Mathematics Grade 12 P1 NSC June 2024 Question Paper (Eng)

Downloaded from hlayiso.com Confidential ey basic education \ D rt t: (a) Basic Education S297 REPUBLIC OF SOUTH AFRICA SENIOR CERTIFICATE EXAMINATIONS/ NATIONAL SENIOR CERTIFICATE EXAMINATIONS MATHEMATICS P1 MAY/JUNE 2024 MATHEMATICS: Paper 1 MARKS: 150 AAA TIME: hour 10611E This question paper consists of 11 pages and 1 information sheet. X@5 ina Copyright reserved pee Please turn over

Downloaded from hlayiso.com Mathematics/P! 2 DBE/May/June 2024 SC/NSC Confidential INSTRUCTIONS AND INFORMATION Read the following instructions carefully before answering the questions. 1. This question paper consists of 11 questions. 2. Answer ALL the questions. 3. Number the answers correctly according to the numbering system used in this question paper. 4, Clearly show ALL calculations, diagrams, graphs, etc. that you have used in determining your answers. 5. Answers only will NOT necessarily be awarded full marks. 6. You may use an approved scientific calculator (non-programmable and non- graphical), unless stated otherwise. 7. If necessary, round off answers to TWO decimal places, unless stated otherwise. 8. Diagrams are NOT necessarily drawn to scale. 9. An information sheet with formulae is included at the end of the question paper. 10. Write neatly and legibly Copyright reserved Ege) Please turn over Bees

Downloaded from hlayiso.com Mathematics/P 1 5 DBE/May/June 2024 SC/NSC Confidential QUESTION 5 In the diagram, the graphs of f(x)=log,x and g are drawn. Graph g is the reflection of f in the line y= x. Graph / passes through the point P(4 ; 2). Q is the x-intercept of f and R is the y-intercept of g. f PO; 2) / 5.1 Write down the coordinates of P/, the image of P on g. (2) 5.2 Show that a = 2. (2) 5.3 Write down the equation of g inthe form y=... qd) 5.4 T isa point on f in the first quadrant where TR is parallel to the x-axis. Calculate the area of ARTP’, (4) [9] Copyright reserved 8 a Please turn over | P fa

Downloaded from hlayiso.com Mathematics/P 1 6 DBE/May/June 2024 SC/NSC Confidential QUESTION 6 The graphs of (x)= x?-2x-3 and g(x)=mx+e are drawn below. D and E are the x-intercepts and P is the y-intercept of The turning point of f is T(1 ;—4). The graphs of f and g intersect at P and E. y f g D oO E x P T 6.1 Write down the range of /- qa) 6.2 Calculate the coordinates of D and E. (3) 6.3 Determine the equation of g. (2) 6.4 Write down the values of x for which /(x)—g(z) > 0. (2) 6.5 Determine the maximum vertical distance between A and g if A(x)=-f(x) for xe@[-2 ; 3]. (5) 6.6 Given: k(x) = g(x)-". Determine 1 if & is a tangent to f (5) [18] Copyright reserved ac Please turn over She)

Downloaded from hlayiso.com Mathematics/P } 7 DBE“MaylJune 2024 SC/NSC Confidential QUESTION 7 O 71 Six years ago, Thabo bought a phone for R13 000. The value of the phone depreciated annually according to the reducing-balance method. The value of the phone is now R8 337,75. Calculate the annual rate of depreciation. (3) 7.2 Eric and Thandi need to save R80000 each to go on a holiday at the end of December 2027. e Thandi decides that she will start saving at the end of January 2025. She will make 36 monthly deposits into a savings account that pays interest at 86% pa, compounded monthly. The deposit will be made at the end of each month. e — Eric calculates that if he makes 48 deposits of R1 402,31, starting at the end of January 2024, he will have enough money to go on holiday. He will make his deposits into a savings account at the end of each month. The savings account pays interest at 8,6% p.a., compounded monthly. Calculate the difference between the total amount that Eric and Thandi will deposit into their respective savings accounts over the given period. (4) 7.3 Lesibana was granted a loan of R225 000. The rate of interest for the loan is 9% p.a., compounded monthly. Lesibana will make monthly payments of R5 500, starting exactly four months after the loan was granted. How many payments will “esibana make to settle the loan? (6) [13] QUESTION 8 1 8.1 Determine f ‘(x) from first principles if f(x) =—. (s) x 8.2 Determine: 8.2.1 (Vax +22") (3) : 3x4 4x7 46 8.2.2 ‘®) if g2)=— 8) if g@) ie @) 8.3 The equation of the tangent to T(x) =3x’ +bxt+e at x=1 isgivenby y=9x-9, Determine the values of b and c. (4) [15] Copyright reserved Re Please turn over

Downloaded from hlayiso.com Mathematics/P 1! 8 DBE/May/June 2024 SC/NSC Confidential QUESTION 9 The graph of f(x) = ax? +x’ +ex—5_ is drawn below. E(—1; 0) and G(5; 0) are the x-intercepts of f- ¥ E G —1 ce) 5 x Sj 9.1 Show that a=1, b=-3 and c=—9. (3) 9.2 Calculate the value of x for which f has a local minimum value. (4) 9.3 Use the graph to determine the values of x for which f "(x)F(X) > 0. (3) 9.4 For which values of ¢ will the graph of p(x) = f(x)+t have two distinct positive roots and one negative root? Q) [13] Copyright reserved a Please turn over aS

Downloaded from hlayiso.com Mathematics/P} g DBEY/ May/June 2024 SC/NSC Confidential QUESTION 10 EHGF is a rectangle. HE is produced x’ cm to N and EH is produced x? cm to P. NF produced intersects PG produced at M to form an isosceles triangle MNP with NM = MP.D lieson NP where MD 1. NP. NP=4cm and MD =3cm. N 4 “ > 10.1 Show that the area of EFGH is given by A(x) = 6x? -3x‘. (4) 10.2 Calculate the maximum area of rectangle EFGH. (4) [8] Copyright reserved OF Ac Please turn over

Downloaded from hlayiso.com Mathematics/P I 10 DBE/May/June 2024 SC/NSC Confidential QUESTION 11 lil Two events, A and B, are such that: e P(A)=0,4 e P(AorB) = 0,52 e A and B are mutually exclusive Calculate P(B). (2) 11.2 The items that a leamer bought at a tuck shop were recorded over a period of time. The probabilities of the leamer buying a sandwich (S), a chocolate (C) and a juice (J) are shown in the Venn diagram below. 11.2.1 What is the probability that the learner will buy a sandwich? (1) 11.2.2 Calculate the probability that the learner will buy at least two of the three items. 2) 11.2.3 Calculate the probability that the leamer would NOT buy any of the three items. Q) Copyright reserved Ot Please turn over

Downloaded from hlayiso.com Mathematics/P 1 I DBE-“May/June 2024 SC/NSC Confidential 11.3 Seven guitar players, each with a different name, participate in a concert. 11.3.1 In how many different ways can the names of the guitar players: be listed, one below the other, in the programme? 11.3.2 After the performance, the guitar players wait backstage. There is a bench with only room for four to sit on. What will be the probability that the four guitar players will be sitting in alphabetical order, from left to right? 11.3.3 During the performance, the seven guitar players sit in a line on stage. Four guitar players are female and three are male. In how many different ways can they be seated if the males may not sit next to each other? TOTAL: Copyright reserved OF 0] eet (1) (3) @) 14] 150

Downloaded from hlayiso.com Mathematics/P 1 DBE/May/.June 2024 SC/NSC Confidential INFORMATION SHEET pe bev? = 400 2a A= P(l+ni) A= P(-ni) A= P(Q-i)" A=P(l4iy” T, =a+(n-l)d s, = 5 Pa+(n-da] T, =ar™ 5-0! sr¥l 8, =o 3-I<r<l -r p+) -1 pio O+iy" t _ S(x+h)-f@) 'Q) = lim 22 £@) hoo h X,+%, Wt d= Ix, -»)? +0, -v) Sy 4% WAV 2 2 yomxte yp-y, =m(x—x) m= 2a m=tan@ X_ —%y (- a) +b) =r a _ 8b ¢ In ABC: sin A sin Be sinc a? =b? +0? —2be.cos A area AABC = 5 asin (es sin(a + 8) =sina.cos B + cosa.sin B cos(a + f) =cosa.cos B-sina.sin 8 for a sin’ a cos2a =/1—2sin’ a 2cos? a-1 Copyright reserved sin(a - B) = sin a.cos B —cosa.sin B cos(a— 8) =cosa.cos B +sina.sin 8 sin 2a = 2sin a@.cosa ¥ (x, - 3) a n P(A or B) = P(A) + P(B) - P(A and B) pe uF) Lex Meal a fay

Downloaded from hlayiso.com -Vertroulik wy basic education | afaes\e Department: G) 4 Basic Education al REPUBLIC OF SOUTH AFRICA SENIORSERTIKAAT-EKSAMEN/ NASIONALE SENIORSERTIFIKAAT-EKSAMEN MEI/JUNIE 2024 | WISKUNDE V1 | L WISKUNDE: Vraestel 1 PUNTE: 150 IM 10:3 war 10611A Xx @ 5 Hierdie vraestel bestaan uit 11 bladsye en 1 inligtingsblad. IM Kopiereg voorbehou ci Blaai om asseblief

Downloaded from hlayiso.com Wiskunde/V1 2 DBE/Mei/Junie/2024 SS/NSS Vertroulik INSTRUKSIES EN INLIGTING Lees die volgende instruksies noukeurig deur voordat jy die vrae beantwoord. 1. 2 10. Kopiereg voorbehou Hierdie vraestel bestaan uit 11 vrae. Beantwoord AL die vrae. Nommer die antwoorde korrek volgens die nommeringstelsel wat in hierdie vraestel gebruik is. Dui ALLE berekeninge, diagramme, grafieke, ens. 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 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. qo} Blaai om asseblief OAC]

Downloaded from hlayiso.com Mathematics/P} 3 DBE/May/June 2024 SC/NSC Confidential QUESTION 1 Ll Solve for x: 1.1.1 3x? 45x =0 1.1.2 4x? +3x-5=0 (answers correct to TWO decimal places) 1.1.3 (x-1)?-920 1.1.4 52 _ 57-9 1.1.5 a =1 1.2 Solve for x and y simultaneously: x+y=9 and 2x?-y? =7 13 Given: P=(I~a) and T=(1+a)t+a7hi+a‘)..+a%) Determine the value of P xT in terms of a. QUESTION 2 2.1 Consider the geometric series: 4+2+1+ ; +o. 2.1.1 Does this series converge? Justify your answer. 2.1.2 Calculate S_. 10 22 Given: 33°" = 29 520. Calculate the value of & pak Copyright reserved &). a) Please turn over (2) (3) (4) (4) 6) (G5) ) [26] (2) (2) (5) (9]

Downloaded from hlayiso.com Mathematics/P } 4 DBE/May/June 2024 SC/NSC Confidential QUESTION 3 3.1 Consider the quadratic number pattern: 3373123... 3.1.1 Show that the general term of this number pattern is given by T,, = Le tin. 2 2 @G) 3.1.2 What number must be added to 7, so that 7, =13 527? (4) 3.2 Given an arithmetic sequence with 7, =8 and 7, =11. 3.2.1 Calculate the value of n if 7, =41. (3) 3.2.2 A new arithmetic sequence P is formed using the term position and the term value of the given arithmetic sequence. For the new sequence, P, =1, P,, =2 and so forth. (a) Write down the value of P,,. eb) (b) Calculate the value of the first term of the new arithmetic sequence. (4) (15] QUESTION 4 1 Given: g(x)=—— +2 x-l 41 Write down the equations of the asymptotes of g. (2) 4.2 Draw a graph of g, indicating any intercepts with the axes and asymptotes. (4) 4.3 Determine the values of x where g(x) > 0. (2) 4.4 Determine the equation of the axis of symmetry of g which has a negative gradient. (2) (10}.. Copyright reserved ng Please turn over a8

Downloaded from hlayiso.com Wiskunde/V] 3 DBE/ Mei/Junie/2024 SS/NSS Vertroulik VRAAG I Ll Los op vir x: 1.11 3x? +5x=0 1.1.2 4x? +3x-5=0 (antwoorde korrek tot TWEE desimale plekke) 113 (x-1)?-9>0 L114 5**_5* = x 1.1.5 => =1 20-x 1.2 Los gelyktydig vir x en y op: xty=9 en 2x?-y?=7 13 Gegee: P=(1-a) en T=(ltaXl+a?)i+a')}..(+a%) Bepaal die waarde van P xT interme van a. VRAAG 2 2.1 Beskou die meetkundige reeks: 4+2+1 + $a 2.1.1 Konvergeer die reeks? Motiveer jou antwoord. 2.1.2 Bereken S,,. 10 22 Gegee: 5°3?' = 29 520. Bereken die waarde van &. pak Kopiereg voorbehou & aot Blaai om asseblief I (O} (2) G) (4) (4) 6) (5) (3) [26] (2) (2) (5) (9]

Downloaded from hlayiso.com Wiskunde/V1 4 DBE/Mei/Junie/2024 SS/NSS Vertroulik VRAAG 3 3.1 Beskou dic kwadratiese getalpatroon: 3; 7;12;... 3.1.1 Dui aan dat die algemene term van hierdie getalpatroon gegee word deur T, = i n+ a ‘ 2 2 3) 3.1.2 Watter getal moet by 7,., getel word sodat 7),= 13 527? (4) 3.2 Gegee 'n rekenkundige reeks met 7, =8 en 7, =11. 3.2.1 Bereken die waarde van n indien T, = 41. GB) 3.2.2 'n Nuwe rekenkundige reeks P word gevorm deur die termposisie en die termwaarde van die gegewe rekenkundige reeks te gebruik. P,=1, P, =2 vir die nuwe reeks, en so aan. (a) Skryfdie waarde van P,, neer. qd) (b) Bereken die waarde van die eerste term van die nuwe rekenkundige reeks. (4) [15] VRAAG 4 1 Gegee: g(x) =- +2 x-l 4.1 Skryf die vergelykings van die asimptote van g neer. (2) 42 Teken 'n grafiek van g en dui enige afsnitte met die asse en asimptote aan. (4) 43 Bepaal die waardes van x waarvoor g(x) > 0. (2) 44 Bepaal die vergelyking van die simmetrie-as van g wat 'n negatiewe gradiént het. (2) [LO]_ Kopiereg voorbehou Blaai om asseblief

Downloaded from hlayiso.com Wiskunde/V] 5 DBE MeilJunie/2024 SS/NSS Vertroulik VRAAG 5 In die diagram is die grafieke van f(x)=log,x en g geskets. Grafiek g is die refleksie van f indie lyn y= x. Grafiek f gaan deur die punt P(4; 2). Q is die x-afsnit van f era R is die y-afsnit van g. £ P(4 ; 2) ee 17 z 5.1 Skryf die koGrdinate van P ’, die beeld van P op &, neer. (2) 5.2 Toon dat a=2. (2) 5.3 Skryf die vergelyking van g indie vorm y=... neer. qd) 5.4 T is‘n punt op f in die eerste kwadrant, met TR ewewydig aan die x-as. Bereken die oppervlakte van ARTP’. (4) 19] Kopiereg voorhehou Sl cl Blaai om asseblief

Downloaded from hlayiso.com Wiskunde/V1 6 DBE/MeifJunie/2024 SS/NSS Vertroulik VRAAG 6 Die grafieke van f(x)=x?—-2x~-3 en g(x)=mx+c_ is hieronder geskets. D en E is die 8! x-afsnitte en P is die y-afsnit van f Die draaipunt van f is T(L ; —4). Die grafieke van f en g sny mekaar by P en E. ¥ f & D (9) E x P T 6.1 Skryf die waardeversameling van f neer. qa) 6.2 Bereken die koérdinate van D en E. (3) 6.3 Bepaal die vergelyking van g. (2) 6.4 Skryf die waardes van x neer waarvoor f(x)~ g(x)>0. (2) 6.5 Bepaal die maksimum vertikale afstand tussen A eng indien A(x)=-f(x) vir xe(-2; 3). (5) 6.6 Gegee: k(x) = g(x) -7. Bepaal » indien & ‘nraaklyn aan /f is. (S) [18] Kopiereg voorbehou al Blaai om asseblief OF O]

Downloaded from hlayiso.com Wiskunde/V} 7 DBE/ Mei/Junie/2024 SS/NSS Vertroulik VRAAG7 71 Thabo het ses jaar gelede 'n foon vir R13 000 gekoop. Die waarde van die foon het jaarliks volgens die verminderdesaldo-metode verminder. Die waarde van die foon is nou R8 337,75. Bereken die jaarlikse depresiasiekoers. 72 Eric en Thandi moet elkeen R80 000 spaar om aan die einde van Desember 2027 met vakansie te gaan. ¢ Thandi besluit dat sy teen die einde van Januarie 2025 sal begin spaar. Sy sal 36 maandelikse deposito's in 'n spaarrekening maak, wat teen 8,6% p,j. rente betaal, maandeliks saamgestel. Die deposito sal aan die einde van elke maand gemaak word, © Eric bereken dat as hy aan die einde van Januarie 2024 begin en 48 deposito's van RI 402,31 maak, hy genoeg geld sal hé om met vakansie te gaan. Hy sal sy deposito's aan die einde van elke maand in 'n spaarrekening maak. Die spaarrekening betaal rente teen 8,6% p.j., maandeliks saamgestel. Bereken die verskil tussen die totale bedrag geld wat Eric en Thandi oor die gegewe tydperk in hulle onderskeie spaarrekenings sal deponeer. 73 ‘n Lening van R225 000 is aan Lesibana toegestaan. Die rentekoers van die lening is 9% p.j., maandeliks saamgestel. Lesibana sal maandelikse betalings van R5 500 maak, met die eerste betaling presies vier maande nadat die lening toegestaan is. Hoeveel betalings sal Lesibana maak om die lening te vereffen? VRAAG 8 8.1 Bepaal f ‘(x) vanuit eerste beginsels indien f( = i x 8.2 Bepaal: 8.2.1 £f 4x6 + 2x?) 4 2 8.2.2 g(x) indien g(x) eet 8.3 Die vergelyking van die raaklyn aan f (x)= 3x? +bxt+e by x=1 word gegee deur y =9x—-9. Bepaal die waardes van 5 en c. Kopiereg voorbehou 4a Blaai om asseblief OPO) Q) (4) (6) (13] 6) Q) (3) (4) [15]

Downloaded from hlayiso.com WiskundelV 1 VRAAG 9 SS/NSS Vertroulik DBE/Mei/Junie/2024 Die grafiek van f(x) = ax? +bx? +ex—-5 is hieronder geskets. E(- 1; 0) en G(5 ; 0) is die x-afsnitte van f- y E G “IN oO 5S ox 9.1 Toon dat a=1, b=-3 enc=-9. QB) 9.2 Bereken die waarde van x waarvoor f ‘n lokale minimum waarde het. (4) 9.3 Gebruik die grafiek om die waardes van x te bepaal waarvoor f "(x) f(x) > 0. Q) 9.4 Vir watter waardes van ¢ sal die grafiek van p(x)= /(x)+# twee verskillende positiewe wortels en een negatiewe wortel hé? (3) [13] Kopiereg voorbehou Blaai om asseblief

Downloaded from hlayiso.com 9 DBE/Mei/Junie/2024 Wiskunde/V1] SS/NSS Vertroutik VRAAG 10 EHGF is 'n reghoek. HE word x? cm verleng na N en EH word x’ cm verleng na P. NF verleng, sny PG verleng by M om‘n gelykbenige drichoek MNP met NM = MP te vorm. D is'npuntop NP waar MD | NP.NP=4em en MD=3cm. M N P + . 10.1 Toon dat die area van EFGH deur A(x) = 6x? —3x' gegee word. (4) 10.2 Bereken die maksimum oppervlakte van reghoek EFGH. (4) [8] dO} Blaai om asseblief Kopiereg voorbehou AO]

Downloaded from hlayiso.com Wiskunde/V1 10 DBE/Mei/Junie/2024 SS/NSS Vertroulik VRAAG 11 lit Twee gebeurtenisse, A en B, is sodanig dat: e P(A) =0,4 « = P(A of B) =0,52 e Aen B is onderling uitsluitend Bepaal P(B). (2) 11.2 Die items wat 'n leerder oor 'n sekere tydperk by ‘n snoepwinkel gekoop het, is aangeteken. Die waarskynlikheid dat die leerder 'n toebroodjie (S), 'n sjokolade (C) en ‘'n vrugtesap (J) koop, word in die Venn-diagram hieronder voorgestel. a J 11.2.1 Wat is die waarskynlikheid dat die leerder 'n toebroodjie sal koop? qd) 14.2.2 Bereken die waarskynlikheid dat die leerder ten minste twee van die drie items sal koop. (2) 11.2.3 Bereken die waarskynlikheid dat die leerder NIE een van die drie items sal koop NIE. (2) Kopiereg voorbehou aS Blaai om asseblief OO]

Downloaded from hlayiso.com Wiskunde/V 1 a DBE/Mei/Junie/2024 SS/NSS Vertroulik 11.3 Sewe kitaarspelers, elkeen met ‘n ander naam, neem aan 'n konsert deel. 11.3.1 Op hoeveel verskillende maniere kan die name van die kitaarspelers onder mekaar in die program gelys word? 11.3.2 Na die vertoning wag die kitaarspelers agter die verhoog. Daar is 'n bankie met plek vir slegs vier om op te sit. Wat sal die waarskynlikheid wees dat die vier kitaarspelers in al fabetiese volgorde, van links na regs, sal sit? 11.3.3 Gedurende die vertoning sit die sewe kitaarspelers in 'n ry op die verhoog. Vier kitaarspelers is vroulik en drie is manlik. Op hoeveel verskillende maniere kan die spelers sit indien die mans nie langs mekaar mag sit nie? TOTAAL: Kopiereg voorbehou OPO] (1) (3) (3) [14] 150

Downloaded from hlayiso.com Wiskunde/V1 DBE/Mei/Junie/2024 SS/NSS Vertroulik INLIGTINGSBLAD ye rb ENO ~ dae 2a : A= P(l+ni) A= P(l-al) A= P(b-i)" A= P(+i)" T, =a+(n-l)d S, = Fa + (nd T, =ar™" s, <2!) sre s.=7* pw-ler<l r- -r ul r f(xrh)-f) x) = li L£@) 10 h + + daJQq mY + - vy M - a2 2 a 2 2 yemx+e yy, =m(x-m) mare m=tan@ xy Xy (x-a) +(y-by =r? in aaBc, 4-6 -_£ sind sinB sinC a’ =b> +7 -2be.cos A area AABC = Sab. sin C sin(a + B)=sina.cos f +cosa.sin B sin(a - 8) =sina.cos B -cosa.sin 8 cos(a + 8) =cosa.cos f -sina.sin B cos(a- 2) =cosa.cos B +sina.sin B cos’ a@-sin’ a cos2a =41—2sin’ a sin 2a = 2sina.cosa 2c0s’ a -1 7 X, -2 ge! g? =! n n _ 4) = P(A) = P(A of B) = P(A) + P(B) - P(4 en B) ofS) a po ee) a " Yw-3y Kopiereg voorbehou ia OF AO]