Mathematics Grade 11 P1 National Exemplar November 2015 QP

basic education Department: Basic Education REPUBLIC OF SOUTH AFRICA NASIONALE SENIOR SERTIFIKAAT GRAAD 11 genase SEP RRRRSR SERRE PERE R RRR RR REESE SSR RR SSSR u WISKUNDE V1 n rl a i NOVEMBER 2015 7 i % PUNTE: 150 TYD; 3 uur Hierdie vraestel bestaan uit 8 bladsye. Kopiereg voorbehou Blaai om asseblief

Downloaded from hlayiso.com Wiskunde/V1 2 DBE/November 2015 KABV - Graad 11 INSTRUKSIES EN INLIGTING Lees die volgende instruksies aandagtig deur voordat jy die vrae beantwoord. 1. 2. Hierdie vraestel bestaan uit 9 vrae. Beantwoord AL die vrae. Nommer die antwoorde korrek volgens die nommeringstelsel wat in hierdie vraestel gebruik is. Dui ALLE berekeninge, diagramme, grafieke ensovoorts wat jy gebruik het om jou antwoorde te bepaal, duidelik aan. Volpunte sal nie noodwendig aan slegs antwoorde toegeken word nie. Jy mag 'n goedgekeurde wetenskaplike sakrekenaar (nieprogrammeerbaar en niegrafies) gebruik, tensy anders aangedui. Indien nodig, rond antwoorde tot TWEE desimale plekke af, tensy anders aangedui. Diagramme is NIE noodwendig volgens skaal geteken NIE. Skryf netjies en leesbaar. Kopiereg voorbchox CUI AAA EA Biealom assebief

Downloaded from hlayiso.com Wiskunde/V1 3 DBE/November 2015 KABV — Graad 11 VRAAG 1 1.1 Los op vir x in elk van die volgende: 111 x +x-12=0 (3) 1.1.2 V2x+1=x-1 (5) 1.1.3 ox 977 (4) 1.14 x? -2x-8 <0 @) 1.2 Gegee: f(x)=5x?+6x-7 1.2.1 Los op vir x as f(x)=0 (korrek tot TWEE desimale plekke). (4) 1.2.2 Vervolgens, of andersins, bereken die waarde van ad waarvoor 5x* +6x-—d=0 gelyke wortels het. G) 1.3 Los gelyktydig vir x en y op: x-2y=-3 en x=20 (6) [28] VRAAG 2 2.1 Vereenvoudig, sonder om 'n sakrekenaar te gebruik: gree qnel 2.1.1 —— (3) 8 2.1.2 \x+¥2x—Lyx—V2x-1 4) 2.2 Gegee: P= = += x+2 3 2.2.1 Vir watter waarde(s) van x sal P 'nreéle getal wees? (2) 2.2.2 Toon aan dat P rasionaal is as x = 3. (2) 2.3 Bereken die som van die syfers van 2”'°x 5719. (4) [15] Kopieres voorbehou FALTER EEN A GAUTENG Blaai om asseblief

Downloaded from hlayiso.com Wiskunde/V1 4 DBE/November 2015 KABV - Graad 11 VRAAG 3 3.1 Die lineére patroon word gegee: 5 ;-2;-9;...;-289 3.1.1 Skryf die eerste gemene verskil neer. qd) 3.1.2 Skryf die waarde van 7, neer. qd) 3.1.3 Bereken die getal terme in die patroon. (3) 3.2 Die verskil tussen opeenvolgende terme in 'n lineére patroon is 3 en die 20°° term is gelyk aan 64 (met ander woorde T,, = 64). 3.2.1 Bepaal die waarde van 7,,. qd) 3.2.2 Watter term in die patroon sal gelyk wees aan 37; —2? (4) 3.3 Beskou die kwadratiese patroon: 5 ; 12 ;29;56;... 3.3.1 Skryf die VOLGENDE TWEE terme van die patroon neer. (2) 3.3.2 Bewys dat die eerste verskille van hierdie patroon altyd onewe sal wees. (3) [15] VRAAG 4 4.1 Beskou die kwadratiese patroon: 3;5;8;12;... Bepaal die waarde van T,,. (6) 4.2 'n Sekere kwadratiese patroon het die volgende eienskappe: e T, =p e T=18 © §67,=47, e 1,-T,=10 Bepaal die waarde van p. (6) [12} Kopiereg voorbehou Blaai om asseblief

Downloaded from hlayiso.com Wiskunde/V1 5 DBE/November 2015 KABV - Graad 11 VRAAG 5 5.1 Die skets hieronder toon die grafiek van f(x) = — 2. x- A is die snypunt van die asimptote van f y . a | \ \ \ i I Sf i i i 1 —_— 0 { x eee yg ee cee meee f | | | \ \ | | 5.1.1 Skryf die kodrdinate van A neer. Q) 5.1.2 Bepaal die koérdinate van die x- en y-afsnitte van f° (5) 5.1.3 Skryf 'n vergelyking van die simmetrie-as van f neer wat 'n negatiewe gradiént het. (2) 5.1.4 Vervolgens, of andersins, bepaal die kodrdinate van 'n punt wat op f in die vierde kwadrant 1é en wat die naaste aan punt A is. (5) 5.1.5 Die grafiek van f word om die x-as gereflekteer om die grafiek van g te vorm. Skryf die vergelyking van g neer, indie vorm y=... (2) 5.2 Gegee: (x)= 42" }41 5.2.1 Bepaal die kodrdinate van die y-afsnit van h. (2) 5.2.2 Verduidelik waarom h geen x-afsnit het nie. (2) 5.2.3 Teken 'n sketsgrafiek van f en dui duidelik alle asimptote, afsnitte met die asse en minstens een ander punt op / aan. (3) 5.2.4 Beskryf die transformasie van h na g as g(x)= 4(2* + 2). 8] Kopiereg voorbehou Blaai om asseblief

Downloaded from hlayiso.com Wiskunde/V1 6 DBE/November 2015 KABV - Graad 11 VRAAG 6 Die skets hieronder verteenwoordig die grafieke van twee parabole, f en g. 1a xy=—x°-8 f() 3 Die draaipunt van g is C (2;9) endie y-afsnit van g is A(0;5). B en D isdie x-afsnitte van f en g onderskeidelik. ¥ C2;9) AQ; 5) f (6) x g 6.1 Toon aan dat g(x)=-x? +4x+5. (4) 6.2 Bereken die gemiddelde gradiént van g tussen A en C. (2) 6.3 Bereken die lengte van BD. (5) 6.4 Gebruik die grafieke om vir x op te los, as: 6.4.1 f(xy20 (2) 6.4.2 f en g albei streng stygend is (2) [15] Kopiereg voorbehou Blaai om asseblief

Downloaded from hlayiso.com Wiskunde/V 1 7 DBE/November 2015 KABV - Graad 11 VRAAG 7 Die skets hieronder toon die grafieke van f(x)=2x+3 en g(x)= -2x7 +14x 4k. C is enige punt op f en D enige punt op g, sodanig dat CD ewewydig aan die y-as is. k is 'n waarde sodanig dat C bokant D lé. 4 y WA x 7A Skryf 'n vereenvoudigde uitdrukking in terme van x en k_ vir die lengte van CD neer. 7.2 As die minimum lengte van CD 5 is, bereken die waarde van k. VRAAG 8 8.1 'n Skool koop tablette teen 'n totale koste van R140000. As die gemiddelde inflasiekoers 6,1% per jaar oor die volgende 4 jaar is, bepaal die koste om hierdie tablette oor 4 jaar te vervang. 8.2 'n Belegging verdien rente teen 'n koers van 7% per jaar, halfjaarliks saamgestel. Bereken die effektiewe jaarlikse rentekoers op hierdie belegging. 8.3 'n Spaarrekening is met 'n aanvanklike deposito van R24000 oopgemaak. Agtien maande later is R7000 uit die rekening onttrek. Bereken hoeveel geld in die spaarrekening aan die einde van 4 jaar sal wees as die rentekoers 10,5% pjj., maandeliks saamgestel, was. 8.4 'n Kar wat R198 000 kos, het 'n boekwaarde van R102 755,34 na 3 jaar. As die waarde van die kar teen r% p.j. op 'n verminderde saldo verminder, bereken r. (3) (4) [7] 6) @) (5) 6) [16] Kopiereg voorbehou Blaai om asseblief

Downloaded from hlayiso.com Wiskunde/V1 8 DBE/November 2015 KABV — Graad 11 VRAAG 9 91 Gegee: P(A) = 0,6 P(B) = 0,3 P(A of B)=0,8 waar A en B twee verskillende gebeurtenisse is Is die gebeurtenisse A en B onderling uitsluitend? Motiveer jou antwoord met toepaslike berekeninge en/of 'n diagram. (4) 9.2 Die tabel hieronder toon data oor die maandelikse inkomste van werkende persone in twee woongebiede. Verteenwoordigende steekproewe is in die versameling van die data gebruik. MAANDELIKSE INKOMSTE (IN RAND) GEBIED 1 GEBIED 2 TOTAAL x<3200 500 460 960 3200 <x <25 600 1 182 340 1522 x = 25600 150 14 164 Totaal 1 832 814 2 646 9.21 Wat is die waarskynlikheid dat 'n persoon wat willekeurig uit die volledige steekproef gekies word: (a) Uit Gebied 1 sal kom Q) (b) Uit Gebied 2 sal kom en minder as R3 200 per maand sal verdien qd) (c) 'n Persoon uit Gebied 2 sal wees wat presies R3200 of meer verdien (2) 9.2.2 Bewys dat om 'n inkomste van minder as R3 200 per maand te verdien, nie onafhanklik is van die gebied waar 'n persoon woon nie. (5) 9.2.3 Wat is meer waarskynlik: 'n persoon uit Gebied 1 wat minder as R3 200 verdien of 'n persoon uit Gebied 2 wat minder as R3 200 verdien? Toon berekeninge om jou antwoord te ondersteun. (3) [17] TOTAAL: 150 Kopiereg voorbehou

basic education Department: Basic Education REPUBLIC OF SOUTH AFRICA NATIONAL SENIOR CERTIFICATE GRADE 11 og a ae | Gall GS FE ae 2 Be 7 MATHEMATICS P1 t "7 a i NOVEMBER 2015 s Nee Ses cet eles ote ote atavetatete a ate a tee atelier Slee = = = = 5 fete = eo” MARKS: 150 TIME: 3 hours This question paper consists of 8 pages. L Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P1 2 DBE/November 2015 CAPS — Grade 11 INSTRUCTIONS AND INFORMATION Read the following instructions carefully before answering the questions. 1. 2. This question paper consists of 9 questions. Answer ALL the questions. Number the answers correctly according to the numbering system used in this question paper. Clearly show ALL calculations, diagrams, graphs et cetera that you have used in determining your answers. Answers only will not necessarily be awarded full marks. You may use an approved scientific calculator (non-programmable and non-graphical), unless stated otherwise. If necessary, round off answers to TWO decimal places, unless stated otherwise. Diagrams are NOT necessarily drawn to scale. Write neatly and legibly. Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P1 3 DBE/November 2015 CAPS — Grade 11 QUESTION 1 11 Solve for x in each of the following: 1.1.1 x’ +x-12=0 (3) 112 V2x+1=x-1 (5) 113 2 977 (4) 1.1.4 x? —2x-8<0 (3) 1.2 Given: f(x)=5x?+6x-7 12.1 Solve for x if f(x)=0 (correct to TWO decimal places). (4) 1.2.2 Hence, or otherwise, calculate the value of d for which 5x?+6x-—d=0 has equal roots. QG) 1.3 Solve for x and y simultaneously: x-2y=-3 and xy=20 (6) [28] QUESTION 2 2.1 Simplify, without using a calculator: n+2 yntl 2.1.1 ia (3) 2.1.2 yxt2x—-1x-J2x-1 (4) 2.2 Given: P= — +2 x+2 3 2.2.1 For what value(s) of x will P be areal number? (2) 2.2.2 Show that P is rational if x =3. Q) 2.3 Calculate the sum of the digits of 27x 5”, (4) [15] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P1 4 DBE/November 2015 CAPS — Grade 11 QUESTION 3 3.1 Given the linear pattern: 5 ;-2;-9;...;—289 3.1.1 Write down the constant first difference. Q) 3.1.2 Write down the value of 7,. (1) 3.1.3 Calculate the number of terms in the pattern. (3) 3.2 A linear pattern has a difference of 3 between consecutive terms and its 20" term is equal to 64 (thatis 7,)= 64). 3.2.1 Determine the value of T,,. (1) 3.2.2 Which term in the pattern will be equal to 37; -2? (4) 3.3 Consider the quadratic pattern: 5; 12 ; 29; 56;... 3.3.1 Write down the NEXT TWO terms of the pattern. (2) 3.3.2 Prove that the first differences of this pattern will always be odd. (3) [15] QUESTION 4 4.1 Consider the quadratic pattern: 3;5;8;12;... Determine the value of 7,,. (6) 4.2 A certain quadratic pattern has the following characteristics: ° qT =p ° %=18 © 1, =47; ° 1-1, =10 Determine the value of p. (6) [12] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P1 5 DBE/November 2015 CAPS — Grade 11 QUESTION 5 5.1 The sketch below shows the graph of f(x) =—— = -2. x- A is the point of intersection of the asymptotes of f y + f \ | | | \ \ \ | f \ | 1 a 9 { x ——-—-—- ~~ EEE bag- acto norco f | | | | 5.4.1 Write down the coordinates of A. (2) 5.1.2 Determine the coordinates of the x- and y-intercepts of f (5) 5.1.3 Write down an equation of the axis of symmetry of f that has a negative gradient. Q) 5.1.4 Hence, or otherwise, determine the coordinates of a point that lies on f in the fourth quadrant, which is the closest to point A. (5) 5.1.5 The graph of f is reflected about the x-axis to obtain the graph of g. Write down the equation of g inthe form y=... (2) 5.2 Given: h(x)=4(2*)+1 5.2.1 Determine the coordinates of the y-intercept of h. Q) 5.2.2 Explain why A does not have an x-intercept. (2) 5.2.3 Draw a sketch graph of h, clearly showing all asymptotes, intercepts with the axes and at least one other point on h. (3) 5.2.4 Describe the transformation from h to g if g(x)=4(2* +2). Oe Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P 1 6 DBE/November 2015 CAPS — Grade 11 QUESTION 6 The sketch below represents the graphs of two parabolas, f and g. 1 =—x’-8 J) 3 The turning point of g is C (2; 9) and the y-intercept of g is A(0; 5). B and D are the x-intercepts of f and g respectively. y C239) A(O; 5) f [e) D x g& 6.1 Show that g(x)=-x? +4x+5. (4) 6.2 Calculate the average gradient of g between A and C. (2) 6.3 Calculate the length of BD. (5) 6.4 Use the graphs to solve for x, if: 6.4.1 f(x)20 Q) 6.4.2 f and g are both strictly increasing (2) [15] Copyright reserved Please turn over

Mathematics/P1 7 DBE/November 2015 Downloaded from hlayiso.com CAPS - Grade 11 QUESTION 7 The sketch below shows the graphs of f(x) =2x+3 and g(x)= 2x7 +14x4h. C is any point on f and D any point on g, such that CD is parallel to the y-axis. k isa value such that C lies above D. a y A TA Write down a simplified expression for the length of CD in terms of x and k. 7.2 If the minimum length of CD is 5, calculate the value of &. QUESTION 8 8.1 A school buys tablets at a total cost of R140 000. If the average rate of inflation is 6,1% per annum over the next 4 years, determine the cost of replacing these tablets in 4 years! time. 8.2 An investment earns interest at a rate of 7% per annum, compounded semi-annually. Calculate the effective annual interest rate on this investment. 8.3 A savings account was opened with an initial deposit of R24 000. Eighteen months later R7 000 was withdrawn from the account. Calculate how much money wiil be in the savings account at the end of 4 years if the interest rate was 10,5% p.a., compounded monthly. 8.4 A car costing R198 000 has a book value of R102 755,34 after 3 years. If the Copyright reserved value of the car depreciates at r% p.a. on a reducing balance, calculate r. Please turn over (3) (4) 7] (3) (3) 6) (6) [16]

Downloaded from hlayiso.com Mathematics/P1 QUESTION 9 9.1 Given: 8 DBE/November 2015 CAPS - Grade 11 P(A) = 0,6 PB) = 0,3 P(A or B)=0,8 where A and B are two different events Are the events A and B mutually exclusive? Justify your answer with appropriate calculations and/or a diagram. 92 The table below shows data on the monthly income of employed people in two residential areas. Representative samples were used in the collection of the data. ee UNEANDS). ME AREA 1 AREA 2 TOTAL x<3200 500 460 960 3 200 <x < 25 600 1 182 340 1522 x > 25600 150 14 164 Total 1 832 814 2 646 9.2.1 What is the probability that a person chosen randomly from the entire 9.2.2 9.2.3 Copyright reserved sample will be: (a) From Area 1 (b) From Area 2 and earn less than R3 200 per month (c) A person from Area 2 who earns more than or equal to R3 200 Prove that earning an income of less than R3200 per month is not independent of the area in which a person resides. Which is more likely: a person from Area | earning less than R3 200 or a person from Area 2 earning less than R3 200? Show calculations to support your answer. TOTAL: (4) (2) (1) (2) 6) (3) [17] 150