SEPTEMBER 016 MATHEMATICS PAPER 1 / WISKUNDE VRAESTEL 1 MEMORANDUM NOTE: If a candidate answers a question TWICE, mark only the first one. Consistent accuracy applies in ALL aspects of the marking memorandum. LET WEL: Indien ʼn kandidaat ʼn vraag TWEE keer beantwoord, merk slegs die eerste poging. Volgehoue akkuraatheid is geurgans op ALLE aspekte van die memorandum van toepassing. QUESTION/ VRAAG 1 () 1.1.1 x(5x + ) = 0 x = 0 or/ of x = 5 1.1. x x 4 = 0 Each root/ Elke wortel Std form/ vorm x = b ± b 4ac a x = ( ) ± ( ) 4( 4) = ± 41 4 x = 0,85 or/ of x =,5 1.1. x x 6 6 x x 1 0 (x + )(x 4) 0 0 0 4 + + x < or/ of x > 4 1. x = y +.(1) Subst (1) into : x + xy y = 0 (y + ) + y(y + ) y = 0 y + 6y + 9 + y(y + ) y = 0 y + 6y + 9 + y + y y = 0 9y + 9 = 0 OR/OF y = 1 x = Substitution/ substitusie Each root/ elke wortel x = 0,85 or/ of x =,5 (-1, if incorrect rounding) (4) std form/ vorm factors/ faktore both critical values/ beide kritieke waardes notation / notasie x subject/ onderwerp subst in simplify / vereenvoudig value of y / Waarde van y value of x/ Waarde van x (4) 1
y = x.(1) y subject/ onderwerp Subst (1) into : 1. x + xy y = 0 x + x(x ) (x ) = 0 x + x x x + 1x 18 = 0 9x 18 = 0 x = y = 1 ± 1 k x = subst in simplify / vereenvoudig value of x/ Waarde van x value of y / Waarde van y = 1 k 0 0 k 1 k 1 If k = 6, is a perfect square and roots rational/ As k = 6 sal n volkome vierkant wees en wortels Rasionaal 1.4 If graphs intersect/ as grafieke sny: f(x) = g(x) x x = x + a x 4x a = 0 If graphs do not intersect/ As grafieke nie sny nie: < 0 ( 4) 4(1)( a) < 0 16 + 1 + 4a < 0 4a < 8 a < 7 value/ waarde k 1 deriving k = 6 / afleiding k = 6 Answer only/ Slegs antwoord = FULL MARKS/VOLPUNTE equating f and g/ stel f en g gelyk < 0 Subst. Simplify/ Vereenv answer/ antw. () [] QUESTION/ VRAAG (15).1.1 1 ; ; 1 Answer/ Antw. (1).1. Arithmetic sequence/ Rekenkundige ry; d = 1 or /of 1 is subtracted each time/ 1 word elke keer afgetrek.1. T n = a + (n 1)d ; a = 1 ; d = 1 ; Tn = - 44,5 ; n =? 44 1 = 1 + (n 1) ( 1 ) 4 1 n = 44 1 AS/ RR; d = 1 Subst into/ in T n T n = - 44,5 1 97 n = n = 97 Answer/ Antw ()
..1 18 + 6 + + GS with r = 1 / MR met r = 1 Thus no negative term / Dus geen term negatief... T 10 = ar n 1 ; a = 18; r = 1 ; n = 10 GS with r = 1 / MR met r = 1 no negative term / geen term negatief = 18 ( 1 )9 Subst into/ in T n.. = 187 S S 10 ; a = 18; r = 1 = a 1 r a(1 rn ) 1 r = 18 18 [1 ( 1 )10 ] 1 1 1 1 ; n = 10 Answer/ Antw Correct sum formulae Subst into sum formulae/ Subst in somformules a = 18; r = 1 ; n = 10 Simplify/ Vereenv = 7 7 [1 ( 1 ) 10] = 4,57 10 4 answer/ antw [15] QUESTION/ VRAAG (11).1 5; 7 65 ; AS/RR: a = -; n = ; d = - ; l = - 65 n =. S n = n [a + (n 1)d] OR/OF S n = n [a + l] S = [( ) + ( 1)( )] S = = [ 6 6] = -1088 [( ) + ( 65)]..1 6 ; 5 + x ; 6 ; 6x x + 5 6; 6 (5 + x); 6x ( 6) x 1; x 11; 6x + 6 ; 1 st diff/ verskil x 11 (x 1) = 6x + 6 ( x 11) ; nd diff/verskil x 10 = 7x + 17 x =.. Sequence/ Ry: 6 ; ; 6 ; 18; 4 ; 8 ; 1 1 st diff/ verskil 4 ; 4 nd diff/ verskil a = 4 a = - a + b = 4 ( ) + b = 4 b = a + b + c = 6 + + c = 6 c = 6 T n = n + n + 6 Subst in sum formula/ Subst in somformule answer/ antw () 1 st diff/ verskille nd diff/verskille nd diff/verskille = Simplify/ Vereenv. (4) a = - b = c = 6 Equation/ Vergelyking (4) [11]
QUESTION/ VRAAG 4 (18) 4.1 OC = 6 units/ eenhede A; B: x-intercepts/ afsnitte: Let/ Stel y = 0 x 4x + 6 = 0 x + x = 0 (x + )(x 1) = 0 x = OR/OF x = 1 A( ; 0) and/ en B(1; 0) AB = 4 units/ eenhede 4. x = b a = [ 4 ( ) ] = 1 OR/OF f (x) = 4x 4 = 0 x = 1 4. Subst. x = 1 in f(x) ST = ( 1) 4( 1) + 6 = 8 units/ eenhede 4.4 m AC = y y 1 x x 1 = 0 6 0 = 4.5 m g = ; // lines/ lyne but/ maar m g = f (x) = 4x 4 = x = OC = 6 Let/ Stel y = 0 Factors/ Faktore Both x-values/ Beide x-waardes answer/ antw Subst x = 1 OR/OF Derivative/ Afgeleide = 0 x = 1 Subst. x = -1 answer/ antw Subst. in m answer/ antw m g = m g = f (x) 4x 4 = x = D( ; 7 1 ) y = ( ) 4 ( ) + 6 = 7 1 OR/OF Let eq. of tangent be/ Laat vgl. van raakl: y = mx + k m g = ; // lines/ lyne y = x + k at D: f(x) = g(x) x 4x + 6 = x + k x 6x + 6 k = 0.(1) For tangent/ Vir raaklyn: = 0 ( 6) 4( )(6 k) = 0 6 + 48 8k = 0 k = 10,5 x 6x + 6 10,5 = 0 Subst k in (1) x + 6x + 4,5 = 0 or/ of 4x + 1x + 9 = 0 (x + ) = 0 x = y = 7 1 m g = eqn/ vgl: x 6x + 6 k = 0 k = 10,5 x = D( ; 7 1 ) y = ( ) + 10,5 = 7 1 4.6 a = 1 ; the axis of symmetry/ die simmetrie-as OR/OF : f(a + t) = f(a t) (a + t) 4(a + t) + 6 = (a t) 4(a t) + 6 a 4at t 4a 4t + 6 = a + 4at t 4a + 4t + 6 8at + 8t = 0 8t(a + 1) = 0 t = o or/of a = 1 y = 7 1 a = 1 Subst. a = 1 4
[18] QUESTION/ VRAAG 5 (10) 5.1 x = 1 answer/ antw (1) 5. x -int/afsn: Let/ Stel y = 0 +x x 1 = 0 x = y = 0 x = A( ; 0) 5. y -int/afsn: Let/ Stel x = 0 +0 0 1 = y y = B(0; ) Area/ Opp AOB = 1 AO OB = 1 = units / eenh B(0; ) Subst. in Area formula answer/ antw () 5.4 f(x) = + x x 1 = x 1 + x 1 +x = x 1 x 1 x 1 x 1 x 1 = x 1 + 1 Simplify to/ Vereenv. tot x 1 x 1 x 1 5.5 (; 1) (CA from/ vanaf 5. - shift units to the right/ skuif eenh regs) 1 [10] QUESTION/ VRAAG 6 (8) 6.1.1 y > 1; y R y > 1; y R 6.1. g(x) = x g 1 : y = log x 6..1 k(x) = x ; x 0 k(x) = x g(x) = x y = log x x 0 6.. (0; 0) OR/OF origin/ oorsprong Answer/ Antw [8] 5
QUESTION/ VRAAG 7 (14) 7.1 A = P(1 i) n ; P = R18 000; A = R45 500; n = 6; i =? 45 500 = 18 000(1 - i) 6 6 45 500 = 1 i 18 000 6 45 500 i = 1 18 000 i = 0,1584 = 15,8% p.a/ p.j 7..1 A = P(1 + i) n ; A =?; P = 50 000; n = 1y = 48 quarters; i = 9,5% p.a = 9,5 per quarter/ per kwartaal 400 A = 50 000(1 + 9,5 400 )48 = R 771 4,67 7.. Home loan/ Huislening = 90 000-771 4,67 = R 148 656, 7.. P v = x[1 (1+i) n ] i = 10, 100 ; P v = 148 656, ; i = 10,% p.a p/m ; n = 0 y = 40 m 10, x [1 (1+ 100 148 656, = ) 40 ] 10, 100 x = R 1 16, 87 OR/OF F v = x[(1 + i)n 1] i 148 656, (1 + 10, 100 )40 = x = R 1 16, 87 10, x [(1+ 100 )40 1] 10, 100 7..4 Interest/ Rente = Amount paid/ bedrag betaal- original loan/ lening = 1 16,87 40-148 656, = R 90 67, 47 Subst. of P = 18 000 into depreciation formula Subst. of 45 500 and/ en n = 6 i the subject/ die onderwerp Answer/ Antw Subst i = 9,5 400 = 0,075 Subst n = 48 in correct formula Answer/ Antw Answer/ Antw Subst into correct formula Subst i = 10, 100 Subst n = 40 Answer/ Antw Calculation/ berekening Answer/ Antw (4) () (1) (4) [14] 6
QUESTION/ VRAAG 8 (15) 8.1 f f(x + h) f(x) (x) h 0 h x+h x h 0 h h 0 h 0 x (x+h) x(x+h) h x x h x +xh h h 1 h 0 x(x+h) h h 0 x +xh f (x) = h 0 x +x(0) x 8..1 h(x) = x 4 x + x 8.. h(x) = x 4x 1 + x 1 h (x) = 6x + 4x + x 1 D x ( 8x 7 x ) = D x ( (x )(4x + 6x + 9) ) x = D x (4x + 6x + 9) = 8x + 6 8. m tan = f (x) = x = (x + ) Which is always/ wat altyd < 0 Subst f(x + h) and/ en f(x) Common denominator/ Gemene noemer Simplify/ Vereenv Subst. h = 0 Answer/ Antw Exponential form/ eksponensiële vorm Answer / Antw (-1 for notation in question / -1 vir notasie in vraag) (4) Simplify fraction/ Vereenv breuk Ans. / Antw () m tan = f (x) = x (x + ) Explain/ Verduidelik () [15] 7
QUESTION/ VRAAG 9 (15) y B A x C D 9.1 f(x) = x + x x 1 (x 1)(x + x + 1) = 0 (x 1)(x + 1)(x + 1) = 0 x = 1 or/of x = 1 AB = units / eenhede 9. f (x) = 0 x + x 1 = 0 (x 1)(x + 1) = 0 x = 1 or / of x = 1 y = 7 (x - 1) (x + x + 1) (x + 1)(x + 1) Both roots/ Beide wortels AB = Derivative/ Afgeleide = 0 Values of x/ Waardes van x D ( 1 ; 7 ) 9. f"(x) = 0 6x + = 0 x = 1 h"(x) < 0 h"(x) > 0 1 Coordinates of D/ Koördinate van D () Linking second derivative with concavity / Verbind tweede afgeleide met konkawiteit 6x + = 0 explanation / verduideliking The concavity changes at x = 1 Die konkawiteit verander by x = 1 () 9.4.1 x > 1 Answer/ Antw. (1) 9.4. x < 1 or/of 1 < x < 1 x < 1 1 < x < 1 Notation (-1) / Notasie (-1) () [15] 8
QUESTION/ VRAAG 10 (7) 10.1 t = 10. D = + 1 1 4 = + = m Answer/Antw. (1) D (t) = t 4 t derivative / afgeleide D () = 4 () =,75 m/h 10. D (t) = 0 t 4 t = 0 4: t 4t = 0 t(t 4) = 0 t = 0 or/of t = 1 1 = 1h 0 m Time/Tyd 10: 0 Answer/Antwoord D (t) = 0 factors / faktore both answers / beide antwoorde Final answer / Finale antw. (4) [7] QUESTION/ VRAAG 11 (14) 11.1.1 P(female and/ green eyes) = 147 540 = 49 180 = 7,% 147 49 or / of or/ of 7,% 540 180 P(vrou en/ groen oë) = 147 540 = 49 180 = 7,% Answer/ Antw 11.1. For events to be independent: P(female and/ green eyes) = P(green eyes) P(female) P(green eyes) P(female) = 0 540 40 540 = 81 = 0,7 P(female and/ green eyes) = 147 540 = 49 180 = 0,7 P(green eyes) = 0/540 P(female) = 40/540 0,7 P(female and/ green eyes) = P(green eyes) P(female) Events are independent and the learner is correct. P(female and/ green eyes) = 0,7 Deduction Vir gebeure om onafhanklik te wees: P(vrou en/ groen oë) = P(groen oë) P(vrou) P(groen oë) P(vrou) = 0 540 40 540 = 81 = 0,7 P(vrou en/ groen oë) = 147 540 = 49 180 = 0,7 9 P(groen oë) = 0/540 P(vrou) = 40/540 0,7
P(vrou en/ groen oë) = P(groen oë) P(vrou) Gebeure is onafhanklik en die leerder is korrek. Alternative/Alternatief: For events to be independent: P(male and/ green eyes) = P(green eyes) P(male) P(green eyes) P(male) = 0 540 00 540 = 55 16 = 0,4 P(male and/ green eyes) = 18 540 = 61 180 = 0,4 P(male and/ green eyes) = P(green eyes) P(male) Events are independent and the learner is correct. Vir gebeure om onafhanklik te wees: P(manlik en/ groen oë) = P(groen oë) P(manlik) P(groen oë) P(manlik) = 0 540 00 540 = 55 16 = 0,4 P(manlik en/ groen oë) = 18 540 = 61 180 = 0,4 P(vrou en/ groen oë) = 0,7 Afleiding P(green eyes) = 0/540 P(male) = 00/540 0,4 P(male and/ green eyes) = 0,4 Deduction P(groen oë) = 0/540 P(manlik) = 00/540 0,4 11..1 11.. P(manlik en/ groen oë) = P(groen oë) P(manlik) P(manlik en/ groen oë) = 0,4 Gebeure is onafhanklik en die leerder is korrek. Afleiding 10! n(s) =!!! = 15100 Numerator/ Teller Denominator/ Noemer Answer / Antw n(e) = 8!!! = 10080 P(E) = 10080 15100 = 1 15 8!!! () 10080 P(E) = 10080 15100 Answer / Antw (4) [15] TOTAL/ TOTAAL: 150 10