MATHEMATICS I & II DIPLOMA COURSE IN ENGINEERING FIRST SEMESTER

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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

108 7) If the slope of one of the lines of h is twie the slope of the other, show tht 8h 9 8) If the eqtion represents two lines perpendilr to eh other, find the vle of nd. 9) Show tht the eqtion - -5 represents pir of lines. Also find the seprte eqtions. ) Show tht the eqtion represents pir of stright lines. Also find the seprte eqtions. ) Write down the seprte eqtions of Also find the ngle etween the lines. ) Show tht the eqtion represents pir of prllel stright lines. Find the seprte eqtions nd the distne etween them. ) Show tht the eqtion represents two prllel stright lines. Find the distne etween them P (i) 8 ANSWERS PART A. k 9. (ii) 6 (iii) 8 (iv) p 5. P X- 99

109 . (i) ( ), (ii), (iii), (iv), 5. P ± 6. P 7. (i) (ii) (iii) 8 PART B () (i) θ 6, (ii)θ 9 7 () tn θ, θ, 8,-5,- () θ,, (5) tn θ, -5, - (6) -, -, tn θ (8), - (9) -, 6- () () -8 - tn θ () 6 dist () 5

110 UNIT IV TRIGONOMETRY.: Trigonometril rtio of llied ngles-epnsion of Sin(A±B) nd os(a±b) prolems sing ove epnsion.: Epnsion of tn(a±b) nd Prolems sing this epnsion.: Trigonometril rtios of mltiple ngles (A onl) nd smltiple ngles. Simple prolems. Introdtion Trigonometr is one of the oldest rnhes of mthemtis. The word trigonometr is derived from Greek words Trigonon nd metron mens mesrement of ngles. In olden ds Trigonometr ws minl sed s tool for stding stronom. In erlier stges Trigonometr ws minl onerned with ngles of tringle. Bt now it hs its pplitions in vrios rnhes of siene sh s srveing, engineering, nvigtions et.,for the std of higher mthemtis, knowledge of trigonometr is essentil. Trigonometril Rtios: In right ngled tringle ABC, Opposite side i) Sine of n ngle θ sinθ Hpotense Adjent side ii) Cosine of n ngle θ osθ Hpotense iii) Tngent of n ngle θ tnθ Opposite side Adjent side

111 iv) Cotngent of n ngle θ otθ Adjent Opposite side side v) Sent of n ngle θ seθ Hpotense Adjent side vi) Cosent of n ngle θ ose θ Note:. Cose θ. Se θ. Cot θ. Tn θ 5. Cot θ sinθ osθ tn θ sinθ osθ osθ sinθ Fndmentl trigonometril identities ) Sin θ os θ ) tn θ se θ ) ot θ ose θ Trigonometrill reties of known Angels Hpotense Opposite side θ Sin θ - Cos θ - Tn θ -

112 Importnt reslts: For ll vles of θ i) sin (-θ) -sinθ ii) os (-θ) osθ iii) tn (-θ) -tnθ iv) ose (-θ) -oseθ v) se (-θ) seθ vi) ot (-θ) -otθ Signs of trigonometril rtios: i) In the first qdrnt, ll trigonometril rtios re positive. ii) iii) In the seond qdrnt, sin θ nd its reiprol ose θ re positive. Other trigonometril rtios re negtive. In the third qdrnt, tn θ nd its reiprol ot θ re positive. Other trigonometril rtios re negtive. iv) In the forth qdrnt, os θ nd its reiprol se θ re positive. Other trigonometril rtios re negtive. v) The signs of trigonometril rtios re sll rememered ode word All Silver Te Cps where the for words eginning with A,S,T,C orrespond to All rtios eing positive in the I

113 qdrnt, Sine in the II qdrnt, Tngent in the III qdrnt nd Cosine in the IV qdrnt respetivel. This is tlted s follows: Fntions Qdrnts I II III IV Sin Cosine Tngent Cotngent Sent osent. TRIGONOMETRICAL RATIOS OF RELATED OR ALLIED ANGLES: The si ngle is θ nd ngles ssoited with θ right ngle (or) its mltiples re lled relted ngles or llied ngles. Ths 9 ±θ, 8 ±θ, 7 ±θ, 6 ±θre known s relted or llied ngles. Working rle: To determine the trigonometril rtios of n ngle, follow the proedre given elow: i) Determine the sign (ve or -ve) of the given T- rtio in the prtilr qdrnt oserving the qdrnt rle ASTC. ii) Determine the mgnitde of the ngle writing in the form (p 9 ± θ). If p is even (like,,6, ) T rtio of llied ngle eomes the sme rtio of θ. i.e. for 8 ±θet., no hnge in rtio If p is n odd (like,,5, ) T.Rtio of the llied ngles eomes the o-rtio of θ. i.e.for9 ±θ,7 ±θet, sin os,tn ot,se ose

114 The trigonometril rtios of relted or llied ngles re tlted s follows. Angle Rtio 9-θ 9θ 8-θ 8 θ 7-θ 7θ 6 -θ 6θ sin os θ os θ sin θ -sin θ -os θ -os θ -sin θ sin θ os sin θ -sin θ -os θ -os θ -sin θ sin θ os θ os θ tn ot θ -ot θ -tn θ tn θ ot θ -ot θ -tn θ tn θ ot tn θ -tn θ -ot θ ot θ tn θ -tn θ -ot θ ot θ se ose θ -ose -seθ -se θ -ose θ ose θ se θ se θ ose se θ se θ ose θ -ose θ -se θ -se θ -ose θ ose θ Note : The trigonometril rtios of ngle n 6 ±θre sme s those of ±θ (e.g.) i) sin 8 sin(6 )sin sin (8 6 )sin6 ii) os 6 os ( 6 ) os (- ) os os (8 6 ) iii) Tn 6 tn (6 6 ) tn tn (6-6 )-tn6-5

115 WORKED EXAMPLES PART A Find the vle of the following withot sing tles: i) Sin 8 ii) os (- ) iii) tn 765 iv) se (- ) i) sin 8 sin(6 ) sin sin (8-6 ) sin 6 ii) os (- ) os os (6-6 ) os 6 iii) tn 765 tn( 6 5 ) tn 5 iv) se (- ) se se (6 6 ) se6. Prove tht sin (- ) sin Soltion LHS -sin sin - sin (6 - ) sin(6 6 ) sin sin6, 6

116 . COMPOUND ANGLES If n ngle is epressed s the lgeri sm or differene of two or more ngles, then it is lled ompond ngle. E.g:AB,A B,ABC,A B C et re ompond ngles. Prove Geometrill tht. sin(ab)sinaosbosa sinb. os(ab)osa osb-sina sinb proof: tht Drw OX nd OX sh XOX A nd XOX B P is n point on OX. Drw perpendilrs PQ to OX, nd PS to OX from the point P. Drw perpendilrs QR to PS nd QT to OX from the point Q. (i) In the right ngled tringle OPS opposite side sin(a B) Hpotense PS OP PR RS PR QT OP OP PR QT OP OP PR PQ QT OQ PQ OP OQ OP osa sinb sina osb sin (AB) sina osbosa sinb (ii) In the right ngled tringle OPS 7

117 Forml: Adjent side os(a B) Hpotense OS OP OT ST OT RQ OP OP OT RQ OP OP OT OQ RQ PQ OQ OP PQ OP osa osb-sina sinb os(ab) osa CosB-sinA sinb ) Sin (A B) Sin A Cos B Cos A Sin B ) Sin (A B) Sin A Cos B Cos A Sin B ) Cos (A B) Cos A Cos B Sin A Sin B ) Cos (A B) Cos A Cos B Sin A Sin B Reslt: (i) Prove tht sin (A B) sin (A B) sin A sin B Soltion LHSsin(AB)sin(A B) (sin A os B os A sin B) (sin A os B os A sin B) (sin A os B) (os A sin B) sin Aos B os Asin B) sin A( sin B) ( sin A) sin B sin A- sin Asin B sin Bsin Asin B sin A-sin B Sin (A B) sin (A B) sin A sin B (ii) Prove tht os (A B) os (A B) os A sin B 8

118 Soltion LHSos(AB)os(A B) (os A os B sin A sin B) (os A os B sin A sin B) (os A os B) (sin A sin B) Cos Aos B sin Asin B Cos A(-sin B) (-os A) sin B Cos A os Asin B sin Bos Asin B Cos A Sin B Cos(AB)os(A B)os A sin B Also Cos (AB)os (A-B) os B-sin A. WORKED EXAMPLES PART A.) Find the vle of sin65 os5 os65 sin 5 Sin65 os5 os65 sin 5 sin(65 5 ) sin9.) Find the vles of sin os os sin Sin os os sin sin( - ) sin.) Wht is the vle of os5 os -sin5 sin Cos5 os sin5 sin os(5 ) os 9.) Wht is the vle of os7 sin sin7 sin 9

119 Cos7 os sin7 sin os(7 - ) os 6 5.) Find the vle of sin5 Sin5 sin(5 - ) sin5 os os5 sin.. 6.) Find the vle of os75 Cos75 os(5 ) os5 os sin5 sin.. 7.) Prove tht os(6 -A) os( A) sin (6 A)sin ( A) We hve os A os B sin A sin B os (A B) LHS os (6 A)os( A) sin(6 A)sin( A) os[(6 A) (A)] os (6 A A) os 9 RHS sin(a B) sin(a B) 8.) Prove tht tn A os(a B) os(a B)

120 sin(a B) sin(a B) LHS os(a B) os(a B) sina osb os A sinb sina osb os A sinb os A osb sina sinb os A osb sina sinb sin A osb sin A tn A R.H. S os A osb os A 9.) Prove tht sin(ab) sin(a-b)sin(bc)sin(b-c) sin(ca) sin(c- A) LHS sin(ab) sin(a-b)sin(bc)sin(b-c) sin(ca) sin(c-a) sin A- sin Bsin B-sin C sin C-sin A RHS PART B ) If A nd B re te nd if sin A nd π sin B prove tht A B 5 Given : sin A nd sin B 5 os A sin 9 A

121 os A osb sin 5 B osb 5 5 Sin (AB) sin A Cos B Cos A sin B Sin (A B) sin (A B) sin 5 A B 5 π A B. 7 ) If A nd B re te nd if os A nd osb, Pr ove tht A Sin A os A π B 6 or 5 9

122 8 9 7 Sin B os B os (A-B) Cos A Cos B sin A sin B Cos (A B) os 6 A B6. COMPOUND ANGLES (CONTINUED) Forml: tn A tnb ) tn (AB) tn A tnb ) tn (A-B) tn A tnb tn A tnb

123 ). WORKED EXAMPLES PART A tn tn5 Find the vle of tn tn5 tn tn5 tn 5 tn tn5 ) If tn A nd tn B, find the vle of tn (AB) Soltion : Given tn A nd tn B, tn A tnb tn (AB) tn A tnb / 6 5 / 6 ) With ot sing tles, find the vle of tn 5 tn 5 tn(6 5 ) tn6 tn 5 tn6 tn 5

124 PART B ) If AB 5, prove (tn A) (tn B) nd hene dede the vle of tn Given : AB 5 Tking tn on oth sides Tn (AB) tn 5 tn A tnb tn A tnb tn A tn B -tn A tn B tn A tn B tn A tn B () RHS (tn A) (tn B) tn A tn B tn A tn B Ths (tn A) (tn B) () Dedtion : Pt B A in AB5 AA 5 A 5 A B ( AB) From () (tn ) (tn (tn tn tn ) ) 5

125 ) If ABC 8,Prove tht tn A tn B tn B tn C tn C tn A Soltion : Given: ABC 8 A B C 9 A B C 9 - Tking tn on oth sides A B C tn tn 9 A tn tn A tn tn tn C A A B tn B tn B tn B tn ot C C tn A B A B tn tn - tn tn tn C tn A tn B tn C -tn A tn B tn A tn B tn B tn C tn C tn A. ) If ABC 8, prove tht A B C A B C ot ot ot ot ot ot 6

126 In the ove emple (), we know tht A B B C C A tn tn tn tn tn tn A B C divided oth sides tn tn tn C A B tn tn tn C A B tn tn tn A B C A B C ot ot ot ot ot ot ) If A&B re te ngles nd if tn A show tht AB π n n nd tn B, n Given: tn A tn (AB) n,tnb n tn A tnb tn A tnb n n n n n n n n(n ) (n ) (n )(n ) (n )(n ) n (n )(n ), n n n n n n n n n n n 7

127 tn (AB) tn (AB) tn 5 AB 5 π AB 5) If tn A tn B p nd ot B ot Aq show tht ot (A-B) p q Given: tn A tn B p () ot B ot A q R.H.S p q tn A tnb otb ot A tn A tnb tn A tnb tn A tnb tn A tnb tn(a B) tnb tn A tn A tnb tn A tnb sing () ot(a-b) L.H.S 8

128 . MULTIPLE ANGLES OF A ONLY AND SUB MULTIPLE ANGLES Trigonometri rtios of mltiple ngles of A in terms to tht of A. ) Consider sin (AB) sin A os B os A sin B PtBA,weget Sin (AA) sina osa os A sin A sinasinaosa ) Consider os (AB) os A os B sin A sin B Pt B A,we get os (AA) os A os A sin A sin A os A os A sin A We hve os A os A sin A -sin A-sin A os A - sin A os A os A sin A os A (-os A) os A os A- Note: () sin A () os A () tn A osa osa osa osa Consider, tn (AB) PtBA,weget tn A tnb tn A tnb tn (AA) tn A tn A tn A tn A tn A tn A tn A 9

129 To epress sin A nd os A in terms of tn A: we hve sin A sin A os A sina os A (mltiple& dividedl osa ) CosA tna se A tn A sina tn A Also, os A os A sin A os A sin A os A os A(-tn A) CosA se A tn tn A A (-tn A) SUB-MULTIPLE ANGLES: If A is n ngle, then A/ is lled s mltiple ngle. i) Sin A sin (A/) SinAsinA/osA/ ii) Cos A os [A/] Cos A os A/ sin A/ -sin A/ (or) os A/- Note : sin os A A/ os os A A/ tn os A A/ os A

130 iii) Tn A tn (.A/) tn A / Tn A tn A / tn A / Similrl, sin A tn A / Cos A tn tn A / A / ) Find the vle of We hve, sina Pt A 5 in (). WORKED EXAMPLES PART A tn5 tn tn 5 tn A tn5 sin (5 ) tn 5 sin ) Find the vle of We hve Cos A Pt A 5 in () tn tn tn tn tn A 5 tn 5 os (5 ) 5 5 A A os () ()

131 sina ) Prove tht tna osa sina L.H.S osa sina os A osa os A os A tna R.H.S ) Prove tht os A sin A osa LHS os A- sin A (os A) -(sin A) (os A sin A) (os A-sin A) () osa osa sina 5) Prove tht ota/ os A sina LHS os A sina / os A / sin A / os A / sina / ota/rhs 6) Prove tht (sin A/- os A/) -sina LHS (sin A/- os A/) sin A/ os A/- sin A/ os A/ -sina RHS

132 PART B ) Prove tht sin A sina os A osa sin A sin A os A LHS os A os A sina( os A) os A( os A) tn A sina tna R.H.S os A ) Prove tht os A 8os A-8os A LHS os (A) os A- (osa) - [os A-] - [os A-os A]- 8os A-8os A- 8os A-8os A RHS π ) If tnα nd tn β show tht αβ 7 Given: tn α, tn β 7 tn α tnα tn α 9

133 tn α / 9. 8 / 9 8 tn (αβ) tnα tnβ tnα tnβ / 8 5 / 8 Tn (αβ) tn 5 αβ 5 π αβ 8 8 osb ) If tn A, prove tht tn A tn B (A nd B re te sinb ngles) Given tn A osb sinb Tn A tn B/ A B/ sin B / sinb / osb / A B Tking tn on oth sides Tn Atn B. sinb / osb /

134 EXERCISE PART A. Show tht os (- )os. Show tht os 78 sin 75. Find the vle of the following: Sin (- )os. Cos (-9 )sin 5. Cos (9 θ) se(-θ) tn (8 - θ) 6. Cot (9 -θ) sin(8 θ) se (6 -θ) 7. Sin os os sin \ 8. Sin 7 os os 7 sin 9. Cos 75 os 5 sin75 sin 5. Cos 75 os 5 sin75 sin 5. Find the vle of sin 75. Find the vle of os 5. Prove tht sin (5 A) (sin A os A). Prove tht os (A 5 ) (os A sin A) 5. Prove tht os (5 A)os(5 B) sin(5 A) sin (5 B)sin(AB) 6. Prove tht sin A sin ( A) sin ( A) tn tn 7. Find tht vle of the following: tn tn 8. tn65 tn65 tn tn 9. If tn A 6 5 nd tn B,findthevleoftn(AB) 5

135 π π. Prove tht tn θ tn θ. Find the vle of the following sin5 os 5. sin 5. os. Cos 5 sin 5 tn 5. tn Prove the following: sina 6. ot A os tn (5 A) 7. sin A tn (5 A) 8. (sin A osa) sina 9. os A sin AosA sina. ot A os A PART B. If sin A,CosB, (A or B e te) find 5 (i) sin(ab) nd (ii) os (AB). If sin A,sinB (A&Brete)findsin(AB). If sin A 7 8,sinB 5, (A&B re te) prove tht 7 sin (AB) 6

136 . 8 If sin A os B (A&B e te ) find os (A-B) Prove tht sin A sin (A) sin (A) 6. Prove tht os A os (A) os (-A) sin( A B) sin( B C) sin( C A) 7. Prove tht sina sinb sinbsinc sincsina sin( A B) sin( B C) sin( C A) 8. Prove tht os A osb osbosc oscos A 9. Prove tht tn tn 5 tn tn 5. If AB5, prove tht (ota-) (ot B-) nd hene dede the vle of ot ot A otb. If AB 5,provetht ( ot A)( otb). If ABC8,prove tht ota otb otb otc otc ot A. If AB C 8,prove tht tna tnb tnc tna tnb tnc. prove tht tn 5A- tn A- tn A tn 5A tn Atn A 5. If tn α nd tn β show tht tn (αβ) osa sina 6. Prove tht tna osa sina 7. Prove tht tn A ot A ose A 8. Prove tht ot A -ot A ose A 9. Show tht sina ot A nd hene dede the vle of osa ot 5 nd ot. If tn θ find the vle of sin θ osθ. Prove tht os A 8sin A- 8Sin A 7

137 . Prove tht os 6 Asin 6 A - sin A. Prove tht os A sin A A tn os A sin A. If tn α nd tnβ 7, show tht αβ5 ) ) ANSWERS PART A 5) sinθ seθ tn θ 6) tnθ sinθ seθ 7) 8) 9) ) ) ) 7) 8) 9) ) ) ) ) 5) Prt B ) i) ii) 65 ) 5 ) ) ) 8

138 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. 5. TRIGONOMETRICAL RATIOS OF MULTIPLE ANGLE OF A. i) sin A sin (AA) sina osaosa sina sina (- sin A)osA(sinAosA) sina-sin AsinA os A sina-sin AsinA (-sin A) sina-sin AsinA-sin A sina sina - sin A ii) CosA Cos(AA) CosACosA-SinA SinA CosA(Cos A-) - Sin A (SinACosA) Cos A-CosA-Sin ACosA Cos A-CosA- (-Cos A) Cos A Cos A-CosA-CosA Cos A CosA Cos A-CosA 9

139 iii) tna tn (A A) tn A tna - tn A tna tna tn A tna - tn A tna - tn A - tn A tn A tn A (- tn ) A tna - tn A - tn A tna - tn A tna - tn tn A tn tn A A A 5. WORKED EXAMPLES PART-A ) Find the vle of sin -sin We hve, sina sin A - sin A sin -sin sin( )(HereA ) sin6 ) Find the vle of os -os We hve, os A os A osa os -os os ( ) (Here A ) os

140 ) If sin θ 5, find the vle of sin θ Given: sin θ 5 We hve, sin θ sinθ sin θ Sin θ 5 ) If os A,findthevleofosA Given : os A We hve os A os A os A Cos A - 7

141 PART B sina osa ) Prove tht sina os A sina osa LHS sina os A sina sin A os A os A sina os A sina( sin A) os A(os A ) sina os A -sin A os A 6 - (sin Aos A) 6 R.H.S os A osa sin A sina ) Prove tht os A sina LHS os A osa sin A sina osa sina os A (os A os A) sin A (sina sin A) os A sina os A os A os A sin A sina sin A os A sina os A os A sina sin A os A sina os A( os A) sina( sin A) os A sina (-os A) (-sin A) sin A os A (sin Aos A) () R.H.S

142 ) Prove tht sina LHS osa sina sinandhenefindthevleofsin5. osa sin A sin ( sin sin A A) sina( sina) A) sina( sin sin A A) sinarhs We hve proved tht sin A pt A 5 sin(5 ) sin5 os (5 ) sina osa sin5 os ( ) ( ) ) Prove tht sina sin (6A) sin (6-A) sina LHS sin A sin (6A) sin (6-A) sina[sin 6 sin A] [sin (AB) sin (A-B) sin A-sin B]

143 sina[( ) sin A] sina[ -sin A] sina sin A sin A -sin A sin A RHS 5) Prove tht os os os 8 8 LHS os os os 8 os os (6 - ) os(6 ) os [os 6 -sin ] [os(ab)os(a-b) os A sin B] os ( os ) os os os os [os -os ] os ( ) os6 RHS 8

144 5. - SUM AND PRODUCT FORMULAE Sm or Differene formle: We know tht sin (AB) sin A os B os A sin B () sin (A-B) sin A os B - os A sin B () Adding () nd (), we get sin (AB) sin (A-B) sin A os B (I) Strting () from (), we get sin(ab)-sin(a-b)osasinb (II) We know tht os (AB) os A os B sin A sin B () os (A-B) os A os B sin A sin B () Adding () nd (), we get os (AB) os (A-B) os A os B (III) Strting () from (), we get os(ab)-os(a-b) -sinasinb (IV) (or) os (A-B) - os (AB) sina sin B sin (A B) sin (A B) sina osb sin (A B) sin (A B) osa sinb os (A B) os (A B) osa osb os (A B) - os (A B) -sina sin B (or) os (A - B) os (A B) sina sinb Prodt forml: LetCABndDA B Then C D A nd C D B A C D B C D 5

145 Ptting these vles of A nd B in I, II, III, IV we get sin C sin D sin sin C sin D os os C os D os os C os D -sin C D C D C D C D os sin os sin (or) os D os C sin C D sin C D C D C D C D C D 5. WORKED EXAMPLES PART A ) Epress the following s sm or differene: i) sinθosθ ii) osasin5a iii) osaosa iv) sinasina i) We hve sina osb sin (A B) sin (A B) sinθosθ sin(θ θ) sin(θ - θ) sinθ sinθ ii) We hve osa sinb sin (A B) - sin (A B) osa sin5a sin (A 5A) - sin (A 5A) sin8a - sin(-a) sin8asina [ sin (-A) -sin A] 6

146 iii) We hve, os A os B os (A B) os (A B) osaos A os (A A) os (A-A) os5aosa iv) WehvesinAsinBos(A-B) os(ab) sin A sin A os (A-A) os (AA) os A os A. ) Epress the following s prodt:. sin A sin A. sin 5A sin A. os A os 7A. os A os A i) We hve sinc sind sin C D os C D sin A sin A sin A A os A A sinaosa C D We hve sinc sind os sin C D ii) sin5a - sina os 5 A A sin 5A A osasina C D We hve osc osd os os C D iii) os A os 7A os A 7A os A 7A os 5A os (-A) os5aosa ( os( θ) osθ) 7

147 iv) We Hve os C - os D -sin os A - os A - sin A A C D sin - sin A sin (-A) -sinasina ) Prove tht sin 5 sin7 sin LHS sin sin5 sin7 sin A A C D sin os 5 7 sin 5 7 sin os6 sin (- ) sin ( ) (-sin ) sin sin RHS ) Prove tht os os os LHS os os os os os os os os os (- ) os os os os RHS. 8

148 5) Prove tht os A os ( A)os( -A) LHS os A os ( A) os ( -A) A A A A osaos os os A os os A osa os A osa osa RHS. 6) Prove tht LHS sina sinb tn(ab) osa osb sina sinb osa osb (A os A os( sin(a B) os(a B) tn(a-b)rhs. B) A B) sin( B A B )os( ) PART-B ) Prove tht (os α osβ) (sinα sinβ) os α β LHS (os α osβ) (sinα sinβ) α β α β α β α β os os sin os 9

149 os α β os α β sin α β os α β os os α α β β α β α β os sin () os α β RHS. ) If sin sin nd os os, Prove tht tn Given: sin sin nd os os Consider (sinsin) (os os ) sin os os os sin os os os os R.H.S sin os os os () sine [sin θ os θ] ( )

150 os ( ) os ( ) ( os ( ) os ( ) sin ( ) tn ( ) L.H. S os ( ) ) If sin sin nd os os Prove tht sin () Consider (sin sin ) (os os ) (sin.()sin os os os,sin os,sin [ sin A os A sina] os [sin()] )(os os os os ) Agin os (refer emple )

151 R.H.S sin()l.h.s os sin( ) os ) Prove tht sina sina sina os A osa osa tna LHS sina (sina sina) osa (osa os A) A A A A sina sin os A A A A osa os os sina sina os A osa osa os A sina( os A) osa( os A) sin A os A tn A RHS. 5. INVERSE TRIGONOMETRIC FUNCTION INVERSE TRIGONOMETRIC FUNCTION : Definition: A Fntion whih does the reverse proess of trigonometri fntion is lled inverse trigonometri fntion. The domin of trigonometri fntion is set of ngles nd rnge is set of rel memers.

152 In se of inverse trigonometri fntion the domin is set of rel nmers nd rnge is set of ngles. Inverse trigonometri fntion of sine is denoted s sin - similrl os -,tn -,os -,se - nd ose - re inverse trigonometri fntions of os, tn, ot, se nd ose fntions respetivel. Emples: - We know sin Sin Note: tn 5 tn - () 5 os os - () i) There is differene etween sin - nd(sin) - sin - isinverse trigonometri fntion of sin wheres (sin ) - is the reiprol of ii) sin. i.e.(sin ) - ose. sin Vle of inverse trigonometri fntion is n ngle. i.e. sin - is n ngle i.e. sin - θ Prinipl vle: Among ll the vles, the nmerill lest vle of the inverse trigonometri fntion is lled prinipl vle. Emples: ) We know sin,sin5 Sin 9, - sin (- ).. Sin -, 5, 9, -

153 The lest positive vle is, whih is lled prinipl vle of sin - ) We know os 6, os (-6 ) Cos, os Cos - 6, -6,,. The prinipl vle of os - is 6 Here 6 nd -6 re nmerill eql nd thogh - 6 is the smller thn 6 onl 6 is tken s prinipl vle. The following tle gives the rnge of prinipl vles of the inverse trigonometri fntions. Fntion Domin Rnge of Prinipl Vle of θ Sin - - π π - θ Cos - - π Tn - θ (-, ) Cot - Se - Cose - (-, ) (-,) eept o (-,) eept o Properties Propert () () sin - (sin ) () os - (os ) - π θ π < θ < π < θ < π,θ π - π < θ< π, θ

154 Proof: () tn - (tn ) (d) ot - (ot ) (e) se - (se ) (f) ose - (ose ) () let sin sin - () () Pt sin in sin - (sin ) Sin - (sin ) sin - () () From () nd (), sin - (sin ) Similrl other reslts n e proved Propert () () sin - ose - () ose - sin - () os - se - (d) se - os - (e) tn - ot - Proof: (f) ot - tn tn θ θ tn - () Let sin

155 Sin sin ose ose () nd () sin - ose - Similrl other reslts n e proved Propert () () sin - (-) -sin - () os - (-) π -os - () tn - (-) -tn - (d) ot - (-) -ot - (e) se - (-) -se - (f) ose - (-) -ose - Proof: () Let sin - (-) () sin - i.e. -sin sin (-) i.e. sin - - -sin - () From () nd () sin - (-) -sin - () Let os - (-) () Cos- i.e. -os 6

156 os (8 -) os os - π -os - () From () nd () os - (-) π -os - Similrl other reslts n e proved Propert () (i) sin - os - π (ii) tn - ot - π (iii) se - ose - π Proof: (i) Let θ sin -. () sinθ i.e. sinθ os(9 -θ) os - 9 -θ os - 9 -sin - (sing ()) sin - os - π 9 Similrl other reslts n e proved Propert (5) If < Prove tht tn - tn - tn - Let A tn - tn A Let B tn - tn B We know tn (AB) tn A tnb tn A tnb 7

157 tn (AB) AB tn - i.e tn - tn - tn - Propert (6) Sin - sin - sin - [ ] Proof: Let A sin - sina Let B sin - sinb We know sin (AB) sin A os B os A sin B sina sin B sin A sin B AB sin - [ ] i.e sin - sin - sin - [ ] 5. WORKED EXAMPLES PART-A ) Find the prinipl vle of (i) os - (ii) sin - (iii) tn - (v) ose - (-) (iv) os - 8

158 i) Let os - os osos os - ii) Let sin - sin sin sin 5 5 sin - 5 iii) Let tn - tn - tn -tn tn tn (- ) - iv) Let os - os os -os 6 9

159 osos(8-6 ) os v. Let ose - (-) ose - ose -ose ose ose (- ) - ose - (-) - ) Prove tht tn tn tn 5 We know tn - tn - tn - tn - ( )tn - ( 5 ) tn - tn tn - RHS ) Show tht tn - tn - π 5 ( )( ) 5 LHS tn - tn - ( ) tn - ot - sing properit () π sing properit () 5

160 PART-B ) Show tht tn - os - Let tn θ θtn - LHS os - os - tn tn θ θ os - [osθ] θ tn - LHS ) Show tht tn - tn - Let tn θ θ tn - LHS tn - tn - tnθ tn tn θ θ tn - [tn θ] θ tn - R.H.S 5

161 ) Show tht tn - - tn 5 LHS tn - tn - - ) tn tn - ( )( ) tn tn sing properit (5) 8 ) Evlte tn os 7 Let os Cos θ 7 8 opp AB tnθ dj BC tn - R.H.S 5 θ () 5

162 5 8 θ tn () From () nd () os Tking tn on oth sides tn AB tn os tn [tn ] EXERCISE PART-A Find the vle of the following:. sin - sin. sin - os. tn tn tn. If sin A 5 (A eing te) find sina 5. If os θ (θeing te) find osθ 5 6. If tn θ, find tnθ 7. Epress in the form of sm or differene. I. sin θ os θ II. os 8θos 6θ 5

163 III. os 6A sin A IV. sin 6θ sin θ A A V. os sin 7A VI. sin os 5A VII. os (6α) sin (6- α) 8. Epress is the form of prodt. I. sin A sin 5A II. sin A - sin 5A III. os A os 5A IV. os A - os 5A V. sin 5 o sin o VI. sin 5 o os 8 o VII. sin o os 5 o VIII. os 5 sin7 Prove the following : 9. sin o sin5 o sin7 o. sin o sin sin8 o. sin 78 o sin8 o os o. Cos 8 o os o os o. Cos 5 o os68 o os 7 o. Cos o os8 o os 6 o Prove the following: 5. Sin 5 o sin o os o 6. Cos o os7 o os o 7. Sin A sin ( A) sin ( o -A) 5

164 8. Sin A sin ( A) sin ( o A) 9. Cos A os ( o -A) os ( o A) sina sina. ot A osa os A sin7a sin5a. tn A os7a os 5A sina sin A. tn A osa sin A osb os A A B. tn sina sinb. Find the prinipl vle of i) Sin ii) Sin iii) Sin ( ) iv) Sin v) Cos ( ) vi) Cos vii) tn ( ) viii) se i) os e ( ) 55

165 Prove the following: 5. Sin - () sin - Π 6. tn - - tn tn - () 7 7. Sin - os - 8. Se - tn - PART B Prove the following: sinθ ) sin θ osθ osθ os θ ) osa sina os A sina -sina sina sin A ) os A osa ota osa ) Prove tht (osa) os A 5) osa Prove tht os A nd hene osa dede the vle os5 6) Prove tht (os o sin o )(os o sin o ) 7) Prove tht sin A sin (6 o A)sin( o A) sin A 8) Prove tht sin o sin o sin 8 o 8 56

166 9) Prove tht sin o sin 5 o sin 7 o 8 ) Prove tht os θ os (6 o -θ) os(6 o θ) osθ ) Prove tht sin o sin o sin 5 o sin 7 o 6 ) Prove tht sin o sin o sin 6 o sin 8 o 6 ) Prove tht Cos o os o os 5 o os 7 o 6 ) Prove tht os o os o os 6 o os 8 o 6 5) Prove tht tnθ tn (6 o θ) tn(6 o -θ) tnθ 6) Prove tht tn o tn o tn 8 o 7) Prove tht sina sin5a sina osa os5a os A tna 8) Prove tht osa os5a os A sina sin5a sina ot A 9) Prove tht sina sina sin5a sin7a tn A os A osa os5a os7a ) Prove tht (Cos α osβ) (sinα -sin β) os α β ) Prove tht (Cos α -osβ) (sinα sinβ) sin α β ) Prove tht (Cos α -osβ) (sinα -sinβ) sin α β ) If sin sin nd os os, prove tht se 57

167 58 ) If sin - sin nd os - os, prove tht se 5) If sin -sin nd os os, prove tht tn 6) Prove tht sin θ sin () sin (-) 7) Prove tht os Aos (A) os (A-) Prove the following: 8) tn - sin - 9) tn - tn - ) tn - tn - tn - ) sin - -sin - sin - ) os - os - os - ) os - -os - os - ) tn - os os 5) tn - tn -

168 5 6) Evlte os sin 7) Evlte os sin sin 5 ANSWERS PART-A ) ) ) ) 5 7 5) 5 9 6) 7) (i) sin 6θ sinθ (ii) os θ osθ (iii) sin 9 A sin A (iv) os θ -os8θ (v) [sina sina] (vi) [sin 6 A sin A] (vii) sin α 8) (i) sin 9A os A (ii) os 9A sin A (iii) os 9A sin A (iv) -sin 9A sin A (v) os sin o (vi)sin os 59

169 (vii)sin os (viii) os 5 7 os ) (i) π ii) π iii) π iv) π v) π vi) π vii) π viii) 6 π i) π PART B 5) 6 6) 7) 65 6

170 MATHEMATICS I MODEL QUESTION PAPER - Time : Hrs (Mimm Mrks: 75) PART A (Mrks:55) I. Answer An 5 Qestions. Solve. Find the Vle 5. If A, B FindAB 7. Find the djoint Mtri of 5. Find the vle of C 7 6. Find the generl term of ( ) 8 7. Epnd ( ) - pto three terms when < 8. Split into prtil frtion withot finding the onstnt ( ) 9. Find the vle of m if the lines m nd 5-6 re perpendilr. Find the omined the eqtion of the lines 5 nd. Show tht the pir of lines re prllel. Write down the ondition for the eqtion h g fto represent pir of stright lines. 6

171 . Show tht sin (-) sin. Find the vle of os5 os - sin5 sin 5. If tn A tn B Find tn ( A B) 6. Find the vle of sin 75 os75 7. If sinθ /, findthe vle of sin θ 8. Find the vle of os - os sina sinb 9. Showtht -ot(ab) osa osb. Show tht tn - tn PART B (Answer An TWO sdivisions in eh qestion) All Qestions rr Eql Mrks 56. Solve sing Crmer s Rle z, -z nd -z Short tht ( ). Find the inverse of 6. Find the middle terms in the epnsion of Find the term independent of in the epnsion of ( /). Resolve in to prtil frtion ( 5)( 6)

172 .. Find the ngle etween the lines 7 nd-5. Find the seprte eqtion of the pir of stright lines 9. Also prove tht the lines re prllel. Show tht the eqtions represented is pir of stright lines π..if sin A nd sin B prove tht A B 5. If AB 5 Prove tht ( tna) ( tnb) nd hene dede the vle of tn sin A sin A.Prove tht tn A os A osa 5..Prove tht sin sin sin6 sin8 6.Prove tht (osα osθβ) (sinα sinβ) sin. Show tht tn - tn tn α β 6

173 MODEL QUESTION PAPER Time : Hrs (Mimm Mrks: 75) I. Answer An 5 Qestions. Find if. Prove tht If A Find A. Find the inverse of PART A (Mrks:55) 5. Find the th term of 6. How mn middle terms re in the epnsion of (5-) 9 7. Write the first three terms in the epnsion of (-) - 8. Withot finding the onstnts split ( )( ) in to prtil frtion 9. Find the perpendilr distnes from the point (,) to the stright line. Write down the ondition for the pir of lines given h to e prllel. Find if the lines represented re perpendilr. Stte the epression for ngle etween pir of line given h g f 6

174 . Find the vle sin 5 withot sing tles or lltor. Find the vle of os os sin sin 5. Simplif tn tn tntn sina 6. Prove tht tna os A 7. Find the vle of sin - sin 8. If Cos A 5 find the vle of osa. 9. Show tht Cos Cos Cos. Show tht sin - tn PART- B (Answer n two sdivision in eh Qestion) All Qestions rr Eql Mrks 56 ) Solve the eqtions z6,--z -6 nd z, sing Crmer s rle ) Prove tht ( ) 5 ) If A show tht A 5A I.) Find the middle terms in the epnsion of (/) ) Find the term independent of in the epnsion of ) Resolve 7 in to prtil fntion ( )( ) 65

175 .) Derive the epression for ngle etween two lines m nd m ) If the slope of one of the stright lines h is twie tht of the other show tht8h 9 ) Find the vle of λ so tht the eqtion 8 8 λ represents pir ofstright lines..) If sin A, os B, 5 find the vles of sin (A-B) nd os (A-B) ) If A B 5 Prove tht (ot A-) (ot B -). Also find the vle of ot sinθosθ ) Show tht tnθ / sin θ osθ os A os A sin A sin A 5.) Prove tht os A sin A ) If sina sinb, osa osb, Show tht tn A B ( ) ) Show tht tn - tn 66

176 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.. CIRCLES Definition: The los or pth of point P(,) whih is t onstnt distne r from fied point C(h,k) is lled irle. The fied point C(h,k) is lled entre nd the onstnt distne is lled the rdis of the irle... Eqtion of irle with entre (h,k) nd rdis r: Let the given entre nd rdis re C(h,k) nd r nits. Let P(,) e n point on the irle. From Fig (..) CP r ie ( h) ( k) r (sing distne forml) 67

177 ( h) ( k) r () Note: When entre is t the origin (,) the eqtion () eomes r. i.e. the eqtion of the irle with entre t the origin nd rdis r nits is r... Generl Eqtion of the irle. The generl eqtion of the irle is g f () Eqtion () n e re written s g g f f g f (or) ( g) ( f) g f [ ( g) ] [ ( f) ] g f Eqtion () is in the form of eqtion () The eqtion () represents irle with entre (-g,-f) nd rdis g f. Note: (i) Coeffiient of oeffiient of () (ii) Centre of the irle oeffiient of, oeffiient of (iii) Rdis g f 68

178 WORKED EXAMPLES PART - A ) Find the eqtion of the irle whose entre is (,-) nd rdis nits. Eqtion of the irle with entre (h,k) nd rdis r is ( h) ( k) r (h,k) (,-) r ( ) ( ) ) Find the entre nd rdis of the irle 6 Here g -6 nd f g- f Centre is (-g,-f) Centre (,-) r ( ) r nits PART B ) If -5 nd 75 re the eqtions of two dimeters of irle of rdis nits write down the eqtion of the irle. Given dimeters re -5 () 75 () 69

179 Solving () nd () we get - nd - entre (-,-) given rdis r Eqtion of the irle is ( h) ( k) r ( ) ( ) i.e. ) Find the eqtion of the irle whose entre is (5,-7) nd pssing throgh the point (,-) Let the entre nd point on the irle e C(5,-7) nd A (,-) Rdis CA (5 ) ( 7 ) r 6 r 7

180 Eqtion of irle is ( h) ( k) r (h,k) (5,-7) r r ( 5) ( 7) 5 9 i.e. 5. CONCYCLIC POINTS If for or more points lie on the sme irle the points re lled onli points... Eqtion of irle with end points of dimeter Let A(, )ndb(, )egiventwoendpointsofdimeter. Let P (,) e n point on the irle. i.e, APB 9 ( Angle in Semi irle is 9 ) AP PB (slope of AP) (slope of PB ) - 7

181 ie. )( ) ( )( ) ( ie )( ) ( )( ) ( is the reqired eqtion of the irle. WORKED EXAMPLES PART A ) Find the eqtion of the irle joining the points (,-) nd (-,) s dimeter Eqtion of the irle is )( ) ( )( ) ( ) (, ) (, ) (,) ( ( )( ) ( )( ) 5 ) Find the eqtion of the irle joining the points (,) nd (,) s dimeter Eqtion of the irle is )( ) ( )( ) (, ) (,)nd(, ) (,) ( ( )( ) ( )( ) 7

182 PART B ) Find the eqtion of the irle pssing throgh the points (,), (,) nd (,) Let the eqtion of the irle e g f () (,) lies on () i.e g() f() gf - () (,) lies on () i.e g() f() g - () (,) lies on () i.e g() f() f - () gf - g- () () f- f- Sstitte f in () f 7

183 Sstitte in () g g Eqtion of the irle is ie. ) Find the eqtion of the irle pssing throgh the points (,), (,) nd hving its entre on the line -5-5 Let the Eqtion of the irle e g f () (,) lies on () i.e g() f() f - () (,) lies on () i.e. g() f() 8g 6f 5 () Centre (-g,-f) lies on the line 5 5 i.e ( g) 5( f) 5 () g 5f 5 f 8g 6f 5 7

184 () () 8 g f g f (5) () (5) gives; g 5f 5 g f 7f 7 7f 7 f - Sstitting f- in (5) g- (-) - g - g - g Sstitting g 5,f-in() 5 8 6( ) 5 i.e., g Eqtion of the reqired irle is 5 ( ) 5 ) Show tht the points (,), (6,5) (,7) nd (,) re onli Let the eqtion of the irle e g f () (,) lies on the irle () g() f() 8g f 7 75 ()

185 (6,5) lies on the irle () 6 5 g(6) f(5) gf -6 () (,7) lies on the irle () 7 g() f(7) gf -5 () () () g 8f - (5) ()-() 8g f -8 (6) Solving (5) nd (6) we get g- nd f- Sstitting g -, f- in () 8(-) (-) -7 C-75 Eqtion of irle pssing throgh the three points (,) (6,5) nd (,7) is ( ) ( ) (7) Sstitting the forth point (,) in (7) 6() 8() (,) lso lies on (7) Hene the given for points re onli... Length of the Tngent to irle from point (, ) Let the eqtion of the irle e g f nd the point A(, ) lies ot side the irle. 76

186 BC r We know tht the entre is C(-g,-f) nd rdis g f From the fig (.) Δ ABC is right ngled tringle. AB BC AC AB AC BC AB ( g) ( f) (g AB g f C AB g f f ) Whih is the length of the tngent from the point (, )tothe irle g f Note: (i) If AB >, A(, ) lies ot side the irle. (ii) If AB <, A(, ) lies inside the irle. (iii) If AB, A (, )liesontheirle. 77

187 .. Eqtion of the Tngent to irle t the point (, )on theirle(resltsonl): Given eqtion of irle is g f The point A (, ) lies on the irle. i.e g f From fig (.) AT is the tngent t A. We know tht entre is C(-g,-f). f Slope of AC g Sine AC is perpendilr to AT ( g) Slope of AT m ( f) Eqtion of the tngent AT t A(, )is m( ) g ( ) f on simplifition, we get g( ) f( ) Note: The eqtion of tngent to the irle r t (, )is otined sstitting g, f nd -r in the ove eqtion to tngent 78

188 r ( ) ( ) r ie.. RESULTS ) E qtionofthet ngenttoirle tpoint(, )is g( ) f( ) ) L engthofthet ngentfromthepoint(, )totheirle Note: g f is g f ()theeqtionofthetngenttotheirle r t(, )is r g f > ()If thenthepoint(, )liesotsidethe irle. ()If g f thenthepoint(, ) lies on the irle. < ()If g f, thenthepoint(, )liesinsidethe irle. WORKED EXAMPLES PART A ) Findthelengthofthetngentfrom(,)totheirle Lengthofthetngent g f () () 9 nits 79

189 ) Show tht the point (9,) lies on the irle 6 Sstitte (9,) on the irle 6 9 6(9) () the point (9,) lies on the irle. ) Find the eqtion of the tngent t (-,) to the irle 5 Eqtion of tngent is r (-) () PART B ) Find the eqtion of the tngent t (,) to the irle 8 6 Eqtion of the tngent t the point (, )is g( ) f( ) Given 8 6 g -8, f -6 (, )(,) g - f - () () ( )[ ] ( )[ ] () ()

190 Eqtion of the tngent is - ) Find the eqtion of the tngent to the irle () (-) 5t (-,-). Eqtion of the irle in ( ) ( ) 5 i.e Here g f - g f- Eqtion of the tngent is g( ) f( ) i.e., ( ) ( ) ( ) ( ) -- EXERCISE PART A. Find the eqtion of the irle whose entre nd rdis re given s (i) (,); nits (ii) (-5,7), nits (iii) (-5,-); 5 nits (iv) (6,-), nits 8

191 . Find the entre nd rdis of the following irles: (i) 8 (ii) 7 5 (iii) 6 (iv). Write down the eqtion of irle whose entre is (h,k) nd rdis r nits.. Write down the entre nd rdis of the irle g f 5. Find the entre nd rdis of the irle ( ) ( ) 6 6. Find the eqtion of the irle desried on line joining the following points s dimeter: (i) (,5) nd (,7) (ii) (-,) nd (,-) (iii) (,) nd (,) (iv) (-6,-) nd (-,-8) 7. Write down the eqtion of the irle whose end points of the dimeter re (, ) nd (, ) 8. Write down the epression to find the length of the tngent to the irle g f from the point (, ) 9. Write down the eqtion of the tngent to the irle g f t the point (, ).. Find the length of the tngent from the point (,) to the irle. Show tht the point (-,-) lies inside the irle 5. Show tht the point (-,-7) lies on the irle 5 8

192 PART B ) nd - re two dimeters of irle of rdis 5 nits. Find the eqtion of the irle. ) Find the eqtion of the irle two of its dimeters re nd - nd pssing throgh (5,-) ) Find the eqtion of the irle pssing throgh the points (5,), (,), (,). ) Find the eqtion of the irle pssing throgh the points (6,) nd (-,-) nd hving its entre on 5 5) Prove tht the points (,), (,5),(-,-) nd (-5,) re onli 6) Find the eqtion of the tngent t (,) to the irle 8 7) Find the eqtion of the tngent t (-7,-) to the irle 5 8) Show tht the point (,-) lies on the irle Also find the eqtion of the tngent t (,-). ANSWER PART A () (i) 6 (ii) 65 (iii) 8 6 (iv) 6 8

193 7 5 () (i) (6,), 5 (ii), ; 78 5 (iii), ; (iv) (,);. (5) (,-), (6) (i) 5 (ii) (iii) () nits () (iv) PART B ) 6) ) 7) 7 7 ) 6 6 8) 5 ) 6 8 8

194 UNIT- II FAMILY OF CIRCLES. Conentri irles ontt of irles (internl nd eternl irles) orthogonl irles ondition for orthogonl irles. (Reslt onl). Simple Prolems Lt n. Limits:Definition of limits - n Lt sin θ Lt tn θ, ( θ in rdin) θ θ θ θ [Reslts onl] Prolems sing the ove reslts. Differentition:. Definition Differentition of n, sin, os, tn, ot, se, n n ose, log, e,± v, v, vw, (Reslts onl). v prolems sing the ove reslts. Simple.. FAMILY OF CIRCLES.. Conentri Cirles. Two or more irles hving the sme entre re lled onentri irles. Eqtion of the onentri irle with the given irle gf is gfk (Eqtion differ onl the onstnt term) 85

195 .. Contt of Cirles. Cse (i) Two irles toh eternll if the distne etween their enters is eql to sm of their rdii. i.e. r r Cse (ii) Two irles toh internll if the distne etween their enters is eql to differene of their rdii. i.e. r -r (or) r -r Orthogonl Cirles Two irles re sid to e orthogonl if the tngents t their point of intersetion re perpendilr to eh other. 86

196 .. Condition for Two irles to t orthogonll.(reslts onl) Let the eqtion of the two irles e X g f X g f The t eh other orthogonll t the point P. The enters nd rdii of the irles re A (-g, f ), B (-g, f ) AP r g f nd BP r g f From fig (.) Δ APB is right ngled tringle, AB AP PB i.e. (-g g ) (-f f ) g f g f Epnding nd simplifing we get, g g f f is the reqired ondition for two irles to t orthogonll. Note: When the enter of n one irle is t the origin then ondition for orthogonl irles is 87

197 . WORKED EXAMPLES PART A. Find the distne etween the entre of the irles 6 8 nd nd - 6 entre: (, -) (5,) Distne ( 5) ( ) 5. Find the eqtion of the irle onentri with the irle 5 nd pssing throgh (,). Eqtion of onentri irle with 5is k whih psses throgh (,) i.e. k k-9 Reqired Eqtion of the irle is 9. Find whether the irles 5nd 5 t orthogonll or not. When n one irle hs entre t origin, orthogonl ondition is i.e. 5-5 Given irles do not t orthogonll. 88

198 PART - B. Find the eqtion of the irle onentri with the irle 8 nd hving rdis nits. Centre of the irle 8is(,-). Centre of onentri irle is (,-) nd rdis r. Eqtion of the reqired irle is ( h) (-k) r ( ) () Show tht the irles 6 9 nd 7 toh eh other. Given irles 69nd -7 entre: (,) (-,-) rdis: r r Distne: 9 ( ) r r 9 r ( ) ( ) r r The irles toh eh other eternll. 89

199 . Find the eqtion of the irle whih psses throgh (,) nd ts orthogonll eh of the irles nd ---. Let the eqtion of irle e gf This psses throgh (,) i.e. g() f() g f - Eqtion () orthogonl with the irle -8-6 g g f f g g f f g -8 f - 6 g - f - orthogonl ondition g g f f i.e. g(-) f(-) 6-8g -f 6 8g f -6 Eqtion () orthogonl with the irle -- i.e. g(-) f (-) -g f - g f ()-() 8g f - 6 g f - 6g - g 6 7 g 9

200 ()-() g f g f g f 7 i.e. pt g in (5) 7 f f f 6 7 Sstitte g nd f in () Simplifing we get Reqired eqtion of irle is or 5 5. LIMITS Introdtion The onept of fntion is one of the most importnt tool in lls. Before, we need the following definitions to std lls. Constnt: A qntit whih retins the sme vle throghot mthemtil proess is lled onstnt, generll denoted,,, 9

201 Vrile: A qntit whih n hve different vles in prtilr mthemtil proess is lled vrile, generll denoted,,z,,v,w. Fntion: A fntion is speil tpe of reltion etween the elements of one set A to those of nother set B Smolill f: A B To denote the fntion we se the letters f,g,h. Ths for fntion eh element of A is ssoited with etl one element in B. The set A is lled the domin of the fntion f nd B is lled odomin of the fntion f.... Limit of fntion. Consider the fntion f: A Bisgiven f( ) when we pt We get f ( ) (Indeterminte form) Bt onstrting tle of vles of nd f() we get X F() From the ove tle, we n see tht s pprohes (nerer) to, f() pprohes to. Lt It is denoted We ll this vle s limiting vle of the fntion... Fndmentl reslts on limits. Lt Lt Lt ) [ f( ) ± g( ) ] f( ) ± g( ) Lt Lt Lt ) [ f( ) g( ) ] f( ) g() 9

202 5 ) Lt f g Lt ( ) ( Lt Lt f ( ) g( ) Lt K f ) Kf( ) ( ) ) If f( ) g( ) then Lt Lt f( ) g( ) Some Stndrd Limits. ) n n Lt n n ( for ll vles of n) When θ isinrdin ) Lt sinθ θ θ Lt tnθ Lt sinnθ θ θ θ θ ( ) Lt tn nθ n θ θ Note: () ( ) n. Evlte: Lt Lt Evlte: Lt 5. WORKED EXAMPLES PART A ( ) ( ) ( ) 6( ) 7 7 9

203 9 ( )( ) ( )( ) 5 Lt 5 Lt ( ) ( ) 6 5 Lt 5 Lt. Evlte: Lt n n n n n n Lt n n n n Lt. Lt Lt Lt 5. Evlte: sin5 Lt sin Lt 5 5 sin5 Lt sin5 Lt sin5 Lt θ θ θ

204 95 PART - B. Evlte: 6 6 Lt Lt Lt 6 6 Lt ( ) ( ) Lt Lt Evlte: θ θ θ sin sin Lt θ θ θ θ θ θ θ θ sin sin Lt sin sin Lt n sinn Lt sin Lt sin Lt θ θ θ θ θ θ θ θ θ. Evlte: os os Lt

205 Lt sin os Lt os sin Lt sin Lt sin sin Lt sin Lt os θ sin θ.. DIFFERENTIATION Consider fntion f() of vrile. Let Δ esmll hnge (positive or negtive) in nd Δ e the orresponding hnge in. The differentition of with respet to is defined s limiting Δ vle of, s Δ Δ d Lt i.e d Δ Δ Δ Lt f( Δ) f() Δ Δ The following re the differentil o-effiient of some simple fntions: 96

206 . d (ons tn t) d. d n n ( ) n d. d () d. d ( ) d 5. d n n n d 6. d d 7. d d ( sin ) os ( os ) sin d 8. d d 9. d d. d d. d d. d d. d ( tn ) ( ot ) ( se ) ( os e) ( e ) e ( log ) se os e se tn os e ot The following re the methods of differentition when fntions re in ddition, mltiplition nd division. If,v nd w re fntions of d d d ( i) Addition Rle : ( v) ( ) ( v) d d d d d d d ( v w) ( ) ( v) ( w) d d d d d d d ( v) ( ) ( v) d d d d d d ( ii) Mltiplition Rle : ( v) ( v) v ( ) d d d (iii) d d d d d d d d ( vw ) v ( w) w ( v) vw ( ) Qotient d d v v Rle : (division) d d ( ) ( v) v d d 97

207 . Find d d - d d if -. WORKED EXAMPLES PART A 6 6 d. Find if e sin d d d e d. FInd d d d e e if sin d d d ( sin) sin ( e ) os sin e d d ( ) ( ) ( ) ( ) d d ( ) ( ) ( )( ) ( ) ( ) d 98

208 d d. Find if ( )( 5)( ) ( )( 5)( ) d ( )( 5) ( ) ( )( ) ( 5) d d d d 5. Find if d d d d FInd d d d ( 5)( ) ( ) 99 d d d d ( )( 5)( ) ( )( )( ) ( 5)( )( ) ( )( 5) ( )( ) ( 5)( ) sin ose sin ot ose PART - B. if ( ) os log ( ) os log d d d ( ) os (log) ( ) log ( os ) d d os d log d d ( ) ( ) os ( ) log [ sin ] os log [ ] ( ) os ( ) log sin os log

209 . d tn Find if d e. tn e tn d se tn d d dv v d d d d v d d d Find d v e dv e d [ ] [ ] ( e )[ se tn ] ( tn )[ e ] ( e ) if v d d d d d d d d dv d ( )[ ] ( )[ ] ( ) [ ] ( ). Find if ( sin)( os ) ( sin)( os )

210 d d d ( sin ) ( os ) ( os ) ( sin) d d ( sin )[ sin] ( os )[ os ] EXERCISE PART - A. Write down the eqtion of the onentri irle with the irle g f. Stte the ondition for two irles to toh eh other eternll.. Stte the ondition for two irles to toh eh other internll.. Stte the ondition for two irles to t eh other orthogonll. 5. Find the eqtion of the irle onentri with the irle 5 nd pssing throgh the point (,-). 6. Find the onstnts g, f nd of the irles 7 nd 6 7. Show tht the irles 8 6 nd 5 6 re orthogonl 8. Evlte the following: (i) Lt (ii) Lt 7 (iii) Lt sinθ θ θ Lt (iv) 9. Differentite the following with respet to : (i) (ii) ( )( ) d

211 (iii) sin e (iv) ( ) tn (v) log (vi) e sin (vii) sin os 7 (viii) PART - B. Find the eqtion of the onentri irle with the irle 7 nd hving rdis 5 Units.. Show tht the following irles toh eh other. (i) 6 6 nd (ii) nd (iii) 6 8 nd 6. Prove tht the two irles 5 nd 8 5 toh eh other.. Find the eqtion of the irle whih psses throgh the points (,-) nd (6,5) nd orthogonl with the irle 8 5. Find the eqtion of the irle throgh the point (,) nd ts orthogonll eh of the irles 8 6 nd 6. Evlte the following: 5 5 Lt Lt 5sin6θ (i) (ii) θ Sinθ Lt Lt Cos (iii) (iv) 5 6 Sin

212 7. Differentite the following: (i) ( )e tn (ii) ( )( 7)( 5) (iii) ot log (iv) e ose (v) ( )ose se os (vi) sin sin (vii) e tn (viii) e ( ) ( ) (i) ( ) ( ) () 5.) K- 6.) g f 7 se tn g f ANSWER PART - A 8.) (i) 9.) (i) (ii) - (ii) 7 (iii) (iv) (iii) sin e os e

213 (iv) ( ) se tn (v) (vi) (vii) log sin e os e sin sin os (viii) ( ) PART - B. 8. (i) Internll (ii) Eternll (iii) Eternll (i) 5 6 (ii) 5 (iii) (iv) 7 7. (i) ( )e se ( )e tn (6 )e tn (ii) ( )( 7) ( )( 5) ( 7)( 5) (iii) Cot( ) log ( oe ) Cot log ( ) (iv) ( ) e ose ot e ose (v) ( )os (se tn ) ( )se ( sin) os se ( sin ) sin ( sin)cos (vi) ( sin ) ( )( ) os sin (vii) ( ) sin

214 (e [ e se ] ( e ) (viii) ) (i) (e ) tn (e ( )[ ] ( )( ) ( ) () os ose ot sin ) 5

215 UNIT- III DIFFERENTIATION METHODS. Differentition of fntion of fntions nd Impliit fntions. Simple Prolems.. 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. DIFFERENTIATION METHODS. DIFFERENTIATION OF FUNCTION OF FUNCTIONS Fntion of Fntions Rle: If is fntion of nd is fntion then d d d. It is lled Fntion of fntion Rle. This rle d d d n e etended. Chin Rle: If is fntion of nd is fntion of v nd v is fntion of then d Find if d d d d. d d. dv dv d. WORKED EXAMPLES PART - A ) Sin( ) ) Cos( ) ) tn( ) ) log( se tn ) 5) tn( 7 ) 6

216 .) Sin( ) Pt sin d Cos d d d. d d d d.) Cos( ) Pt Cos d Sin d d d d. d d d.) tn( ) Pt tn d Se d d d Cos() Cos( ) d d ( sin) Sin( ) d d d se se ( ) d.) log( se tn ) Pt se tn log d d d d se tn d se tn se d se (tn se ) [ se ( se tn ) ] ( se tn ) se se tn se 7

217 5.) ( 7 ) Pt 7 d d d 7 d 7 d d ( 7) ( 7 ) ( 7) PART - B Differentite the following w.r.t.to.) ( ) 8 Sin e Sin.).) Sin.) logsin Sin 5.) e 6.) log(sin5) tn 7.) e 8.) sin 9.) log( se ).) Let sin( ) Pt sin d d os d d d d d. os () d d d Cos( ) Sin.) e Pt sin e d e d d d d d sin d os d d e.os e d 5.) os( e ) sin.os

218 .) sin Pt Sin sin d d os d d d.cos Sin.Cos d.) log(sin ) Pt Sin log sin d d os d d d Cos Cos Cot d sin sin 5.) e Pt Sin d d d e. e. ( sin ) e d sin d e 6.) log(sin5) d d 7.) e d e d sin 5 sin 5 tn e tn tn ( ) tn d ( sin Cos) e d d ( sin 5) ( 5 os 5) 5 ot 5 d d. ( tn ) ( ) ( ) 9

219 8.) Sin (sin) d d ( sin) ( sin) d ( Cos) sin sin 9.) Y log( se ) d d se se d Cos d (se d 5.) Cos( e ) d d sin e sin e 5e ) ( se )( se tn ) tn 5 d 5 ( ) ( e ) d 5 5 ( ).5e ( 5 sin e ) 5... Differentition of impliit fntions If the vriles nd re onneted the reltion of the form f(,) nd it is not possile to epress s fntion of in the d form f() then is sid to e n impliit fntion of. To find in d this se, we differentite oth sides of the given reltion with respet φ to. keeping in mind tht derivtive of ( ) dφ d d d d d w.r.t s. i.e ( φ( ) ) d d For Emple ( sin ) d Cos d dφ d. d d d d nd ( ) d d

220 WORKED EXAMPLES PART A d Find for the following fntions d ) ) os ( ) ) ) 5) k ) Differentite oth sides w.r.t d d d d d d ) os() Differentite oth sides w.r.t d d sin( ) d d d sin d sin d ( ) sin( ) d d ( ) sin( ) d d [ sin( ) ] sin( ) d d sin( ) d sin( ) ) Differentite oth sides w.r.t d d d d

221 ) d d d d d d 5) k d d d d d d PART - B d Find of the following d ) ) g f ) ) h 5) sin( ) ) Differentite oth sides w.r.t d d d d

222 d d. ) g f Differentite oth sides w.r.t ( f ) d d ( g) d ( f) g f d d g f d d d g ) Differentite oth sides w.r.t d d d d d d d d d d d ( ) d d ( ) d ( )

223 ) h Differentite oth sides w.r.t d d h d d d d h h d d d ( h ) h d d h d h ( h) h 5) sin( ) Differentite oth sides w.r.t d d os( ) sin( ) d d d d os( ) sin( ) d d d [ os( ) ] sin( ) d d sin( ) d os( )... Differentition of Inverse Trigonometri Fntions If sin then sin - sin -, os -, tn -, ose -, se - nd ot - re inverse trigonometri fntions. Emple: () - d If sin, find d sin sin

224 Differentite oth sides w.r.t d os d d d os os sin Emple: () Find the differentition of os - Let os - os Differentite oth sides w.r.t d - sin d d d sin sin os Emple: () Find the differentition of tn -. Let tn - tn Differentite oth sides w.r.t d se d d d se tn Emple: () Find the differentition of ot -. Let ot - ot Differentite oth sides w.r.t d -ose d d d ose ot 5

225 Emple: (5) (Differentition of se - ) Let se - se Differentite oth sides w.r.t d se tn d d d se tn se se Emple: (6) Differentition of ose - Let ose - ose Differentite oth sides w.r.t d - ose ot d d d ose ot ose ose FORMULA d sin d d os d d tn d d ot d d se d d ose d ) ( ) ) ( ) ) ( ) ) ( ) 5) ( ) 6) ( ) 6

226 WORKED EXAMPLES PART - A Differentite the following w.r.t () sin ( ) () os ( ) () ( ) sin (5) sin () ot (6) sin (7) e tn ) sin ( ) ( ) (8) tn ( ) d d d d ( ) ( ) ) os ( ) d d ( ) d d ( ) ) ot d d d d ) ( sin ) d d sin sin d d ( sin ) sin ( ) 7

227 5) sin d d sin sin 6) sin d d 7) e tn d e d e 8) tn ( ) d d ( ) tn tn d d sin ( ) ( e ) ( ) ( ) sin PART - B Differentite the following w.r.t ) sin ) tn ).) Let sin Pt tn θ sin - θ tn tnθ tn θ - sina sina sin A os 8

228 9 ( ) - d d tn sin sin θ θ.) tn Let ( ) d d tn tn tn tn tn tn tn tn Pt θ θ θ θ θ θ A tn A tn tna.) Cos Let ( ) d d tn Cos Cos tn tn Cos tn tn Pt θ θ θ θ θ θ A tn A tn - CosA

229 ... Differentition of prmetri fntion: Sometimes nd re fntions of third vrile lled prmeter withot eliminting the prmeter, we n find d d s follows. Let f( θ) nd F( θ) so tht θ is prmeter. d d d dθ. dθ d d dθ d dθ WORKED EXAMPLES PART-A.) If d Cos θ nd Sin θ, find d d sin θ dθ d os θ dθ d d dθ osθ ot θ d d dθ sinθ d.) Find if t nd t d d t dt d dt d d dt d d dt t t

230 d.) Find if d d se θ tn θ dθ d se θ dθ Cosθ sinθ Cosθ se θ nd tnθ d d dθ se θ se θ d d dθ se θtnθ tnθ sin d.) Find if t nd d d dt d dt t d t d t θ t Cose θ PART - B d.) Find if ost tsint nd sint- tost d d sint t os t sin t t ost dt d os t t sin t os t t sint dt d t sint tn t d t ost

231 d.) Find if sint nd ost d d ost ost() dt sint sint d sint ( sint) dt ost ost d d sint ost sint ost ost sint ( sint) ( ost) ( tnt) sint ost d if ( ) ( ).) Find θ sinθ nd osθ d d ( osθ) dθ d ( sinθ) ( sinθ) dθ d sinθ sinθ d osθ os ( ) θ sinθ osθ θ tn os θ d if ( ).) Find log se θ tnθ nd se θ d d ( se θtnθ se θ) dθ se θ tnθ se θ(tnθ se θ) se θ se θ tnθ

232 d se θtnθ dθ d se θ ot θ. d se θ tnθ tnθ.. Sessive differentition: d If f(), then, the derivtive of w.r. to is fntion of d d nd n e differentited one gin. To fi the ide, we shll ll d s the first order derivtive of with respet to nd the derivtive of d w.r. to s seond order derivtive of w.r. to nd will e d denoted d d. Similrl the derivtive of third order derivtive nd will e denoted Note: Alterntive nottions for d d d d n d n d n ' f'() D() '' (n) '' f () D () f (n) n () D () d w.r.t will e lled d d d nd so on.

233 .) If tn, tn, d d se se find WORKED EXAMPLES PART - A d d d se (se tn ) d tn.) If log(sin ),find os ot sin ose..) If os sin,find sin os os sin..) If 5 Ae Be,find D () D() A e D () 9 A e 5.) If,find 5 B e 5 5 B e 5

234 .) If sin, PART - B prove tht ( 6) sin ( os ) sin os sin ( sin) os os sin - sin os sin [ ( 6) sin os sin ] 8 sin os sin sin 6 sin ( os sin) ( sin os 6).) If oslog sinlog, show tht, oslog sinlog sinlog oslog Mltipl on oth sides sinlog oslog Differentite oth sides w.r.to oslog sinlog Agin mltipl on oth sides ( oslog sinlog ) oslog sinlog sin 5

235 .) If e sin e, prove tht ( ) e sin sin. e sin Differentite oth sides w.r.t ( ) e sin Mltipl on oth sides ( ).) If ( tn ), prove tht ( ) ( ) ( tn ) tn Mltipl ( ) ( ) tn on oth sides Differentite oth sides w.r.t. ( ) ( ) Mltipl ( ) ( ) ( ) on oth sides 6

236 .. Formtion of differentil eqtion: Definition: An eqtion involving differentil oeffiients is lled differentil eqtion. Emples: d.) d d.) d d.) 5 d d d 6 d d.) d d Order of the differentil eqtion: The order of the differentil eqtion is the order of the highest derivtive ppering in the eqtion, fter removing the rdil sign nd frtion involved in the eqtion. e.g:.) d d e (order is ) d d d d.) 6 (order is ) d d Degree of the differentil eqtion: The degree of the differentil eqtion is the power (or) degree of the highest derivtive oring in the differentil eqtion. Emple: d d.) e (degree ) d d d d.) sin d d 5 (degree ) 7

237 WORKED EXAMPLES PART - A.) If, form the differentil eqtion eliminting the Constnt d d PART - B.) If A Cos Bsin, form the differentil eqtion eliminting the onstnts A nd B. A Cos B sin d A sin BCos d d A Cos Bsin d d is d [ ACos Bsin] the reqired differentil eqtion ) Form the differentil eqtion eliminting the onstnt nd from the eqtion. e e Given e e Differentite oth sides w.r.to e e Differentite gin w.r. to, e e is the reqired differentil eqtion. 8

238 EXERCISE PART - A I. Differentite the following w.r. to.) ( 5) (.) ( 5 ).) ( ) tn (5.) sin (.) os e sin e (6.) log( tn ) II. d Find if d.) (.) sin.) k (.) III..) sin 5 sin os.) sin (.) ( ) (.) ( ) tn PART - B I) Differentite the following w.r.t 5.) ( ).) os sin Cos5 sin.) e tn os 6.) e tn5 e 5.) tn 6.) Cos () 7.) ( ) tn 8.) sin II) d Find of the following d.) 6.) sin C.) e.) 5.) m n m n 9

239 III) Differentite the following w.r.t.) tn.) sin.) sin ( ).) os ( ) 5.) 6.) sin 7.) se 8.) sin IV) d Find if d.) sin t sin t nd os t - os t.) ( θ sin θ) nd ( os θ).) os θ nd sin θ V).) If os, Pr ove tht ( ) ( 6).) ( sin ), Show tht ( ) If tn.) e, Pr ove tht ( ) ( ) If If.) log, Show tht ( ) os 5.) If e, Pr ove tht ( ) ANSWER PART - A I.) ( X 5) 9.) sin( 5 ) os( 5 ) os 6 tn.) e Cos sin.) 9 5.) e sin () 6.) Cose se

240 II..) III..) 5 5 ( tn ).) 6.) tn.).) -.) PART - B 9.) e 5 I.) [ os os] sin sin 5os sin.) os5 5( ) sin5 sin os 6.) e 6 sin6 tn tn5.) e 5 se 5 5.) tn 6.) os sin 7.) tn 8.) e II..).) tn.) e.) III..) 5.) 5.) ( ).) 6.) IV..) tn t.) m n.) 7.) ot θ.) tnθ.) 8.)

241 UNIT- IV APPLICATION OF DIFFERENTIATION I. Derivtive s rte mesre-simple Prolems.. Veloit nd Aelertion-simple Prolems. Tngents nd Normls-simple Prolems Δ.. DERIVATIVE AS A RATE MEASURE: Let f() is differentile fntion of in n intervl of. Let e smll inrement in nd the orresponding inrement in Δ is Δ. Then represents the verge hnge in the intervl nd Δ Δ Δ. When Δ tends to zero, the verge rte of hnge will Δ eome nerer nd nerer to the tl rte of hnge of t. Ths Lt Δ d represents the tl rte of hnge of t. Δ Δ d dr dv Note: If r, v nd s re rdis volme nd srfe re, then, dt dt ds nd re their rte of hnge with respet to time t. dt. WORKED EXAMPLES PART - B. The rdis of metl plte is inresing niforml t the rte of.5 m per seond on heting. At wht rte the re is inresing when the rdis is m. Let r e the rdis Given : dr.5 dt da To find when r.m. dt

242 We know tht re A πr da dr πr dt dt dr Pt r nd.5. dt da π()(.5) π sq.m. / se dt The Are is inresing t the rte of π sq.m / se. The re of sqre is inresing t the rte of m² / se. How fst its side is inresing when it is 8m long? Let e the side of the sqre. da Given : m² / se dt d To find when 8m. dt A² da d dt dt d (8) dt d m / se. dt 6 The side is inresing t the rte of m / se.. The rdis of sphere is inresing t the rte of m. per seond How fst its volme is inresing when the rdis is m? Let r e the rdis of the sphere nd V e the volme. dr Given : dt m /se

243 dv To find: when r m. dt V πr dv dt dr πr dt dr πr dt π m / se. π() The volme is inresing t the rte of πm / se. The rdis of sphere is inresing t the rte of m/se. Find the rte t whih its srfe re is inresing when the rdis is m. Let r e the rdis nd S e the srfe re dr Given : m /se dt ds To find : when r m dt S πr ds dr dr πr 8πr dt dt dt π m / se 8π() The srfe re is inresing t the rte of π m / se 5. A lloon whih remins spheril is eing inflted pmping in m³ of gs per seond Find the rte t whih the rdis of the lloon is inresing when the rdis is 5m. Let r e the rdis nd V e the volme. dv Given : m /se dt dr To find : when r 5m dt

244 V πr dv dr πr dt dt dr π(5) dt dr dt 9π 9π πr dr dt The rdis is inresing t the rte of m / se 9π 6. A stone thrown in to still wter ses series of onentri ripples. If the rdis of the oter ripple is inresing t the rte of meter / se, how fst is the re is inresing when the oter ripple hs rdis of meters. Let r e the rdis dr Given : meter /se dt da To find : when r dt A πr da dr πr π() πm / se dt dt The oter re of the ripple is inresing t the rte of πm / se 7. The side of n eqilterl tringle is inresing t the rte of 5m/se. Find the rte of hnge of its re when the side is 8m. Let e the side of the eqilterl tringle d Given : 5 m /se dt da To find : when 8m dt 5

245 Are da dt A d dt (8)5 The rte of hnge of its re is m / se m² /se. 8. A rod m long moves with its ends A nd B lws on the es of nd respetivel. If A is 8 m from the origin nd is moving w t the rte of m per se. Find t wht rte the end B is moving. Let AB e the rod. At time t se the rod m e the ove position. Let OA nd OB. Given: d dt d To find when 8 m dt We know tht 6

246 d d ( ) dt dt d d dt dt 8 () m se 6 The end of the rod is moving t the rte of 8 m se 9.) A mn of height 6 feet wlks w from lmp post of height 5 feet t onstnt speed of feet per seond Find the rte t whih his shdow on the grond is inresing when the mn is feet from the lmp post. Let AB Lmp post CD mn, CE length of the shdow Let e the distne etween the lmp post nd mn nd e the length of the shdow. 7

247 d Given: dt d To find: when feet dt Here Δ ABE similr to Δ CDE AE AB CE CD , d d dt dt d d dt dt d () dt 9 6, The shdow is lengthening t the rte of feet / se Note:. VELOCITY AND ACCELERATION Let the distne s e fntion of time t Veloit is the rte of hnge of displement ds Veloit v dt Aelertion is the rte of hnge of veloit. dv d ds d s elertion is. dt dt dt dt. At time t, we get the initil veloit.. When the prtile omes to rest, then veloit v. When the elertion is Zero, veloit is Uniform. 8

248 . WORKED EXAMPLE PART - A.) If s 5t t t find the veloit nd elertion fter t seonds. s 5t t t ds Veloit v 5t 6t dt dv elertion t 6 dt.) The distne s trvelled prtile in t se is given s t 5t 6 Find the veloit t the end of seonds. s t ds v t dt When t, 5t 6 5 v () m / se.) The distne time forml of moving prtile is given s 7t 5t t Find the initil elertion s 7t v elertion 5t ds t dt dv dt t t t Initil elertion () -- Unit /se. 9

249 .) If the distne s meters trvelled od in t seonds is given s 5t t.find t wht time the veloit vnishes. s 5t t ds v t dt Veloit vnishes, v t, t t se s 5.) Find the initil veloit of the od whose distne time reltion is s 8ost sint s 8ost sint ds v 6sint os t dt Initil veloit 6 sin os 6.) The distne time forml of moving prtile is given t t s e e. Show tht the elertion is lws eql to the distne trvelled. t t s e e Veloit v ds dt Aelertion s. t t e e dv dt t t e e Aelertion distne trvelled.

250 PART - B.) The distne s meters t time t seonds trvelled prtile is given s t 9t t 8 ) Find the veloit when the elertion is zero. ) Find the elertion when the veloit is zero. s t 9t t 8 ds v t 8t dt d s 6t 8 dt (i) Find veloit when elertion is zero d s i.e., dt 6t 8, 6t 8 t se Veloit () 8() m/se (ii) Find the elertion when the veloit is zero. v t 8t t 6t8 (t ) (t -) tort. At t se, Aelertion 6() 8 8-6m/se At t ses, Aelertion 6() 8 8 6m/se

251 . TANGENTS AND NORMALS: Let the grph of the fntion f() e represented the rve AB, Let P(,) e n point on the rve let Q( Δ, Δ) e neighoring point to P on the rve. Let the hord PQ mkes n ngle θ with is. Let the tngent t P mkes n ngle ψ with X is. Δ Δ Slope of the Chord PQ Δ Δ Now let the point Q moves long the rve towrds the point P nd oinide with P, so tht Δ ο nd Δ ο nd in the limiting position PQ eome the tngent t P. Slope of the tngent t Pslope of the hord when Lt Lt Δ tn ψ tnθ Q P Δ Δ d d Q P d The geometril mening of is the Slope of the tngent t d point (,) on it nd it is denoted m Note:. Eqtion of the line pssing throgh (, ) m is m( ) nd hving slope

252 . If the slope of the tngent is m the slope of the norml is m. Slope of the line prllel to is is zero. Slope of the line prllel to is is. WORKED EXAMPLES PART - A.) Find the slope of the tngent t the point (,) on the rve Differentite oth sides w.r.t d d d d d d d At, d slope of the tngent t (,) is.) Find the slope of the tngent t the point to the rve 5 d 6 d At 5, d d Slope of the tngent t is -

253 .) Find the slope of the norml t (,-) on the rve At d d d d 6 (, ) 6 Slope of the tngent Slope of the norml.) Find the grdient of the rve t (,) d d d d d At (,) d Grdient of the rve 5.) Find the point on the rve prllel to is d d d Sine the tngent is prllel to is d - When, - The point is (,) t whih the tngent is

254 6.) Find the slope of the tngent t the point ( t,t) prol At ( t,t) d d d d d d t t The slope of the tngent is t on the PART - B.) Find the eqtion of the tngent nd norml to the rve 6 t (,). 6 d d d At (,) d Slope of the tngent - Eqtion of the tngent t (,) is m( ) ( ) slope of the norml 6 Eqtion of the norml is m( ) 5

255 ( ).) Find the eqtion of the tngent nd norml to the rve t (,) d ( )( ) ( ) d At (,) ( ) ( ) d d 6 ( ) m 6 Eqtion of the tngent is ( ) 5 5( ) ( ) 5 ( ) Slope of the norml 5 Eqtion of the norml is m( ) 5 ( )

256 .) Find the eqtion of the tngent to the rve 5 t the point where the rve ts the -is. The rve ts -is The point is (,5) d 6 d d At(,5) d Slope of the tngent m Eqtion of the tngent is ( ) 5 ( ) 5 5 ) Find the eqtion of the tngent to the rve ( )( ) the point where the rve ts the -is. If the rve ts the is then ( )( ) nd The points re (,) nd (.) At (,) ( )( ) d d d 6 7 d ( ) ( ) 7 t 7

257 Slope of the tngent t (,) is - Eqtion of the tngent t (,) ( ) is d At (,) 8 7 d Slope of the tngent t (,) is. Eqtion of the tngent t (,) is ( ) 5.) Find the eqtion of the tngent nd norml to the prol t ( t ),t d d d d d At (t,t) d t t Is slope of the tngent t Eqtion of the tngent is m( ) t t t t t t t t Slope of the norml -t Eqtion of the norml is ( t ) t t 8

258 m t t ( ) 9 ( t ) t t t t t t 6.) Find the eqtion of the tngent to the prol mkesnngle5 withx- is. d d slope of the tn gent d d Tngent mkes n ngle 5 with X - is. Slope of the tngent tn5 Pt in () The point is (,) Slope Eqtion of the tngent is m(- ) ( ) whih 7.) Find the Eqtion of the tngent to the rve 6 t the point where the tngent is prllel to the line () d 6 6 d

259 6 Slopeofthegivenline 6 If the tngent is prllel to the given line then Pt in () () 6() The point is (,) Eqtion of the tngent is m( ) 6 ( ) 6 6 EXERCISE PART - A.) If A², m., d da.6 m Find dt dt da dr.) Find if r.m. m nd A π r² dt dt ds dr.) If S π r², r m, π.m Find dt dt dr dv.) If V π r,r.m..m Find dt dt PART - B.) The re of sqre is inresing t the rte of m²/se. How fst its side is inresing when it is 8m long?.) The rdis of irlr plte is inresing t the rte of. m/se. At wht rte the re is inresing when the rdis m. 5

260 .) The irlr pth of oil spreds on wter. The re is inresing t the rte of 6 m²/se. How fst is the rdis inresing when the rdis is 8m..) The distne s meters trvelled od in t se is given s8t-6t². Find the veloit nd elertion fter seonds. 5.) The distne time forml of moving prtile is given sost sint. Prove tht the elertion vries s its distne. 6.) Find the slope of the tngent to the rve 6 t the point (,8). 7.) Find the slope of the norml to the rve ²- t (,-). 8.) The rdis of sphere is epnding niforml t the rte of m/se. Find the rte t whih its srfe re is inresing when the rdis is m. 9.) The rdis of spheril lloon is inresing t the rte of 5m/se. Find the rte of inresing in the volme of the lloon, when its rdis is m..) A stone thrown in to still wter ses series of onentri ripples. If the rdis of oter ripple is inresing t the rte of 5m/se how fst is the re inresing when the oter ripple hs rdis of m.) An inverted one hs depth of m nd se of rdis 5m. Wter is pored into it t the rte of.5./se. Find the rte t whih the level of the wter in the one is rising when the depth is.m..) A. mn m tll, wlks diretl w from light 5m ove the grond t the rte of m/se. Find the rte t whih his shdow lengthens..) The distne trvelled prtile in time t seonds is s t 6t 8. Find the veloit when the elertion vnishes. Find lso the elertion when the veloit vnishes..) The distne time forml of moving prtile is given s t t 7t. Find the elertion when the veloit vnishes. Find lso the initil veloit. 5

261 5.) A Prtile is moving in st. line. Its distne t time t is given s t 5t 6t 7 ) Find the initil veloit ) Findthetimewhentheveloitiszero ) Findthetimewhentheelertioniszero 6.) The Veloit v of prtile moving long stright line when t distne s from the origin is given s²mnv². Show tht the s elertion of the prtile is n 7.) A od moves in stright line in sh mnner tht s vt s eing the spe trvelled in time t nd v is the veloit t the end of time t prove tht the elertion is onstnt. 8.) Find the eqtion of the tngent nd norml to the following rves..) t.) 6 t (,8 ).) 6 t (,) 6 d.) t (, ) e.) t (, ) f.) t ( Cosθ, sinθ) g.) t ( se θ, tn θ) 9.) Find the eqtion of the tngents t the points where the rve meets the -is..) Find the eqtion of the tngent to the rve 5 7 where it ts the is. 5

262 ANSWERS PART - A.).m².).πm.) m.) 6.πm PART - B.).5.).6πm / se.) π.) 6m/s,-m/se² 6.) - 7.) 9.) 8 π m³/se.) π m²/se 8.)6 π.5.) m/se.) m/se.) π 9.),-7 5.) () 6 () t or () 5 8.).) 5, 5 6.), 6.), d.) 5, 5 7 e.) os θ sinθ, se θ tn θ f.) se θ tnθ, se θ se θ 9.) 6,.) 5 7 5

263 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 orders onl. Simple prolems. 5. Definition of Homogeneos fntions Elers theorem Simple Prolems. 5. APPLICATION OF DIFFERENTIATION-II Inresing fntion: A fntion f() is sid to e inresing fntion if vle of inreses s inreses or vle of dereses s dereses In the ove figre f() is inresing fntion nd Δ >, Δ > d Lt Δ d Δ Δ ve d At ever point on theinresing fntion the vle of d is positive 5

264 Deresing fntion: Afntionf()issidtoederesingfntionifthevleof deresess inresesorv leofinresess dereses Intheovefigref()isderesingfntionnd Δ > nd Δ < d Lt Δ ve d Δ Δ At ever point on the deresing fntion the vle of d d is Trning points: negtive Afntionneednotlwseinresingorderesing.Inmost sesthefntionisinresinginsomeintervlndderesingin someotherintervl. 55

265 nd [, ] In the ove figre the fntion is inresing in the intervl [,] It is deresing in the intervl[,] Definition of trning points: Tring point is the point t whih the fntion hnges either from deresing to inresing or from inresing to deresing. In the ove figre P nd Q re trning points. Sine t the point P, the fntion hnges from inresing to d deresing, vleof hngesfrom positive to negtive. Hene t the d d point P, d d Similrl t the point Q, d Mimm of fntion: The mimm vle of fntion f() is the ordinte ( oordinte) of the trning point t whih the fntion hnges from inresing to deresing fntion. d At mimm, d 56

266 Minimm of fntion: The minimm vle of fntion f() is the ordinte ( oordinte) of the trning point t whih the fntion hnges from deresing to inresing fntion. d At minimm, d Conditions for mimm: d d At mimm, nd is negtive d d Conditions for minimm: d d At minimm, nd is positive d d 5. WORKED EXAMPLES PART-A.) Write the Condition for the fntion f() to e mimm At, d (i) d d (ii) < d.) Write the ondition for the fntion () to e minimm At, d (i) d d (ii) > d 57

267 .) For wht vle of the fntion will hve mimm or minimm vle d d d d At, the fntion will hve mimm or minimm vle..) For wht vle of the fntion will hve mimm or minimm vle Let d d d > d 5 At 5, the fntion will hve mimm or minimm vle. 5.) For wht vle of the fntion sin will hve mimm or minimm vle d sin, os d d os os 9 d 9 or π / At π the fntion will hve mimm or minimm vle., 58

268 6.) Find the minimm vle of d d d Pt d d When, > d t the fntion Pt in () () 8 is minimm - is the minimm vle. 7.) Find the mimm vle of d d d Pt d / d When, d when the fntion is mimm 59

269 Pt in, is the mimm vle. 8.) Find the mimm or minimm vle of d d d d d Pt d d when, d At, the fntion is mimm. When, () () is the m imm 6 vle 9.) Find the mimm or minimm vle of d d 6 d 6 d

270 d Pt d 6 6 d, 6 d When /,the fntion Pt in, is 9 9 PART - B mimm. is themimm vle. 9.) Find the mimm nd minimm vles of the fntion d d 6 6 d 6 d d Pt d i.e., ( )( ) 6 6

271 d When, d When, the fntion is minimm Pt in, ( ) ( ) is the minimm vle d When, ( ) 6 d When, the fntion is mimm Pt in, 6 ( ) ( ) 6( ) is the minimm vle.) Find the mimm nd minimm vles of ( ) ( ) Let ( ) ( ) d d d d d d ( ) () ( ) ( )() ( ) ( )( ) ( ) [ ( ) ] ( )( 5) ( )( ) ( 5)( )

272 d Pt d ( )( 5) 5 5 d (i) When, 6() 8 d when Pt in, the ( ) ( ) fntion is is the mimm vle mimm 5 d 5 (ii) When, d 5 when 5 Pt in, 5 the 5 ( ) fntion. 9 7 is the minimm vle 7 is minimm 6

273 .) Find the mimm vle of Let d d d d log log log. () log ( ) ( log )( ).log log d Pt d log log log log e e e e d loge When e, d e When e the fntion is m imm Pt e in, log loge e e e is the m imm vle for positivevle of. 6

274 .) Find the minimm vle of log. Let log d log. () log d d d d Pt d When At, e log, e log ; log d d e the fntion Pt in, e.log log e e e e [ log loge] e [ loge] [ ] e e e is e e e is the minimm vle minimm 65

275 5.) Show tht the fntion f ( ) sin( os ) t π Let sin(os) d Pt d d d sin sin os ( sin ) ( os ) os os sin os os d sin sin d is mimm os os os os os.os π os os π π C D C D ( osc osd os os ) π os os π π π d π π π When, Sin Cos Sin d The fntion is mimm t π 66

276 6.) AB is 8m long, find point C on AB so tht minimm AC BC m e Let: AC so tht BC8- i.e AC BC ( 8 ) d 8 6 d d 8 d d Pt d 8 6 > 8 6 > when d, 8 d When the fntion is the minimm The point C is t m froma 7.) A retngle sheet of metl is m m. Eql sqres re t off t the for orners nd the reminder is folded p to form n open retnglr o. Find the side of the sqre ot off theso tht the volme of the o m e mimm. 67

277 Let the side of the sqre e Length of the o - Bredth of the o- Height of the o Volme of the o V(-)(-)() dv d ( 6 8 ) [ volme l h] ( 88 ) V V d V 76 d dv Pt d i.e., ( ) 5( ) 5 5 d V when, () 76 d 7 76 At the volme is mimm Hene the volme is mimm when the side of the sqre t from eh of the orner is m. 68

278 8.) Find two nmers whose sm is nd whose prodt is mimm. let the nmers e nd Given, sm i.e.,, - Let p e the prodt of the nmers p ( ) dp d p, d d dp Pt,, 5 d d p At 5, d Pismimmwhen5 The Nmers re 5,5 9.) The ending moment t B t distne from one end of em of length L niforml loded is given M wl w where lod per nit length. Show tht the mimm ending moment is t the entre on the em. Bending moment M dm d wl d M w d dm Pt d wl w w wl wl w 69 w

279 L L d M When, w d L t, M is mimm. The mimm ending moment is t the entre of the em..) A od moves so tht its distne from given point t time t seonds, is given S t t t Find when the od ttins its mimm veloit. S t t t ds Veloit v.t.t dt ds veloit v t t dt dv t dt d v dt dv pt dt t t d v When t, dt At t seonds v is mimm The od ttins its mimm veloit when t se. 7

280 5.. PARTIAL DIFFERENTIATION Fntions of two or more vriles In mn pplitions, we ome ross fntion involving more thn one independent vrile. For emple, the re of retngle is fntion of two vriles, the length nd redth of the retngle. Definition: Let f(, ) e fntion of two independent vriles nd. The Derivtive of with respet to when vries nd remins onstnt is lled the prtil derivtive of with respet to nd is denoted Lim f( Δ,) f(, ) Δ Δ f is lso written s (,) f or Similrl when remins onstnt nd vries, the derivtive of with respet to is lled the prtil derivtive of with respet to nd is denoted. Lim f(, Δ) f(, ) Δ Δ f or Seond order Prtil derivtives In Generl, nd re lso fntion of nd. The n e frther differentited prtill w.r.t. nd s follows. f Hene (i) (or) (or)f f is lso written s (,) 7

281 7 (ii) (or)f f (or) (iii) (or)f f (or) (iv) (or)f f (or) Generll, for ll ordinr fntions, f f 5. WORKED EXAMPLES PART A.) If,. Find,,,,,nd.) 6 6.) 6.) ( ) d.) ( ) e.) ( ) 6 6 f.) ( ) 6 6 6

282 .) If. sin. sin, find ll seond order prtil derivtives.. sin. sin, ( ) ( sin ) sin...) sin. Cos.). ( sin) ( ) sin..os sin..) os os (.os sin) d.) os os ( sin os ) e.).sin ( sin os ) f.).sin ( os sin).) If, find (i) (iv) (v), (vi). (ii.). (iii.). 7

283 7 (i) (ii) (iii) 6 (iv) 6 (v) (vi).) Find When ( ) log ( ). log 5.). Find When sin.. 6.) If sin. os Find nd If sin os osos sin ( sin.) -sinsin

284 75 7.) If e Find nd e e, e 8.) Find if tn tn.. 9.) If, find nd Here

285 76 PART - B.) If, Show tht ( ) ( ) ( ) ( ) ) If, Show tht ( ) ( ) ( ) ( ) ( )

286 77 5. HOMOGENEOUS FUNCTIONS: An epression in whih eh term ontins eql powers is lled homogeneos epression. A fntion of severle vriles is sid to e homogeneos of degree n if mltipling eh vrile t(where t>) hs the sme effet s mltipling the originl fntion n t. Ths f(,) is homogeneos fntion of degree n if f(t, t ) t n f(, ). Emple: If find the degree Pt t, nd t ( ) ( ) t t t t t t is homogeneos fntion of degree -. Eler s theorem on homogeneos fntions: If is homogeneos fntion of, of degree n, then n... Proof: If is homogeneos fntion of, ( ) φ., f n Differentiting () Prtill wrt φ φ φ φ ' n n n n ' n n

287 78 Differentiting () prtill w.r.t φ φ ' n ' n Mltipling () nd () nd dding. () n ' n. n n n φ φ φ 5. WORKED EXAMPLES PART A.) Find the degree of the fntion Pt t,t ( ) ( ) ( ) t t t t t t t degree of the fntion is

288 .) Find the degree of the fntion 9 Pt t, nd t t t ( t) 9( t)(. t) ( t) 9t t [ 9 ] degree of the fntion is 9 PART-B.) Verif Eler s theorem when ( t,t) t ( t) t t t (t)(t) t t tt [ ] t (t) Therefore (,) is homogeneos fntion of degree. B Eler s theorem n To prove tht 6 [ ] Hene Eler s theorem is verified. 6 79

289 8.) Verif Eler s theorem for (, ) ( ) t t t t t (t) (t, t) (, ) is homogeneos fntion of order -. B Eler s theorem ( ). Verifition: ( ) ) ( ) ( ) ( ) ( () ) ( ) ( ) ).( ( ) ( ) ( ) ( ) (... From () nd ( ) Eler s theorem is verified

290 .) Using Eler s theorem, for sin prove tht. tn. U is not homogeneos fntion. Bt sin is homogeneos fntion. Define sin f(t,t) f(,) sin t ( t) ( t) t t ( ) ( ) t t t f(, ) sin is homogeneos fntion of degree. B Eler s theorem ( sin) ( sin) os sin.. tn os. os sin sin 8

291 8. Using Eler s theoremif tn, prove tht. sin.. is not homogeneos fntion. Bt tn is homogeneos fntion. tn ( ) ( ) ( ) ( ) t t t t t t t t t t t tn tn is homogeneos fntion of degree. B Eler s theorem, tn (tn). (tn). tn se tn se.se se tn, se sin sin sinos os sinos 5.) Using Eler s theorem Prove tht tn If sin

292 is not homogeneos nd hene tke the fntion sin sin sin Pt f(t,t) t, t t t t t ( ) t t ( ) t ( ) Sin is homogeneos fntion of order Using Eler s theorem. (sin) ( sin).. (os) Cos) os, sin sin tn. 6.) Using Eler s theorem if.. sin Pt t, t sin show tht 8

293 t ( t) t sin t t sin homogeneosfntion of order. Using Eler s theorem, we hve.. z z 7.) If z e Prove tht.. zlogz z e logz loge logz f ( ) loge f(, ) ( t,t) t. t. t ( Y ) t (,) t [ f(,) ] f is homogeneos fntion of order. (i.e) logz is homogeneosfntion of order. B Eler s theorem. ( logz) ( logz). logz z z.... logz z z z z.. z.log z 8

294 85 8.) If 5 5 log show tht log ( ) ( ) ( ) ( ) ( ), z t t ) ( t ) ( t t t t t z(t,t) z(, ) (s) z e ( ), z is homogeneos fntion of order. B Eler s Theorem, z z. z. ( ) ( ) e.e i.e,.e e e. e i.e,. Dividing e..

295 EXERCISE PART-A.) Find mimm (or) minimm Point of.) Find mimm (or) minimm Point of.) Find mimm (or) minimm Point of Cos.) When nd d verif the point is mimm or d minimmpoint 5.) When nd d verif the point is mimm or d minimm point 6.) If - is the minimm point of wht is the minimm vle? 7.) If is the minimm point of wht is the minimm vle? 8.) Find the mimm vle of 9.) Find the mimm vle of.) Find the mimm vle of.) Write down the onditions for mimm vle of f( ).) Write down the onditions for mimm vle of f( ).) Find nd for the following..) ) 5sin tn.) e 7.sin 8 d.) e.) f.).sin.se sin.cos t t 86

296 g.) tn h.) log( e e ) i.) sin j.).cos.) Find the order of the following homogeneos eqtion.) z () ( ).) ( ) e.) 87 (d) 6 PART-B.) Find the mimm nd minimm vles of the following fntions: () 6 () () 5 (d) 7 (e) 6 9 (f) 8 7 (g) 9 (h) ( )( ) (i) (j)

297 .) Find the nmers whose sm is nd whose prodt is the mimm..) Find two nmers whose prodt is for nd the sm of sqre is minimm..) ABCD is retngle in whih AB 9m nd BC6m Find point P in CD sh tht PA PB is minimm 5.) The perimeter of retngle is meters. Find the side when thereismimm. 6.) A sqre sheet of metl hs eh edge 8 m long. Eql sqres re t off t eh of the Corner, nd the flps re then folded p to form n open retnglr o. Find the side of the sqre t off so tht the volme of the o m e mimm. 7.) Find the mimm vle of log for positive vle of. 8.) Find the minimm re of retngle insried is semiirle. 9.) Show tht the mimm retngle insried in irle is sqre..) Find the frtion, the differene etween whih nd its sqre is mimm.) Find nmers whose prodt is for nd the sm of their sqre is minimm..) Verif tht for the following fntions (.) (.) tn (.) (d.).sin. Sin.) Verif Eler s theorem for the fntions. (.) log (.) sin (.) (d.) 88

298 89 (e.) (f.) (g.) (h.).sin (i).) If, Show tht.. 5.) If sin nd Prove tht.. 6.) If sin, Prove tht.. 7.) If log, Prove tht.. 8.) If tn sin, Prove tht.. 9.) If sin, Prove tht tn...) If, Prove tht...) If sin,provetht tn...) If tn,provetht sin...) If, Show tht..

299 .) If sin 5.) If, Show tht.. tn, Show tht.. 6.) If Sin, Show tht.. 7.) If tn, Show tht.. ANSWERS PART-A () () 5 () () (5) (6) (7) (8) () () 5, () () (d) (e) 5 os, se.e, 7 os sin.,.os se.tn, se. (f) os.os., sin sin. 9

300 (g) (h) (i) (j) e e...e os. () () () (d) (e) - sin. os () PART-B ) ) 8,- )6,5 (), - 5 (d) 8, 8 (e)5, (f), (g)-5,7 (h) 5 7, 7 (i) -6, (j), ) 5,5 ), ),5 5) 5 6) ) 7) e ), 8) 9) Nil 9

301 MATHEMATICS II MODEL QUESTION PAPER Time : Hrs M Mrks : 75 PART A (Mrks:55). ANSWER ANY 5 QUESTIONS: ) Find the entre nd rdis of the irle - ) Find the eqtion of the irle with entre (-, -) nd rdis 5 Units. ) Write down the eqtion of the irle with end points of dimeter (, )nd(, ) ) Show tht the point (5, -) lies otside the irle --6 5) Stte the ondition for two irles to t orthogonll 6) Evlte Lt sin 7) Find d 7os d 8) Find d { tn} d d d 9) Find [ os(log ) ] ) Find d [ sin ( )] d d ) Find if tn d ) Find the differentil eqtion eliminting onstnt r, from r 9

302 ) If A nd d da find when 5 d dt ) If the distne s given s t 5t7, find the veloit when t seonds. 5) Find the slope the tngent to the rve -5 t the point (,-). 6) Find the slope of norml to the rve t (,-). 7) Show tht the fntion - 7 is the mimm t. 8) If 5 find, 9) Iflog( )find ) Stte Eler s Theorem. PART B (Answer An TWO sdivisions in eh qestion) All Qestions rr Eql Mrks 56 ) Find the eqtion of the irle pssing throgh the point (-9,) nd hving entre t (,5) ) Find the eqtion of the irle pssing throgh the points (,),(,)nd (-,5) ) Find the eqtion of the tngent t (5, -) to the irle ) Show tht the irles 68nd - 6 toh eh other. Lt ) Evlte 5 7 ) Differentite the following:- (i) e log sin. (ii) sin Cos 9

303 .) d Find if (i) log (setn) (ii) h d ) d - Find if (i) os d (ii) (tost), (sint) ) if Cos, prove tht - ( 6).) The rdis of sphere is inresing t the rte of m/se. How fst the volme will e inresing when the rdis is m ) A missile is fired from the grond level rises meters vertill pwrds in time seonds nd t - 5 t.find the initil veloit nd mimm height of the missile 5.) ) Find the eqtion of the tngent nd norml to the rve - t (,). Find the mimm nd minimm vles of ) If, Find ) If Prove tht nd 9

304 MATHEMATICS - II MODEL QUESTION PAPER - Time:Hors m.mrks:75 PART - A I. Answer n 5 Qestions:. Find the eqtion of the irle with entre (,) nd rdis nits. Find the entre nd rdis of the irle. Find the eqtion of the irle with the points (, -) nd (, ) joining s dimeter.. Find the length of tngent from the point (5,7) to the irle Show tht this irles -- nd --9 re onentri irles. 6. Evlte Lt 7. Find d d if d 8. Find If e log d d 9. Find If os d d. Find if tn - ( ) d. Find d d if sin ().. Find the differentil eqtion eliminting the onstnts from. If V d nd, find dt d when 5 dt 95

305 . If Se t e -t, show tht elertion is lws eql is to distne 5 If the distne time forml is given e st -5t 7t-, find the initil veloit. 6. Find the slope of the norml to the rve 7 t (,8) 7. Find the minimm vle of 8. If - find 9. If tn () find. Show tht is homogeneos. Stte the order of the fntion. PART - B Answer n TWO s division from eh qestion: All Qestions rr Eql Mrks 5X6.) Find the eqtion of the irle, two of whose dimeters re 6 nd nd whose rdis is Units. ) Find the eqtion of the irle pssing throgh (, ) nd (, ) nd hving its entre on the line ) Find the eqtion of the tngent t (, ) on the irle -6-5 ) Find the eqtion of the irle whih psses throgh the origin nd ts Orthogonll with irles 8 nd 6 Lt sin ) Evlte sin7 d ) Find if (i) d (ii) Y (X -5)Coslog 96

306 d ) Find d if (i) sin (e log ) (ii) ) d - Find if (i) tn d (ii) t, t ) if os (log) sin (log) prove tht ) The se rdis nd height of onil fnnel re m nd m respetivel. Wter is rnning ot of the fnnel t /se. Find the rte t whih the level of wter is deresing when the level is m. ) If the distne time forml is given st -5t 6t7, find the time when the veloit eomes zero. ) Find the eqtion of the tngent nd norml to the rve 6 - t (, ) 5 ) Find the mimm nd minimm vle of ) Iflog( )find ) If tn - show tht sin 97