Teisteanais Nàiseanta
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1 Teisteanais Nàiseanta AH08 X774/77/ Matamataig DIARDAOIN, CÈITEAN 9:00 M :00 F Comharran gu lèir 00 Feuch na ceistean UILE. Faodaidh tu àireamhair a chleachdadh. Gus na comharran gu lèir fhaighinn, feumaidh tu d obrachadh a-mach a shealltainn. Cuir na h-aonadan anns na freagairtean agad far a bheil sin iomchaidh. Chan fhaighear comharraidhean idir airson freagairtean air an togail bho dhealbhan-sgèile. Sgrìobh do fhreagairtean gu soilleir ann an leabhran nam freagairtean. Ann an leabhran nam freagairtean feumaidh tu àireamh na ceiste a tha thu a freagairt a chomharrachadh gu soilleir. Cleachd inc gorm no dubh. Mus fàg thu seòmar na deuchainne, feumaidh tu leabhran nam freagairtean a thoirt don Fhreiceadan; mura dèan thu sin, dh fhaodadh tu na comharran gu lèir airson a phàipeir a chall. *X77477* A/SQA
2 LIOSTA FHOIRMLEAN Deribheatan cumanta Iontagralan cumanta f ( ) f ( ) f ( ) f ( ) d sin cos ( ) sec a tan( a) + c a a sin + c a tan + a + tan + c a a tan sec ln + c cot cosec a e a e a + c sec sec tan cosec ln cosec cot e e Suimean Sn n a n d (Sreath airitmeatrach) = + ( ) (Sreath geoimeatrach) S n n ( r ), a = r r nn ( + ) nn ( + )( n+ ) n( n+ ) r =, r =, r = 6 4 n n n r= r= r= Teorem dà-theirmeach n n n n r r a+ b = a b r= 0 r far a bheil ( ) n n n! = C = r r r!( n r)! Leudachadh Maclaurin iv 4 f ( 0) f ( 0) f ( 0) f( ) = f( 0) + f ( 0) !! 4! duilleag 0
3 LIOSTA FHOIRMLEAN (a leantainn) Teorem De Moivre [ (cos sin )] n n r θ + i θ = r ( cos nθ + isin nθ ) Iomadachadh bheactoran i j k a a a a a a a b= absin θ nˆ = a a a = i j + k b b b b b b b b b Cruth-atharrachadh matraigs Cuartachadh tuathal tro cheàrn, θ, mun origin, cosθ sin θ sin θ cosθ [Tionndaidh an duilleag duilleag 0
4 Comharran gu lèir 00 Feuch na ceistean UILE COMHARRAN. (a) Ma tha f( ) = sin, lorg f ( ). 5 e (b) Diofaraich y = 7 +. (c) Ma tha ycos + y = 6, lorg dy d le diofarachadh fhillteach. 4. Le bloighean pàirteach lorg 7 d (a) Sgrìobh sìos agus sìmplich an teirm coitcheann anns a leudachadh dà-theirmeach 9 aig 5 +. (b) Leis an fhiosrachadh seo, no ann an dòigh eile, lorg an teirm nach eil an eisimeil air. 4. Ma tha z = + i agus z = p 6 i, p, lorg: (a) zz ; (b) luach p far a bheil z z na fìor àireamh. 5. Cleachd algoritim Euclid airson nan iontaidsearan a agus b a lorg ma tha 06a+ 9b= 7. 4 duilleag 04
5 6. Air raon freagarrach tha lùb co-chomharraichte gu parameatrach le = t + agus y = t+. ln ( ) COMHARRAN Obraich a-mach co-aontar a bheantainn ris a lùb far a bheil t = Tha na maitraigsean C agus D air am mìneachadh le: C = 0 0 agus D= k+ 0, far a bheil k. (a) Lorg C D far an e bheil C an transpòs aig C. (b) (i) Lorg agus sìmplich abairt airson an deteirmeanant aig D. (ii) Sgrìobh sìos luach k far nach eil D idir ann. 8. Leis an ionadachadh u = sinθ, no ann an dòigh eile, lorg luach π 4 sin θcosθdθ. π Dearbh gu bheil: (a) sùim trì iontaidsearan leantainneach sam bith so-roinnte le ; (b) e comasach àireamh còrr sam bith a sgrìobhadh mar sùim dà iontaidsear leantainneach. [Tionndaidh an duilleag duilleag 05
6 0. Ma tha z = + iy, dèan sgeidse dhen locus anns an raon coimpleacs z = z + i. COMHARRAN. (a) Lorg am matraigs, A, mar thoradh air cuairt tuathal de π radianan mun oirigin. (b) Lorg am matraigs, B, mar thoradh air faileas-sgàthain anns an -ais. (c) Leis an fhiosrachadh seo lorg a matraigs, P, mar thoradh air cuairt tuathail de π radianan mun oirigin agus faileas-sgàthain anns an -ais. Sgrìobh do fhreagairtean ann an luachan mionaideach. (d) Mìnich carson nach eil ceangal sam bith eadar matraigs P agus cuairt mun oirigin.. Airson n, nan iontaidsearan dhearbhte air fad, dearbh le ionducsean gu bheil n = ( ). n r r= 5 duilleag 06
7 COMHARRAN. Tha einnseanair air inneal-togaidh a dhealbh. Tha sgriubha a tionndadh agus bidh seo a giorrachadh faid an inneil agus a meudachadh an àirde. Tha an t-inneal air a mhodaileadh air rhombus, le gach taobh 5 cm a dh fhaid. Tha an fhaid cm, agus an àirde h cm anns an diagram seo. 5 cm h (a) Dearbh gu bheil h = 500. (b) Tha an fhaid a lùghdachadh aig reit 0 cm gach diog nuair a tha an sgriubha air a thionndadh. Obraich a-mach an reit aig a bheil an àirde ag atharrachadh nuair a tha = 0. 5 [Tionndaidh an duilleag duilleag 07
8 COMHARRAN 4. Ann an òrdugh geoimeatrach tha a chiad teirm 80 agus an co-mheas coitcheann. (a) Airson an òrdugh seo, obraich a-mach: (i) an 7 mh teirm; (ii) an sùim gu neo-chrìochnachd dhen t-sreath geoimeatrach seo. Tha a chiad teirm dhen òrdugh geoimeatrach seo co-ionann ris a chiad teirm de òrdugh airitmeatrach. Tha sùim a chiad còig teirmean dhen òrdugh airitmeatrach seo 40. (b) (i) Obraich a-mach diofar coitcheann an òrdugh seo. (ii) Sgrìobh sìos agus sìmplich abairt airson an nmh teirm. Tha S n a riochdachadh sùim a chiad n teirmean dhen òrdugh airitmeatrach seo. (c) Obraich a-mach luachan n far a bheil S n = (a) Cleachd iontagrachadh pàirteach agus lorg sin d. (b) Leis an fhiosrachadh seo lorg fuasgladh sònraichte dy y = sin, 0 d ma tha =π nuair a tha y = 0. Sgrìobh do fhreagairt anns an riochd y = f( ). 7 duilleag 08
9 6. Seo na co-aontaran aig na raointean π, π agus π : COMHARRAN π : y+ z = 4 π : 5y z = π : 7 + y + az = far a bheil a. (a) Cleachd gearradh-às Gauss agus lorg luach a gus am bi trasnadh na raointean π, π agus π na loidhne dhìreach. (b) Lorg co-aontar a loidhne trasnaidh nuair a tha an luach seo aig a. 4 Tha raon eile π 4 ann le co-aontar 9+ 5y+ 6z = 0. (c) Obraich a-mach an ceàrn caol eadar π agus π 4. (d) Mìnich an dàimh geoimeatrach eadar π agus π 4. Dearbh do fhreagairt. 7. (a) Ma tha f ( ) = e, obraich a-mach leudachadh Maclaurin airson f ( ) agus a toirt a-steach, an teirm ann an. suas gu, (b) Air raon freagarrach tha g( ) = tan. (i) Dearbh gu bheil an treasamh deribheataibh aig g( ) air a riochdachadh le 4 g ( ) = sec + 4 tan sec. (ii) Leis an fhiosrachadh seo, lorg leudachadh Maclaurin airson g( ) suas gu, agus a toirt a-steach an teirm ann an. (c) Leis an fhiosrachadh seo, no ann an dòigh eile, lorg leudachadh Maclaurin airson e tan suas gu, agus a toirt a-steach, an teirm ann an. (d) Sgrìobh sìos a chiad trì teirmean, nach eil aig neoni, ann an leudachadh Maclaurin airson e tan + e sec. [CRÌOCH A PHÀIPEIR] duilleag 09
10 [DUILLEAG BHÀN] NA SGRÌOBH AIR AN DUILLEIG SEO duilleag 0
11 [DUILLEAG BHÀN] NA SGRÌOBH AIR AN DUILLEIG SEO duilleag
12 [DUILLEAG BHÀN] NA SGRÌOBH AIR AN DUILLEIG SEO duilleag
13 Teisteanais Nàiseanta AH08 X774/77/ Matamataig Briathrachas DIARDAOIN, CÈITEAN 9:00 M :00 F *X77477* A/SQA
14 Gàidhlig Diofarachadh fhillteach Teirm coitcheann Leudachadh dà-theirmeach Algoritim Euclid Transpòs Ionadachadh Iontaidsearan leantainneach So-roinnte Àireamh còrr Raon coimpleacs Oirigin Faileas-sgàthain Dearbh le ionducsean Co-mheas coitcheann Òrdugh geoimeatrach Sreath geoimeatrach Òrdugh airitmeatrach Diofar coitcheann Iontagrachadh pàirteach Fuasgladh sònraichte Raointean Trasnadh Loidhne trasnaidh Beurla Implicit differentiation General term Binomial epansion Euclidean algorithm Transpose Substitution Consecutive integers Divisible by Odd number Comple plane Origin Reflection Prove by induction Common ratio Geometric sequence Geometric series Arithmetric sequence Common difference Integration by parts Particular solution Planes Intersection Line of intersection duilleag 0
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