Mathematics P2 Sept2021 QP SESOTHO ONLY D hlayiso.com

Downloaded from hlayiso.com SENIOR CERTIFICATE/ NATIONAL SENIOR CERTIFICATE KEREITI 12 DIPALO P2 LOETSE 2021(2) MATSHWAO: 150 NAKO: Dihora tse 3 Pampiri ena ena le maqephe a 15 le leqephe le 1 la tlhahisoleseding. *MATSES2* Ditokelo di sireleditswe Phetla

Downloaded from hlayiso.com Dipalo/P2 2. DBE/Loetse 2021(2) SC/NSC DITAELO LE TLHAHISOLESEDING Bala ditaelo tse latelang ka hloko. 1. Ena pampiri ena le dipotso tse 10. 2. Araba dipotso TSOHLE ho BUKANA E IKGETHILENG YA HO ARABELA. 3. Ka makgethe BONTSHA tsohle dipalo, ditshwantsho, dikerafo, jwalojwalo. tseo o di sebedisitseng ho fumana dikarabo tsa hao. 4. Dikarabo feela di KEKE tsa fumana matshwao kaofela ka nako tsohle. 5. Oka sebedisa khalukhetara e dumelletsweng ya saense (esa porogeremuang ebile ena le dikerafo), kantle le haeba ho boletswe ka tsela e nngwe. 6. Moo ho hlokahalang teng, atametsa dikarabo tsa hao ho desimale tse PEDI, kantle le haeba ho boletswe ka tsela e nngwe. 7. HAHO bolele hore ditswantsho tsohle di takuwe ho latela ditekanyo tse nepahetseng. 8. Leqephe la tlhahisoleding le nang le di-fomula le teng moo pampiri e fellang teng. 9. Ngola ka makgethe leka tsela e bonahalang. Ditokelo di sireleditswe Phetla

Downloaded from hlayiso.com Dipalo/P2 3. DBE/Loetse 2021(2) SC/NSC POTSO 1 Mokete wa selemo wa dipapadi o tshwerwe nako ya matsatsi a 11. Lebenkele le rekisa dino tse tjhesang moketeng ona. Ho letsatsi le leng lele leng ho matsatsi a pele a 10, monga lebenkele o ile a ngola mofuthu ka 13:00 le palo ya dino tse tjhesang tse rekisitsweng. Tlhahisoleseding ena e bontshitswe tafoleng leho sekhatha poloto ka tlase. Mofuthu (in °C) 14 24 26 18 20 28 22 15 12 19 Palo ya dino tse tjhesang tse 410 258 192 324 328 156 280 384 230 280 rekisitsweng 450 SEKHATHA POLOTO 400 Lenane la dino tse tjhesang tse rekisitsweng 350 300 250 200 150 10 15 20 25 30 Motjheso ka °C 1.1 Hlalosa mokgwa oo data eo bontshang. (1) 1.2 Fumana ekweshini ya mola wa disekwere tse nyane tsa rekereshene tsa datha. (3) 1.3 Monga lebenkele o hlokometse hore o sebedisitse litara ele nngwe ya lebese ho dikopi tse 8 tsa dino tse tjhesang tse rekisitsweng. Haeba mofuthu ka 13:00 ka letsatsi labo 11 ene lebelletswe hoba 17 °C, akanya lenane la lebokoso la litara ele nngwe ya lebese monga lebenkele a tlamehang ho e reka ka letsatsi la bo 11. (3) 1.4 Kgetha outlier ho datha. (1) [8] Ditokelo di sireleditswe Phetla

Downloaded from hlayiso.com Dipalo/P2 4. DBE/Loetse 2021(2) SC/NSC POTSO2 2.1 Baithuti hotswa dikolong tse fapaneng ba ngotse teko tsa mahlale hore ba kgone hore ba khwalifaye ho fumana basari. Matshwao a bona (ka peresente) a bontshitswe ho ogive (cumulative frequency kerafo) ka fatshe. OGIVE 190 180 170 160 150 140 Cumulative frequency 130 120 110 100 90 80 70 60 50 40 30 20 10 0 0 10 20 30 40 50 60 70 80 90 100 110 Matshwao (as a %) 2.1.1 Ho ngotse baithuti ba bakae teko? (1) 2.1.2 Ngola modal class ya data. (1) 2.1.3 Matshwao a tlase a ho khwalifayela basari ke 75%. Ke ba bake baithuti batla khwalifayela basari? (2) Ditokelo di sireleditswe Phetla

Downloaded from hlayiso.com Dipalo/P2 5. DBE/Loetse 2021(2) SC/NSC 2.2 Tafole e latelang e bontsha matshwao a baithuti ba 15 hotswa sekolong se itseng a fumanweng ho teko ya mahlale. Matshwao 62 58 78 85 74 48 74 84 100 46 80 92 60 90 92 (ka%) Fumana: 2.2.1 Letshwao la mini le fumanweng ke baithuti. (2) 2.2.2 Setandade devieshene ya matshwao a baithuti. (1) 2.2.3 Lenane la baithuti bao matshwao a bona a fumanehang ho feta setandathe devieshene sele seng ka hodimo ho mini. (2) 2.3 Matshwao a Kereiti 11 (ka persente) aho qetela a fumanweng ke baithuti a lekotswe. Setandathe devieshene sele seng ho inthavale ya mini e fumanwe ele (82,7; 94,1). Fumana setandathe devieshene ya matshwao aho qetela a Kereiti 11. (3) [12] Ditokelo di sireleditswe Phetla

Downloaded from hlayiso.com Dipalo/P2 6. DBE/Loetse 2021(2) SC/NSC POTSO 3 Ho setshwantsho se latelang, mola wa BF o takilwe hotswa ho B(8 ; –2) ho kgaola x-axis ho P 1 le y-axis ho F. Engele ya inclination BF ke β le ekweshene ya BF y  x  2 . Hotswa ho 2 A(4 ; k), mola o mong o takilwe o pharalele ho BF le ho kgaola y-axis ho E. Mola o fetang ho A le ho B ena le inclination ya 45° le ho kgaola x-axis ho C. AB̂F T . y F β 45° O P C x T B(8 ; –2) E A(4 ; k) 3.1 Batla keradiente ya AB. (1) 3.2 Bontsha hore velu ya k ke -6. (2) 3.3 Fumana ekweshini ya EA ka tsela ya y mx  c . (3) 3.4 Fumana: 3.4.1 Boholo ba T (3) 3.4.2 Bolelele ba BF (3) 3.4.3 Eria ya 'ABF (4) 3.5 Haeba G ke ntlha e khwaderanteng ya bone hore APBG ebe pharalelokereme. Fum a na b oh olo b a PÂG. (4) [20] Ditokelo di sireleditswe Phetla

Downloaded from hlayiso.com Dipalo/P2 7. DBE/Loetse 2021(2) SC/NSC POTSO 4 Ho setshwantsho se latelang, S ke ntlha ho x-axis. Sedikadikwe se nang le bohare S le sedikadikwe se nang le bohare M di takilwe. Didikadikwe tse pedi di kopana kantle ho E. FE ke thanjente ya didikadikwe tse pedi ho E. Sedikadikwe sa bohare ba M, ena le ER ele diametha, e kopana le x-axis ho P. y F R M E x S P O 4.1 Ekweshini ya sedikadikwe se nang le bohare S k e ( x  8) 2  y 2 4. 4.1.1 Fumana dikhoodinetse tsa S. (2) 4.1.2 Bontsha hore diametha ya sedikadikwe sa bohare ba S ke diyuniti tse 4. (1) § 8 24 · 4.2 Haeba re boela re nehwa hore SR = 8 diyuniti le R ¨  ; ¸ , bala:: © 5 5 ¹ 4.2.1 Bolelele ba EM (2) 4.2.2 Keradiente ya thanjente FE (3) 4.2.3 Khoodinetse tsa M (4) 4.2.4 Khoodinetse tsa E (2) 4.3 Sedikadikwe sa bohare ba M(–4 ; 3) e s h i f o t i l e u n i t i e 1 hoya ka letsohong le leqele le ho refolekotiwa ho x-axis ho fumana sedikadikwe se setjha ho K. Fumana hore ntlha (–8 ; 0) eka hare kapa kantle ho sedikadikwe ho K. Bontsha mesebetsi YOHLE. (5) [19] Ditokelo di sireleditswe Phetla

Downloaded from hlayiso.com Dipalo/P2 8. DBE/Loetse 2021(2) SC/NSC POTSO 5 5.1 Ntlha T(–1; –2) re e nehilwe setshwantshong se ka fatshe. A ke ntlha ho x-axis hore reflex AÔT D . y A x O x T(–1; –2) Fumana, ntle le tshebediso ya khalukhetara, velu ya tse latelang: 5.1.1 tan D (1) 5.1.2 cos D (2) 5.1.3 cos(D  45q) ho foromo e bobebe. (4) 5.2 Fumana, ntle le tshebediso ya khalukhetara, velu ya expression tse latelang: 2 sin( 20q). sin 160q  cos 40q (4) 5.3 Ela hloko: 3 cos x. sin x  tan x. cos 2 (180 q  x) 5.3.1 Etsa expression ena ebe bobebe hoya ho trigonometric reshio ele nngwe. (4) 5.3.2 Ebe, o ngola renje ya: f ( x) 3 cos x. sin x  tan x. cos 2 (180 q  x) (2) cos 3x 5.4 Pruva identity: 4 cos 2 x  3 (5) cos x 5.5 Fumana jeneral solushene ya x ho ekweshini e latelang: 32 tan x  3tan x1 54 (5) [27] Ditokelo di sireleditswe Phetla

Downloaded from hlayiso.com Dipalo/P2 9. DBE/Loetse 2021(2) SC/NSC POTSO 6 Setshwantshong se latelang, kerafo ya f ( x)  cos x  1 le g ( x) 2 sin( x  30q) di takilwe ho inthavale x [180q; 90q] . y f –180° –150° –120° –90° –60° –30° 0° 30° 60° 90° x g 6.1 Ke velu efe ya x, x [180q ; 90q] , moo: 6.1.1 f ( x).g ( x) t 0 (2) 6.1.2 g ( x) 1 (2) 6.2 y-axis e tsamaisitswe 90° hoya ka letsohong le letona. Fumana ekweshini e ntjha ya kerafo eo qalong e ile ya bitswa f , ho foromo e bobebe. (2) [6] Ditokelo di sireleditswe Phetla

Downloaded from hlayiso.com Dipalo/P2 10. DBE/Loetse 2021(2) SC/NSC POTSO 7 Setshwantshong se latelang, B, C and D di sebakeng se tshwanang se tshekaletseng. AD ke palo e tsepameng e tsheheditsweng ke dithapo tse pedi, AB and AC. Engele tsa eleveishene ho qala ho B le C hoya ho A, tsullung ya palo, ke T le 2T ka ho latellana. AB 5 diyuniti le AC x diyuniti. A 5 x D B C 5 7.1 Bontsha hore x 2 cos T (5) 7.2 Bala bolelele ba BC haeba re nehilwe hore BÂC 112q le T 30q . (3) [8] Ditokelo di sireleditswe Phetla

Downloaded from hlayiso.com Dipalo/P2 11. DBE/Loetse 2021(2) SC/NSC POTSO 8 8.1 Setshwantshong se latelang, dikhodo KL, LP and KP di takilwe ka hare ho sedikadikwe, se nang le bohare ba O. SLR ke thanjente ya sedikadikwe ho L. S L K O R P Pruva theoreme e bolelang hore engele pakeng tsa thanjente SLR le khodo KL e lekana le englele e fumanehang ho althanete segmente, ka hoo pruva hore SL̂K P̂. (6) Ditokelo di sireleditswe Phetla

Downloaded from hlayiso.com Dipalo/P2 12. DBE/Loetse 2021(2) SC/NSC 8.2 Ho setshwantsho se latelang, PWRS ke saetliliki khwaderilatherale ka hare ho sedikadikwe, bohareng ho O. ∆PSW ke ekhwilaterale teraengele. TW ke thanjente ya sedikadikwe ho W. Radii OR le OW di takilwe. Ŵ5 25q . P O 1 2 1 2 3 S 1 4 W 5 25° 1 2 R T 8.2.1 Fumana, o fana ka mabaka, boholo ba: (a) Ŝ1 (2) (b) Ô 1 (2) (c) R̂ 1 (5) 8.2.2 Pruva hore SP || TW. (3) Ditokelo di sireleditswe Phetla

Downloaded from hlayiso.com Dipalo/P2 13. DBE/Loetse 2021(2) SC/NSC 8.3 Setshwantshong se latelang, sedikadikwe se nang le bohare O se takilwe. H, J, G and L ke dintlha tse hodima sedikadikwe. ∆HJL e takilwe. HOG e kgaola JL mahareng ho M. HJ 12 5 diyuniti and JM = 12 diyuniti. H O J 12 M L G 8.3.1 Haeba MG = 6 diyuniti le OM = x, ngola HM ka mokgwa wa x. (2) 8.3.2 Bala, o fana ka mabaka, bolelele ba rediase ya sedikadikwe. (5) [25] Ditokelo di sireleditswe Phetla

Downloaded from hlayiso.com Dipalo/P2 14. DBE/Loetse 2021(2) SC/NSC POTSO 9 Setshwantshong se latelang, P, Q, R and S ke dintlha tse hodima sedikadikwe. PS, PQ le PR di lelefaditswe hoya ho V, T le W ka ho latelanang. VT || SQ le SR || VW. T Q 2 1 W R 1 2 P 2 1 3 S V Pruva, o fana ka mabaka, hore: TQ WR 9.1 (3) QP RP 9.2 TPVW ke setliliki khwadilatherale (5) [8] Ditokelo di sireleditswe Phetla

Downloaded from hlayiso.com Dipalo/P2 15. DBE/Loetse 2021(2) SC/NSC POTSO 10 Setshwantshong se latelang, B, C le D ke dintlha tse hodima sedikadikwe hore BC = CD. EC le ED ke di thanjente hodima sedikadikwe ho C le D ka ho latellana. BC e lelefaditswe ho kopana le thanjente ho DE e lelefaditsweng hoya ho F. B̂ x . B x 3 2 4 1 D 3 C 2 1 1 2 E F 10.1 Pruva, o neha mabaka, hore: 10.1.1 Ê 1 180q  2 x (5) 10.1.2 ∆ECD ||| ∆CBD (3) 10.2 Pruva, o fan aka mabaka, hore: 10.2.1 CD 2 CE.BD (3) CF 2 BD 10.2.2 EF 2 DE (6) [17] KAOFELA: 150 a Copyright reserved

Downloaded from hlayiso.com Dipalo/P2 DBE/Loetse 2021(2) SC/NSC LEQEPHE LA TLHAHISOLESEDING  b r b 2  4ac x 2a A P(1  ni) A P(1  ni) A P(1  i) n A P(1  i) n n Tn a  (n  1)d Sn >2a  (n  1)d @ 2 a r n 1 a Tn ar n 1 Sn ; r z1 Sf ; 1  r  1 r 1 1 r F > x (1  i ) n  1 @ P > x 1  (1  i )  n @ i i f ( x  h)  f ( x ) f / ( x) lim h o0 h § x  x 2 y1  y 2 · d x2  x1 2  y2  y1 2 M¨ 1 ; ¸ © 2 2 ¹ y 2  y1 y mx  c y  y1 m x  x1 m m tan T x 2  x1 2 2 xa  y b r2 a b c In 'ABC: sin A sin B sin C a2 b 2  c 2  2bc. cos A 1 area 'ABC ab. sin C 2 sin( D  E ) sin D cos E  cosD sin E sin( D  E ) sin D cos E  cosD sin E cos(D  E ) cosD cos E  sin D sin E cos(D  E ) cosD cos E  sin D sin E ­cos 2 D  sin 2 D ° cos 2D ®1  2 sin D 2 sin 2D 2 sin D cos D °2 cos 2 D  1 ¯ n x ¦x ¦ x x i 2 n V2 i 1 n n( A ) P(A) P(A or B) P(A)  P(B)  P(A and B) n(S) yˆ a  bx b ¦ xx y y ¦ xx 2 Ditokelo di sireleditswe