FS Gr 12 Sept 2024 Afr en Eng P1_hlayiso.com_.pdf

Downloaded from hlayiso.com ——— = ———————————————————— education Department of Education FREE STATE PROVINCE VOORBEREIDENDE EKSAMEN GRAAD 12 WISKUNDE V1 SEPTEMBER 2024 MATHEMATICS P1 TTC 126118 TYD: 3 UUR PUNTE: 150 41) 5 Hierdie vraestel bestaan uit 8 bladsye en 1 inligtingsblad. Kopiereg voorbehou i s

Downloaded from hlayiso.com Wiskunde V1 2 VS/September 2024 Graad 12 Voorb. Eksam. INSTRUKSIES EN INLIGTING Lees die volgende instruksies aandagtig deur voordat jy die vrae beantwoord. 1. 2. 10. Hierdie vraestel bestaan uit 10 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. Slegs antwoorde beteken nie noodwendig volpunte nie. Jy mag ‘'n goedgekeurde, wetenskaplike sakrekenaar (nieprogrammeerbaar en niegrafies) gebruik, tensy anders vermeld. Indien nodig, rond antwoorde tot TWEE desimale plekke af, tensy anders vermeld. Diagramme is NIE noodwendig volgens skaal geteken NIE. 'n Inligtingsblad met formules is aan die einde van die vraestel ingesluit. Skryf netjies en leesbaar. Kopiereg veorbehou Blaai om asseblief

Downloaded from hlayiso.com Wiskunde V1 3 VS/September 2024 Graad 12 Voorb. Eksam. VRAAG 1 11 Los op vir x: lll (7-—x)(0+x)=0 Q) 1.1.2 3x(2x+1)=1 (korrek tot TWEE desimale plekke) (4) 113 6x? +7x+220 (3) 114 V2x+x=2 (5) 1.2 Los gelyktydig vir x en y op: —2y+x=4 en x?+xy-—2y?=0 (5) 13 Gegee: 4" = p(2’"')+ p. Toon dat vir p #2 die vergelyking hierbo in die vorm m= ; log, [ 2p } 2-p geskryf kan word. (4) [23] VRAAG 2 Gegee: 0;7 en 12 is die derde, vierde en vyfde terme van 'n kwadratiese getalpatroon. 2.1 Bereken dic cerste term van die ry. (2) 2.2 Bepaal 'n uitdrukking vir die n“ term van die patroon. (4) 2.3 Bepaal watter term van die patroon die hoogste waarde sal hé. (3) [9] VRAAG 3 3.1 Gegee die rekenkundige ry: 10x + 6; 2x + 4;4x —8 3.1.1 Bepaal die waarde van x. (2) 3.1.2 Bepaal die 10® term van die ry. (3) 3.1.3. Bepaal die som van die eerste 99 terme van die ry. (2) Kopiereg voorbehou Blaai om asseblief

Downloaded from hlayiso.com Wiskunde V1 4 VS/September 2024 Graad 12 Voorb. Eksam. 3.2. Beskou die oneindige meetkundige reeks: 2(k —5) + 2(k —5)? + 2(k-—5)3 ++ 3.2.1 Vir watter waarde(s) van k is die reeks konvergent? (3) 3.2.2 Ask = 45, bereken Sn @) 3.3. Drie getalle is in die verhouding 1: 3:10. As 20 van die derde getal afgetrek word, vorm die getalle 'n meetkundige ry. Bepaal die drie getalle. (4) [17] VRAAG 4 Gegewe die funksies gedefinieer deur: f (x) = — = -—len g(x) =k’. Die punt (1; 3)16 op g. 4.1 Bepaal die waarde van k. (2) 4.2 Skryf die vergelykings van die asimptote van f neer. (2) 43 Skryf die vergelyking van g~* neer, die omgekeerde van g. (2) 4.4 Bepaal die x-afsnit(te) van f. (2) 4.5 Teken netjiese sketse van f en g~‘ op dieselfde assestelsel, wat alle asimptote en afsnitte met die asse duidelik aandui. (5) 4.6 Vir watter waardes van x sai die simmetrie-as van f, wat 'n negatiewe gradiént het, met die grafick van fsny? (4) 4.7 Beskou die grafiek van h, waar h(x) = g"'(x +2). Bepaal die waarde(s) van x waar gl (x+2)<1. (2) {19} Kopiereg voorbehou Blaai om asseblief

Downloaded from hlayiso.com Wiskunde V1 5 VS/September 2024 Graad 12 Voorb. Eksam. VRAAG 5 Die skets toon die grafieke van die funksies f(x) = x? —x + 12 en g(xyax+4, Aen B is die x-afsnitte van f, terwy! C die draaipunt van f is. Die funksies sny by Aen T. y Cc g T A B re) x f 5.1 Bepaal die x-kodrdinate van A en B. (3) 5.2 Bepaal die waardeversameling van f(x). (3) 5.3 Bepaal die maksimum waarde van k(x) = 3/' (x12 (2) 54 Bepaal vir watter waarde(s) van x sal x. f (x) > 0. (2) 5.5 Vir watter werklike waarde(s) van k sal —x? — x + 12 = k twee negatiewe onewe wortels hé? (2) 5.6 Skryf neer die vergelyking van h in die vorm h(x) = a(x + p)? + q as h die refleksie van f in die reguitlyn x = 1 is. (2) [14] Kopiereg voorbehou Blaai om asseblief

Downloaded from hlayiso.com Wiskunde V1 6 VS/September 2024 Graad 12 Voorb. Eksam. VRAAG 6 6.1 Die waarde van 'n voertuig ter waarde van R150 000 depresieer teen 13% pj. Bereken die waarde van die voertuig na 6 jaar as waardevermindering op die verminderdesaldo-metode bereken word. (3) 6.2 'n Lening van R300 000 word aangegaan teen 'n rentekoers van 5,3% p.j. kwartaalliks saamgestel. Die lening is op 1 Maart 2016 aangegaan. Die eerste betaling is op 1 Desember 2016 gemaak en word in 72 gelyke kwartaallikse paaiemente terugbetaal. 6.2.1 Wat is die uitstaande saldo van die lening op 1 September 2016? (3) 6.2.2 Bepaal die kwartaallikse betalings wat nodig is om die lening terug te betaal. G3) 6.3 Gert het 'n pos gekry wat hom kwartaalliks R27 562,50 vergoed het. Hy het toe besluit om 'n beleggingsrekening te open en 11% van sy salaris aan die einde van elke kwartaal in die beleggingsrekening te deponeer, wat 'n rente van 7,9% p.j. verdien, maandeliks saamgestel vir agt jaar. Wat is die bedrag aan die einde van agt jaar in die rekening? (6) [15] Kopiereg voorbehou Blaai om asseblief

Downloaded from hlayiso.com Wiskunde V1 7 VS/September 2024 Graad 12 Voorb. Eksam. VRAAG 7 TA Bepaal f/(x) van eerste beginsels as dit gegee word dat f(x) = —1 + 4x?. (5) 72 Bepaal: 7.2.1 D,[(2x3 +5)?] (3) dy 7 3 722 —— = 3xt--+2.V x? x as y=3x ot 2.x? (4) 73 Bepaal die koérdinate van die punt op die kromme van y = 2x? + 3x +1, waar die raaklyn by die punt loodreg op y + 5x = 4 is. (4) [16] VRAAG 8 Die vergelyking van die derdegraadse funksie f is gegee as f(x) = -3x3 + 15x? - 21x +9 8.1 Bepaal die x- en y-afsnitte van die grafiek. (4) 8.2 Bepaal die kodrdinate van die draaipunte van f- (4) 8.3 Skets die grafiek van f wat die snypunte met die asse en die draaipunte duidelik aandui. (4) 8.4 Vir watter waarde(s) van: 8.4.1 xis f(x) > 02 (2) 8.4.2 ksal f(x) = k presies drie verskillende werklike wortels hé? (2) [16] VRAAG 9 Daar word 200 sakke teen R400 per sak verkoop. Vir elke R20 verhoging in die prys van die sakke word vier minder sakke verkoop. Teen watter prys moet die sakke verkoop word om die wins te maksimeer? [7] Kopiereg voorbehou Blaai om asseblief

Downloaded from hlayiso.com Wiskunde V1 8 VS/September 2024 Graad 12 Voorb. Eksam. VRAAG 10 10.1 AenB is twee gebeurtenisse in 'n steckproefruimte. P(A of B) = 0,8 en P(B) = 0,4. 10.1.1 Bepaal die P(A) as gebeurtenisse A en B mekaar uitsluit. (2) 10.1.2 Bepaal die P(A) as gebeurtenisse A en B onafhanklik is. (2) 10.2. 'n Mediese sifting word uitgevoer om te toets vir die teenwoordigheid van 'n siekte in 'n bevolking. Die toets is nie 100% akkuraat nie. 10% van diegene wat die siekte het, sal negatief toets en 5% van diegene wat nie die siekte het nie, sal positief toets. As dic toets op die hele bevolking uitgevoer word en 6,7% toets positief, watter persentasie van die bevolking het die siekte? (6) 10.3. In die Vrystaat word nommerplate ontwerp met drie letters van die alfabet, vokale uitgesluit, langs mekaar en dan enige drie syfers van 0 tot 9 langs mekaar. FS is konstant in al die Vrystaatse nommerplate, byvoorbeeld DNV 295 FS. Letters en syfers mag op 'n nommerplaat herhaal word. 10.3.1 Hoeveel unieke nommerplate sal beskikbaar wees as dic letters en die nommers nie herhaal word nie? (1) 10.3.2 Bepaal gevolglik die waarskynlikheid dat die nommerplate met 'n 6 sal begin, in ooreenstemming met die volgorde hierbo. (3) [14] TOTAAL: 150 Kopiereg voorbehou

Downloaded from hlayiso.com Wiskunde V1 Graad 12 Voorb, Eksam. INLIGTINGSBLAD: WISKUNDE ~b+v52 —4ac 2a A=P(l+ni) A=P(l-ni) T, =at(n—Nd T, =ar S pai =1 Le+h)- FQ) A a t J'G)= lim d=, -x,)? +0,-»)” m( ma22221 *2 7% In AABC: 4. < sinA sinB sinc yy, =mM(x—-x,) Opp AABC = =ab. sin sin(a + 8) =sina.cos f + cosa.sin B cos(a + 8) =cosa.cos B-sina.sin B cos? a —sin’? @ cos2a =41-2sin’a@ 2cos*a—-1 x= o= A=P(-i)" _ ar" -1 r-l Ht Mt V2 2° 2 VS/September 2024 A=P(+i)" S, = 5 [2a+(n—1)d] 3 rél pawll=ds iy "] I } y=mxt+e (x—a)’ +(y-bP =r? m=tan@ a? =b2 402 —2be.cos A sin(a - B) = sina.cos B —cosa.sin B cos(a— 8)=cosa.cos B +sina.sin B sin2@ = 2sina@.cosa n(A) Kopiereg voorbehou P(A) = P(A of B) = P(A) + P(B) — P(A en B) n(S) p= a FY- 9) YVi@-xy

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Downloaded from hlayiso.com wy education Department of Education FREE STATE PROVINCE PREPARATORY EXAMINATION GRADE 12 MATHEMATICS P1 SEPTEMBER 2024 MATHEMATICS P1 ANITA 126118 TIME: 3 HOURS MARKS: 150 xo 5 This question paper consists of 8 pages and 1 information sheet. N2 Copyright reserved

Downloaded from hlayiso.com Mathematics P1 2 FS/September 2024 Grade 12 Prep. Exam. INSTRUCTIONS AND INFORMATION Read the following instructions carefully before answering the questions. 1. 2. 10. This question paper consists of 10 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. An information sheet with formulae is included at the end of this question paper. Write neatly and legibly. Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics P1 3 FS/ Grade 12 Prep. Exam. QUESTION 1 1.1 Solve for x: Lil (7-x)(10+x) =0 1.1.2 3x(@2x+D=1 (correct to TWO decimal places) 1.13 6x?+7x+2>0 114 9 yv2x+x=2 1.2 Solve simultaneously for x and y: -2y+x=4 and x?+xy-2y?=0 13 Given: 4” = p(2"*)+ p. Show that for p #2 the above equation can be written in the form m=Hiog, (24) QUESTION 2 Given: 0;7 and 12 are the third, fourth and fifth terms of a quadratic number pattern. 2.1 Calculate the first term of the sequence. 2.2 Determine an expression for the n™ term of the pattern. 2.3 Determine which term of the pattern will have the highest value. QUESTION 3 3.1 Given the arithmetic sequence: 10x + 6; 2x + 4;4x —8 3.1.1 Determine the value of x. 3.1.2 Determine the 10" term of the sequence. 3.1.3. Determine the sum of the first 99 terms of the sequence. Copyright reserved ‘September 2024 2) 4) (3) (5) (5) (4) [23] (2) (4) (3) 19] (2) (3) (2) Please turn over

Downloaded from hlayiso.com Mathematics P1 4 FS/September 2024 Grade 12 Prep. Exam. 3.2 Consider the infinite geometric series: 2(k —5) + 2(k —5)? + 2(k —5)3 ++ 3.2.1 For which value(s) of k, is the series convergent? (G) 3.2.2 Ifk= 45, calculate Soo Q) 3.3. Three numbers are in the ratio 1:3:10. If 20 is subtracted from the third number, the numbers form a geometric sequence. Determine the three numbers. (4) [17] QUESTION 4 Given the functions defined by: f(x) = -=- land g(x) =k’*. The point (1; 3) lies on g. 4.1. Determine the value of k. (2) 4.2 Write down the equations of the asymptotes of f. (2) 4.3. Write down the equation of g~*, the inverse of g. Q) 4.4 Determine the x-intercept(s) of f. (2) 4.5 Draw neat sketches of f and g~* on the same system of the axes, clearly indicating all asymptotes and intercepts with the axes. (5) 4.6 For which values of x will the axis of symmetry of ff which has a negative gradient, intersect with the graph of f? (4) 4.7 Consider the graph of h, where h(x) = g7'(x+2). Determine the value(s) of x for which g!(x+2)<1. (2) {19} Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics P1 5 FS/September 2024 Grade 12 Prep. Exam. QUESTION 5 The sketch shows the graphs of the functions f(x) = -x’-« + 12 and g(x)=xt+4. A and B are the x-intercepts of f, while C is the turning point of f. The functions intersect at A and T. 5.1 5.2 5.3 5.4 5.5 5.6 y Cc g T A B 7) x f Determine the x-coordinates of A and B. (3) Determine the range of f (x). @) Determine the maximum value of k(x) = 34-12 (2) Determine for which value(s) of x will x. f(x) > 0. (2) For which real value(s) of k will —x? — x + 12 = k have two negative unequal roots? (2) Write down the equation of h in the form h(x) = a(x + p)? + q if his the reflection of f in the straight line x = 1. (2) [14] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics P1 6 FS/September 2024 Grade 12 Prep. Exam. QUESTION 6 6.1 The value of a vehicle worth R150 000 depreciate at 13% p.a. Calculate the value of the vehicle in 6 years if depreciation is calculated on the reducing- balance method. (3) 6.2 A loan of R300 000 is taken out at an interest rate of 5,3% p.a. compounded quarterly. The loan was taken out on the 1“ of March 2016. The first payment was made on 1 December 2016 and is repaid in 72 equal quarterly payments. 6.2.1 What is the outstanding balance of the loan on 1 September 2016? (3) 6.2.2 Determine the quarterly repayments required to pay back the loan. () 6.3 Gert landed a job which remunerated him R27 562,50 quarterly. He then decided to open an investment account and deposit 11% of his salary at the end of every quarter into the investment account, earning an interest of 7,9% p.a. compounded monthly for eight years. What amount will be in the account at the end of eight years? (6) [15] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics P1 7 FS/September 2024 Grade 12 Prep. Exam. QUESTION 7 71 Determine f/(x) from first principles if it is given that f(x) = ~1 + 4x2. (5) 72 Determine: 721 Dy[(2x? +5)?] (3) dy 7 3 722 = j = 3x4 —--42. x? a if y=3x ot 2.Vx? (4) 73 Determine the coordinates of the point on the curve of y = 2x? + 3x +1, where the tangent at the point is perpendicular to y + 5x = 4. 4 [16] QUESTION 8 The equation of the cubic function f is given as fd) = -3x3 +: 15x? - 21x +9 8.1 Determine the x- and y-intercepts of the graph. (4) 8.2 Determine the coordinates of the turning points of f. (4) 8.3 Sketch the graph of f clearly indicating the intercepts with the axes and the turning points. (4) 8.4 For which value(s) of: 8.4.1 xis f'(x) > 0? (2) 8.4.2 k will f(x) = k have exactly three different real roots? (2) [16] QUESTION 9 At a price of R400 per bag, 200 bags are sold. For every R20 increase in the price of the bags, four fewer bags are sold. At what price must the bags be sold to maximise the profit? 7] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics P1 8 FS/September 2024 Grade 12 Prep. Exam. QUESTION 10 10.1 10.2 10.3 A and B are two events in a sample space. P(A or B) = 0.8 and P(B) = 0.4. 10.1.1 Determine the P(A) if events A and B are mutually exclusive. (2) 10.1.2 Determine the P(A) if events A and B are independent. (2) A medical screening is performed to test for the presence of a disease in a population. The test is not 100% accurate. 10% of those who have the disease will test negative, and 5% of those who don't have the disease will test positive. If the test is performed on the entire population and 6,7% test positive, what percentage of the population has the disease? (6) In the Free State, license plates are designed with three letters of the alphabet, excluding vowels, next to one another and then any three digits from 0 to 9 next to one another. FS is constant in all Free State license plates, for example, DNV 295 ES. Letters and digits may be repeated in a license plate. 10.3.1 How many unique license plates will be available if the letters and the numbers are not repeated? (1) 10.3.2 Hence, determine the probability that the license plates will start with a 6, in keeping with the order above. (3) [14] TOTAL: 150 Copyright reserved

Downloaded from hlayiso.com Mathematics P1 FS/September 2024 Grade 12 Prep. Exam. INFORMATION SHEET: MATHEMATICS _ bt Vb? —4ae 2a A=P(l+ni) A=P(-ni) A=P(1-i)" A=P(1+i)" T, =a+(n-l)d 8, =5[2a+(n-1)d] r= ar 5, =a ; 5 rel $,= 3-I<r<l r- —r p 2+ =I polar") L x) = Yim LEPO-LO Po py 1 > d=y(x,-x,)? +0 —»)? mt yoyam(x—y) — m=22 71 XY In AABC: ab _¢ sind sinB sinC area AABC = 5 ab.sinc sin(a + B)=sina.cos f + cosa.sin B cos(a + 2)=cosa@.cos 8 —sina.sin B cos’ asin’ a cos 2a =31-2sin’ a 2cos? a1 x 7 ¥,-3) yoo o Copyright reserved x, +Xy al + Yo wa, 215%2) t y=mx+e m=tan@ (x—-a) +(y-by =r? 2 a? =b* +07 —2be.cos A sin(a- B) = sina.cos 8 —cosa.sin B cos(a— £)=cosa.cos B+sina.sin B sin 2a = 2sina@.cosa P(A) = na} P(A or B) = P(A) + P(B) — P(A and B) ( »Le-Do—D Y@-3yP

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