Mathematics Grade 12 Paper 1 MP Mock 2023 Paper & Memo

education MPUMALANGA PROVINCE REPUBLIC OF SOUTH AFRICA NATIONAL SENIOR CERTIFICATE MARKS: 150 TIME: 3 hours This question paper consists of 10 pages and a one-page information sheet

Mathematics P| 3 Downloaded from Stanmorephysics.com QUESTION 1 1.1 Solve for x LL. @-x)2-x)=0 Tele ax? 2 (Correct to 2 decimal places) 113 44 6x° 1.1.4 9° +9=10,3* 1.2 Solve for x and y: y-2x=-1 y? —3xy=-2 12 ZS uf , calculate two values of cm y =x 4 y QUESTION 2 2.1. Consider the following quadratic sequence: 319 ; 280 ; 243 ; 208; 175; 144;.... 2.1.1. Show that the n' term of the sequence is T, =n” —42n +360. 2.1.2 Determine which terms has a value of 0)? 2.1.3. Which term in the sequence will have the lowest value? 2.2 3r: 41-1; 23 are the first three terms of an arithmetic sequence. 2.2.1 Prove that 1=5. 2.2.2 Calculate the sum of the first 50 terms of the series. Copyright reserved MDE/September 2023 Please turn over (2) (4) (4) (4) (6) G) [25] (4) (3) (2) (2) (4) (15)

Mapeihloaded from Stanmorephysics.com aaa QUESTION 3 3.1 The following geometric series is given: 44+12+36+...to 15 terms 3.1.1 Write the series in sigma notation. (2) 349 ate the series. (2) 3.1.3 'S Series convergent? Provide a reason for your answer. (2) 3.2. The sum of an infinite, geometric series is 13,5(r #1). The sum of the same series, calculated from the third term is 1,5. 3.2.1 Calculate r if r>0. (4) 3.2.2 Hence, determine the first THREE terms of the series. Q) [12] Copyright reserved Please turn over

MahenPigyfhloaded from Stanmore Raysics. com MDE/September 2023 QUESTION 4 In the diagram, A(—2; 0) is the x-intercept of h(x) =< 2. = di -....------. h ee ee ee fone nna Ee ! _4 i oO ! 1 H 4.1 Calculate a. (2) 4.2 Calculate the av een X=—4 and x=-2. (3) 4.3. For which values (2) 44 Determine the e (2) your answer in 4.5 Write down the equations, S i i i (2) the x-axis. [11] Copyright reserved Please turn over

Mahe Bwnloaded from Stanmorgphysics.com sania QUESTION 5 Sketched below is f(x) = “58 shrek and h(x) = q*. The x-intercepts of f are BandC. h and f intersect the y-axis . Aisa point on h vertically above C witha y-value of % PR is perpendicular to the x-axis. | “y P x R 5.1 Calculate the coordinates of D. (lb 52 Write down the value of k. () 53 Calculate the length of BC. (4) 5.4 Calculate the length of PR if OP is 2 units. (2) ; 5.5 Consider h(x) = q* §.5.1 Calculate the value of g. (3) 5.5.2 Hence, determine the equation of A‘, in the form y=. (2) 5.6 Write down the range of h. () 5.7 Determine the equation of |g, the tangent to f at B. (5) 5.8 For which values of x is x.f/(x) <0? (2) (21] Copyright reserved Please turn over

Mathematics P| ownloaded from Stanmorephysics.com MDE Sepeniera722 QUESTION 6 6.1 Jane deposits Rx into an investment account. How long will it take for the value of the investme' double if the interest rate is 5,4% p.a. compounded annually? (3) 6.2 Thabo a printing company and needs to borrow money for start-up costs. He can make equal monthly payments of R3 300. What amount can Thabo borrow if the interest rate on the loan is 12% p.a. compounded monthly and the loan is granted over 5 years? (4) 6.3 A group of investors consider investing in a fund that promises growth at a rate of 5% p.a. compounded quarterly. Calculate the effective annual percentage rate of the growth promised. (3) 6.4 Sarah is 18 years old and wishes to accumulate R10 000 000 by the month before her 50" birthday. She will deposit equal monthly payments into an account that pays 15% p.a. compounded monthly. The first payment starts on her 18" birthday and the last payment one month before her 50" birthday. Calculate the monthly instalment that Sarah will make. (3) {13} QUESTION 7 7.1 Given: f(x)=-2x7 +1 Determine f'(x) from first principles. (5) 7.2 Determine: 721 Fist f@)=5* : (3) wer gle 2x? + 4x a ek (3) {11} Copyright reserved Please turn over

Mathenf¥igyfhloaded from Stanmore } sics.com MDE/September 2023 QUESTION 8 In the diagram, the graph of f(x) =— x? +5x" +8x-12 is drawn. A,B and C are the x-intercepts of f. a turning point of f. T is a point on fand G is a point on the x-axis such that TG is perpe lar to the x-axis. D is the y-intercept of f- ny E T, A lo) B(I ; 0) G D 8.1 Calculate the coordinates of C if B( ; 0). (4) 8.2 Determine the coordinates of E. (5) 8.3 For which values of x will f be concave up? (2) 8.4 Calculate the length of OG if the tangent to the curve at T is parallel to the tangent to (5) the curve at D. 8.5 Determine the value of m if y=mx+c intersects f perpendicularly at x=5. GB) [19] Copyright reserved Please turn over

MathemSSWnloaded from Stanmoreghysics.com MDE/September 2023 QUESTION 9 A farmer has a piece of land in the shape of a right-angled triangle OMN, as shown in the figure below. He allocates a rectangular piece of land PTOR to his daughter, giving her the freedom to choos nywhere along the boundary MN. Let OM=a, ON=b and P(x; y) be any point on A y (0 :b) 1 P(x;y) < > fe) R M(a;0) * Y 9.1 Determine an equation for MN in terms of a and b. (2) 9.2 Prove that the daughter’s land will have a maximum area if she chooses P at the (6) midpoint of MN. [3] Copyright reserved Please turn over

moeaWitloaded from StanmorepHysics.com NSC QUESTION 10 10.1 A language survey was done with 45 grade 12 learners in Mbombela Secondary School. The ai e study was to find out what language learners speak in their homes. The re f the survey are listed below: e 22 speak Afrikaans e 24 speak English e 11 speak SiSwati and Afrikaans © 5 speak all three languages e 3 speak only Afrikaans e 38 learners speak at least one of the languages at home 18 speak SiSwati 10.1.1 Determine the number of learners that do not speak any of the named languages at home. () 10.1.2 If x learners speak English and SiSwati but not Afrikaans, draw a Venn diagram to represent the information, (4) 10.1.3. Solve for x. (2) 10.1.4 What is the probability that a learner speaks only two of the three languages at home? (2) 10.2 Three brothers Owen, Richard and Robert are to run a race which has 8 runners in total. The eight competitors line up one to a lane in the lanes numbered 1 to 8. 10.2.1 Write down the total number of possible arrangements of the runners at the (D starting line. 10.2.2 Calculate the total number of arrangements in which the three brothers are all (3) next to each other. 10.2.3 Calculate the probability that Owen is in lane 1, Robe! lane 2 and Richard (2) is in lane 3. (15) TOTAL: 150 Copyright reserved Please turn over

Downloaded fapmggganporephysics.com education MPUMALANGA PROVINCE NATIONAL SENIOR CERTIFICATE MARKS: 150 This marking guideline consists of 13 pages.

Mathmpoawnteaded from Stanmorephysics.com MDE/MDOSeptember 2023 SC/NSS QUESTION 1 / VRAAG 1 Lil (3-x)(2-x)=0 v Yanswers x=3 or daar? nnn Q) 1.1.2 2x? + "i 2x’ +7x=2% 0 Y standard form —(N+ V7? -4@2)\(-2 x= ~M#NT ~4QV2) Y substitution into correct formula 2(2) _ -74V65 4 Penalise 1 mark for || v V answers (4) x=0,27 or x =—3,77 incorrect rounding 1.1.3 445x > 6x’ 6x? —5x-4<0 (3x 4)(2x11)<0 Y factors On ©) <+—_}+——_+—> _ OR it KOR -= 4 2 3 Y critical values 2 3 1 -a<x<n 2 3 vv answer OR 1.4 xel—5 55 (4) 1.1.4 9*+9 =10.3* 3" -10.3" +9 =0 vs Y fact (3° -9\3"-1)=0 “ses ; , v exponential law 3* -3° or 3* -3 Y answ (4) x=2 or x=0 Q Copyright reserved/ Kopiereg voorbehou Please turn over / Blaai om asseblief

Mathmpoawnteaded from Stanmorep hysics.com MDE/MDOSeptember 2023 SC/ NSS 12 y-2x=-] y= 2x-1....() v equation | y —3xy=-2.... (2) (2x1) —3x(@x-1) =-2 v substitution 4x° —4x 06x? +3x=-2 fanny — 2x" — xk O Y standard form 2x +x v (2x+3)(x-1)=0 factors 3 v x-values x=-> or x=1 2 v y-values (6) y=f-3 1 or y=2(1)-1 y=-4 or y=1 OR y-2x=-1 \ v equation | yt ve ) y’ —3xy=-2.... (2) V cubstiut substitution yo (21\y =-2 2 2yv -3y? -3y+4=0 ma ~sy+4=0 Y standard form y +3y-4=0 ¥ factors (y+4)(y-1=0 y=-4 or y= v y-values Vx _ -; or x=l x-values (6) Multiply every term by LCD 4xy 13 4x? +4y’ =17xy v¥ multiply by LCD 4x’ -1Txy+4y’ =0 Y standard form 4x — y)(x-4y) =0 ‘ p\Mx-4y) Y 5B x-y=0 xl Y apver y 4 y 5 x—4y=0 answer (5) *_4 y Copyright reserved/ Kopiereg voorbehou Please turn over / Blaai om asseblief

Mathmpoawnteaded from Stanmorephysics.com MDE/MDOSeptember 2023 SC/ NSS OR xy a y * Y fraction x tL ~yis y x y x 17x Y standard form = -==.4_.1=0 x 4y 2 7 + (2) -4(1)(1) Y substitution into correct formula Em. 20) 17 . 225 Y simplification _A4 ~\V 16 2 v x x 1 answers (5) ~=4 or X= y y 4 [25] QUESTION 2 / VRAAG 2 2.1 | 319; 280; 243 ; 208; 175; 144;.... 2.1.1 319 280 243 208 -39 -37 —35 2 2 2a=2 3a+b=-39 a+b+c=319 a=1 3(1)+b=-39 1-42 +e =319 ¥ 3(1)+b=-39 b=-42 c= 360 T, =n’ —42n+360 Copyright reserved/ Kopiereg voorbehou Please turn over / Blaai om asseblief

Mathmpoawnteaded from Stanmorep hysics.com MDE/MDOSeptember 2023 SC/NSS v 1-42+c=319 (4) 2.1.2 n? —42n+360 =0 Y T,=0 (n-12\n-30)= 0 v factors n=12 or 0 v answer (3) On — Y derivative = 2.1.3 on 42 rim oR n= (42) derivative = 0 2()) v answer (2) n=21 n=21 OR Tar is lowest Y substitution into x = oa a Y answer (2) 2.2 3t 3 4r—-1; 23 2.2.1 3t 3 4t—-1; 23 4t -1—3t = 23-(4t-1) ¥ subtraction 4t-1-3t=23-4¢+1 5t=25 Y simplification (2) t=5 2.2.2 15:19; 23 v a=15 a=15 d=4 v value of d 500 Sy = “7 PCS) +(49)4] ¥Y substitution into correct formula =5650 v answer (4) [15] Copyright reserved/ Kopiereg voorbehou Please turn over / Blaai om asseblief

Mathembownieaded from Stanmorephysics. com MDE/MDOSeptember 2023 SC/ NSS QUESTION 3 3.1 x=4+12+36+....to 15 terms T, =ar" | —~ 3.1.1 a ir iat T, =43"" T v T, = 43" _—— (43 at v answer (2) n=l 3.1.2 | S,= ale" =) rl Y — substitution into the correct 15 S = 4G -1) formula " 3-1 =28697812 V answer (2) 3.1.3 r=3 and not between —] and 1 ¥ r=3notin-I<r<l .. not converging Y answer (2) 301 S.= 133 Y — substitution into correct formula -r a =13,5(I-r) S,= _ =1,5 Y a=135(1-r) ar? =1,5(1-r) 1,5(1-r) = 15(1-r 13,5(1-r) = oe v equating pea AS” 135 3,5 -l 9 r= +l 3 Butr>0 1 lear Yo pe = r= tal 4) Copyright reserved/ Kopiereg voorbehou Please turn over / Blaai om asseblief

Mathmpoawnteaded from Stanmorephysics.com MDE/MDOSeptember 2023 SC/NSS 3.2.2 v value of a v answer Q) [12] QUESTION 4 4.1 0= a@_ 45 ¥ substitution —2+1 v answer (2) a=2 —2)- Y h(-4) and A(-2 4.2 Average gradient = M2) h-4) (A) and A-2) -2-=(-4) 2 4+2- 2 +2) Y substitution oe i —4+1 2 ew v answer (3) 3 43 -l<x<0 ¥ critical values v answers (2) 44 y=xt+l+2 Y gradient ; ypax+3 v answer (2) oR OR yexte 2=-1he Y value of c c=3 ¥ answer (2) yaxt+3 -2 4.5 | g@=—- 2 ; xt ¥ horizontal asymptote x=-1 _ 2 y=-2 (2) ay Copyright reserved/ Kopiereg voorbehou Please turn over / Blaai om asseblief

Mathemabawnieadedt from Stanmorephysics.com MDE/MDOSeptember 2023 SC/ NSS QUESTION 5 hay=ge f(x)=—-48 45x +k g(x) =mx+e S| AG) =" =1 => D(0 3 1) v answer (1) 2 fran? v 1 5. k=1 fact answer (1) 53° | fa)=-deh tes v =0 x -x-2=0 y Je) (x-2)(x+1)= 0 factors x=2 orx=-l v¥ both x values BC = 3units v answer (4) 5.4 f2= 5027 +hC2+1 ¥ substitution of —2 =-2 PR =2 units v answer (2) 5.5.1 | hx) = va(t] Y substitution of (5] q=t 4 2 ¥ value of g (3) 5.5.2 1\ v equation of h we A(x) -(5} . 4 2 Y equation of h (2) h7'(x) =log, x 5.6 y>o v answer (1) M(x) --x 41 v derivative 7 P@)=-e+ 2 v derivative x =-1 feyn=-Cy+h=3 Y f(=3 vr = m(x—x,) B(-1 ; 0) ¥ substitution of B 3 yo0= 3 -CD) Y tangent__ parryi Oa] J 7 x+ im (5) 5.8 x<-lor x>2 vv ansiaa (2) OR xe(-0 5; IUQ;%) [21] Copyright reserved/ Kopiereg voorbehou Please turn over / Blaai om asseblief

Mathmpoawnteaded from Stanmorephysics.com MDE/MDOSeptember 2023 SC/NSS QUESTION 6 6.1 | 4=P(l+i)’ dy = x(1 +0 ¥ substitution into correct formula 2=(1, Y correct use of logs log(2) a2 ~ Tog(,054) v answer (3) n=13,18 62 |p-% [b+ i) i v . 012 60 value of i 3300 | 1 1+ 1D v value of n p=—4—__*_t 0,12 Y substitution into correct formula 12 = R148351,63 Y answer (4) ; 0,05)" Y value of n 6.3 ig =| 1+ 4 -l ¥ substitution into correct formula — 0,05094533691 Rate = 5,09% Y rate (3) 0.15)" x Ito al v value of n 6.4 10000000 = Js v substitution into correct formula 12 125000 = x(116,9484518) v answer (3) x =R1068,85 [13] Copyright reserved/ Kopiereg voorbehou =) Please turn over / Blaai om asseblief

Mathmabawnieaded from Stanmorephysics.com SC/ NSS MDE/MDOSeptember 2023 QUESTION 7 Penalty for notation error in 7.1 only. 71 | f(x)=— 2x7 41 ¥ —2x? —4xh- 2h? +1 S(x+A)= +hy +1 S (x + h) =/20#? — 4xh-2h? +1 He) — fin SERS =F (2) (0)= fig EL) ' . —2x? — 4xh— 2h? 4+1-(-2x? +1 f (x) = him I ( ) Y correct substitution into formula _ 2 _ 2 2 (x)= lim 2x° —4xh —2h? +14+2x° -1 0 h Y simplification , . —4xh = 2h? 1()= fig 2 , . h(-4x—2h S')= fim, Ce) h ) ¥ common factor 2 py, v answer (5) 721 | Se)= 42" 5x7 5x7 vx Sf'(x) =x 45x? v 5x? (3) 244 7.2.2 | D 2x? +4x x x 2x? x8 ¥ eax! =D) = Paps a xX ae -2 ¥-2+x7 =D,|-2+x* | 7 -u = = -—x" v i 4 x cere (3) Copyright reserved/ Kopiereg voorbehou i m oJ Please turn over / Blaai om asseblief

Mathmpoawnteaded from Stanmorephysics.com MDE/MDOSeptember 2023 SC/ NSS QUESTION 8 8.1 1 _|-l 5 8 —12 00 \-a [4 [2 ( i im 2 4 0 |e ¥ quadratic factor x1 - hANX + = : Y factors (x-1(x?—4x-12)=0 , “e he (x-1x -6\x+2)=0 “ewaiues x=1or x=6 or x=-2 Vv" answer (4) - (650) 8.2 —3x? +10x+8=0 ¥ derivative 3x? -10x-8 =0 S'(x)=0 (3x+2\x-4)=0 ¥ factors x=-2 orx=4 3 ¥ x-values Ff (4) =-(4)° +5(4)* +8(4) -12 = 36 v answer (5) 83 | -—6x+10>0 Y inequality 5 *s 3 Y answer (2) 8.4 | f(x) =-3x? +10x4+8 f'()=8 Y f'(0)=8 —3x? +10x+8=8 v . —3x? +10x = 0 equating x(—3x +10)=0 Y factors x=0 or x= » v x-values v¥ value of OG (5) 5 _10 “OG= 3 8.5 | f(x) =-3x? +10x+8 Y¥ substitution S/(5) =-3(5)° +1065) +8 ; =-17 Vv f'6)s=17 1 raat me TT v answer (3) loon 19] + Copyright reserved/ Kopiereg voorbehou Please turn over / Blaai om asseblief

Mathemabawnieadedt from Stanmorephysics.com MDE/MDOSeptember 2023 SC/ NSS QUESTION 9 9.1 Mim = a Y m., b yo-x a v answer (2) 9.2 | Area of PTOR =TP x PR =o. + bx a x= 2 --2(2)+0=$ a\2 2 O+a 0+b) (ab Midpoint of MN: |~9 °° J (92 «. Max at midpoint v substitution into area formula v simplification v derivative =0 v value of x Y value of y Y midpoint (6) [8] Copyright reserved/ Kopiereg voorbehou Please turn over / Blaai om asseblief

Mathmpoawnteaded from Stanmorephysics.com MDE/MDOSeptember 2023 SC7/ NSS QUESTION 10 ; 0ar 10.1.1 | 45-38 = 7 Hla] Janay v answer dd) 10.1.2 S= v8 v6 Vil-x ¥7-x (4) 10.1.3 | 3+84+64+5+11—x+x+7—-x+7=45 Y calculation x=2 v answer Q) 6+8+2 10.1.4 | P(Qlanguages) = 7 Y 16 _16 v 16 (2) a5 45 10.2.1 | 8! = 40320 v8 () 10.2.2 | 3! x 5! x 6 = 4320 v3! OR v 5! 3! x 6! = 4320 v 3x Sx 6 (3) v5! 10.2.3 = = = a ; " v= OR — (2) 8! 336 [15] TOTAL: 150 Copyright reserved/ Kopiereg voorbehou