MATHEMATICS I & II DIPLOMA COURSE IN ENGINEERING FIRST SEMESTER

Transcription

1 MATHEMATICS I & II DIPLOMA COURSE IN ENGINEERING FIRST SEMESTER A Plition nder Government of Tmilnd Distrition of Free Tetook Progrmme ( NOT FOR SALE ) Untohilit is sin Untohilit is rime Untohilit is inhmn DIRECTORATE OF TECHNICAL EDUCATION GOVERNMENT OF TAMIL NADU

2 Government of Tmilnd First Edition Chirperson Thir Kmr Jnth I.A.S Commissioner of Tehnil Edtion Diretorte of Tehnil Edtion, Chenni 5 Co-ordintor Convener Er. S. Govindrjn P.L. Snkr Prinipl Dr. Dhrmml Government Poltehni College Thrmni, Chenni Letrer (Seletion Grde) Rjgopl Poltehni College Gdithm-66 Reviewer Dr. S. Pl Rj Assoite Professor, Dept of Mthemtis Ann Universit, MIT Cmps, Chenni -. Athors R. Rmdoss Letrer, (Seletion Grde) TPEVR Govt. Poltehni College Vellore-6 B.R. Nrsimhn Letrer (Seletion Grde) Arlmig Plnindvr Poltehni College Plni-66 M. Devrjn Letrer (Seletion Grde) Dr. Dhrmml Government Poltehni College for Women Trmni, Chenni-6 Dr. L. Rmppilli Letrer (Seletion Grde) Thigrjr Poltehni College Selm-665 K. Shnmgm Letrer (Seletion Grde) Government Poltehni College Prswlkm, Chenni-6 Dr.A. Shnmgsndrm Letrer (Seletion Grde) Vlivlm Desikr Poltehni College Ngppttinm-6 M. Rmlingm Letrer (Seletion Grde) Government Poltehni College Ttiorin-688 R. Srmnin Letrer (Seletion Grde) Arsn Gnesn Poltehni College Sivksi-66 Y. Anton Leo Letrer Mothill Nehr Government Poltehni College Pondiherr-658 C. Srvnn Letrer (Senior Sle) Annmli Poltehni College Chettind-6 This ook hs een prepred the Diretorte of Tehnil Edtion This ook hs een printed on 6 G.S.M Pper Throgh the Tmil Nd Tet Book Corportion

3 FOREWORD We tke gret plesre in presenting this ook of mthemtis to the stdents of Poltehni Colleges. This ook is prepred in ordne with the new slls frmed the Diretorte of Tehnil Edtion, Chenni. This ook hs een prepred keeping in mind, the ptitde nd ttitde of the stdents nd modern methods of edtion. The lid mnner in whih the onepts re eplined, mke the tehing lerning proess more es nd effetive. Eh hpter in this ook is prepred with strenos efforts to present the priniples of the sjet in the most es-to-nderstnd nd the most es-to-workot mnner. Eh hpter is presented with n introdtion, definition, theorems, eplntion, worked emples nd eerises given re for etter nderstnding of onepts nd in the eerises, prolems hve een given in view of enogh prtie for mstering the onept. We hope tht this ook serves the prpose i.e., the rrilm whih is revised DTE, keeping in mind the hnging needs of the soiet, to mke it livel nd virting. The lngge sed is ver ler nd simple whih is p to the level of omprehension of stdents. List of referene ooks provided will e of mh helpfl for frther referene nd enrihment of the vrios topis. We etend or deep sense of grtitde to Thir.S.Govindrjn, Co-ordintor nd Prinipl, Dr. Dhrmml Government poltehni College for women, Chenni nd Thir. P.L. Snkr, onvener, Rjgopl poltehni College, Gdithm who took sinere efforts in prepring nd reviewing this ook. Vlle sggestions nd onstrtive ritiisms for improvement of this ook will e thnkfll knowledged. Wishing o ll sess. Athors iii

4 SYLLABUS FIRST SEMESTER MATHEMATICS - I UNIT - I DETERMINANTS. Definition nd epnsion of determinnts of order nd.properties of determinnts.crmer's rle to solve simltneos eqtions in nd nknowns-simple prolems.. Prolems involving properties of determinnts. Mtries :Definition of mtri.tpes of mtries.alger of mtries sh s eqlit, ddition, strtion, slr mltiplition nd mltiplition of mtries. Trnspose of mtri, djoint mtri nd inverse mtri-simple prolems. UNIT - II BINOMIAL THEOREM. Definition of ftoril nottion, definition of Permttion nd Comintions with forml. Binomil theorem for positive integrl inde (sttement onl), finding of generl nd middle terms. Simple prolems.. Prolems finding o-effiient of n, independent terms. Simple prolems. Binomil Theorem for rtionl inde, epnsions, onl pto for negtive integers. Simple Epnsions. Prtil Frtions :Definition of Polnomil frtion, proper nd improper frtions nd definition of prtil frtions. To resolve proper frtion into prtil frtion with denomintor ontining non repeted liner ftors, repeted liner ftors nd irredile non repeted qdrti ftors. Simple prolems. iv

5 UNIT - III STRAIGHT LINES. Length of perpendilr distne from point to the line nd perpendilr distne etween prllel lines. Simple prolems. Angle etween two stright lines nd ondition for prllel nd perpendilr lines. Simple prolems. Pir of stright lines Throgh origin :Pir of lines pssing throgh the origin h epressed in the form (-m)(-m). Derivtion of ngle etween pir of stright lines. Condition for prllel nd perpendilr lines. Simple prolems. Pir of stright lines not throgh origin: Condition for generl eqtion of the seond degree hgf to represent pir of lines.(sttement onl) Angle etween them, ondition for prllel nd perpendilr lines. Simple prolems.. UNIT - IV TRIGONOMETRY. Trigonometril rtio of llied ngles-epnsion of Sin(AB) nd os(ab)- prolems sing ove epnsion. Epnsion of tn(ab) nd Prolems sing this epnsion. Trigonometril rtios of mltiple ngles (A onl) nd s-mltiple ngles. Simple prolems. UNIT - V TRIGONOMETRY 5. Trigonometril rtios of mltiple ngels (A onl) Simple prolems. 5. Sm nd Prodt formle-simple Prolems. 5. Definition of inverse trigonometri rtios, reltion etween inverse trigonometri rtios-simple Prolems v

6 FIRST SEMESTER MATHEMATICS II UNIT - I CIRCLES. Eqtion of irle given entre nd rdis. Generl Eqtion of irle finding enter nd rdis. Simple prolems.. Eqtion of irle throgh three non olliner points onli points. Eqtion of irle on the line joining the points (,) nd (,) s dimeter. Simple prolems.. Length of the tngent-position of point with respet to irle. Eqtion of tngent (Derivtion not reqired). Simple prolems. UNIT-II FAMILY OF CIRCLES:. Conentri irles ontt of irles (internl nd eternl irles) orthogonl irles ondition for orthogonl irles.(reslt onl). Simple Prolems. Limits:Definition of limits n - n n n- - Lt sin q, q q Lt Lt q tn q q (q in rdin) [Reslts onl] Prolems sing the ove reslts.. Differentition:Definition Differentition of n, sin, os, tn, v ot, se, ose, log, e, v, v, vw, (Reslts onl). Simple prolems sing the ove reslts. UNIT- III. Differentition of fntion of fntions nd Impliit fntions. Simple Prolems. vi

7 . Differentition of inverse trigonometri fntions nd prmetri fntions. Simple prolems.. Sessive differentition p to seond order (prmetri form not inlded) Definition of differentil eqtion, formtion of differentil eqtion. Simple Prolems UNIT- IV APPLICATION OF DIFFERENTIATION I. Derivtive s rte mesre-simple Prolems.. Veloit nd Aelertion-simple Prolems. Tngents nd Normls-simple Prolems UNIT-V APPLICATION OF DIFFERENTIATION II 5. Definition of Inresing fntion, Deresing fntion nd trning points. Mim nd Minim (for single vrile onl) Simple Prolems. 5. Prtil Differentition: Prtil differentition of two vrile p to seond order onl. Simple prolems. 5. Definition of Homogeneos fntions-elers theorem-simple Prolems. vii

8 FIRST SEMESTER MATHEMATICS - I Contents Pge No Unit DETERMINANTS.... Introdtion.... Prolems Involving Properties of Determinnts.... Mtries... 9 Unit BINOMIAL THEOREM.... Introdtion Binomil Theorem Prtil Frtions Unit STRAIGHT LINES Introdtion Pir of stright lines throgh origin Pir of stright lines not throgh origin Unit TRIGONOMETRY.... Trigonometril Rtios of Relted or Allied Angles.... Compond Angles (Contined) Mltiple Angles of A Onl nd S Mltiple Angles... 9 Unit - 5 TRIGONOMETRY Trigonometril Rtios of Mltiple Angle of A Sm nd Prodt Formle Inverse Trigonometri Fntion... MODEL QUESTION PAPER... 6 viii

9 MATHEMATICS II Contents Pge No Unit CIRCLES Cirles Conli Points Length of the Tngent to irle from point(,) Unit FAMILY OF CIRCLES Fmil of irles Definition of limits Differentition Unit Differentition Methods Differentition of fntion of fntions Differentition of Inverse Trigonometri Fntions.... sessive differentition... Unit APPLICATION OF DIFFERENTIATION.... Derivtive s Rte Mesre.... Veloit nd Aelertion Tngents nd Normls... Unit 5 APPLICATION OF DIFFERENTIATION-II Introdtion Prtil derivtives Homogeneos Fntions MODEL QUESTION PAPER... 9 i

10 SEMESTER I MATHEMATICS I UNIT I DETERMINANTS. Definition nd epnsion of determinnts of order nd Properties of determinnts Crmer s rle to solve simltneos eqtions in nd nknowns-simple prolems.. Prolems involving properties of determinnts. Mtries Definition of mtri. Tpes of mtries. Alger of mtries sh s eqlit, ddition, strtion, slr mltiplition nd mltiplition of mtries. Trnspose of mtri, djoint mtri nd inverse mtri-simple prolems... DETERMINANT The redit for the disover of the sjet of determinnt goes to the Germn mthemtiin, Gss. After the introdtion of determinnts, solving sstem of simltneos liner eqtions eomes mh simpler. Definition: Determinnt is sqre rrngement of nmers (rel or omple) within two vertil lines. Emple : is determinnt Determinnt of seond order: The smol onsisting of nmers,, nd rrnged in d two rows nd two olms is lled determinnt of seond order. The nmers,,, nd d re lled elements of the determinnt The vle of the determinnt is Δ d-

11 Emples:.. 5 () () (5) () 5-6 () (-5) (6) () Determent of third order: The epression onsisting of nine elements rrnged in three rows nd three olmns is lled determinnt of third order The vle of the determinnt is otined epnding the determinnt long the first row Δ - ( - ) - ( - ) ( - ) Note: The determinnt n e epnded long n row or olmn. Emples: ( 8) ( ) ( 5) ( 7) ( 8) ( ) () (6 ) ( ) () () () Minor of n element Definition : Minor of n element is determinnt otined deleting the row nd olmn in whih tht element ors. The Minor of I th row J th Colmn element is denoted m ij

12 Emple: 5 Minor of Minor of Coftor of n element Definition : Co-ftor of n element in i th row,j th olmn is the signed minor of I th row J th Colmn element nd is denoted A ij. (i.e) A ij (-) ij m ij i j Thesignistthedtherle (-) Emple 7 6 Co-ftor of - (-) 7 6 Co-ftor of 7 (-) (-) () - (-) (-) - Properties of Determinnts: Propert : The vle of determinnt is nltered when the rows nd olmns re interhnged. (i.e) If Δ nd Δ T then Δ T Δ,

13 Propert : If n two rows or olmns of determinnt re interhnged the vle of the determinnt is hnged in its sign. If Δ then Δ -Δ nd Δ Note: R nd R re interhnged. Propert : If n two rows or olmns of determinnt re identil, then the vle of the determinnt is zero. (i.e) The vle of, is zero Sine R R Propert : If eh element of row or olmn of determinnt is mltiplied n nmer K /, then the vle of the determinnt is mltiplied the sme nmer K. If Δ nd K K K Δ, then Δ KΔ Propert 5: If eh element of row or olmn is epressed s the sm of two elements, then the determinnt n e epressed s the sm of two determinnts of the sme order.

14 5 (i.e) If Δ d d d, then Δ d d d Propert 6: If eh element of row or olmn of determinnt is mltiplied onstnt K / nd then dded to or strted from the orresponding elements of n other row or olmn then the vle of the determinnt is nltered. Let Δ Δ n m n m n m m m m n n n Δ m n Δ Δ m()n()δ Propert 7: In given determinnt if two rows or olmns re identil for, then (-) is ftor of the determinnt.

15 Let Δ For, Δ [ C, nd C re identil] (-) is ftor of Δ Nottion : Usll the three rows of the determinnt first row, seond row nd third row re denoted R, R nd R respetivel nd the olmns C,C nd C If we hve to interhnge two rows s R nd R the smol dole sided rrow will e sed. We will write like this R R it shold e red s is interhnged with similrl for olmns C C. If the elements of R re strted from the orresponding elements of R, then we write R -R similrl for olmns lso. If the elements of one olmn s C, m times the element of C nd n times tht of C re dded, we write like this C Cm C n C. Here one sided rrow is to e red s is hnged to Soltion of simltneos eqtions sing Crmer s rle: Consider the liner eqtions. let Δ 6

16 7 Δ Δ Then Δ Δ nd Δ Δ,provided Δ, re niqe soltions of the given eqtions. This method of solving the line eqtions is lled Crmer s rle. Similrl for set of three simltneos eqtions in, nd z zd zd nd zd, the soltion of the sstem of eqtions, rmer s rle is given, Δ Δ, Δ Δ nd z Δ Δ z, provided Δ where Δ d d d Δ d d d Δ nd z d d d Δ

17 . WORKED EXAMPLES PART A. Solve or ( ). Solve ±. Find the vle of m when 7 m Given 7 Epnding the determinnt long, R we hve m m(-6)- () (98) m(-6) - () (7) 8

18 -6m m 9-6m -9 9 m 6. Find the Co-ftor of element in the determinnt Coftor of A (-) 5 7 (-) (7-) 7 PART B. Using rmer s rle, solve the following simltneos eqtions z - z - z Δ (-) - (-) (-) (-) - () (-) ---6 Δ 9

19 Δ (-) - () () (-) - () () Δ z () - (-) () () - () () 6 (--) - () (-) B Crmer s rle, Δ 6 Δ 6 Δ Δ Δz z Δ 6. Using Crmer s rle solve: -z- --z5 -z - Rerrnge the given eqtions in order --z -; --z -5; --z Δ

20 Δ Δ Δ z -(-) (-) (-) - (-) (-) () (-) () (-5-) () (-) (--) -() (-) (-) (5) (--) - (-) Δ Δ,, nd Δ 7 Δ 7. Using Crmer s rle solve Δz 69 z Δ 7

21 - 5-8 Δ () (-) (-) () Δ (5) (-) (-) (8) 8-5 Δ B Crmer s rle Δ - Δ 5 5 Δ Δ 5-5. PROBLEMS INVOLVING PROPERTIES OF DETERMINANTS PART-A ) Evlte 7 9 Δ C C C Δ sine C C

22 ) Withot epnding, find the vle of 9 6 Let Δ 9 6 () ) ( () (), sine R R ) Evlte Let Δ C C C () () (), sine C C

23 ) Prove tht z z z z z z L.H.S z z z z z z z z z z z z z z z z C C C C z z z z R.H.S PART B ) Prove tht z z (-) (-z) (z-) L.H.S z z z z z z R R,R R R R

24 5 (-) (-z) z z z (-) (-z) z (epnded long the first olmn) (-) (-z) [(z) ()] (-) (-z) (z-) L.H.S R.H.S ) Prove tht ( ) (-) (-) (-) L.H.S o o Δ C C,C C C C Δ ) )( ( ) )( ( Δ (-) (-) Δ (-) (-) (epnded long the first row )

25 ( - ) ( - ) ( - ) ( - ) [ ( - ) ( - ) [ ( - ) ( - ) [ [ ( )] ] ( - )] ( - ) ( - ) [( ) ( - ) ( - )] ( - ) ( - ) ( - ) [ ] - - ] R.H.S ) Prove tht ²() L.H.S R R R R () C C - C, C C - C () () (²-) ²() R.H.S 6

26 7 ) Prove tht () L.H.S R R R R () () ) ( C - C C, C - C C () ) ( ) ( () [() ] () R.H.S 5) Prove tht Let Δ ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (

27 8 R R R R ) ( ) ( C - C C, C - C C ) ( ) ( [-] Δ ) (

28 . MATRICES Introdtion: The term mtri ws first introded Frenh mthemtiin Cle in the er 857. The theor of mtries is one of the powerfl tools of mthemtis not onl in the field of higher mthemtis t lso in other rnhes sh s pplied sienes, nler phsis, proilit nd sttistis, eonomis nd eletril irits. Definition: A Mtri is retnglr rr of nmers rrnged in to rows nd olmns enlosed prenthesis or sqre rkets. Emple:. A. B 5 Usll the mtries re denoted pitl letters of English lphets A,B,C,et nd the elements of the mtries re represented smll letters,,,.et. Order of mtri If there re m rows nd n olmns in mtri, then the order of themtriismnormn. Emple: A A hs two rows nd three olmns. We s tht A is mtri of order Tpes of mtries Row mtri: A mtri hving onl one row nd n nmer of olmns is lled row mtri. Eg: A ( -)

29 Colmn mtri Mtri hving onl one olmn nd n nmer of rows is lled olmn mtri. Eg: B Sqre mtri A mtri whih hs eql nmer of rows nd olmns is lled sqre mtri. 9 Eg: A ; B 6 A is sqre mtri of order B is sqre mtri of order Nll mtri (or) zero mtri or, void mtri: If ll the elements of mtri re zero, then the mtri is lled nll or zero mtri or void mtri it is denoted. Eg: () Digonl mtri: A sqre mtri with ll the elements eql to zero eept those in the leding digonl is lled digonl mtri Eg: 5 Unit mtri: Unit mtri is sqre mtri in whih the prinipl digonl elements re ll ones nd ll the other elements re zeros.

30 It is denoted I. Eg: I Here I is nit mtri of order. I I is nit mtri of order. Alger of mtries: Eqlit of two mtries: Two mtries A nd B re sid to e eql if nd onl if order of A nd order of B re eql nd the orresponding elements of A nd B re eql. 5 Eg:ifA nd B z then A B mens 5 - z Addition of mtries: If A nd B re n two mtries of the sme order, then their sm AB is of the sme order nd is otined dding the orresponding elements of A nd B. Eg: If A, B then AB Note : If the mtries re different order, ddition is not possile.

31 Strtion of mtries: If A nd B re n two mtries of the sme order, then their differene A-B is of the sme order nd is otined strting the elements of B from the orresponding elements of A. Eg: If A B then A-B Slr mltiplition of mtri If A is given mtri, K is nmer rel or omple nd K then KA is otined mltipling eh element of A K. It is lled slr mltiplition of the mtri. Eg:ifA 6 5 nd K KA Mltiplition of two mtries: Two mtries A nd B re onformle for mltiplition if nd onl if the nmer of olmns in A is eql to the nmer of rows in B. Note: If A is mn mtri nd B is np mtri then AB eists nd is of order mp. Method of mltiplition: Let A nd B z z z Here the nmer of olmns of mtri A is eql to the nmer of rows of mtri B. Hene AB n e fond nd the order is.

32 Eh element of the first row of AB is got dding the prodt of the elements of first row of A with the orresponding elements of first, seond nd third olmns of B. On similr lines, we n lso get the seond row of AB. (ie) AB z z z z z z z z z Note: () If A is of order nd B is of order then AB is of order t BA does not eist. () If AB nd BA re of sme order the need not e eql. In generl AB BA. Emple: If A nd B then AB Trnspose of mtri If the rows nd olmns of mtri re interhnged then the resltnt mtri is lled the trnspose of the given mtri. It is denoted ' A (or) A T Emple: If A then T A 6 5 7

33 Note: (i) If mtri A is of order mn, then the order of (ii) (A T ) T A T A is nm. Co-ftor mtri: In mtri, if ll the elements re repled the orresponding o-ftors is lled the o-ftor mtri. Emple: The o-ftor mtri of the mtri. 8 Minors o-ftors m (-) m (-) () is s follows m 8 (-) m (-) (8) -8 m - (-) m (-) (-) m (-) m (-) () o-ftor mtri is 8 Adjoint mtri (or) djgte mtri: The trnspose of the o-ftor mtri is lled the djoint mtri. or djgte mtri. It is denoted dj A. Emple Let A Coftor of (-) () Coftor of (-) (-) Coftor of - (-) () - Coftor of (-) ()

34 Coftor mtri Adj A Singlr nd Non-singlr mtries: A sqre mtri A is sid to e singlr if A. If the determinnt vle of the sqre mtri A is not zero it is nonsinglr mtri. Emple: Let A A A (-)-5(-7-)7(-8-5) 685- The given mtri A is singlr Inverse of mtri: Let A e non-singlr sqre mtri if there eists sqre mtri B, sh tht ABBAI where I is the nit mtri of the sme order s tht of A, then B is lled the inverse of mtri A nd it is denoted A -. (to e red s A inverse). This n e determined sing the forml. A - dj A A 5

35 Note:. if A, then there is no inverse for the mtri. A - A AA - I,. (AB) - B - A -. (A T ) - (A - ) T WorkingrletofindA - : ) Find the determinnt of A ) Find the o-ftor of ll elements in A nd form the o-ftor mtri of A. ) Find the djoint of A. ) A - Adj A A provided A Note: For seond order mtri, the djoint n esil e got interhnging the prinipl digonl elements nd hnging the signs of the seondr digonl elements. Emple A d A d d - Adjoint d A A d d 6

36 7 WORKED EXAMPLES: PART A. If A 5 wht is the order of the mtri nd find T A Soltion : 5 A The order of the mtri is 5 A T. If A 7 6 B, 5 find A B A-B If f() nd A find f(a) A ( ) f nd f(a) AI,

37 8 8 f(a). If XY 6 nd X-Y 8 find X nd Y soltion: Given XY 6.() nd X-Y 8.() Adding X 8 8 X Sstitte mtri X in () 6 Y 5 Y 5 6 Y 5. Find the vle of so tht the mtri is singlr

38 9 soltion: Let A The mtri A is singlr, then A Epnding throgh first row (-) (-8) If A nd B find AB 6 AB 7. Find A,ifA 5 A.A A

39 8. Find the djoint of Let A A A A A (-) (-) (-) (-) () (-) (-) (-) () (-) Coftor mtri A (-) () - () (-) () Adj A Aliter: Inter Chnging elements in the prinipl digonl nd hnging sign of elements in the other digonl Adj A 9. Find the inverse of 5 Step : let A 5 Now A A - eists.

40 dj A 5 5 AdjA A A - PART-B ) If A nd B Show tht AB BA Now AB AB BA

41 5 5 5 AB BA ) Show tht AB BA if A nd B Now AB 6 5 Similrl BA 8 6 AB BA ) If A Find I A A T A

42 Now A A T I ) If A showtht(a-i)(a-i) A A-I ( )( ) O I A - A -I I A -

43 5) If A show tht A -A 5I nd hene find A Let A A A -A I RHS To find A We hve proved tht A -A 5I

44 A A5I Mltipling oth sides A, we get A A 5AI A 5A ) If A, B Show tht (AB) - B - A - AB 7 AB 7-7- (AB) - eists Adj (AB) 7 5

45 6 (AB) - 7 AB Adj(AB) () B B --- B - eist Adj (B) B - B AdjB A A A - eist Adj A A - A AdjA B - A () From () nd () (AB) - B - A -

46 7. Find the inverse of the mtri Soltion : Let A A (-) - (6-) - () () - (6) - (5) A - eists A - A - - (6-) -6 A ()5 A - - () -5 A (-) A - - () - 7

47 8 A () - A - () - A (-) - Co-ftor mtri A Adj A A - 8 A AdjA EXERCISE PART-A. Find the Vle of z. Find the vle of the determinnt θ θ θ θ sin os os sin. Find the vle of minor 5 in the determinnt 6 5. Find the vle of m so tht m

48 5. Find the vle of if 6. Evlte withot epnding 7. Show tht 8. Prove tht 9. Show tht z. Find the vle of. Show tht z z If A nd B 6 find A B. Find mtri A if ij ij. If f() nd A find f(a) 5. If f () -5 nd A find f(a) 9

49 6. Show tht the mtri 6 is singlr 7. Prove tht the mtri is non-singlr 8. If A nd B find AB 9. If X nd Y 5 find XY.. If A find A. Find the o-ftor mtri of. Find the djoint of 6. Find the in inverse of. Find the in inverse of PART B. Solve Crmer s rle. z 8, - z nd z. z,5z-nd z. z,z9nd z d. z, z, z e. z, - z, -7 z -6 f. z -, z, -z 6

50 . Prove tht ( )( )( ). Prove tht z z z z z. Prove tht 5. Prove tht ( ) 6. If A nd B show tht AB BA 7. If A 6 nd B 5 verif tht (AB) - B - A - 8. Find the inverse of the following i) ii) iii) 9 8 iv) 6

51 -z - M ANSWERS PART - A

52 PART - B. ),, ),,- ),, d),, e),,- f),-, 8. i) ii) 6 iii) iv)

53 UNIT II BINOMIAL THEOREM. Definition of ftoril nottion, definition of Permttion nd Comintions with forml. Binomil theorem for positive integrl inde (sttement onl), finding of generl nd middle terms. Simple prolems.. Prolems finding o-effiient of n, independent terms. Simple prolems. Binomil Theorem for rtionl inde, epnsions, onl pto for negtive integers. Simple Epnsions.. Prtil Frtions Definition of Polnomil frtion, proper nd improper frtions nd definition of prtil frtions. To resolve proper frtion into prtil frtion with denomintor ontining non repeted liner ftors, repeted liner ftors nd irredile non repeted qdrti ftors. Simple prolems.. BINOMIAL THEOREM Definition of Ftoril Nottion: The ontined prodt of first n ntrl nmers is lled n ftoril nd is denoted n! or n ie n!..... (n-).n 5!...5 Zero ftoril: we will reqire zero ftoril for llting n vle whih ontins zero ftoril. It does not mke n sense to define it s the prodt of the integers from to zero. So we define! Dedtion: n!.....(n-)n [.....(n-)]n (n-)!n Ths, n! n[(n-)!] For Emple, 9! 9(8!)

54 Binomil Theorem Introdtion: Before introding Binomil theorem, first we introde some si ides nd nottion. First we egin with the following prolem. We wnt to selet rom plers from mong 5 good rrom plers. Let s denote them the letters A,B,C, D nd E. To Selet plers, we shll first tke A nd then with him, ssoite B,C,D nd E. Tht is AB, AC, AD nd AE re for tpes of seletion of plers. Also strting with B, we hve BC, BD nd BE with C, CD nd CE nd finll strting with D, we hve DE onl. So totll there re ws, for seleting plers ot of 5 plers, If we denote the nmer of ws of seletion of persons ot of 5 persons, smolill 5C, then we hve 5C Also let s ssme tht the seleted plers re going to Pne for ntionl level ompetition. Sine ttending sh ntionl level ompetition itself is dmirle, we wnt to give pose for grop photo. Two hirs re roght s shown elow, there re two tpes of rrngements Arrngement Arrngement A B Chir Chir Chir Chir Ths we see tht for one seletion, there re two different rrngements nd so for the totl of seletion, the totl nmer of rrngements is. Tht is if the nmer of ws of rrngement of persons ot of 5 persons, is denoted 5P, we hve 5P. The ove rrngement nd seletion re sll lled permttion nd omintion. The men the rrngement the work permttion nd the seletion, the work omintion. So the nmer of ws of rrngement nd the nmer of ws of seletion in the ove emple re respetivel denoted 5P nd 5C. 5 B A

55 Henewehve5P nd 5C, s seen in the ove emple of rom plers, nd the method of llting 5P or 5C n e rememered s elow: 5P 5(5-) 5 5(5 ) 5 5C Emples: ) 7P nd7c.. 5 ) p And C Note : Seletionofriketplersotof7plersnedonein 7C ws nd ( ) 7C ( ) C Tht is, seletion of ot of 7 is sme s seletion of 6 ot of 7. In generl, n r n n r Also it mst e noted tht the nmer of ws of seletion or rrngement (omintion or permttion) will lws e positive integer nd it n never e frtion. Definition: np r is the no.of ws of rrngement (or permttion) of r things ot of n things. n r is the no. of ws of seletion (or omintion) of r things ot of n things. (or) np r mens nmer of permttion of n things, tken r t time nd n r mens the no.of ws omining n things, tken r t time. 6

56 Note: The vles of np r nd n Cr re given elow n! np r n(n - ) (n - )...(n - r ) (n r)! n(n )(n )...(n r ) n! n r...r (n r)!r! where n!...n Emples: np n,np, n(n-), np n(n-)(n-) n n(n ) n(n )(n ) n, n, n et.,.... BINOMIAL THEOREM Binomil mens n epression, whih onsists of two nmers or grop of nmers onneted pls sign or mins sign Emple:, -,, - ( ), - et., In inomil theorem, we del with the powers of inomil epressions. From shool stdies, we know tht () nd (). After stding ot the vles of n r (n,n,n,...et), we n nderstnd nd write the epnsion of () s elow ().. Similrl for () nd () 5,wenwriteselow: ()... nd ()

57 Sttement: Binomil theorem for positive integrl inde If n is n positive integer, then () n n n. n-.n. n-. n. n-... n r. n-r. r... n. Notes : ) The totl nmer of terms in the epnsion is (n) ) In eh term, sm of the powers (eponents) of nd is eql to n. ) The generl term nr. n-r. r is (r) th term. Ie t r n r. n-r. r ) n,n,n...et re lled inomil o-effiient 5) Sine n r n n-r We hve n n n,n n n-,n n n-, et., It mst e notied tht n n n 6) If n is n even integer, there is onl one middle term whih will e th n t ple nd if n is odd nmer there re two middle th n n terms, whih re t nd ples. 7) To find the term independent of in the inomil epnsion, pt the power of in the generl term s zero 8) In finding the term independent of, if the vle of r omes to e frtion, then it mens tht other is no term independent of. Binomil Theorem for rtionl inde: If is nmerill less thn one nd n, n rtionl nmer, then, n n(n ) ( ) n. Note : 8 th n(n )(n )..... ) The no. of tems in the epnsion is infinite ) Here the nottions n,n,n...et re meningless; sine n is rtionl nmer ) Also here we onsider, onl epnsions of negtive integers pto-

58 When the vles of n re -,-,- the epnsions re ) (-) -... ) () ) (-) -... ) () ) (-) ) () WORKED EXAMPLES PART A.Epnd (-) sing inomil therorem () n n n. n-. n. n-....n r. n-r. r... n (-) [(-)] () () (-).() (-)..(-) (-) Epnd 5 5 sing inomil theorem

59 .Find the generl term in the epnsion of Soltion : The generl term, t r n. r nr r. r r r.().,sine here n, nd r. r r.( ). r.( ). r 5 r r. Find the 5 th term in the epnsion of The generl term, Here r 5 rweget5 th term t 5 Also n8, nd 8 t t r n r. n-r. r PART B 8 8.Find the middle term in the epnsion of Here n is even nmer nd so there is onl one middle term th 8 nd tht is term. So, 5 th term is the middle term. Now, generl term is t r n r. n-r. r To find t 5, pt r

60 8 t5 t 8.() Find the middle terms in the epnsion of Sine here n is odd nmer, the totl no. of terms in the th n epnsion is even nd so there re two middle terms nd th n terms re the middle terms. i.e, nd th terms, tht is 6 th nd 7 th term re the middle terms nr r Now, generl term is t n. t t 6 7 t ( 5 r ( ) ( ). 5 t ( ) ) (. 6 6 ) r. Χ th 5

61 .B sing Binomil theorem, find the 6 th power of Soltion : 6 ( ) () ,, 6,,,5,,,5 6,77,56.Using Binomil theorem, find the vle of (.) 5 orret to deiml ples. Soltion : 5 (.) (.) (.) 5..(.)..(.) 5 (.) 5 5. (.) (.). 5.5.(.)... 5 (.) , Corret to deiml ples...(.)

62 . WORKED EXAMPLES PART A. Find the epnsion of () - We know tht ( ) n ( ) n! ( ) n n. n(n )(n ).. ( )( ) ( )( ) ( ) ( )... ( ) ( )( ) X. ( )( )( )... Find the epnsion of ( ) -... ( ) ( )( )( ) ( )( ). ( )... ( )( )( ) PART B.Find the o-effiient of in the epnsion of t Now, generl term is n. r r r nr r. 5 t r 5.( r 5r ). 5

63 5 6 r r. r r r ( ) 67r r 5.( ). () To find the o-effiient of, pt 6 7r 8 7 r 6 8 ndr 7 Appling r in the eqtion (), we get t 68 5.( ). i.e., t so, o-effiient of is 5.. Find the term independent of in the epnsion of Now, generl term is t r r n. r nr r. r r r.() t.() r r r.( ) r. r r ( ).. r () To find independent term, pt -r r nd r 5 Using the vle of r5 in eqtion (), we get the independent term s t ( ) 5.( )

64 .Epnd (-) - sing Binomil theorem (-) - [.] ( )( ) ( )( )( 5) ( )... (6 ) PARTIAL FRACTIONS Definition of Polnomil Frtion: p() An epression of the form where p() nd q() o re q() polnomils in is lled polnomil frtion. 5 The epressions, re emples for rtionl or polnomil frtion. Proper Frtion; A proper frtion is one in whih the degree of the nmertor is less thn degree of the denomintor. The epressions proper frtion. Improper frtion:, 7 9 re emples for 5 An improper frtion is frtion in whih the degree of the nmertor is greter thn or eql to the degree of the denomintor 55

65 The epressions frtions. Prtil Frtion Consider the sm of We simplif it s follows: 5, 5 nd 5 5( ) ( ) ( )( ) ( )( ) re emples for improper 8 ( )( ) Conversel the proess of writing the given frtion 8 5 s is known s splitting into prtil frtions ( )( ) or epressing s prtil frtion. A given proper frtion n e epressed s the sm of other simple frtions orresponding to the ftors of the denomintor of the given proper frtion. This proess is lled splitting into Prtil p() Frtion. If the given frtion is improper then onvert into sm q() of polnomil epression nd proper rtionl frtion dividing p() q(). Working rle: p() Given the proper frtion. Ftorise q() into prime ftors. q() Tpe To resolve proper frtion into prtil frtion with denomintor ontining non-repeted liner ftors. If is liner ftor of the denomintor q(), then A orresponding to this ftor ssoite simple ftor, where A is onstnt (A ) ie., when the ftors of the denomintor of the 56

66 given frtion re ll liner ftors none of whih is repeted, we write the prtil frtion s follows. A B where A nd B re onstnts to e ( )( ) determined. Tpe : Repeted liner ftors If liner ftor ors n times s ftors of the denomintor of the given frtion, then orresponding to these ftors ssoite the sm of n simple frtions, A A ( ) A ( ) An... ( ) Where A, A, A,.An re onstnts. Tpe Irredile non repeted qdrti ftors If qdrti ftor whih is not ftorle into liner ftors ors onl one s ftor of the denomintor of the gives frtion, then orresponding to this ftor A B where A nd B re onstnts whih re not oth zeros. Consider ( )( ) We n write this proper frtion in the form n ( )( ) A B C The first ftor of the denomintor (-) is of first degree, so we ssme its nmertor s onstnt A. The seond ftor of the denomintor is of nd degree nd whih is not redile into liner ftors. We ssme its nmertor s generl first-degree epression B. 57

67 . WORKED EXAMPLES PART A.Split p ( ) into prtil frtion withot finding the onstnt ( ) A B.Withot finding the onstnts split ( )( ) A B C ( )( ) ( )( )( ) where A, B nd C nd re onstnts..split withot finding the onstnts 5 6 A B 5 6 ( )( ) where A nd B re onstnts.withot finding the onstnts split 5.Split 5 ( )( ) ( )( ( )( A B C ( ) 58 5 ( )( ) into prtil frtion. into prtil frtion withot finding the onstnts. ) A B C where A, B nd C re onstnts. )

68 PART B. Resolve Let into prtil frtion ( 5)( ) A B ( 5)( ) 5 A( ) B( 5) ( 5)( ) ( 5)( ) A ( ) B( 5) () Eqting the o-effiient of the like powers of, We get, Co-effiient s of : A B () Constnt term: A 5B () Solving () nd () we get 5 A nd B ( 5)( ) 5 Note: The onstnts A nd B n lso e fond sessivel giving sitle vles of..resolve ( )( ) into prtil frtion. Let ( )( ) A B C ( ) 59

69 ( )( ) A( ) B( )( ) C( ) ( )( ) A( ) To find C, pt in () B( )( ) C( ) () A( ) B X C C / To find A pt - in () A( ) A( ) 9A A 9 B( )( ) C( ) B( )( ) C( ) To find B, eqting o-effiient of on oth sides, A B B 9 B 9 ( )( ) 9 9 ( ).Resolve : ( )( into prtil frtion ) Let A B C ( )( ) 6

70 ( ) A( ( )( ) ) (B C)( ) ( )( ) A( ) (B C)( ) () To Find A, pt - in () A[ ] ( B C)() A A / To find C, pt in () A( A C / C C / To find B, pt in () ) (B C)( ) A( ) (B )( ) A() (B C). A B C B B B B B ( )( ) 6

71 EXERCISE PART A.Stte Binomil theorem for positive integrl inde.epnd [] sing inomil theorem..epnd [-] 5 sing inomil theorem..epnd (-) sing inomil theorem. 5.Epnd [5-] sing inomil theorem 6.Find the generl term in the epnsion of 7.Find the generl term in the epnsion of 8.Find the generl term in the epnsion of 9 9.Find the generl term in the epnsion of 8. Find the generl term in the epnsion of. Epnd (-) - sing inomil theorem.. Write the first terms of () -. Write the first three terms of (-) -. Epnd ( ) pto terms 5. Epnd (- ) - inomill 6. Write the first terms of (-) - 7. Split into prtil frtion withot finding the onstnts. ( ) 6

72 8. Split into prtil frtion withot finding the onstnts. ( ) 9. Split ( )( ) onstnts. into prtil frtion withot finding the. Split into prtil frtion withot finding the ( )( ) onstnts. PART B.Find the middle term in the epnsion of.find the middle term in the epnsion of 5.Find the 6 th term in the epnsion of.find the 5 th term in the epnsion of 5.Find the middle terms in the epnsion of 6.Find the middle terms in the epnsion of 7.Find the middle terms in the epnsion of Find the o-effiient of -5 in the epnsion of 5 7 6

73 9.Find the o-effiient of -7 in the epnsion of 5. Find the o-effiient of in the epnsion of 7. Find the term independent of in the epnsion of. Find the term independent of in the epnsion of. Find the term independent of in the epnsion of. Find the onstnt term in the epnsion of 5. Using inomil theorem, find the vle of Resolve into prtil frtion 8 7. Resolve 8. Resolve 9. Resolve. Resolve. Resolve into prtil frtion ( )( )( ) into prtil frtion ( )( ) ( )( ) ( )( ) into prtil frtion. into prtil frtion. into prtil frtion. ( ) ( ) 6

74 . Resolve. Resolve. Resolve 6 into prtil frtion. ( ) ( )( ( )( ) into prtil frtion. ) into prtil frtion. ANSWERS PART A.Refer sttement. ( ).().().().() () 5 5.().( ) ( ) 5.( ) 5.( ) () 5. ( ). ( ).() ( ).()( ) ( ) 5. ( 5).(5).( ).(5).( ).(5).( ) ( ) 6. T 7. T 8. T r 9r.( r ). 8r r r r r...( ) r.(). r r r r r T r. r r r.. 5r ( )( ) ( ).( ) ( )... ( )( ) ( )

75 ) )( )( (! ) )( ( ) ( ) ( ) (. ) )( (! ) )( ( ) (. ) )( (! ) ( 7 B A 8 C B A 9 ) ( C B A C B A PART B ; 7

76 ( ) 57 nd ; , 7, ( 7) 9 9 ( ) ( ) 67

77 68. ) ( ) (

78 UNIT III STRAIGHT LINES. Length of perpendilr distne from point to the line nd perpendilr distne etween prllel lines. Simple prolems. Angle etween two stright lines nd ondition for prllel nd perpendilr lines. Simple prolems. Pir of stright lines Throgh origin Pir of lines pssing throgh the origin h epressed in the form (-m )(-m ). Derivtion of h tnθ ondition for prllel nd perpendilr lines. Simple prolems.. Pir of stright lines not throgh origin Condition for generl eqtion of the seond degree h gf to represent pir of lines. h g h f (Sttement onl) g f Angle etween them, ondition for prllel nd perpendilr lines simple prolems. STRAIGHT LINES Introdtion Anltil Geometr is rnh of Mthemtis whih dels with soltions of geometril prolems Algeri methods. It ws developed the fmos Frenh mthemtiin lled Rne Desrtes. 69

79 Aes of o-ordintes: Tke two stright lines XOX nd YOY t right ngles to eh other. The horizontl line XOX is lled the X is nd the vertil line YOY is lled the Y-is. These two es interset t O, lled the origin. Crtesin Retnglr Co-ordintes: Digrm M (-) 7

80 Let XOX nd YOY e the es of o-ordintes. Let P e n point in the plne. Drw PM perpendilr to OX. Then the position of P is niqel determined the distnes OM nd MP. These distnes OM nd MP re lled the Crtesin retnglr oordintes of the point P with respet to X-is nd Y is respetivel. It is to e noted tht the X o-ordinte mst e in first ple nd the Y o-ordinte mst e in the seond ple. This order mst e stritl followed. Stright Line: When vrile point moves in ordne with geometril lw, the point will tre some rve. This rve is known s the los of the vrile point. If reltion in nd represent rve then (i) The o-ordintes of ever point on the rve will stisf the reltion. (ii) An point whose o-ordintes stisf the reltion will lie on the rve. Stright line is los of point. Digrm Let the line AB t the X-is t A nd -is t B. The ngle mde the line AB with the positive diretion of the -is is lled 7

81 the ngle of inlintion of the line AB with the -is nd it is denoted θ. Hene XAB θ.the ngle n tke n vles from to 8. Slope or grdient of stright line: The tngent of the ngle of inlintion of the stright line is lled slope or grdient of the line. If θ is the ngle of inlintion then slope tn θ nd is denoted m. Emple: If line mkes n ngle of 5 with the X-is in the positive diretion then the slope of the line is tn5. i.e m tn5 In shool stdies stdents hve lernt, the distne etween two points setion forml, mid point of the line joining two points, vrios form of eqtion of the stright line, point of intersetion of two lines, et., in nltil Geometr. Stndrd forms of the eqtion of stright line. (i) Slope interept form: When is the interept nd slope is m, the eqtion of the stright line is m (ii) Slope point form: When m is the slope of the stright line nd (, )ispointon the stright line its eqtion is - m(- ) (iii) Two point form: Eqtion of the line joining the two points (, )nd(, )is 7

82 (iv) Interept form: When the nd interepts of stright line re given s nd respetivel, the eqtion of the stright line is ie., (v). Generl form: The generl form of the eqtion of stright line is. If is the eqtion of stright line then oeffiient of Slope m oeffiient of -interept -interept ons tn t term oeffiient of ons tn t term oeffiient of Some Importnt Formle: (i) The length of the perpendilr from (, ) to the line is ± (ii) The length of the perpendilr from origin to the line is ± (iii) The distne etween the prllel lines nd is ± 7

83 ANGLE BETWEEN TWO STRAIGHT LINES Book Work: Find the ngle etween the lines m nd m. Dede the onditions for the lines to e (i) prllel (ii) perpendilr θ Proof: Let θ e the ngle of inlintion of the line m. Slope of this line is m tnθ. Let θ e the ngle of inlintion of the line m. Slope of this line is m tnθ. Let θ e the ngle etween the two lines, then θ θ θθθ -θ tn θ tn(θ -θ ) tnθ tnθ tnθ tnθ m m tn θ m m θ tn m m mm 7

84 (i) Condition for two lines to e prllel: If the two lines re prllel then the ngle etween the two lines is zero (i.e) tn θ tn m m m m m m m m For prllel lines, slopes re eql. (ii) Condition for two lines to e perpendilr: If the two lines re perpendilr then the ngle etween them θ 9 tn θ tn9 m m m m m m m m - For perpendilr lines, prodt of the slopes will e - Note : ) The te ngle etween the lines Y m nd m is tn θ m m m m ) If the slope of line is m then the slope of prllel line is lso m. ) If the slope of line is m then the slope of n line perpendilr to the line is m 75

85 ) An line prllel to the line will e of the form d (differ onl onstnt term) 5) An line perpendilr the line will e of the form d. WORKED EXAMPLES PART A ) Find the perpendilr distne from the point (,) to the stright line The length of the perpendilr from the point (, ) to the line is Give stright line is Given point (, )(,) (i.e) () () () () 5 ) Find the length of the perpendilr to the line 67 from the origin The length of the perpendilr is Here, 6, 7 (i.e.) () 7 (6)

86 ) Find the distne etween the line nd - Now The distne etween the prllel lines is Here nd - () 5 ( ) ) Find the ngle etween the lines nd - (i.e.) () nd - () oeffiient of Slope of () oeffiient of tn θ θ 6 slope of () oeffiient of oeffiient of tnθ θ 5 Let θ e the ngle etween () nd () θ θ -θ θ

87 5) Show tht the lines 6- nd re prllel 6- () () 6 Slope of the line () m 6 Slope of the line () m 6 m m The lines re prllel. 6) Find p sh tht the lines 7- nd p 6 re prllel. 7- () p--6 () Slope of the line () m 7 7 Slope of the line () m p p Sine () nd () re prllel lines m m 7 p p 8 8 p 7 p7 78

88 7) Show tht the lines -7 nd - re perpendilr. -7 () - () Slope of the line () m Slope of the line () m Now m m m m - The lines () nd () re perpendilr 8) Find the vle of m if the lines m nd 5-6 re perpendilr m- () 5-6 () Slope of the line () m - m Slope of the line () m - 5 Sine the liner re perpendilr m - m m 5m -5m m

89 PART B ) Find the ngle etween the lines 68 nd () Slope of the line () m 6-5 () Slope of the line () m Let θ e the ngle etween two lines tn θ m m m m ( ). 75 θtn - (.75) θ 6 5 ) Find the eqtion of the stright line pssing throgh (-, ) nd prllel to. Let the eqtion of line prllel to - () is k () Eqtion () psses throgh (-,) pt -, in eqtion () (-) () k -8k k-7 Reqired line is 7 8

90 ) Find the eqtion to the line throgh the point (,-) nd perpendilr to -- Reqired stright line is perpendilr to -- () nd pssing throgh (,-). Reqired eqtion of the stright line is - k () Reqired line psses throgh (,-) Pt, - in eqtion () -() (-) k -9k k k- S in eqtion () - - Reqired eqtion of stright line is. PAIR OF STRAIGHT LINES THROUGH ORIGIN An line pssing throgh the origin is of the form Let () nd () e the two lines pssing throgh the origin. The omined eqtion of () nd () is ( ) ( ) ( ) () Tking, h,nd We get h () 8

91 whih is homogenos eqtion of seond degree in nd. It represents pir of stright lines pssing throgh the origin. Let m nd m re the slopes of the lines given (). Then the seprte eqtions re m nd m (i.e.) m (5) m (6) ( -m ) ( -m ) (m m )m m (i.e.) m m (m m ) (7) Eqtion () nd (7) represent the sme prt of stright lines. Hene the rtios of the orresponding o-effiient of like terms re proportionl. m m (m m ) h The reltion (8) gives h m m i.e., Sm of the slopes nd m m h (8) (9) i.e., prodt of the slopes BOOK WORK : Find the ngle etween the pir of stright lines h pssing throgh origin. Also derive the onditions for the two seprte lines to e (i) perpendilr (ii) oinident (or prllel). 8

92 Proof: We know ngle etween two stright lines is given tn θ m m m m tn θ ± (m m ) m m m m ± ± ± ± h h h (h ) 8

93 tn θ ± h (i.e.) θ tn ± h is the ngle etween the pir of stright lines. (iii) Condition for the two stright lines to e perpendilr If the two lines re perpendilr, then θ 9 tn θ tn 9 ± h (i.e.) oeffiient of oeffiient of (iv) Condition for the two stright lines to e oinident If the two stright lines re oinident then θ tn θ tn h ± (i.e.) h - (i.e.) h 8

94 . WORKED EXAMPLES PART - A ) Write down the omined eqtion of the pir of lines - nd The omined eqtion is (-) () (i.e.) 6 (i.e.) ) Write down the seprte eqtions of the pir of lines (-) 5 (-) (-) (5) The seprte eqtions re - nd 5 ) Show tht the two lines represented re prllel to eh other. () This is of the form h Here, h, h, If the lines re prllel then h - h ()() pir of lines re prllel 85

95 ) Find the vle of p if the pir of lines p9 re prllel to eh other. p9 This is of the form h Here, h p, h p,9 If the lines re prllel h p ( )( 9) p 6 p p ± 5) Prove tht the lines represented re perpendilr to eh other This is of the form h Here 7, h-8, h-, -7 If the lines re perpendilr (i.e.) 7-7 6) If the two stright lines represented the eqtion p 57 re perpendilr to eh other, find the vle of p. p 87 p 87 This is of the form h 86

96 Here p, 7 If the lines re perpendilr (i.e.) p7 p-7 PART B ) Find the seprte eqtions of the line 7.Also find the ngle etween them. 7 This is of the form h (, h -7, h-7/, ) 6 ( ) ( ) ( )( ) The seprte eqtions re -nd- Let θ e the ngle etween the two stright lines tn θ ± ( 7 ) h ± ()() ±

97 ± 9 5 ± ± 5 tn θ ± tn θ tn 5, θ5 ) The slope of one of the lines h is thrie tht of the other. Show tht h h () Let m nd m e the seprte eqtions of eqtion () m m h m m Slope of one of the line thrie slope of the other line (i.e.) m m () () Eqtion () eomes m m m m h h h h 88

98 Sstitte m m in eqtion () m.m ( m ) h h h h (ie)h. PAIR OF STRAIGHT LINES NOT PASSING THROUGH THE ORIGIN Consider the seond degree eqtion ( l m n)(l' m' n' ) If (,) lies on l m n () then l m n. Hene (, ) stisfies eqtion (). Similrl n point on l ' m' n' () lso stisfies () onversel, n point whih stisfies () mst e on n of the stright lines () nd (). Ths ( l m n) (l' m' n' ) represent pir of lines. Epnding eqtion () we get II' II' Im' In' I'm mm' Im' () mn' I' n nm' nn' ( I'm) mm' ( In' I'n) ( mn' m'n) nn' 89

99 Tking II' h lm' l' m, mm' g ln' l'n f mn' m' n nn' We get h g f Condition for the seond degree eqtion h hf to represent pir of stright lines is fgh-f -g -h (or) h g h f g f or h g h f g f ) Angle etween pir of lines h g f is tnθ ± h ) The ondition for the pir of lines to e prllel is h ) The ondition for the pir of lines to e perpendilr is.. WORKED EXAMPLES PART - A ) Find the omined eqtion of the lines whose seprte eqtions re nd The two seprte lines re nd The omined eqtion of the given line is ( ) ( ) (i.e.)

100 ) Show tht the pir of lines given re prllel. This is of the form h g f Here 9, h, 6 h If the lines re prllel h (ie) () (9)(6) Hene the lines re prllel. ) Show tht the pir of lines given re perpendilr This is of the form h h g f Here 6-6 If the lines re perpendilr (i.e) 6 (-6) Hene the pir of lines re perpendilr. PART B ) Prove tht eqtion represents pir of stright lines. Given eqtion () (i.e.) This is of the form h g f l Hene 6 6 h, g 8, f 7 9

101 If the eqtion () represents pir of stright lies then h g h f g f LHS ( 8 9) ( 5 56) 8( 9 96) RHS Hene the given eqtion represents pir of stright lines. ) Show tht the eqtion represents pir of stright lines. Also find the seprte eqtion of the lines. Given This is of the form h g f h7 g5 f5 If the eqtion () represents pir of stright lies then LHS h g h f 5 5 g f ( 6 5) 7( 8 5) 5( 5 ) 6 9

102 -5 75 RHS The given eqtion represents pir of stright lines. Net we find seprte lines. Ftorise the seond degree terms Let 7 6 ( ) ( ) ( )( ) ( l)( m) (s) Eqting the oeffiient of, lm5 () Eqting the oeffiient of, lm5 () Solving () nd () l6m lm 5 5m5 m S in (), l () 5 l The seprte eqtion re nd ) Find k if k represents pir of stright lines. Find the ngle etween them. Soltion : k This is of the form h g f Hene k h-7 g5 f-5 9

103 Sine the given eqtion represents pir of stright lines h g h f g f k ( k 5) 7( k 5) 5( 5 ) 8k 98k k -5k - K It θ is the ngle etween the given lines then tn θ ± h ± ± ± 5 5 ± 5 tn θ tn θ tn 5 π θ 7 ()() 9

104 ) Show tht the pir stright lines 6 represents pir of prllel stright lines nd find the distne etween them. Given: 6 () This is of the form h g f Here h h g-6 g- f- f-/ - If the lines re prllel h (ie) () () () - The given eqtion () represents pir of prllel stright lines. To find the seprte lines of () Ftorise ( ) ( ) ( ) ( )( ) ( ) 6 ( ) Let z, then (i.e.) z z ( z )(z ) z nd z (i.e.) nd re the seprte eqtions Distne etween prllel lines nd is 95

105 ( ) ( ) EXERCISE PART A ) Find the perpendilr distne from the point (,-) to the line ) Find the perpendilr distne from the point (,6) to the line -6 ) Find the length of the perpendilr to the line -65 from the origin. ) Find the distne etween the line nd - 5) Find the distne etween the line -9 nd - 6) Show tht the lines -5 nd 6-8 re prllel. 7) Show tht the lines -66 nd -7 re prllel. 8) Find the vle of k if the lines 7- nd k8 re prllel. 9) Find the vle of p if the lines 5 6 nd p 7 re prllel. ) Show tht the lines - nd 8 re perpendilr. ) Show tht the lines -6 nd - re perpendilr ) Find the vle of p if the lines -p- nd 7 re perpendilr. ) Find the vle of p if the lines -p6 nd -8 re perpendilr. 96

106 ) Find the slope of the line prllel to the line joining the points (,) nd (-,6) 5) Find the slope of the line perpendilr to the line joining the points (,) nd (-,) 6) Show tht the line joining the points (,-5) nd (-5,-) is prllel to the line joining (7,) nd (5,9) 7) Show tht the line joining the points (,-) nd (,) is perpendilr to the line joining (,) nd (,). 8) Find the eqtion of the line pssing throgh (,) nd prllel to the line 7 9) Find the eqtion of the line pssing throgh (-,5) nd perpendilr to 5-8 ) Write down the omined eqtion of the lines whose seprte eqtion re (i) nd- (ii) nd (iii) nd (iv)nd ) Find the seprte eqtion of eh of the stright liens represents (i) 9 6 (ii) 5 (iii) 6 - (iv) 5 7 ) Show tht the two lines represented 9 6 re prllel to eh other. ) Show tht the eqtion 9 represents pir of prllel stright lines. ) Find the vles of p if the two stright lines represented p 5 re prllel to eh other. 97

107 5) Show tht the pir of stright lines given -- is perpendilr. 6) Find the vle of p so tht the two stright lines represented p 6- re perpendilr to eh other. 7) Write down the omined eqtion of the lines whose seprte eqtions re (i) nd - (ii) nd -- (iii) - nd 8) Show tht the eqtion -6-- represents two prllel lines. 9) Show tht the eqtion 69-5 represents two prllel stright lines ) Show tht the eqtion -- represents perpendilr pir of stright lines PART B ) Show tht the following eqtion represents pir of stright line (i) (ii) (iii) 6-- ) Find the ngle etween the pir of stright lines ) Find the ngle etween the pir of lines given 6-5. Find lso the seprte eqtions. ) Find the ngle etween the pir of lines given 9. Find lso the seprte eqtions. 5) Find the ngle etween the pir of lines given -85. Find lso the seprte eqtion. 6) Find the seprte eqtions of the pir of lines -. Find lso the ngle etween the lines. 98