Mathematics Grade 12 Paper 1 Limpopo Pre June 2024 Memo

Downloaded from hlayiso.com yea LIMPOPO PROVINCIAL GOVERNMENT REPUBLIC OF SOUTH ARRICA DEPARTMENT OF EDUCATION VHEMBE WEST DISTRICT MARKS: 120 This question paper consists of 7 pages including the cover page Copyright reserved/Kopiereg voorbehou Please tum over/Blaai om asseblief

Downloaded from hlayiso.com Mathematics P1 2 Marking Guidelines/Nasienriglyne NOTE: . If a candidate answers a question TWICE, only mark the FIRST attempt. . Consistent Accuracy applies in all aspects of the marking memorandum. LET WEL: . Indien ’n kandidaat ’n vraag TWEE keer beantwoord, merk slegs die EERSTE poging. . Volgehoue akkuraatheid is DEURGAANS op ALLE aspekte van die memorandum van toepassing. QUESTION/VRAAG 1 1.1.1 | x2-5x-6=0 (x — 6)(x +1) =0 x=6 or x=-l v factors Y both answers (2) OR/OF OR/OF xo Sx 6=0 Y correct subst into correct 5 — | ul x= St yy -4MC9) formula 2(/) ya 5t V49 2 x=6 or x=-l Y both answers (2) 1.12 | Gx-1)(r-4)=16 y / 3x2 —13x 12-0 standard form x= 14) C13) 4@)C12) Y correct subst into correct 2(3) formula xe 13313 6 x =5,12 or x=-0,78 v Vv answers (4) OR/OF OR/OF 3x? -13x-12=0 Y standard form > 13 x*—-__x=4 3, 2 3 2 eo yy _13) -a{-2] v adding (- 2) both 3 6 J 6 } 6 13 2 313 sides (.-2) 28 6) 36 _ 13+V313 6 x=5,12 or x=-0,78 V ¥ answers (4) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Mathematics P1 3 Marking Guidelines/Nasienriglyne 11.3 | x(4-x)20 Y factorisation -x(x-4)20 or -x(x-4)>0 x(x - 4) <0 0 4 0 4 O<x<4 or xe [0:4] vv OSxS4 (3) 1.1.4 52x —] Sé+1 v . (+=) _ 4 factors in numerator 5* +1 5*-1=4 VY 5*-1=4 5* = x=l Y answer (3) OR/OF OR/OF 521 S*4+l 5 —1=4.5* +44 5 —4.5*-5=0 Y standard form 5*—5 )(5*+1)=0 ( ¢ ) Y¥ factors S*=5 or S*#-1 x- v answer (3) D2 | 2A BY ccccccccccsvvssvsvssssssvsvvssssvsssvsssssssee () x=2—3y x? AY 5 HO cece eecteceeteceeee (2) Substitute (1) in (2): 2 = (2—3y)y +4y(2—3y)-5=0 Y correct subst into correct 4-12y+9y? +8y-12y? -5=0 formula —3y? -4y-1=0 3y? +4y +1=0 v either standard form By+1l(y+1)=0 1 =—~ or y=-l y 3 y v y—values x=3 orx=5 v¥ x—values (5) OR/OF OR/OF Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Mathematics P1 Marking Guidelines/Nasienriglyne 4 eC. x2+ax}?—*|_5=0 F 3x? + 8x—4x7 -15=0 —x? +8x-15=0 x?-8x+15=0 («—5)(x-3)=0 x=3 orx=5 Substitup (1) a (2): 1 y=- or y=-1 3 Y correct subst into correct formula Y either standard form v x—values v y—values (5) 1.3 ab =2V10 be =3V2 ac = 6V5 ab.bc.ac = 2V10.6V5.3v2 (abc) =36V100 abc = 1360 = 6V10 OR/OF ac =6,/5 anew be =3,/2 ob = 37 ab = 2/10 (85 3.2) 9 Le oe ) 18V10 =2/10.c? e=9 c=3 Volume = abc = 2V10.3 = V360 = 6V10 0 volume = abc | ab.be.ac=2 40.6530 (abe)? =36100 answer 65) OR/OF a= oP bai Cc value of c 0 Volume = abc answer (5) [22] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Mathematics P1 5 Marking Guidelines/Nasienriglyne QUESTION/VRAAG 2 2.1.1 | 59 answer e) 2.1.2 15 29 41 51 \ f \ sf \ fF 14 12 10 \ Ss 0\ 2 2 2a=-2 ¥ second difference of — 2 a= -l Ya 3(-1)+b=14 b=17 Vb (-1)+(17)+e=15 c=-l Ye T, =n? +17n-1 (4) 2.13 | 7 =-(27)'+17(27)-1 substitution 27 answer 0) 221 | -18 1 "36.22 v answer (1) 2.2.2 nl nl 1 (1) T, =36| —— T, = 36| — —| 2 2) 9 1 n-1 —_ =36{ —— 4096 2) 1 ( a. 1 yy 16384 \ 2) 16384 (2) aaa L2J 4 2) 14=n-l answer n=15 (3) OR/OF OR/OF 36;-18:9; 228s 2 4 8 4096 If you look only at the denominator: 2; 4; 8; ... ; 4096 2* = 4096 2* = 4096 pk — 912 k=12 k=12 “.n=15 terms answer (3) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Mathematics P1 Marking Guidelines/Nasienriglyne 2.2.3 S = a 2 Jop . 36 Y correct subst into correct = 1 formula with -—1<r<1 1-/-— 2 =24 v answer if—1<r<1 (2) 2.2.4 250 -1s\(1)> -1 1 q v r=_ andn=250 So50even = 1 4 ~-] 4 =-24 v S50 even = —24 250 36f(1Y" J 4 so aN J 250 odd 1 1 4 v Sys0.0dd = 48 =48 Soa _ 48 Seven — 24 =2 v answer (4) OR/OF OR/OF T, +7; +T; +T, +...+ Tago T, +1, +15 +71, +...+T 500 _ atar? tart +...+ar%8 Y atar2 +ar4+..+ar8 art+ar> +ar>+...+ar*® Vart+ar +ar+...+ar _ atar? +ar*+...+.ar%8 (atar?+ar*+..+ar%) “4 I a+ar? +ar4+..+ar®*) oF =2 v answer (4) [17] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Mathematics P1 7 Marking Guidelines/Nasienriglyne QUESTION/VRAAG 3 3.1.1 | p+6-(2p+3)=p-2-(p+6) Y equating i.t.o p —p+3=-8 v simplifying p=ll (2) 3.12 | 7, =25+(n-1)(-8) =33-8n Y subst into 7, formula 33-8n <-55 —8n <-88 nol von>il “. Term 12 will be the first term smaller than —55 vn=12 3 «. Term 12 sal die eerste term kleiner as — 55 wees. @) 320 |g =" ta+n= (x3) +(x-18)] 6 2 2 = 6x - 63 ¥ 6x—63 s =" a+r = [e—-3)+@-2) 5 2 2 = 9x -135 v 9x -135 6x — 63 = 9x -135 3x = 72 y mas is “ S = atl = > [(x—-3)+(x—45)] V7 [en 3) +r 45)] 2 2 = oy — 4g] 2 =b 2(24) -48]= 0 = RHS ¥ substitution of x [2(24) — 48} 2 (5) OR/OF OR/OF 9 Y'-3k)=0 a Y expansion (x-21)+(x—24)+ (x-27)=0 V 3x_72=0 “.3x-72=0 ~ 3x = 72 x=24 v 24 15 Y'(24-3k) k=l = 21418415 4.....+-21. Y substitution of x Ss ="[a+)] "2 15 v sum of 15 terms =—}21-21 at ] (S) =0=RHS OR/OF OR/OF Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Mathematics P1 8 Marking Guidelines/Nasienriglyne (x —3)+(x- 6) + (x—9) + («-12) + («—-15) + (-18) Vv expansion =(x—3)+(x-6)+(x—9) + (x—12) + (x -15) + (x -18) +(x—21)+(x—-24)+(x-27) v —72= -3x-72=0 3x—72=0 3x =72 V x=24 24 15 (24 - 3k) k=1 = 21418415 +.....4-21. Y substitution of x S,="[a+]] 2 15 =—|21-21 2 [ ] ¥ sum of 15 terms (5) =0=RHS [10] QUESTION/VRAAG 4 41 y>0 v answer qd) OR/OF OR/OF ye(0; 0) v answer Q) 42 6 g:y=| = 2 1. fl y ere(3) y=log i or y= —log | x or ys log, | ¥ equation (2) 2 x 43 Yes. The vertical line test cuts g? once v yes Ja. Die vertikale lyn toets sny g' slegs eenkeer. Y valid reason (2) OR/OF OR/OF Yes. For every x-value there is a unique y-value v yes Ja. Vir elke x-waarde is daar ‘n unieke y-waarde Y valid reason (2) OR/OF OR/OF Yes. g is a one-to-one function / Ja. g is ‘n een-tot-een funksie | ¥ yes Y valid reason (2) OR/OF OR/OF Yes. The horizontal line cuts g only once v yes Ja. Die horisontale lyn sny g slegs een keer Y valid reason 2) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Mathematics P1 9 Marking Guidelines/Nasienriglyne 441 y =-logo x 2=-log: a ¥ correct subst into correct a1 ay 1 formula (a ; 2) =2 = = = “ 4 “4 Lo 4 v answer (2) 442° |ulo:'l or mM '(2;a) Y answer a) () 45 M/l_4-71 v4 a 9 ry Vv2 4 (3) [1] QUESTION/VRAAG 5 5.1.1 x=-2 v answer = v answer y=3 (2) 5.1.2 ( 0: 7) [ 3 5 v answer Gr Q) 5.1.3 43=0 x+2 vy=0 143(x+2)=0 3x=—-7 xa ~ 3 v answer x-intercept (_ 1 ; 0) Q) Ls) 3.14 4 y | v asymptotes at y=3and =-2 3,5 * a _. Y intercepts at a Bp y=3,5 and x=-2,3 ¥ shape x (reasonable —2,3\ -2 0) representation in correct quadrants) | (3) , Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Mathematics P1 10 Marking Guidelines/Nasienriglyne 521 | —2x+4=0 vy=0 2x=4 x=2 vx=2 “.S(2; 0) (2) $.2.2 Equation of k: y=a(x+1) +18 Y y=a(x+l) 418 O=a(241V +18 = or O=a(-4+1) +18 7 substitute (2,0) or 9a=-18 €4;9) a=2 Ya y=-2(x +1) 418 GB) 5.2.3 — 2x? — 4x +16 =-2x+4 Y equating —2x? -2x+12=0 x? +x-6=0 Y standard form (: * 3)(« 7 2) =o 3 =) v choosing x =-3 ¥= >? OF X= v answer y=-2(-3)+4=10 T(-3 ; 10) (4) 5.2.4 x<-3 or x>2 vv answer (2) OR/OF OR/OF vv (0; -3)U(2; w) answer (2) 5.2.5(a) | x<-l ¥| answer (2) OR/OF OR/OF (-o ; -1) v| answer (2) 5.2.5(b) 0 shape of cubic with local min tp moving to local max tp turning points at x=2 and x =-4 O point of inflection at x=-l (3) [24] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Mathematics P1 Marking Guidelines/Nasienriglyne QUESTION/VRAAG 6 61 | f@=x +2 fe th) =(x+hy +2 =x 4+2xh+h? +2 = 2xh+h* ho =lm h0 h hoo h hod =2x OR/OF f (x) = lim Let = £0) =li foe+h)— f(x) = + 2xh + #2 4+2—(2 +2) £1 () = tim LD = £0) 2xh +h? =lim h(2x +h) = lim(2x + A) Vv x eben th b2-(x, +2) m2 2 2 hoo h li 2xh + h? =limst oo" h0 h = fim Hx +4) h>0 h = lim(2x + h) hv =2x V x? +2xh+h? +2 2xh + I? v lim hoo h v lim 222 +4) h0 h v answer 6) OR/OF Y x 42xh+h? +2 2 v im eth hoo h v lim 222+) h>0 h v answer (5) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Mathematics P1 Marking Guidelines/Nasienriglyne 6.2.1 y=4x5 + 2x7! VY 42x71 dy _ 2 _ 94-2 e =12x 2x VY 1x? v —2x7 (3) 6.2.2 y=4¥x + 333)? 1 = axe + 9x8 V 4x3 v 9x8 dw 42 4 5 nis Ste vot v 54x (4) 6.3 Point of contact: (1; 5) m=2 ¥ m=2 y-y,=m(x-x,) or y=2x+e Y substitution of (1 ; 5) y—S5=2(x-1) S=2+e¢ c=3 y=2x+3 y=2x+3 Y answer (3) [14] QUESTION/VRAAG 7 7A 3 Ix) =-2+ )r—1)Gr+3) vv¥—2¢ +e —De +3) Y correct simplification A(x) = -(2x +3)(2 + 2x-3 P ()=-2r+9(e +2x-3) 6) h(x) = -2x3 —7x? +9 OR/OF OR/OF A(x) = —(2x + 3)(x -1)(x + 3) VV —(2x + 3)(x-1)(x + 3) Y correct simplification = 2 W(x) =-(0x+3)(2 +2x-3) 6) h(x) = -2x3 — 7x? +9 7.2 h'(x) =—6x? -14x y 2 _ first derivative —6x* -14x =0 v=0 —2x(3x+7)=0 x=0 or x=—~ 3 Y both answers (3) 73 x<—l or x>0 vv answer 3 (2) OR/OF OR/OF ences T Yu(0s) — > vv answer ( 3 } (2) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Mathematics P1 13 Marking Guidelines/Nasienriglyne TA y=4x+7 v y=4x+7 — 6x? -14x = 4 v W(x)=4 0 = 6x? +14x+4 Q=3x2 + 7x42 0 standard form 0=(3x+1)(x +2) x=-_ or x=-2 Y both answers 3 (5) [12] QUESTION/VRAAG 9 8.1 Volume of Sphere -4 a(8)° = ABT = 214,66 3 3 v answer e) 8.2 rt? =8? (Pythagoras) Y substitution or reason r?=64—x? Pythagoras () 83 Veone = 1 ah 3 1 (6a )(g ) Vh=8+x =" 64-x J(8+x 3 v 1 (64-22 )(8+x) z 2 3) 3 = 3 512 +64x—8x° —x Y expansion av 040 lon, 32 yi _O4n lor 3a, ax 3 3 3 a 3 3 3 0 = 64 -16x — 3x7 0=(8-3x)(x+8) x= 8 x#-8 Vx= 8 3 3 1 of 512 32) Veone 3 L. he ) v volume of the cone Venere 2048 8 3 “3 Vv > oop 7 7 (7) [9] Copyright reserved/Kopiereg voorbehou TOTAL/TOTAAL: 120 Please turn over/Blaai om asseblief