Mathematics Grade 12 P1 North West Mock 2023 Paper & Memo

education Depariment: Education North West Provincial Government REPUBLIC OF SOUTH AFRICA NATIONAL SENIOR CERTIFICATE 12611E X10 This question paper consists of 9 pages and 1 information sheet. Copyright reserved Please turn over

PRawnleaded from Stanmorephysics.com mEemeeerzis NSC INSTRUCTIONS AND INFORMATION Read the following instructions carefully before answering the questions. 1. Thisquestion 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 Please turn over

Dawnleaded from Stanmorephysics.com NSC QUESTION 1 1 Solve for x: 1.192022 — fa 5x con, 1. Bipp x’ -2x-13=0 (correct to TWO decimal places.) L13 (x-2)i-x) <0 LL4 2¥2x-l=x-ll 1.2 Solve for x and y simultaneously: 3x-y=4 and x +xy=24 1 1 i 1 1) 130 Let S=)14+—] 14=}}1+- ) 14+—}...)1+— | where meN and TO 8 OJ 10 m) 7T<m<30. Calculate all the value(s) of #2 for which S will be a natural number. (Show ALL calculations) QUESTION 2 2.1 Given a quadratic number pattern: -120 ; -99 ; -80; -63:... 2.1.1 Write down the next TWO terms of the pattern. 2.1.2 Determine the 7“"term.of the number pattern in the form T,=an' +bn+c. 2.1.3. What valué must be added to T, for the sequence to have only one value of w for whith TZ, = 0? 2.2 Given a finite arithmetic series: 9+ 14+ 19+... + 124 2.2.1 Determine the generabterm of this series in the form T, =dn+c. 2.2.2 Write the series in sigma notation. nny 4000 2.3 Given the arithmetic series: 2° +2°" +3.2° +2"? +... danny 3 Copyright reserved Calculate the value of x if the sum of the first 30 terms add up to 3 720. NW/September 2023 (3) (3) Q) (5) (6) (5) [25] 2) (4) ) (4) [19] Please turn over

Rawnleaded from Stanmorephysics.com —swseetemnber 2023 NSC QUESTION 3 3.1 Given: 5; 10; 20: ... a geometric sequence. 3.1.1Metermine the n” term. 3.1.2-—€alculate the sum of the first 18 terms. 3.2 The first and second term of a geometric series is given as (2x—4) and (4x° -16) respectively. Determine the value(s) of x for which the series will converge. 3.3. Aconvergent geometric series has a first term of 2 and r= Calculate So : 2 QUESTION 4 Given: f(x)= 7 _» x~1 4.1 Write down the equations of the asymptotes of f- 4.2. Draw the graph of /f. Clearly label ALL the intercepts with the axes and the asymptotes on your graph. 4.3. Determine the equation of the line of symmetry of f for m<0. 44 — For which values of x will f(x) <—4? QUESTION 5 The graph of g(x) = af 3) —5 passes through A(—2 ; —4). 5.1 Show that ae. 25 5.2. Determine the coordinates of the x-intercept of g. 5.3. Given: ms)-(4) : 5.3.1 Determine the equation of A(x) , the inverse of h, inthe form yo... 5.3.2. Describe the translation from g to h. qd) (2) (4) (3) [20] @) (A) (2) (2) 10} 2) (3) Q) (2) 19 Copyright reserved Please turn over

Oawaleaded from Stanmorephysics.com NW/September 2023 NSC QUESTION 6 The sketch below represents the following graphs: f(x) = ax’ 4 bx+¢ and x)=-xdQT gtx) 2 “in * Tha of symmetry of f cuts the x-axis at B, which is also the x ceptof zg. ® =27'0) is an x-intercept of f. e PO; -12) is the »-intercept of /. eh 6.1 Calculate the coordinates of B. @) 6.2 Determine the equation of f in the form f(x) =ax’ +bx+c. (4) 6.3 If f(x)=-x’ -8x—12, determine the coordinates of G if FH, the vertical length between the points of intersection of f and g, is a maximum. 4) 64 For which values of x will x. f'(x)>0 ? (2) Saar [12] dena? ura Copyright reserved Please turn over

Rewnload ed from Stanmorep: hysics.com NW/September 2023 NSC QUESTION 7 7.1 Convert an effective interest rate of 11,3% p.a. to its equivalent nominal rate per annum;-compounded quarterly. G) 7.2 Lisa opened a savings account and deposited R10 000 immediately into the account=The account paid interest at 5,3% per annum, compounded monthly. She Started making additional monthly deposits of R500 into the account three months after the account was opened. Her last monthly deposit of R500 was made 5 years after the account was opened. How much money was in the account 5 years after the account was opened? (6) 7.3 Sam wants to buy a house and takes out a loan of R860 000. He can only afford to pay R7 200 per month starting one month after the loan is granted. The interest rate is compounded monthly at 9,5% p.a. 7.3.1 Calculate the number of payments that Sam will make to repay the loan. 4) 7.3.2. How much will Sam pay in the last month to settle the loan? (5) 17] QUESTION 8 8.1 Determine f(x) from first principles if f(x) =1-2" (5) 8.2 Determine: 82.1. D, [se°-2] @) x 8.2.2 2 if y=Vx(Vx-sx) “) [12] Copyright reserved Please turn over

Rownloaded from Stanmorgphysics.com NSC QUESTION 9 The graph of _f(x) = ax® + bx* has stationary points at p(2 ; -4) and O(0; 0) iN NW/September 2023 9.1 Prove that qe and pei. 12 4 9.2 For which values of x will f be concave down? 9.3 Determine the equation of the tangent to f at x=—2 in the form J, =mx+n. 9.4 Use the graph to determine the values of k for which Jeg 12 =0 will have three real roots. 12 4 Copyright reserved (A) (3) (4) 2) [13] Please turn over

Pewnloaded from Stanmorephysics.com NSC QUESTION 16 NW/September 2023 In the sketch below, AABC is an equilateral triangle with side AB = m units. DEGF isa reptangle with BF = GC = x units. Qo A D E B F G Cc 10.1 Prove that the area of the rectangle DEGF is /3x(m—2x) (3) 10.2 Determine, in terms of m , the maximum area of the rectangle. (4) 7) QUESTION it 11.1 Given: P(A)=0,3, P(B)=0,43 and P(C)=0,18. The events A and B are independent. The events A and C are mutually exclusive. 11.1.1. What is the probability that A and C will happen simultancously? (1) 11.1.2 What is the probability that at least one of A or B will take place? GB) 11.2 A group of 250 Grade 12 learners participated in a survey. They were asked if they used the social media app TikTok. The results are represented in the table below: Use TikTok Don’t use TikTok Fotal Girls 105 25 130 Boys 68 52 120 Total 173 77 cei? AUD] 11.2.1 A name is drawn randomly out of the group. What is thelQ] probability that the name drawn will be a girl and using! Tok? 1) 11.2.2 Are the events being a girl and using TikTok independent? Motivate your answer by doing appropriate calculations. (4) Copyright reserved Please turn over

Rawnleaded from Sta nm orgp h ysics -com NW/September 2023 NSC 11.3 A group consisting of 5 boys, 6 girls and a teacher goes to the movies. They buy 12 tickets in a row next to one another. One of the seats is next to an aisle. 11.3.1_In how many different ways can they be seated? qd) To avoid disruptive behaviour, the teacher considers to place the following restriction on how they may sit: ~The teacher at the end next to the aisle, and the children in any order as long as Peter and John do not sit together. Calculate in how many different ways they can be seated. GB) 11.3.3. Duncan wants to sit next to Suzie. If the whole group of twelve are seated at random, what is the probability that the two of them will sit together? (3) [16] TOTAL: 150 Copyright reserved

Rewnleaded from Stanmorephysics.com ema NW/September 2023 INFORMATION SHEET: MATHEMATICS _ -btvb* —4ac A= P(i A= P(i—ni) A=P(t~iy” A= P(+i)" T, =a Da 8, => Qa+(n-d) T, =ar"™" Ss _ ar’ -1) 3 rel S,=-;-ler<l . r-l l-r pansy -1 poal-+iy") F —_— 2 F(xth)— f(x) yy = lim Lets) Le) h>0 h XX Yt+y. dae —%) +0, -HY Mj 2 ; bo 2 2 y=mxte yoy, =m(x-x,) m=227*1 m=tan@ xX) —X, (x-a¥ +(y—-bY =? mnaspce; —2_=>_._© sn A sinBo sinc area AABC = 5 ab. sm C sin(a + 8) = sina.cos f + cosa.sin 8 cos(a + h)= cosa.cos # —sina.sin (cos? a-sin’ a cos 2a =<1-2sin? @ [2c08* 1 _ x a aie n n(A) P(A) = (A) lS) poathx Copyright reserved a? =b? +¢7 —2be.cos A sin(@ - £) = sin a.cos 8 - cosaz.sin B cos(a - f)= cosa@.cos § +sin a.sin § sin 2a = 2sina.cosa a 2 XG _ x) o = 4} W P(A or B) = P(A) )— P(A and B) pe ee FP) dSi@-xY

anmorephysics.com education Department: Education North West Provincial Government REPUBLIC OF SOUTH AFRICA These marking guidelines consists of 16 pages. Copyright reserved Please turn over

Bownleaded from Stanmorephysics.com Nw/september 2023 NOTE: NSC-Marking guidelines e Ifa candidate answers a question TWICE, only mark the FIRST attempt. e Consistent Accuracy applies in ALL aspects of the marking memorandum. QUESTION? Ll | x Ls¢ x LSQk o ¥ standard form x(¥=5) -0) Y factors 0 ~ v x-values (3) x=0 or x=5 1.1.2 | x?-2x-13=0 2 Y substitution into the ~(-2)+ {2-413 su x= (2) tVC2) OC!) correct formula 2(1) Vx = 4,74 x=4,74 or x =—2,74 vx =-2,74 (3) 1.1.3. | (x-2)(1-x)<0 ae Y critical values xSlorx22 / xé(-;lJor[2;0) Yx<l Vv x>2 (3) OR (x-2)(x-1)20 xSlorx22 / xe(—c;ljor[2;0) 114 | 2/2x-t=x-11 4(2x-1)=x° -22x+121 ¥ square on both sides 8x-4=x° —22x4121 x? 30x +125 =0 Y standard form (x—25)(x-5)=0 ¥ factors/formula x=25orx¥5 ¥ both answers ¥ reject x=5 OR (5) 2V2x-1l=x-11 x1 = 2 2x-1= x= 2244121 v square on both sides 4 8x—4=x° —22x +121 x’ -30x+125=0 Y standard form (x-25)(x-5)=0 Y¥ factors/formula -2 Y both answers x= 25 orx#5 Y reject x=5 (5) Copyright reserved Please turn over

Bownleaded from Stanmorephysics.com Nw/september 2023 NSC-Marking guidelines 1.2 3x-y=4 y=3x-4 x +xy =24 x cE 4)=24 —4x =24 “R x-24=0 x56 =0 (x-3)(x+2)=0 x=3 or x=-2 y =3(3)-4 or y=3(-2)-4 Y y=3x-4 ¥ substitution Y standard form Y factors/formula Y both x-values y=5 y=-10 Y¥ both p-values (6) OR 3x=y+4 -2+4 VY xa t4 3 3 x +xy = 24 2 (4) +y(234)-24 ¥ substitution 2 2 y +8yt16 y +4Y 44 9 3 y +8y+1643y? +12y =216 Ay’ +20y—200 =0 Y standard form y +5y-50=0 (v+10)(y-5)=0 Y factors/formula y=-10 or y=5 Y both y-values _(10)+4 +4 3 3 xy=2 x=3 Y both x-values (6) * [se(ea ras} le) a8 “O)ale}- Gh (s}-C5) i S= —_ —_ mt+1 7 9 \ 7 ry For S to be a natural number, m+1 must be a multiple of 7 v fae m+1=14 or m+1=21 or m+1=28 les 14,21,28 m=13 m=20 m=27 v answer (5) [25] Copyright reserved Please turn over

Bownleaded from Stanmorephysics.com Nw/september 2023 NSC-Marking guidelines QUESTION 2 2.1.1 | -120 -—99 Ww Wy ~19 17 -2 -2 Y 48 t TWO terms: —48 ; —35 y -35 (2) 2.1.2 3a+b=21 a+b+c=-120 v 2”4 diff = — 3(-l)+b=21 (-1) +(24)+e=-120 Ya=-l b=24 c=-143 vb =24 T, =-n? +24n-143 Y c= 143 (4) 2.1.3 | T'n=-2n+24=0 ¥ method n=12 v¥ n=12 T, =—(12)? +24(12)-143 1,=1 A maximum of | v maximum | Add -1 to Tn v-l (4) OR _ 74) _ “21 v¥ method T, =-(12)? +24(12)—143 ¥ n=12 [,=1 A maximum of 1 ¥ maximum | Add -1 to Tn y-l (4) OR T, =-W +24n-143+k Y method = (24) —4(-1)(k -143) =576+4k —572 =4k+4 v A=4k+4 but A=0 YA=0 4k+4=0 k=-l ¥ _ (4) 2.2.1 | 94144+19+4...4124 Answer only: full marks T, =(9)+ (2-5) T, =5n+4 v avon into the rétt formula 7 asne4 (2) Copyright reserved Please turn over

Bownleaded from Stanmorephysics.com Nw/september 2023 NSC-Marking guidelines Copyright reserved 22.2 | 7 =5n+4=124 Y =124 5n=120 n=24 ¥ n=24 Ds 4) Y answer (3) 23 | 2 a o Ea VY a=2*and d=2* 30 + x Soo =F )+29(2 | Y substitution into the 3720 = 15(31.2") correct formula 248 = 31.2" se24 Y 2° =8 2=2* x=3 Y x=3 (4) [19] QUESTION 3 3.1.1 5;10;20;.... T,=ar"' T, =()(2)"7 v answer (1) 3.1.2 _a(r"=1) " r-l 5] (2)"*-1 f= s[e"=1] Y substitution into the 2-1 correct formula Sig = 1310715 v answer (2) 3.2 22x44) (2x4) 4g Y r=2x+4 ed Converge: -1< r< 1 Y -ler<l -l< 2x+4<1 Y substitution -5< 2x< -3 -2 <x< -5 Y answer (4) anal dona] py Please turn over

Bownleaded from Stanmorephysics.com Nw/september 2023 NSC-Marking guidelines Copyright reserved 3.3 v8, v S$, Y dividing Y answer (3) [10] QUESTION 4 4.1 x=l Y x=l y=-2 Vv ye? 0) 42 ‘ | . v x-asymptote and iV y-asymptote v x-intercept v y-intercept v form-decreasing (4) 43 ya-xte OR y-y=-I(a-x) —2=-(I)+e y=-x+1-2 v¥ method re yeoxet v answer (2) y=-x-l Answer only: full marks 4.4 Osx<l Y Oandl v inequalities (2) OR ¥) id 1 xeL0;1) ackets (2) gy [10] — Please turn over

Bownleaded from Stanmorephysics.com Nw/september 2023 NSC-Marking guidelines QUESTION 5 5.1 1y" =a|—| —5 y o( 2] 2 -4 4 -5 Y substitution 5) Y simplification (2) 5 5.2 x lf 0=—|=] -5 = #(5) ¥ y=0 s-3(3) 25\5 Y simplifying 125= (3 53 =5-* x=-3 (-3:0) v answer G) 5.3.1 hive ly Dye 5 LY Answer only: «=-(4] full marks Y swop x and y y=log, x or y=—logsx Y answer (2) 5 gy 3) (5 ny? ec=(4] -5 Y rewriting g 2 units right and 5 units up ¥ 2 units right and 5 units up (2) 19] Copyright reserved Please turn over

Bownleaded from Stanmorephysics.com Nw/september 2023 NSC-Marking guidelines QUESTION 6 6.1 I = 2x42 vase Y y=0 ¥ answer (2) 6.2 x, )(x-x)) yaales 2)(x+6) “D=c(0 +2)(0+6) -12=12a a=-l ye-(x? +8x+12) y=-x’ -8x-12 OR (-2;0): 0 = a(-2)? + b(-2)-12 12=4a—2b 6=2a—b.......{1) and ~_ 4 2a —b=-8a b =8a.....(2) 6=2a-8a 6=-6a a=- b=8(-1) =-8 y=-x’ -8x-12 Y coordinates of A(-6;0) Y substitution ¥Ya=-l v answer (4) ¥ substitution Y substitution Ya=-l1 v answer (4) Copyright reserved Please turn over

Bownleaded from Stanmorephysics.com Nw/sepiember 2023 NSC-Marking guidelines 63 FH =x 8x-12-( 3702] Y FO)- 9%) —-x* ~8x-12-1x-2 2 Taal Y FHitox di -oy—17 _9 or yb v¥ method 2 2a yy eT (85) 2 2(-l) 17 17 x=-— x=-— 4 4 17 o(-Z0) v answer (4) 6.4 4<x<0 v answer Y inequility (2) OR v -4and0 xe(—4;0) Y inequility Q) [12] QUESTION 7 TA i” ” 1+i, =| 14+— om m i')’ 1+0,113 [4] Y substitution i 1,027...=1+7 Y 4" root it 0,027...=— 4 0,1085..=i' r=10,85% Y answer (3) Copyright reserved Please turn over

Bownleaded from Stanmarephysics.com Nwiseptember 2023 NSC-Marking guidelines 7.2 Y substitution into the 0,054..= (14°95 12 —N = 104 097.) 9,054... —n =—369, 212... Sam will have 370 installments 0,053)" A= 100001 + oe) correct formula A=13026,71 ¥ n= 60 and (= 22> Y substitution into the F fanny formula TT ¥ n=58 lon Ee 970,51 Total =13026,71+32970,51 = R45997,22 Y answer (6) OR 500] (14 0,053)" -] Y¥ substitution into the 0,053)” 12 correct formula =10000{ } + 0.053 0,053 U9 v¥ n=60 and i= 12 Y substitution into the F formula v n=58 = R45997,22 ¥ answer (6) =n Y substitution into the 7.3.1 ram) 1 | correct formula -—__t_ . 0,095 860000 0,095 v i<~ 12 0,945... -1-(14 228) 12 —0, 054... = (i + os | 12 Y correct use of log v answer (4) Copyright reserved dana? foan! fag Please turn over

Bownleaded from Stanmarephysics.com Nwiseptember 2023 NSC-Marking guidelines 7.3.2 309 A= 860.000{1 + oe) A= R15782859,31 12 F = R15781334,69 Balance = R1524,62 (after 369 installments) 1 Last installment = 1524, 62 [ + o, 7) = R1536,69 OR > 0,2127679... 7200 i-{1+ "| 12 P= 0,095 12 P= R1524,62 (after 369 installments) i Last installment = 1524, 62 [ + o, 7) = R1536,69 Y¥ loan and interest ¥ n= 369 Y payment and interest v method v answer (5) v¥ method v¥ n=-0,2127679... ¥ balance ¥ method v answer (5) [17] Copyright reserved Please turn over

Bownleaded from Stanmarephysics.com Nw/september 2023 NSC-Marking guidelines QUESTION 8 8.1 S@=l-x =1-(x+h) =1-(x° +2xh+h?) =1-x?-2xh-h? ¥ =1—-x?-2xh—h? f (x+h)-f (x) h 1—x° —2xh-h’ -(1-x°) = lim h Y substitution —2xh—W* _ -2xh-h? ne) h v “h - h(-2x-h) — lim F Y -2x-h = lim(—2x—h) f= 2x v —2x (5) 8.2.1 bax =2] x Y —2x71 D. [3x7 -2x""] v 6x =6x+2x? ¥ 2x7? @) 82.2 | yavx (ir —5r) it - yar? fi *4 > 3 yH=xo=Sx? 1 T 35 N53 dx 6 2 Y change from surd to exponential form Copyright reserved Please turn over

Bownleaded from Stanmarephysics.com Nwiseptember 2023 NSC-Marking guidelines QUESTION 9 9.1 f (x)=ax’ +bx* -5 pay +60) SF (x) = 3ax’ + 2bx f'(2)=3a(2)? +25(2) = 0 12a+4b=0 4b =-12a b =-3a.....(2) 1 —==8a+4(-3 3 oe (—3a) v substitution (2 3 -1) Y f(x)=0 Y substitution x=2 Y solve simultancously 1 =8a-124 3 14g 3 1 a-— 12 1 b=—7 (4) " 1 1 9.2 f (x)J=5%-5 Y fe) 1 1 3775 <0 Y f"(ay<0 x<l / xEe(-0 31) v answer (3) Copyright reserved Please turn over

Bownleaded from Stanmoarephysics.com Nwiseptember 2023 NSC-Marking guidelines 93 ~t. la FO D* 4 Ion, 15 5 Y y=? —?2) =—(-?)’ ——(-2)° =-= y= S-) a! ) 4! ) 3 3 :-3) > 3 (xjste—ty v gradient 4 2 1 | ¥ substitution of xy and (2) =—(-2)? —=(-2 substitution of x,y and m S'(-2)= 402-52) m=2 y=2xt+e 32-2) +e 3 7 c== 3 y=2xt— v answer (4) 9.4 1 1 -<k<0 / ke -330 vv answer (2) [13] QUESTION 10 DF 10.1 tan 60° = — = v3 an x 8 Y tan60° = V3 DF = 3x Y DF =%3x «. Area rectangle = DF x DE Y DE=(m-2x) =V3x(m—2x) GB) Copyright reserved Please turn over

Bownleaded from Stanmavephysics.com Nwiseptember 2023 NSC-Marking guidelines 10.2 Area =V3mx —2V3x2 “ = 3m -4V3x =0 x=— 4 M a= V3x(m —2x) -5(F("--()) Be m 8 OR dA _ 4 = Bm—aV3x=0 a Nee m x=— 4 V3mx —2V3(x) 2 m m — Bm? 3m? 4. 8 _ 23m? —8V3m? 7 8 _ 3m’ 8 Y fiy=0 v substitution N32 vm (4) Y f(x)=0 vy” 4 v substitution m (4) [7] Copyright reserved Please turn over

Bownleaded from Stanmorephysics.com Nwisepiember 2023 NSC-Marking guidelines QUESTION 11 11.1.1 | P(AandC)=0 v answer (1) 11.1.2 | P(AandB) = P(A)*P(B) P(AgndB) = 0,3x 0,43 P(Adid B) = 0,129 Y P(AandB) =0,129 P(AGEB) = P(A) + P(B)—P(A and B) P(AorB) = 0.3 +0,43-0,129 ¥ substitution P(A or B) = 0,6 v answer (3) 11.2.1 P(GandT) = 105 = 21 < gv, Y answer (1) 250 50 11.2.2 | Independent: P(T and G) = P(T)x P(G) P(T)xP(G) P(GandT) Y PT)= 173 173 _ 130 105 250 550 ~ 950 550. 130 250 250 250 Y P(G)=—— = 0.36 = 0.42 250 P(GandT) # P(G) x P(T) Y P(T)xP(G)=0,36 Events are not independent. ¥_answer (4) 11.3.1] 12! Y answer qd) 11.3.2 | Pieter and John next to one another =10!2! v 10!2! 11!-10!2! v 11!10!2! = 32659200 ¥ answer (3) 11.3.3) 1112! 12! v 11!2! 1 v 12! = 6 Y answer (3) [16] TOTAL: 150 son] Copyright reserved