MATHEMATICIA GENERALLI

Transcription

1 MATHEMATICIA GENERALLI (y Mhmmed Abbs) Lgithmi Reltis lgb ) lg lg ) b b) lg lg lg m lg m d) lg m. lg m lg m e) lg lg m lg g) lg lg h) f) lg lg f ( ) f ( ). Eetil Reltis ). lge. lge.... lge...!! b) e f ll.!!! Lgithmi Seies 4 ) lge..., 4 4 lge...,. 4 b) imil Esis b C0 b C b C b... C b... C b if ) 0 0 Z. C C... C... C if is sitive itege. b) ) d) suh tht d!! Tigmeti Seies 5 7 ) si...! 5! 7! 4 6 b) s...! 4! 6! 5 ) t Sum f Seil Sequees ( ) ) Sum f fist tul umbes:.... Z ( )( ) b) Sum f sques f fist tul umbes:... 6 ( ) ) Sum f ubes f fist tul umbes:... d) Sum f y stt k t times:... t times k k k k k k. Q. Mthemtii Geelli by Mhmmed Abbs

2 MATHEMATICIA Vl. f CLASS by MOHAMMED AAS (II) List f fmule f Mesuti f usig i the blems f Mim d Miim 0. Cile: Peimete i.e. iumfeee Ae 0. Equiltel tigle: Peimete Ae 4 0. Retgle: l b Peimete Ae lb 05. Cubid: Ltel sufe e l b. h Ttl sufe e lb bh hl Vlume lbh 04. Sque: Peimete 4 Ae 06. Cube: Ltel sufe e 4 Ttl sufe e 6 Vlume 07. Shee: 08. Hemishee: Sufe e 4 Cuved sufe e Vlume 4 Ttl sufe e Vlume 09. Cylide: 0. Ce: Cuved sufe e h Cuved sufe e l, whee l h Ttl sufe e h Ttl sufe e l Vlume h Vlume h Fllwigs e ls f imte, thugh questis them e ely fud i mim d miim:. Fustum f e: Cuved sufe e l R, whee l h R Ttl sufe e l R R, whee l h R h R R Vlume. Set d segmet f Cile: Ae f the set f gle 60 Legth f the f set f gle 60 Ae f the segmet f ile Ae f the esdig set Ae f esdig tigle si s si Ciumfeee i.e. legth f semiile f dius Peimete f semiile f dius. Ae f tezium (Sum f llel sides) (Diste betwee llel sides) 4. Ae f hmbus (Pdut f digls) 5. Ae f tigle AC bsic bsi A si b 6. Ae f tigle by He s fmul s( s )( s b)( s ) whee, s. Mthemtii Geelli by Mhmmed Abbs

3 MATHEMATICIA Vl. f CLASS by MOHAMMED AAS (III) Algebi Idetities ) b b b b) b b b ) b b b b d) b b b b e) b b b b f) b b b b b b b b g) h) b b b b b b. Cet f Ifiity We side the eistee f tw symbls d utside the set f el umbes R d wuld ll them mius ifiity d lus ifiity esetively with the ft f evey R. Thus d e t el umbes but just the symbls (like we use, y et.). Whe we wite, we me tht: ) is lge th y el umbe hweve lge. b) is t fied umbe. Als, if 0, if 0., if Symbls d thei meigs S. N. Symbl Meig 0. N Set f tul umbes 0. I Z Set f iteges 0. Q Set f til umbes 04. T Set f itil umbes 05. R Set f el umbes 06. C Set f mle umbes 07. is elemet f ( belgs t) 08. is t elemet f ( des t belg t) 09. S ξ U Uivesl set 0. : / Suh tht. Emty set Null set. is subset f. is sueset f 4. is e subset f 5. is e sueset f 6. Ui 7. Iteseti 8. F ll 9. Imlies 0. if d ly if Mthemtii Geelli by Mhmmed Abbs

4 MATHEMATICIA Vl. f CLASS by MOHAMMED AAS (IV) TRIGONOMETRIC FORMULAE Relti betwee tigmeti tis si ) t b) t ) t t s t s d) t e) se f) se si si s Tigmeti idetities ) si s b) t se ) t se Additi / subtti fmule & sme elted esults si A si As s Asi ) s A s As si Asi b) s s s si s si A A A A ) si si si si s s A A A A d) e) t A f) t A t A t t At t t A t t A Tsfmti f sums / diffeees it duts & vie-ves C D C D ) sic si D si s C D C D b) sic si D s si C D C D ) sc s D s s C D C D d) sc s D si si si As si A si A s Asi si A si A e) f) g) s As s A s A h) si Asi s A s A Multile gle fmule ivlvig A d A ) si A si As A A A b) si A si s ) sa s A si A A A d) s A s si e) s A s A f) s A sa g) s A si A h) si A s A t A i) si A t A t A j) s A t A t A k) ta t A l) si A si A 4si A m) s A 4s A s A t A t A ) ta t A Reltis i Diffeet Mesues f Agle Agle i Rdi Mesue = Agle i Degee Mesue Agle i Degee Mesue = Agle i Rdi Mesue ( i di mesue) l Als fllwigs e f imte s well: Right gle 90 = 60, = = = dis imtely di = seds. Mthemtii Geelli by Mhmmed Abbs

5 MATHEMATICIA Vl. f CLASS by MOHAMMED AAS (V) Geel Slutis ) si si y ( ) y, whee Z. b) s s y y, whee Z. ) t t y y, whee Z. Relti i Degee & Rdi Mesues Agles i Degee Agles i Rdi I tul tie, we mit the eet d isted f witig we simly wite d similly f thes. Tigmeti Rti f Stdd Agles Degee /Rdi T Rtis 0 si 0 s t se se t 0 0 Tigmeti Rtis f Allied Agles Agles T- Rtis OR si s s si si s s si si s si si s s si si s s t t t t t t t t t t t t t t t t t t se se se se se se se se se se se se se se se se se se Mthemtii Geelli by Mhmmed Abbs

6 MATHEMATICIA Vl. f CLASS by MOHAMMED AAS - MG (VI) NUMER SYSTEM 0. Ntul umbes: The umbes used i diy utig i.e.,,,..., e lled tul umbes (d sitive iteges s well). The set f tul s. is deted by N. Als if we ilude 0 t the set f tul umbes, we get set f the whle umbes whih is deted by the symbl W. W= 0,,,,.... Theefe N=,,,... d, 0. Iteges: The umbes...,,, 0,,,,... e lled iteges. The set f iteges is deted by I Z. Thugh w we use Z isted f I t symblize the set f iteges. Theefe, I Z=...,,,0,,,,.... Clely N Z. Als fm the bve disussi, it is evidet tht iteges e f thee tyes viz.: + Z =,,,... ) Psitive iteges i.e. b) Negtive iteges i.e. Z =,,,... ) Ze itege i.e. -sitive d -egtive itege. 0. Rtil umbes: A umbe f the fm, whee d q e iteges d 0 q q, is lled til. The set f til s. is deted by Q. Theefe Q= ;, q Z d q 0 q Clely N Z Q. Ze beig itege, is ls til umbe. 04. Itil umbes: A itil umbe hs -temitig d -eetig deiml eesetti i.e. it t be eessed i the fm f. The set f itil s. is deted by T. q Few emles f itil umbes e, 5 7, 8, 5,,,... e et. Nte tht is itil while is til Rel umbes: The set f ll umbes eithe til itil, is lled el umbe. The set f el s. is deted by R. Clely N Z Q R. Slvig f Qudti Equti: Cside qudti equti f the fm, b 0 the, its ts e give by 4 b b d, 4 b b whee D b 4. Mthemtii Geelli by Mhmmed Abbs