Mathematics Grade 12 P2 Eastern Cape Prelim 2024 QP

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE KEREITI YA 12 LOETSE 2024 MMETSE P2 MATSWAO: 150 NAKO: Dihora tse 3 Pampiri ena e na maqephe a 14 ho kenyelletsa le leqephe le 1 la tlhahisoleseding mmoho le buka ya dikarabo e nang le maqephe a 25. Hosted by www.ecexams.co.za.

Downloaded from hlayiso.com 2 MMETSE P2 (EC/LOETSE 2024) DITAELO LE TLHAHISOLESEDING Bala ditaelo tse latelang ka hloko pele o ka araba dipotso. 1. Pampiri ena e na le dipotso tse 10. 2. Arabela dipotso KAOFELA ho BUKA E KGETHEHILENG YA HO ARABELA o e filweng. 3. Bontsha ka ho hlakileng KAOFELA dikhaltjhuleishene, didayakeramo, dikerafo j.j. tseo o di sebedisitseng ho fumana dikarabo. 4. Dikarabo fela di ke ke tsa abelwa matshwao a felletseng. 5. O ka sebedisa khaltjhuleitha ya saentifiki (e sa phorekeremuwang, le e se nang dikerafo), ntle le ha ho boletswe. 6. Ebang ho hlokeha, atametsa dikarabo tsa hao ho didesimale tse PEDI, ntle le ha ho boletswe. 7. Didayakeramo HA DI latele tekanyetso e nepahetseng. 8. Pampiri ya tlhahisoleseding e nang le difomula e kenyelleditswe qetellong ya pampiri ena. 9. Ngola ka mongolo o hlakileng mme o makgethe. Difuperweng tsa pampiri ena di se sebediswe ntle ho tumello Phetla Hosted by www.ecexams.co.za.

Downloaded from hlayiso.com (EC/ LOETSE 2024) MMETSE P2 3 POTSO YA 1 1.1 Lenane la dilitara tsa diesel tse rekuweng ke bakganni ba dilori ba 15 seteisheneng sa peterole le rekhotilwe ka mokgwa ona o latelang. 82 64 55 50 41 71 78 88 98 96 63 66 80 84 88 1.1.1 Ngola mode. (1) 1.1.2 Ngola range. (1) 1.1.3 Khaltjhuleitha mean. (2) 1.1.4 Khaltjhuleitha standard deviation ya mean. (1) 1.1.5 Fumana lenane la bakganni ba dilori ba rekileng dilitara tsa diesel tse ka tlase ho one standard deviation ya mean. (3) 1.2 Mean ya boima ba batho ba 8 ba kenang ka hara lifti ke 75 kg. Lifti ha e nke boima ba fetang 1 000 kg. Ke batho ba bakae ka lenane ba ka nnang ba kena ka lifting ebang re nahanela hore mean e dula e le 75 kg? (4) [12] POTSO YA 2 Diphetho tsa kereiti ya 8 ya diteko tse pedi moo e le nngwe e ngolletsweng hodima matshwao a 50 di ngotswe ka tlase. TEKO YA A (x) 39 33 35 44 37 40 24 31 30 5 TEKO YA B (y) 41 45 48 40 47 42 37 44 43 24 2.1 Fumana outlier ho tswa tafoleng e ka hodimo. (1) 2.2 Fumana ekhweishene ya least squares regression line. (3) 2.3 Sebedisa ikhweishene ya least squares regression line ho nahanela matshwao a TEKO YA B ebang moithuti o fumane matshwao a 14 ho TEKO YA A. Atametsa karabo ya hao ho whole namba e haufi. (2) 2.4 Tshwaela ka strength sa correlation dipakeng tsa TEKO YA A le TEKO YA B. (2) [8] Difuperweng tsa pampiri ena di se sebediswe ntle ho tumello Phetla Hosted by www.ecexams.co.za.

Downloaded from hlayiso.com 4 MMETSE P2 (EC/LOETSE 2024) POTSO YA 3 Khwadrilatherale AEBC e teroilwe. Dikhoodineithi tsa B ke (3; 4). G, O le F(5; 0) ke x- inthesepthe tsa dilaene AC, AB le BC ka ho latellana. H (1 ; – 4) ke poente e ho laene AC. AB̂C =  . Area of ΔOBF = 12 square units le inclination ya laene AB ke 45o . HC = 2AH y B (3 ; 4)  E G O 450 F (5 ; 0) x A • H (1 ;– 4) C 3.1 Khaltjhuleitha bolelele ba BF. O siye karabo ya hao e le ho surd form e (2) bebofaditsweng. 3.2 Khaltjhuleitha gradient ya BF. (2) 3.3 Khaltjhuleitha saeze ya  . (3) 3.4 Pruva hore HF AB . (4) 3.5 Ho boetse ho fuwe hore, EC o baesektha AB perpendicularly. AEBC ke mofuta ofe wa khwadrilaterale? (1) 3.6 O nto, kapa ka tsela e nngwe o khaltjhuleithe bolelele ba AC. (4) 3.7 Khaltjhuleitha eriya ya khwadrilaterale AOFC. (3) [19] Difuperweng tsa pampiri ena di se sebediswe ntle ho tumello Phetla Hosted by www.ecexams.co.za.

Downloaded from hlayiso.com (EC/ LOETSE 2024) MMETSE P2 5 POTSO YA 4 4.1 Ho dayakeramo e ka tlase, R ke mokgubu wa sekel OPQ. Poente Q ke y-inthasepthe ya 3 sekele. SQT ke thanjente ya sekele ho Q. Ikhweishene ya SQT ke y = − x − 8 . 4 y S O P x R Q T 4.1.1 Khaltjhuleitha dikhoodineithi tsa Q. (2) 4.1.2 Fumana ikhweishene ya QR ka mokgwa ona y = mx + c . (3) 4.1.3 Khaltjhuleitha dikhoodineithi tsa P, x-inthasepthe ya laene QR. (2) 4.1.4 Khaltjhuleitha dikhoodineithi tsa R mokgubu wa sekele. (3) 4.1.5 Ngola ikhweishene ya sekele e nang le mogubu ho R ka mokgwa: (x − a )2 + ( y − b )2 = r 2 . (3) 4.1.6 Ebang y = k ke thanjente ho sekele, fumana di/velu tsa/ya k. (3) 4.2 Khaltjhuleitha bolelele bo hodimodimo ba radius ya sekele e nang le ikhweishene x 2 + y 2 − 2 x sin  − 4 y sin  = −2 . (5) [21] Difuperweng tsa pampiri ena di se sebediswe ntle ho tumello Phetla Hosted by www.ecexams.co.za.

Downloaded from hlayiso.com 6 MMETSE P2 (EC/LOETSE 2024) POTSO YA 5 5.1 Dayakeramong e katlsase, poente S(-4 ; q) le reflex angle  di bontshitswe. O ke poent e ho origin. OS = 3 2 . y  x O 3 2 S(− 4; q ) Ntle le ho sebedisa khaltjhuleitha, fumana velu ya: 5.1.1 q (2) 5.1.2 sin ( + 45o ) (4) 5.1.3 cos ( 2 − 360o ) (4) 5.2 Simplifaya tse latelang ntle le ho sebedisa khaltjhuleitha: sin ( 90o −  ) .cos 480o + cos (180o −  ) cos  .sin150o − tan180o (5) cos x cos 2 x 5.3 Pruva hore − = sin x sin 2 x 2 sin x (5) cos 60o sin 60o 5.4 O fuwe: − =2 sin x cos x cos 60o sin 60o 5.4.1 Bontsha hore ikhweishene − = 2 e ka ngolwa e le sin x cos x cos ( x + 60o ) = cos ( 90o − 2 x ) (3) 5.4.2 O nto , kapa ka mokgwa o mong, o fumana general solution ya cos 60o sin 60o − =2 sin x cos x (4) a O fuwe hore cos 22,5 = o 5.5 le a 2 + b2 = c 2 . c 2ab 2 Ka thuso ya dayakeramo, kapa ka mokgwa o mong, bontsha hore 2 = . c 2 (5) [32] Difuperweng tsa pampiri ena di se sebediswe ntle ho tumello Phetla Hosted by www.ecexams.co.za.

Downloaded from hlayiso.com (EC/ LOETSE 2024) MMETSE P2 7 POTSO YA 6 Kerafo ya f ( x) = − sin( x − 30o ) e teroilwe ho inthavale ya x   −60 ; 180  . o o Sebedisa kerafo ho araba dipotso tse latelang. 6.1 Ngola period ya f. (1) 6.2 Ngola minimum velu ya f. (1) 6.3 Fumana range ya f ( x) + 1. (2) Ke divelu dife tsa x moo kerafo ya f e inkhrisang , moo x   −60 ; 180  ? o o 6.4 (2) 6.5 Kerafo ya f e shiftile 600 ho ya ho le letona mme ya reflekhtwa ka ho x-axis ho etsa kerafo e ntjha ya g. Fumana ekhweishene ya g ka mokgwa o bobebe. (3) 6.6 Teroya kerafo ya g ho sete e le nngwe ya di-axes. Bontsha ka ho hlakileng diinthasepthe ho di-axis le turning point ya x   −60 ; 180  . o o (3) [12] Difuperweng tsa pampiri ena di se sebediswe ntle ho tumello Phetla Hosted by www.ecexams.co.za.

Downloaded from hlayiso.com 8 MMETSE P2 (EC/LOETSE 2024) POTSO YA 7 ̂ = 120o , ̂ Ho ∆ABC e ka tlase, AC = 2x, BC = (16 – 4x), C B = 𝜃. A B θ 2x (16 − 4x) 120o C 7.1 Fumana eriya ya ∆ABC ka mokgwa wa x, ntle le ho sebedisa khaltjhuleitha. (3) 7.2 Ke di/velu dife/efe tsa/ya x moo eriya ya ∆ABC e tla ba maximum? (3) [6] Difuperweng tsa pampiri ena di se sebediswe ntle ho tumello Phetla Hosted by www.ecexams.co.za.

Downloaded from hlayiso.com (EC/ LOETSE 2024) MMETSE P2 9 POTSO YA 8 8.1 Dayakeramong e ka tlase, O ke mokgubu wa sekele. BD ke chord ya sekele. E ke midpoente ya chord BD. O 1 2 B E D Sebedisa tayakeramo o e filweng ho HO BUKA YA DIKARABO ho pruva theorem e reng: Laene e teroilweng ho tloha mokgubung wa sekele e baesekthang chord e perpendicular ho chord. Ka mantswe a mang, pruva hore: OE⊥ BD . (5) Difuperweng tsa pampiri ena di se sebediswe ntle ho tumello Phetla Hosted by www.ecexams.co.za.

Downloaded from hlayiso.com 10 MMETSE P2 (EC/LOETSE 2024) 8.2 Dayakeramong e ka tlase, O ke mokgubu wa sekele. Q ke midpoente ya chord PR. OQS ke rediyase ya sekele. PR = 8 di-unithi, OQ = x di-unithi and QS = 2 di-unithi. O x P Q R 2 S 8.2.1 Fumana, o fana ka mabaka, saeze ya OQ̂P . (2) 8.2.2 Khaltjhuleitha bolelele ba PO. (5) [12] Difuperweng tsa pampiri ena di se sebediswe ntle ho tumello Phetla Hosted by www.ecexams.co.za.

Downloaded from hlayiso.com (EC/ LOETSE 2024) MMETSE P2 11 POTSO YA 9 9.1 A, B, C le D ke dipoente tse ho sekhamfarense ya sekele e nang le mokgubu ho O. AOB ke dayametha ya sekele. AOD ˆ = 70o . D C A 70o 1 O B 2 ̂ , o fana ka mabaka. Khaltjhuleitha saeze ya C (5) Difuperweng tsa pampiri ena di se sebediswe ntle ho tumello Phetla Hosted by www.ecexams.co.za.

Downloaded from hlayiso.com 12 MMETSE P2 (EC/LOETSE 2024) 9.2 PT ke thangente ya sekele ho T. PR TS mme PQR ke straight laene. Q, R le S ke dipoente tse ho sekhamfarense ya sekele. R̂ 2 = y mme Ŝ = x . QR = TS. P Q R 1 2 2 y 1 1 2 3 T x S 9.2.1 Bolela, o fana ka mabaka, di-engele tse ding tse PEDI moo engele ka nngwe e lekanang le y. (4) 9.2.2 Fumana, o fana ka mabaka , engele e nngwe e lekanang le T̂2 . (2) 9.2.3 Pruva hore TR ke dayametha ya sekele. (4) [15] Difuperweng tsa pampiri ena di se sebediswe ntle ho tumello Phetla Hosted by www.ecexams.co.za.

Downloaded from hlayiso.com (EC/ LOETSE 2024) MMETSE P2 13 POTSO YA 10 BCDM ke cyclic quadrilateral. Chords MD le BC di ekeditswe mme tsa kopana ho poente A. N ke poente ho CD. AC MN mme AM = CD . AC = 36 di-unithi, AD = 48 di-unithi le BM = 24 di-unithi. B C 2 1 A 3 2 1 N 2 M 1 D 10.1 Pruva hore ΔABM ||| ΔADC . (4) 10.2 Pruva hore CD2 = BM  AC . (3) 10.3 Khaltjhuleitha bolelele ba CN. (6) [13] MATSHWAO KAOFELA: 150 Difuperweng tsa pampiri ena di se sebediswe ntle ho tumello Phetla Hosted by www.ecexams.co.za.

Downloaded from hlayiso.com 14 MMETSE P2 (EC/LOETSE 2024) PAMPIRI YA TLHAHISOLESEDING: MMETSE − b  b 2 − 4ac x= 2a A = P(1 + ni) A = P(1 − ni) A = P(1 − i) n A = P(1 + i) n x (1 + i ) − 1 x 1 − (1 + i ) −n  n F=   P=    i i n Tn = a + (n − 1)d Sn = (2a + (n − 1)d ) 2 Tn = ar n −1 Sn = ( a r n −1 ) ; r 1 S = a ; −1  r  1 r −1 1− r f ( x + h) − f ( x ) f ' ( x) = lim h→ 0 h  x + x2 y1 + y 2  d = ( x 2 − x1 ) 2 + ( y 2 − y1 ) 2 M  1 ;   2 2  y 2 − y1 y = mx + c y − y1 = m( x − x1 ) m= m = tan  x 2 − x1 ( x − a )2 + ( y − b ) 2 = r 2 1 In ABC: a = b = c a 2 = b 2 + c 2 − 2bc. cos A area ABC = ab. sin C sin A sin B sin C 2 sin ( +  ) = sin  . cos  + cos .sin  sin ( −  ) = sin  . cos  − cos .sin  cos( +  ) = cos . cos  − sin  . sin  cos( −  ) = cos . cos  + sin  . sin  cos 2  − sin 2   cos 2 = 1 − 2 sin 2  sin 2 = 2 sin  . cos  2 cos 2  − 1  n 2  (x − x ) x= x i n( A)  = i =1 2 P ( A) = P(A or B) = P(A) + P(B) – P(A and B) n n n (S ) yˆ = a + bx b=  (x − x )( y − y ) (x − x) 2 Difuperweng tsa pampiri ena di se sebediswe ntle ho tumello Phetla Hosted by www.ecexams.co.za.