CA y-intercepts/afsnitte CA shape clearly showing the horizontal major axis/vorm wys duidelik die horisontale hoofas (5) [1]
8 TECHNICAL MATHEMATICS P/TEGNIESE WISKUNDE V (EC/JUNE 018) QUESTION/VRAAG 3 3.1 3.1.1 13sin 1 1 sin 13 x x 5 y = 1 P(-5;1) 13 1 (Pyth) 3.1. 1 13 tan sec 5 5 5 5 5 M sin θ the subject/die onderwerp M Pythagoras A x-value/waarde A y-value/waarde (4) CA correct value/korrekte waarde: tan CA correct value/korrekte waarde: sec CA answer/antwoord (3) 3.1.3 1 sin 13 1 13 180 67,38... 1 Ref sin 67,38... 11,6 M 180 - CA 11,6 ()
(EC/JUNIE 018) TECHNICAL MATHEMATICS P/TEGNIESE WISKUNDE V 9 3. tan 180 1sin cos 180 sin 360 tancos cos sin sin cos cos 1 cos sin sin 1 sin A tan A cos θ A cos A sin A A sin cos cos sin 1 CA sin (7) 3.3 RHS = sin 1 cos sin LHS = sin 1 cos 1cos sin sin 1 cos 1 cos cos sin sin 1 cos 1cos 1 sin 1 cos cos sin 1 cos 1 cos sin 1 cos sin RHS A RHS A denominator/noemer A numerator/teller CA expansion/ uitbreiding A cos sin 1 CA factorising/ faktorisering (6) []
10 TECHNICAL MATHEMATICS P/TEGNIESE WISKUNDE V (EC/JUNE 018) QUESTION/VRAAG 4 S R 4.1 9, 48 sin 0,606... 15,64 37,3 A trig ratio/verhouding CA value for θ /waarde vir θ () 4. 1 Area PSR = PS SQsin PSQ ˆ 0,5 4,9515, 64 sin 5, 69 30, 79 cm A area-rule/area-reël CA value for PSQ ˆ /waarde vir PSQ ˆ A correct substitution/ vervanging CA Area of DEF (4) 4.3 PSQ ˆ 90 37, 31 5, 69 PQ PS SQ PS SQ cos PSQ PQ 4,95 15, 64 4,9515, 64 cos 5, 69 175,61... 175,61... 13,4 m ˆ S A cos-rule/reël A correct substitution/ vervanging CA simplification/ vereenvoudiging CA value of/waarde van PQ (5) [11]
(EC/JUNIE 018) TECHNICAL MATHEMATICS P/TEGNIESE WISKUNDE V 11 QUESTION/VRAAG 5 5.1. A x-int s 10 and/en 300 A TP (30 ;1) A TP (10 ;-1) A shape/vorm (4) 5..1 Amplitude of f = A amplitude = (1) 5.. Range g: 1 y 1 A y- values/waardes CA interval () 5..3 x 90 CA (1) 5..4 f ( x) g( x) x10 ;360 OR/OF 10 x 360 CA x = 10 CA x = 360 A notation/notasie (3) [11]
1 TECHNICAL MATHEMATICS P/TEGNIESE WISKUNDE V (EC/JUNE 018) QUESTION/VRAAG 6 6.1 Proportionally/Eweredig (1) 6. Let QS = x S SC = 4 x SC = 4 x CS CR SA RB 4 x 1 x 3 1 3x x 10 4x x,5 (line to one side of ) S R S SA = x + A Setup equation/stel vergelyking op QS =,5 units/eenhede CA Simplify equation/vereenvoudig vergelyking CA value / waarde x (7) [8]
(EC/JUNIE 018) TECHNICAL MATHEMATICS P/TEGNIESE WISKUNDE V 13 QUESTION/VRAAG 7 7.1 7.1.1 AB = 8,36 cm (sides opp = s) BD = 5,91 cm S R S (3) 7.1. AB 8,36 1,41 BD 5,91 BC 8,36 1,41 DC 5,91 AC 11,8 1,41 BC 8,36 S S S (3) 7.1.3 triangles are similar/driehoeke is gelykvorming S similar triangles / gelykvormige driehoeke (1) 7. 7..1(a) R ˆ T ˆ 56 ( s in same segment) 3 S R ()
14 TECHNICAL MATHEMATICS P/TEGNIESE WISKUNDE V (EC/JUNE 018) 7..1(b) R ˆ T ˆ 56 (tan-chord) /(tan-koord) S R 4 1 () 7.. Aˆ Mˆ ( s in same segment) Aˆ Rˆ (tan-chord) 1 S S () 7..3 AP PR s /// MP PT 8, 9 11,11 MP 1,018, 9 8, 93,7 MP 11,11,78 cm S S S PT = 1,01 8,9 = 3,7 S S value of / waarde van MP (5) [18]
(EC/JUNIE 018) TECHNICAL MATHEMATICS P/TEGNIESE WISKUNDE V 15 QUESTION/VRAAG 8 8.1 PTR ˆ 90 (line from centre to midpt of chord) TR PR PT 9,47 8,43 18, 616 TR 4,31... WR 8,63 units (Pythagoras) S R S Pythagoras A Simplification/ Vereenvoudiging CA Value of/waarde van TR CA Value of/waarde van WR (6) 8. ABO ˆ 40 (radii; equal s opp equal sides) AOB ˆ 100 (int s of ) Dˆ 50 ( at centre = at circumf.) x 130 (opp s of cyclic quad) S S S R S R (6) [1]
16 TECHNICAL MATHEMATICS P/TEGNIESE WISKUNDE V (EC/JUNE 018) QUESTION/VRAAG 9 9.1.1 108,5 108,5 180 17 360 1,89 radians A Multiply with factor/maal met faktor CA Answer/antwoord () 9.1. Radius = 10,84 5,4 units /eenhede A Answer/antwoord (1) 9.1.3 s r d 5, 41,89 10,4 units A correct/korrekte formula CA Substitution/vervanging CA answer/antwoord (3) 9.1.4 r Area = 5, 4 1,89 7,76 units 9. 4h 4dh x 0 x d h 4h 8,76 3,15 43,15 9,4 units A Correct formula / korrekte formule CA Substitution/ vervanging CA answer/antwoord (3) A Correct formula / korrekte formule A Making d the subject/maak d die onderwerp A Substitution/ vervanging CA answer/antwoord (4)
(EC/JUNIE 018) TECHNICAL MATHEMATICS P/TEGNIESE WISKUNDE V 17 9.3 o o 10 18 6 14 19 0,8 0 6 109,8 1 n AT a o o3 o4... on 1 658,8 cm A Correct formula / korrekte formule A Convert to cm /herlei na cm A Substitution/ vervanging CA 109,8 CA 658,8 (5) [18]
18 TECHNICAL MATHEMATICS P/TEGNIESE WISKUNDE V (EC/JUNE 018) QUESTION/VRAAG 10 10.1.1 10.1. 10.1.3 10.1.4 n 15 rev/s n 15 30 94,5 rad/s d 570mm 0,57m Dn 0,5715 6,86 m/s st 6,86... 60 33 m t 30 0,3 =9 rad A Correct formula / korrekte formule A Substitution/vervanging CA value of/waarde van ω (3) A Convert diameter to metre/herlei middelyn na meter A Correct formula / korrekte formule A Substitution/ vervanging CA value of/waarde van ν (4) A Correct formula / korrekte formule A Multiply t =, with 60 / Vermenigvuldig t =, met 60 CA value of/waarde van s CA rounded answer /afgeronde antwoord (4) A Correct/korrekte formula A Substitution/vervanging CA value of/waarde van θ (3)
(EC/JUNIE 018) TECHNICAL MATHEMATICS P/TEGNIESE WISKUNDE V 19 10. Vol of cylinder = rh 50 100 6500 196350 mm 1 Vol of hemisphere = Vol of sphere 1 4 3 r 3 3 5 3 3150 3 3 375 mm Total/Totale volume = 9 075 mm 3 3 A Formula of cylinder/formule vir silinder A Divide diameter by / deel middellyn deur A Substitution/ vervanging CA Vol of cylinder/silinder A Formula of hemisphere/formule vir hemisfeer A Substitution/vervanging CA Vol of hemisphere/hemisfeer CA Adding the Volumes/tel die volumes bymekaar (7) [1] TOTAL/TOTAAL: 150