fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES fiziks
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1 Istitute o NET/JRF, GATE, IIT-JAM, JEST, TIFR GRE i PHYSICAL SCIENCES iziks Foum o CSIR-UGC JRF/NET, GATE, IIT-JAM/IIS, JEST, TIFR GRE i PHYSICAL SCIENCES Bsi Mthemtis Fomul Sheet o Physil Siees He oie iziks, H.No., G.F, Ji Si, Ne IIT, Huz Khs, New Delhi-6 Phoe: / Bh oie A Istitute o Mthemtis, 8-B/6, Ji Si, Ne IIT Huz Khs, New Delhi-6 Emil: iziks.physis@gmil.om
2 Istitute o NET/JRF, GATE, IIT-JAM, JEST, TIFR GRE i PHYSICAL SCIENCES Bsi Mthemtis Fomul Sheet o Physil Siees. Tigoomety (-9). Tigoometil Rtios Ietities... (-7). Ivese Ciul Futios.... (8-9). Dieetil itegl Clulus. (0-0). Dieetitio (0-). Limits.. (-4). Tgets Noml...(5-6).4 Mim Miim..(6).5 Itegtio...(7-9).5. Gmm itegl..(9). Dieetil Equtios....(0-) 4. Vetos....(-5) 5. Alge....(6-) 5. Theoy o Quti equtios...(6) 5. Logithms..(7) 5. Pemuttios Comitios...(8-9) 5.4 Biomil Theoem...(0) 5.5 Detemits...(-) 6. Coi Setio....() 7. Poility....(4-5) He oie iziks, H.No., G.F, Ji Si, Ne IIT, Huz Khs, New Delhi-6 Phoe: / Bh oie A Istitute o Mthemtis, 8-B/6, Ji Si, Ne IIT Huz Khs, New Delhi-6 Emil: iziks.physis@gmil.om
3 Istitute o NET/JRF, GATE, IIT-JAM, JEST, TIFR GRE i PHYSICAL SCIENCES. Tigoometil Rtios Ietities. Tigoomety. si os. se t. ose ot si t os os ot 6. si si ose os 8. t se ot Aitio Suttio Fomule Fo y two gles A B. SiA B si Aos B os Asi B. SiA B si Aos B os Asi B. osa B os Aos B si Asi B 4. osa B os Aos B si Asi B t A t B t A.t B 5. ta B 6. ta B Doule Agle Fomule t A t B t A.t B. si si os,. os os si si os. t t t Tiple gle Fomule. si si 4si. os 4os os. t t t t Tigoometi Rtios o θ/. si si os,.. t t t os os si os si He oie iziks, H.No., G.F, Ji Si, Ne IIT, Huz Khs, New Delhi-6 Phoe: / Bh oie A Istitute o Mthemtis, 8-B/6, Ji Si, Ne IIT Huz Khs, New Delhi-6 Emil: iziks.physis@gmil.om
4 Istitute o NET/JRF, GATE, IIT-JAM, JEST, TIFR GRE i PHYSICAL SCIENCES Fomule o siθ & osθ i tems o tθ. si t. t os t t Fomule o siθ & osθ i tems o tθ/ t. si. t t os t Tsomtio o sum/ieees ito Pouts C D C D. si C si D si os C D C D. si C si D os si C D C D. osc os D os os C D C D C D D C 4. osc os D si si si si Tsomtios o Pouts ito sum/ieee. SiAos B Si( A B) Si( A B). os Asi B Si( A B) Si( A B). os Aos B os( A B) os( A B) 4. si Asi B os( A B) os( A B) Tigoometi Rtios o (-θ). si si. os os. t t 4. ot 5. se ot 6. ose ose He oie iziks, H.No., G.F, Ji Si, Ne IIT, Huz Khs, New Delhi-6 Phoe: / Bh oie A Istitute o Mthemtis, 8-B/6, Ji Si, Ne IIT Huz Khs, New Delhi-6 Emil: iziks.physis@gmil.om 4
5 Istitute o NET/JRF, GATE, IIT-JAM, JEST, TIFR GRE i PHYSICAL SCIENCES Tigoometi Rtio o : (All Positive). os si. si os. t ot 4. ot t 5. ose se 6. se ose Tigoometi Rtio o :( Oly si os e is Positive). os si. si os. t ot 4. ot t 5. ose se 6. se ose Tigoometi Rtios o :( Olysi os e is Positive). os os. si si. t t 4. ot ot 5. ose ose 6. se se Tigoometi Rtios o :( Oly t ot is Positive). os os. si si. t t 4. ot ot 5. ose ose 6. se se He oie iziks, H.No., G.F, Ji Si, Ne IIT, Huz Khs, New Delhi-6 Phoe: / Bh oie A Istitute o Mthemtis, 8-B/6, Ji Si, Ne IIT Huz Khs, New Delhi-6 Emil: iziks.physis@gmil.om 5
6 Istitute o NET/JRF, GATE, IIT-JAM, JEST, TIFR GRE i PHYSICAL SCIENCES Tigoometi Rtio o :( Oly t ot is Positive). os si. si os. t ot 4. ot t 5. ose se 6. se ose Tigoometi Rtio o :( Oly os se is Positive). os si. si os. t ot 4. ot t 5. ose se 6. se ose Tigoometi Rtios o :( Oly os se is Positive). os os. si si. t t 4. ot ot 5. ose ose 6. se se Tigoometi Rtios o : (All Positive). os os. si si. t t 4. ot ot 5. ose ose 6. se se He oie iziks, H.No., G.F, Ji Si, Ne IIT, Huz Khs, New Delhi-6 Phoe: / Bh oie A Istitute o Mthemtis, 8-B/6, Ji Si, Ne IIT Huz Khs, New Delhi-6 Emil: iziks.physis@gmil.om 6
7 Istitute o NET/JRF, GATE, IIT-JAM, JEST, TIFR GRE i PHYSICAL SCIENCES Shot-ut metho to ememe the Tigoometi tios. si si. os os. t t 5. os si 6. t ot whe is eve itege 4. si os whe is o itege θ 0 o 0 o 45 o 60 o 90 o 0 o 5 o 50 o 80 o 70 o 60 o si θ 0 os θ t θ 0 ot θ se θ ose θ He oie iziks, H.No., G.F, Ji Si, Ne IIT, Huz Khs, New Delhi-6 Phoe: / Bh oie A Istitute o Mthemtis, 8-B/6, Ji Si, Ne IIT Huz Khs, New Delhi-6 Emil: iziks.physis@gmil.om 7
8 Istitute o NET/JRF, GATE, IIT-JAM, JEST, TIFR GRE i PHYSICAL SCIENCES. Ivese Ciul Futios. si si. os os. t t 4. ot ot 5. se se 6. ose os e 7. si si 8. os os 9. se se 0. os eos e. si ose. os se. t ot 4. ot t 5. se os 6. ose si 7. si si 8. os os 9. si 9. t t 0. t ot. se os ose. si os. os si 4. t se 5. ot ose 6. se t 7. ose ot 8. si si 9. si 4 si. t t 0. os 4 os. t t y. t t t y y y 4. t t t y y He oie iziks, H.No., G.F, Ji Si, Ne IIT, Huz Khs, New Delhi-6 Phoe: / Bh oie A Istitute o Mthemtis, 8-B/6, Ji Si, Ne IIT Huz Khs, New Delhi-6 Emil: iziks.physis@gmil.om 8
9 Istitute o NET/JRF, GATE, IIT-JAM, JEST, TIFR GRE i PHYSICAL SCIENCES 5 Some Impott Epsios:. si...! 5! 5 4. sih.... os...! 5!! 4! osh t...! 4! 5 6. e e...!!!! Some useul sustitutios:- Epessios Sustitutio Fomul Result 4 si si 4si Siθ 4 os 4os os osθ = t θ = t θ = t θ = t θ t t t t t t t t t = si θ si tθ siθ osθ tθ osθ = os θ os osθ = si θ si os θ = os θ os si θ = se θ se t θ = ose θ ose ot θ = t θ t se θ = ot θ ot ose θ He oie iziks, H.No., G.F, Ji Si, Ne IIT, Huz Khs, New Delhi-6 Phoe: / Bh oie A Istitute o Mthemtis, 8-B/6, Ji Si, Ne IIT Huz Khs, New Delhi-6 Emil: iziks.physis@gmil.om 9
10 Istitute o NET/JRF, GATE, IIT-JAM, JEST, TIFR GRE i PHYSICAL SCIENCES. Dieetitio.. y. Dieetil itegl Clulus h lim. ' h0 y lim 0 h lim 4. k h0 h h ; k is ostt utio 8. ; N 0. si os. os si. t se. se se.t 4. os e os e.ot 5. log ; 0, e e log 8. si ; 9. os ; ot. ; R. os e ; y t 0. ; R. se : He oie iziks, H.No., G.F, Ji Si, Ne IIT, Huz Khs, New Delhi-6 Phoe: / Bh oie A Istitute o Mthemtis, 8-B/6, Ji Si, Ne IIT Huz Khs, New Delhi-6 Emil: iziks.physis@gmil.om 0
11 Istitute o NET/JRF, GATE, IIT-JAM, JEST, TIFR GRE i PHYSICAL SCIENCES Rules o Dieetitios. Aitio Rule: I y = (u + v) the. Susttios Rule: I y = (u - v) the. Pout Rule: I y = uv the 4. Quotiet Rule: I 5. I y = (u) is u = g() the y y y v u u v u y y the v v y u u y. y u. u y 6. I u = (y), the y y 7.. y o y whee 0 y y y u u u v u v v v Deivtives o omposite utios si os. 5. os si. 6. t se. 7. ot os e. 8. se se t. 9. os e os e ot. log.. 0. log. He oie iziks, H.No., G.F, Ji Si, Ne IIT, Huz Khs, New Delhi-6 Phoe: / Bh oie A Istitute o Mthemtis, 8-B/6, Ji Si, Ne IIT Huz Khs, New Delhi-6 Emil: iziks.physis@gmil.om
12 . iziks Istitute o NET/JRF, GATE, IIT-JAM, JEST, TIFR GRE i PHYSICAL SCIENCES e e. g g g g Deivtives o omposite utios. si. os. t. 4. ot. 5. se 6. ose Impliit utios:-.... Tke the eivtives o these utios ietly i y/ Pmeti utios:- I t y gt & the y y / whee 0 / t t Logithemi Dieetitio:- I the utio is i the om o The tkig Logithm o oth sies 5 & the i y/ Highe oe Deivtives o omposite utios:- y y " II oe, y "' I Geel; y y th oe g y III oe He oie iziks, H.No., G.F, Ji Si, Ne IIT, Huz Khs, New Delhi-6 Phoe: / Bh oie A Istitute o Mthemtis, 8-B/6, Ji Si, Ne IIT Huz Khs, New Delhi-6 Emil: iziks.physis@gmil.om
13 Istitute o NET/JRF, GATE, IIT-JAM, JEST, TIFR GRE i PHYSICAL SCIENCES. Limits Limits o utio I o evey 0 thee eist δ > 0 suh tht i l the we sy lim o () s is l Theoem o limits i.e. lim l I () g() e two utios the. lim g lim lim g. lim g lim lim g. lim. g lim 4. lim g lim lim g. lim g 5. k k lim lim whee k is ostt 6. lim lim p / q 7. lim lim p / q : whee p & q e iteges Some Impott st limits. lim. lim : whee is ostt wheeve 0 R. lim ; R 4. lim ; N, 0 si 5. lim 0 t 7. lim 0 si k 9. lim k 0 6. lim 0 si 8. lim 0 t t k 0. lim k 0 He oie iziks, H.No., G.F, Ji Si, Ne IIT, Huz Khs, New Delhi-6 Phoe: / Bh oie A Istitute o Mthemtis, 8-B/6, Ji Si, Ne IIT Huz Khs, New Delhi-6 Emil: iziks.physis@gmil.om
14 Istitute o NET/JRF, GATE, IIT-JAM, JEST, TIFR GRE i PHYSICAL SCIENCES os. lim 0 0 si si m. lim ;lim si 5. limos ; lim os 0 0 / 0 0. lim 0 os m osk k 4. lim 0 6. limsi 0; lim si / 0 e 7. lim log whee > 0 8. lim log 9. lim log ; lim log e 0 / / / 0. k lim 0 log k. lim 0 e; k lim k 0 e os os. lim 0 os os. os os lim 0 4. lim 0 / e / 5. lim e 6. lim 0; lim lim 0; lim 0 8. lim 0 whee k >0 k 9. lim k k; lim k k whee k is ostt 0. limsi si. limos os. lim.... lim log ;, lim se os e ;lim lim e ;lim e 6. lim h h h e He oie iziks, H.No., G.F, Ji Si, Ne IIT, Huz Khs, New Delhi-6 Phoe: / Bh oie A Istitute o Mthemtis, 8-B/6, Ji Si, Ne IIT Huz Khs, New Delhi-6 Emil: iziks.physis@gmil.om 4
15 Istitute o NET/JRF, GATE, IIT-JAM, JEST, TIFR GRE i PHYSICAL SCIENCES. Tgets Noml Tget t y y Let, to y e give uve y P, Q, y y e two eighouig poits o it. Equtio o the lie PQ is Y y y y y X y o Y y X This lie will e tget to the give uve t P i Q P whih i te mes tht, y y P, y Q 0 we kow tht lim 0 y y y Theeoe the equtio o the tget is Y y X Noml t, y The oml t, y eig pepeiul to tget will hve its slope s hee its equtio is Y y y X y Geometil meig o y y epesets the slope o the tget to the give uve He oie iziks, H.No., G.F, Ji Si, Ne IIT, Huz Khs, New Delhi-6 Phoe: / y t y t y poit, y whee is the gle whih the tget to the uve mkes with +ve ietio o -is. y I se we e to i the tget t y poit, y the Bh oie A Istitute o Mthemtis, 8-B/6, Ji Si, Ne IIT Huz Khs, New Delhi-6 Emil: iziks.physis@gmil.om 5, y y i.e. the vlue o t, y will epeset the slope o the tget hee its equtio i this se will e
16 Istitute o NET/JRF, GATE, IIT-JAM, JEST, TIFR GRE i PHYSICAL SCIENCES Noml y y y y y, y y, y Coitio o tget to e pllel o pepeiul to -is I tget is pllel to -is o oml is pepeiul to -is the I tget is pepeiul to -is o oml is pllel to -is the.4 Mim Miim Fo the utio y y 0 o 0. y y t the mimum s well s miimum poit the tget is pllel to -is so tht its slope is zeo. y y Clulte 0 solve o. Suppose oe oot o 0 is t =. I y ve o =, the mimum t =. I y ve o =, the miimum t =. y y I 0 t =, the i. y I 0 t =, eithe mimum o miimum t =. y I 0 t =, the i 4 4 y. 4 4 y y I 0 i.e +ve t =, the y is miimum t = i is mimum t = so o. i.e -ve t =, the y He oie iziks, H.No., G.F, Ji Si, Ne IIT, Huz Khs, New Delhi-6 Phoe: / Bh oie A Istitute o Mthemtis, 8-B/6, Ji Si, Ne IIT Huz Khs, New Delhi-6 Emil: iziks.physis@gmil.om 6
17 Istitute o NET/JRF, GATE, IIT-JAM, JEST, TIFR GRE i PHYSICAL SCIENCES.5 Itegtio Ieiite Itegtio I F, the we sy tht tieivtive o Some st Itegls we wite F F is ieiite itegl o log 5. e e 6. log 7. os si 8. si os 9.se t 0. os e ot.se t se. os eot ose. si 5. se 4. t 6. osh sih 7. sih osh 8. se h th 9. os eh oth 0. se h th se h. ehoth oseh os. t logse. ot logsi 4. se logse t 5. os e log t 6. t He oie iziks, H.No., G.F, Ji Si, Ne IIT, Huz Khs, New Delhi-6 Phoe: / Bh oie A Istitute o Mthemtis, 8-B/6, Ji Si, Ne IIT Huz Khs, New Delhi-6 Emil: iziks.physis@gmil.om 7
18 Istitute o NET/JRF, GATE, IIT-JAM, JEST, TIFR GRE i PHYSICAL SCIENCES 7. log ; i 8. log ; i si log o sih log o osh. log. log 4. si 5. se ose m 6. si si m L L 0 Rules o Itegtio 0,, He oie iziks, H.No., G.F, Ji Si, Ne IIT, Huz Khs, New Delhi-6 Phoe: / m m.. k k, whee k is ostt. k k k k, whee k k e ostts Bh oie A Istitute o Mthemtis, 8-B/6, Ji Si, Ne IIT Huz Khs, New Delhi-6 Emil: iziks.physis@gmil.om 8
19 Istitute o NET/JRF, GATE, IIT-JAM, JEST, TIFR GRE i PHYSICAL SCIENCES Rule o itegtio y sustitutio t. I t, t t ' t t.. ' 4. ' g log : Rules o itegtio y ptil tio This metho e use to evlute itegl o the type whee (i) P() & Q() e Polyomils i (ii) Degee o P() < egee o Q() (iii) Q() otis two/moe istit lie/quti tos i.e. P Q A B C uv u v v u P Q.5. Gmm itegl (i) Gmm itegl is give y e ( ) = ( ). 0 (ii) B e 0 B He oie iziks, H.No., G.F, Ji Si, Ne IIT, Huz Khs, New Delhi-6 Phoe: / Bh oie A Istitute o Mthemtis, 8-B/6, Ji Si, Ne IIT Huz Khs, New Delhi-6 Emil: iziks.physis@gmil.om 9
20 Istitute o NET/JRF, GATE, IIT-JAM, JEST, TIFR GRE i PHYSICAL SCIENCES. Dieetil Equtios Oe egee o ieetil equtio The oe o ieetil equtio is the oe o the highest ieetil o-eiiet peset i the equtio. y y Emple: y 0 is seo oe ieetil equtio. The egee o ieetil equtio is the egee o the highest eivtive te emovig the il sig tio. y y Emple: 0 y hs egee o. D.E. o the ist oe ist egee. Septio o the viles:. Homogeeous Equtio y y y, y, y i eh tem o (,y), y. Equtios euile to homogeeous om y y, A By C X h let y Y k y h k 0 Choose h, k so tht Ah Bk C 0 y y Y X is o the sme egee. X Y h k AX BY Ah Bk X Y AX BY y Cse o ilue: A B m m y C 4. Lie Dieetil Equtios y Py Q whee P Q e utio o (ut ot y) o ostt. I F. e P. y I. F. Q I. F. He oie iziks, H.No., G.F, Ji Si, Ne IIT, Huz Khs, New Delhi-6 Phoe: / Bh oie A Istitute o Mthemtis, 8-B/6, Ji Si, Ne IIT Huz Khs, New Delhi-6 Emil: iziks.physis@gmil.om 0
21 Istitute o NET/JRF, GATE, IIT-JAM, JEST, TIFR GRE i PHYSICAL SCIENCES 5. Equtios euile to the lie om y Py Qy ivie y y put z y y y y P Q z Pz Q 6. Et ieetil Equtio yostt M + Ny = 0 i M M y N y y tems o N ot otiig y C 7. Equtios euile to the et om ) I M N y N ieet equtio. ) I M N y N z is utio o loe, sy () the is utio o y loe, sy (y) the ) I M = y (y) N = (y), the. F. M Ny. F. e multiply with yy. F. e. He oie iziks, H.No., G.F, Ji Si, Ne IIT, Huz Khs, New Delhi-6 Phoe: / Bh oie A Istitute o Mthemtis, 8-B/6, Ji Si, Ne IIT Huz Khs, New Delhi-6 Emil: iziks.physis@gmil.om
22 Istitute o NET/JRF, GATE, IIT-JAM, JEST, TIFR GRE i PHYSICAL SCIENCES Lie D.E. o seo oe with ostt oeiiets y y P Qy R whee P, Q R e utio o o ostt. C.F. y = C.F. + P.I. ) oots, el ieet m m y Ce Ce y C C e m ) oots, el equl ) oots imgiy y C e Ce e Aos B si P.I. e ) D e i 0 the e D e ) D D ) si D si os D si I (- ) = 0 the D ) D D e e e) e e si Im e D ) D D si os He oie iziks, H.No., G.F, Ji Si, Ne IIT, Huz Khs, New Delhi-6 Phoe: / Bh oie A Istitute o Mthemtis, 8-B/6, Ji Si, Ne IIT Huz Khs, New Delhi-6 Emil: iziks.physis@gmil.om
23 Istitute o NET/JRF, GATE, IIT-JAM, JEST, TIFR GRE i PHYSICAL SCIENCES 4. Vetos Ctesi ooite system Iiitesiml isplemet l ˆ y yˆ z zˆ Volume elemet yz Giet: ˆ yˆ zˆ y z A A y Az Divegee:. A y z A A z y A A Az y A Cul: A ˆ yˆ zˆ y z z y Lpli: y z Spheil Pol Cooite System(,, ) si os, y si si, z os z y y z, os, t Iiitesiml isplemet l ˆ ˆ siˆ Volume elemet si ge om 0 to, om 0 to, om 0 to. Giet: ˆ ˆ ˆ si A si si Divegee:. A A si A ˆ ˆ siˆ Cul: A si A A si A He oie iziks, H.No., G.F, Ji Si, Ne IIT, Huz Khs, New Delhi-6 Phoe: / Bh oie A Istitute o Mthemtis, 8-B/6, Ji Si, Ne IIT Huz Khs, New Delhi-6 Emil: iziks.physis@gmil.om
24 Istitute o NET/JRF, GATE, IIT-JAM, JEST, TIFR GRE i PHYSICAL SCIENCES Lpli: si si si Cyliil Cooite System (,, z ) os, y si, z z y, t Iiitesiml isplemet l ˆ ˆ z zˆ Volume elemet z ge om 0 to, om 0 to, z om to. Giet: ˆ ˆ zˆ z A Az. z Divegee: A A y Cul: Lpli: ˆ ˆ zˆ A z A A A z z He oie iziks, H.No., G.F, Ji Si, Ne IIT, Huz Khs, New Delhi-6 Phoe: / Bh oie A Istitute o Mthemtis, 8-B/6, Ji Si, Ne IIT Huz Khs, New Delhi-6 Emil: iziks.physis@gmil.om 4
25 Istitute o NET/JRF, GATE, IIT-JAM, JEST, TIFR GRE i PHYSICAL SCIENCES VECTOR IDENTITIES Tiple Pout () A.( B C) B.( C A) C.( A B) () A( B C) B( AC. ) C( A. B) Pout Rules () ( g) ( g) g( ) A. B A ( B) B ( A) A. B ( B. ) A (4) (5).( A) (. A) A.( ) (6).( A B) B.( A) A.( B) (7) ( A) ( A) A ( ) A B B A A B A B B A (8) (. ) (. ) (. ) (. ) Seo Deivtive (9).( A) 0 i.e. ivegee o ul is lwys zeo. (0) ( ) 0 i.e. ul o giet is lwys zeo. () ( A). A A FUNDAMENTAL THEOREMS Giet Theoem:. l Divegee Theoem:. A A. Cul Theoem: A. A. l He oie iziks, H.No., G.F, Ji Si, Ne IIT, Huz Khs, New Delhi-6 Phoe: / Bh oie A Istitute o Mthemtis, 8-B/6, Ji Si, Ne IIT Huz Khs, New Delhi-6 Emil: iziks.physis@gmil.om 5
26 Istitute o NET/JRF, GATE, IIT-JAM, JEST, TIFR GRE i PHYSICAL SCIENCES. Roots o the equtio 5. Alge 5. Theoy o Quti equtios 0 e Sum Pout o the oots I e the oots, the 4.. To i the equtio whose oots e. The equie equtio will e 0 o. 0 o S P 0 whee S is the sum P is the pout o the oot.. Ntue o the oots. Roots o the equtio 0 e The epessio 4 is lle isimit. () I 4 0, oots e el. (i) I 4 0, the oots e el uequl. 4. (ii) I 4 0, the oots e el equl. () I 4 0, the 4 is imgiy. Theeoe oots e imgiy uequl. He oie iziks, H.No., G.F, Ji Si, Ne IIT, Huz Khs, New Delhi-6 Phoe: / Bh oie A Istitute o Mthemtis, 8-B/6, Ji Si, Ne IIT Huz Khs, New Delhi-6 Emil: iziks.physis@gmil.om 6
27 Istitute o NET/JRF, GATE, IIT-JAM, JEST, TIFR GRE i PHYSICAL SCIENCES 5. Logithms Popeties o Logithms ( 0,, m 0, 0 ). y the log y. log. log 0 4. log o log.log log 5. Bse hgig omul log log.log log log 6. log m log m log, log log m log m 7. log m log m O i ptiul log q p 8. log log O i ptiul q q log q q p q log 9. Rules o iies. m m. m m m m. m m 4. m m m 5. m m 6. m 7. 0 He oie iziks, H.No., G.F, Ji Si, Ne IIT, Huz Khs, New Delhi-6 Phoe: / Bh oie A Istitute o Mthemtis, 8-B/6, Ji Si, Ne IIT Huz Khs, New Delhi-6 Emil: iziks.physis@gmil.om 7
28 Istitute o NET/JRF, GATE, IIT-JAM, JEST, TIFR GRE i PHYSICAL SCIENCES Pemuttio 5. Pemuttios Comitios Eh o the ieet gemets whih e me y tkig some o ll o ume o thigs is lle pemuttio. Comitio Eh o the ieet goups o seletios whih e me y tkig some o ll o ume o thigs (iespetive o the oe) is lle omitio. Fumetl Theoem I thee e m wys o oig thig o eh o the m wys thee e ssoite wys o oig seo thig the the totl ume o wys o oig the two thigs will e m. Impott Results () Nume o pemuttios o issimil thigs tke t time. P!! whee! Note tht!.!..! () Nume o pemuttios o issimil thigs tke ll t time. P!!......! () Nume o omitios o issimil thigs tke t time. C! P!!! () Nume o omitios o issimil thigs tke ll t time. C!!! 0! 0! He oie iziks, H.No., G.F, Ji Si, Ne IIT, Huz Khs, New Delhi-6 Phoe: / Bh oie A Istitute o Mthemtis, 8-B/6, Ji Si, Ne IIT Huz Khs, New Delhi-6 Emil: iziks.physis@gmil.om 8
29 Istitute o NET/JRF, GATE, IIT-JAM, JEST, TIFR GRE i PHYSICAL SCIENCES (e) I out o thigs p e etly like o oe ki, q etly like o seo ki etly like o thi ki the est ll ieet, the the ume o pemuttios o thigs tke ll t time! p!. q!.! () I some o ll o thigs e tke t time the the ume o omitios will e C C... C (g) C C (h) C C o. (i) C C C He oie iziks, H.No., G.F, Ji Si, Ne IIT, Huz Khs, New Delhi-6 Phoe: / Bh oie A Istitute o Mthemtis, 8-B/6, Ji Si, Ne IIT Huz Khs, New Delhi-6 Emil: iziks.physis@gmil.om 9
30 Istitute o NET/JRF, GATE, IIT-JAM, JEST, TIFR GRE i PHYSICAL SCIENCES 5.4 Biomil Theoem () Sttemet o iomil theoem o positive egtive itegl ie C C... C C C... C C... C... () Nume o tems mile tem The ume o tems i the epsio o is. I is eve thee will e oly oe mile tem i.e. th. I is o thee will e two mile tems i.e. th Epsio th Ftoiztio Stelig s omul Usig summtio ottio, iomil epsio e witte s y y k 0 k k k Stelig s ppoimtio (o Stelig s omul) is ppoimtio o lge toils. l l whee is vey lge He oie iziks, H.No., G.F, Ji Si, Ne IIT, Huz Khs, New Delhi-6 Phoe: / Bh oie A Istitute o Mthemtis, 8-B/6, Ji Si, Ne IIT Huz Khs, New Delhi-6 Emil: iziks.physis@gmil.om 0
31 Istitute o NET/JRF, GATE, IIT-JAM, JEST, TIFR GRE i PHYSICAL SCIENCES 5.5 Detemits I lie lge the etemit is vlue ssoite with sque mti. The etemit o mti A is eote yet( A ), o A. Fo iste, the etemit o the mti I A I the A e the et( A ) = g h i Popeties e e e h i g i g h g h i () The vlues o etemit is ot ltee y hgig ows ito olums olums ito ows. e.g. y z = y z y z z z () I y two jet ows o two jet olums o etemit e itehge the etemit etis its solute vlue ut hges its sig. e.g. y z = y z y z y z () I y two ows o two olums o etemit e ietil the the etemit vishes. Thus He oie iziks, H.No., G.F, Ji Si, Ne IIT, Huz Khs, New Delhi-6 Phoe: / Bh oie A Istitute o Mthemtis, 8-B/6, Ji Si, Ne IIT Huz Khs, New Delhi-6 Emil: iziks.physis@gmil.om
32 Istitute o NET/JRF, GATE, IIT-JAM, JEST, TIFR GRE i PHYSICAL SCIENCES () I eh ostituet i y ow o i y olum e multiplie y the sme to the the etemit is multiplie y tht to p p p p q q q q (e) I eh ostituet i y ow o i y olum osists o tems the the etemit e epesse s the sum o etemits. Thus () I oe ow o olum is k times the othe ow o olums espetively the etemit o mti will e 0. e.g. k. k. 0 k. k. k. 0 g k. g i g h i Some si popeties o etemits e:. et( I) I whee I is the T. et( A ) et( A) whee ietity mti. T A is tspose o A.. et( A ) whee A is ivese o A. et( A) 4. Fo sque mties A B o equl size, et( AB) et( A)et( B) 5. et( A) et( A) o mti He oie iziks, H.No., G.F, Ji Si, Ne IIT, Huz Khs, New Delhi-6 Phoe: / Bh oie A Istitute o Mthemtis, 8-B/6, Ji Si, Ne IIT Huz Khs, New Delhi-6 Emil: iziks.physis@gmil.om
33 Istitute o NET/JRF, GATE, IIT-JAM, JEST, TIFR GRE i PHYSICAL SCIENCES 6. Coi Setio I the Ctesi ooite system the gph o quti equtio o two viles epeset oi setio whih is give y A By Cy D Ey F 0. The oi setios esie y this equtio e lssiie with the isimit D B 4AC i D 0, the equtio epesets ellipse i D 0, A C B 0 speil se o ellipse; i D 0, the equtio epesets pol i D 0 the equtio epesets hypeol, the equtio epesets ile whih is i we lso hve D 0, A C 0, the equtio epesets etgul hypeol Note tht A B e polyomil oeiiets, ot the legths o semi-mjo/mio is s eie i some soues. Coi Equtio Eetiity Semi-ltus Pol equtio Pmeti om setio etum Cile y 0 os, y si Ellipse y e l l eos 0 e os, y si Pol y 4 e l os t, y t Hypeol y e l l eos e t, y se He oie iziks, H.No., G.F, Ji Si, Ne IIT, Huz Khs, New Delhi-6 Phoe: / Bh oie A Istitute o Mthemtis, 8-B/6, Ji Si, Ne IIT Huz Khs, New Delhi-6 Emil: iziks.physis@gmil.om
34 Istitute o NET/JRF, GATE, IIT-JAM, JEST, TIFR GRE i PHYSICAL SCIENCES 7. Poility Poility The poility p o ouee o evet i system is eie with espet to sttistil esemle o N suh systems. I the Poility esity p N N N systems i the esemle ehiit the evet The poility esity ( u) is eie y the popety tht ( u) u yiels the poility o iig the otiuous vile u i the ge etwee u u u. Me vlue The me vlue o u is eote y u s eie s u pu whee the sum is ove ll possile vlue vluesu o the vile u p is eotes the poility o ouee o the ptiul vlueu.aove eiitio is o isete vile. Fo otiuous vile u, u = u( u) u Dispesios o vie The ispesio o u is eie s to ( u) ( u u ) ( u) p ( u u) whih is equivlet He oie iziks, H.No., G.F, Ji Si, Ne IIT, Huz Khs, New Delhi-6 Phoe: / Bh oie A Istitute o Mthemtis, 8-B/6, Ji Si, Ne IIT Huz Khs, New Delhi-6 Emil: iziks.physis@gmil.om 4
35 Istitute o NET/JRF, GATE, IIT-JAM, JEST, TIFR GRE i PHYSICAL SCIENCES Joit poility I oth evets A B ou o sigle peome o epeimet, this is lle the itesetio o joit poility o A B, eote s p( A B). Iepeet poility I two evets, A B e iepeet the the joit poility is Mutully elusive p( A B) p( A). P( B) I eithe evet A o evet B o oth evets ou o sigle peome o epeimet this is lle the uio o the evets A B eote s p( A B). I two evets e mutully elusive the the poility o eithe ouig is p( A B) p( A) P( B) Not mutully elusive I the evets e ot mutully elusive the p( A B) p( A) P( B) p( A B) Coitiol poility Coitiol poility is the poility o some evet A, give the ouee o some othe evet B. Coitiol poility is witte p( A / B ), is e "the poility o A, give B". It is eie y p( A B) p( A/ B) p( B) He oie iziks, H.No., G.F, Ji Si, Ne IIT, Huz Khs, New Delhi-6 Phoe: / Bh oie A Istitute o Mthemtis, 8-B/6, Ji Si, Ne IIT Huz Khs, New Delhi-6 Emil: iziks.physis@gmil.om 5
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