Unit 1 Chapter-3 Partial Fractions, Algebraic Relationships, Surds, Indices, Logarithms

Transcription

1 Uit Chpter- Prtil Frctios, Algeric Reltioships, Surds, Idices, Logriths. Prtil Frctios: A frctio of the for 7 where the degree of the uertor is less th the degree of the deoitor is referred to s proper frctio. If the degree of the uertor is greter th or equl to the degree of the deoitor the frctio is referred to s iproper frctio. A iproper frctio of the for e writte s This process is clled epressig or decoposig sigle frctio s su of two (or severl) seprte frctios or Prtil frctios. Eple Cosider the rtiol epressio. This epressio c e epressed i the copct for s =. Eple Cosider the epressio. This epressio c e epressed s the su of two Prtil frctios s = B A. Here the vlues of A d B re to e deteried i the followig w, = B A. B A

2 Sustitutig gives =.. Suppleetr Proles A d sustitutig gives. B. So ) ( )( ) ) ( )( ) ) ( )( ) ) ( )( ) ) ( ) ) ( )( ) 7) ( )( )( ) ) ( )( ) 9) ( )( ) 0) ( )( ) ) ( )( ) ) ( ) ( )( ) ) ( )( ) Multiple choice Eercise Tpe p q ) If, ( )( c) the vlue of p d q re: () p, q () p, q (c) p, q (d) p, q (e) p, q. If p 0 hs equl roots d p 0, p is : () ()0 (c) (d) (e). If p ( q), the vlues of p d q re: () p, q () p, q (c) p, q (d) p, q (e) p 0, q. Algeric Reltioships Solvig qudrtic equtio c 0.

3 c Let 0 c 0 c c c. Hece the two vlues for re c d c Note: i) whe c 0, hs two distict roots. Eple: ii) whe c 0, hs two equl roots. iii) whe c 0, hs o rel roots. Solve; 0 But, whe is rel uer. Note: The uer is ot rel uer. It is cosidered s igir uer d deoted with the letter i. i.e. i. So, i the ove eple the solutios of the give equtio re i d i.

4 .. Solve the followig qudrtic equtios copletig the squre ) 0 ) 0 ) 7 0 ) 0 ) 0 ) c 0 Deterie the ture of the roots of the followig equtio ut do ot solve the equtios. 7) 9 0 ) 0 0 9) 0 0) 0 ) 9 0 ) 9 0 ) For wht vlues of K is 9 k perfect squre? ) The roots of k 0 re equl.fid K. )Fid if 0 hs equl roots.. Surds.. Geerl rules of Surds A uer which c e epressed s frctio of itegers (ssuig the deoitor is ever 0) is clled rtiol uer. Eples of rtiol uers re, d. A uer which cot e epressed s frctio of two itegers is clled irrtiol uer. Eple of irrtiol uers re, 7 d Π. A irrtiol uer ivolvig root is clled Surd... Multiplictio of Surds: For eple: i. ii... Divisio of Surds: For eple:

5 i. 7 7 ii. 9.. Additio d Sutrctio of Surds: Eple (): Siplif () 7 d () 0 () () 0

6 .. Rtioliztio of the Deoitor ( 9 )( 9 ) ( 9 For eple: ( 7 )( 7 ) ( 7) ( ) 7 ( )( Eple (): ) ( 9) ) ( ( ) ) Siplif Eple (): Siplif ( 7 7 ) 7 Eple () Fid, without usig tles or clcultors, the vlue of ( ( ) ( )( ) ) Suppleetr Proles. Siplif ech of the followig: i. 7 vi. 7 7 ii. iii. 9 iv. 7 v. vii. viii. i. 7 7

7 7. Epress the followig i the for of c : i. ii. ( ( ) ) iv. vi. 7 7 iii. ( ) v.. Rtiolize the deoitors of the followig, givig our swer i the siplest for possile: i. iii. 7 ii. 7. Fid the vlue of ech of the followig: i. iii. ( ) ( ) ii. 0 7

8 . Idices If positive iteger is ultiplied itself three ties.we get, i.e. is clled the se d, the ide or power.thus.here es the th power of, I geerl, es the power of, where is positive ide of the positive iteger... Rules of idices There re severl iportt rules to reeer whe delig with idices. If,, d re positive itegers, the () () () () () e.g. e.g. ( ) e.g. ) ( e.g. e.g. ( ) 0 ( ) () 0 e.g. 0 (7) e.g. () e.g. (9) ) ( e.g. ( ) Eple Evlute: (i) (ii) (iii) (iv) i. iii. ii. iv.

9 9 9 Eple Evlute: (i) (ii) (iii) i ii. iii Solvig Epoetil Equtios Eple Solve the followig epoetil equtios: (i) (ii) 0. i. ii. 0. Eple Solve the equtio: Let or Whe

10 0 0 Whe ) ( le idissi Hece Eple If 7 9 d, clculte the vlue of d Fro (): Fro (): () (): Sustitute = ito (): d

11 .. Suppleetr Proles. Evlute ech of the followig without usig clcultor: i. ii iii. 9 vii. viii. 7 i. ii. iv. v. vi. i. 7.. Siplif ech of the followig givig our swer i ide for: i. vi. i. ii. vii. ii. 9 iii. viii. 7 iv. i. v.. 9. Solve the followig equtios: i. ii. iii. iv. v.

12 vi. 7 9 i. ii. vii.. viii. 7 i B usig pproprite sustitutio, or otherwise, solve the followig equtios: i. ii. t t iii. 9 0 iv. 9. Solve the followig pirs of siulteous equtios: i. 7 9, 7 v. 9 vi. vii. 9 0 viii. ii. 7, 7 iii., iv., v. 7,

13 . Logriths For uer such tht to e d is deoted log. (>0 d ), the logrith of to the se is defied Thus log = es tht For eple, 00 0 log log 0 00 Note: The logrith of to se is 0, i.e. log 0 Eple The logrith of uer to se of the se uer is, i.e. log The logrith of egtive uer is ot defied. Fid the vlue of (i) log (ii) log 9 (iii) i. Let log log (iv) log 0. 9 ii. Let log 9 9 iii. Let log iv. Let log 0.

14 Eple 0. Fid the logrith of the followig to the se idicted i rckets: (i) 7 () (ii) () (iii) 000 (0) (iv) 0. () i. ii. iii. iv. 7 log log log log

15 .. Lws of Logriths () log log log () log log log Proof: () Let log d log () log log d Multipl : log () Let log d log d log log Divide : log log log () Let log log log Eple Without usig tles, evlute log0 log log0 log0 :

16 log 0 log log log 0 70 log 00 log log Eple log Give tht log 0 log 0 p, Clculte the vlue of p without usig tles or Clcultors. log log 0 p log 0 ( ) = 0 p 0 p 0 p = 00 P= Sice p cot e - ecuse log 0 (-) is ot defied, p=.. Suppleetr Proles Write the followig i logrithic fro: Write the followig i ide for: 7. log. log 9. log

17 7 0. log 9 0. log. log Solve the followig equtios:. log. log. log. log log. log log 0. Sipl the followig logriths: log. log. log 0. log.. log 7. log. 0. log log. log log log log. log log log 9. log 7 log 7 7. log log 0 log. log 0 log 0 log 0 log If log. 09 d log., evlute the followig without the use of clcultors or logrith tles:. log 0. log. log 7. log. log log log. log 00. log. log 0.. log log. Evlute the followig without usig clcultors: () log log log

18 () log 0 7 log 0 9log 0 9 log 0 7 log 0 () log log log () 0 log log log 9 () log 0 7 log 0 9 log 0 () 9 log log log log Coo logriths Eple Solve the Equtio. 7.7 Tkig logriths to se 0, lg lg lg.7 lg.7 = lg.7 lg =.09 Eple Give tht =, fid correct to two decil plces. = = Tkig logriths to se 0, lg = lg lg lg

19 9 = lg lg =.(correct to decil plces)... Suppleetr Proles. Solve the followig equtios, givig our swer correct to sigifict figures, where ecessr : () 7 () 0. 7 () 00 () Siplif lg 7 lg lg lg lg 0. Give tht, () Fid the vlue of ( ) i ters of. ()Fid the vlue of d if. Sur Surds: () () () d re cojugte surds. The product of Cojugte surds is rtiol uer. Idices: () () () ( ) () 0 () () (7) ( ) Logriths () log es. () log log log ( ) 9

20 0 () log log log () log log () log 0 () log 0