MATHEMATICS (860) CLASS XI

Transcription

1 MATHEMATICS (860) Aims:. To ele didtes to quire kowledge d to develop uderstdig of the terms, oepts, symols, defiitios, priiples, proesses d formule of Mthemtis t the Seior Seodry stge.. To develop the ility to pply the kowledge d uderstdig of Mthemtis to ufmilir situtios or to ew prolems.. To develop skills of - () omputtio. () redig tles, hrts, grphs, et. 4. To develop ppreitio of the role of Mthemtis i dy-to-dy life. 5. To develop iterest i Mthemtis. 6. To develop sietifi ttitude through the study of Mthemtis. A kowledge of Arithmeti d Pure Geometry is ssumed. The prts of Geometry whih re of hief importe i other rhes of Mthemtis re the fudmetls oerig gles, prllels (iludig lies d ples i spe), similr trigles (iludig the theorem of Pythgors) the symmetry properties of hords d tgets of irle, d the theorem tht lie perpediulr to two o-prllel lies i ple is perpediulr to every lie therei. The emitio my ilude questios with geometril otet. As regrds the stdrd of lgeri mipultio, studets should e tught: (i) To hek every step efore proeedig to the et prtiulrly where mius sigs re ivolved. (ii) To ttk simplifitio pieemel rther th e lok, e.g. ever to keep ommo ftor whih e elled. (iii) To oserve d t o y speil fetures of lgeri form tht my e oviously preset. The stdrd s regrds (iii) is diffiult to defie; iitil prtie should e o the esiest ses, 'trik' emples should e voided d it should e kept i mid tht (iii) is susidiry i importe to (i) d (ii) Tehers should e srupulous i settig stdrd of etess d i voidig the slovely hit of omittig rkets or replig them y dots. CLASS XI The syllus is divided ito three setios A, B d C. Setio A is ompulsory for ll didtes. Cdidtes will hve hoie of ttemptig questios from either Setio B or Setio C. There will e oe pper of three hours durtio of 00 mrks. Setio A (80 mrks) will osist of ie questios. Cdidtes will e required to swer Questio (ompulsory) d five out of the rest of the eight questios. Setio B / C (0 mrks) Cdidtes will e required to swer two questios out of three from either Setio B or Setio C. SECTIO A. Mthemtil Resoig Mthemtilly eptle sttemets. Coetig words / phrses osolidtig the uderstdig of if d oly if (eessry d suffiiet) oditio, implies, d/or, implied y, d, or, there eists d their use through vriety of emples relted to rel life d Mthemtis. Vlidtig the sttemets ivolvig the oetig words, differees etwee otrditio, overse d otrpositive. 7

2 . Alger (i) Comple umers Comple umers s ordered pir of rel umers i the form + i, (, ) Geometril represettio i omple ple - Argd digrm for z ( omple umer), /z, z d z ; equlity of two omple umers; solute vlue (modulus). OTE: Rel d imgiry prts of omple umer d equlity of two omple umers re required to e overed. (ii) Qudrti Equtios Use of the formul: 4 i solvig qudrti equtios. Equtios reduile to qudrti form. ture of roots Produt d sum of roots. Roots re rtiol, irrtiol, equl, reiprol, oe squre of the other. Imgiry umers. Comple roots. Frmig qudrti equtios with give root. OTE: Questios o equtios hvig ommo roots re to e overed. Qudrti Futios. (i) Give, s roots the fid the equtio whose roots re of the form,, et. Rel roots (ii) Cse I: > 0 Comple roots (iii) Cse II: < 0 Equl roots Rel roots Comple roots Equl roots (iii) where is the oeffiiet of i the equtios of the form + + = 0. Uderstdig the ft tht qudrti epressio (whe plotted o grph) is prol. - Sig of qudrti Sig whe the roots re rel d whe they re omple. - Qudrti iequlities. Usig method of itervls for solvig prolems of the type: () () A perfet squre e.g () (d) Fiite d Ifiite Sequees () Arithmeti Progressio (A.P.) T = + ( - )d S = { ( ) d} Arithmeti me: = + - Isertig or rithmeti me etwee y two umers. - Three umers i A.P..: - d,, + d - Four i A.P.: - d, - d, + d, + d () Geometri Progressio (G.P.) T = r -, S ; r r S ( r ), r 8

3 - Geometri Me, - Isertig or Geometri Me etwee y two umers. Three umers re i G.P. r,, r - Four r, r, r -, r - () Hrmoi Progressio,, re i H.P the /, /, / re i A.P. Hrmoi me = (d) Arithmetio Geometri Series Idetifyig series s A.P. G.P. (whe we sustitute d = 0 i the series, we get G.P. d whe we sustitute r = the A.P.) (e) Speil sums,, Usig these summtios to sum up other relted epressio. (iv) Permuttios Comitios Ftoril ottio!,! =(-)! Fudmetl priiple of outig. () Permuttios P r. Restrited permuttio. Certi thigs lwys our together. Certi thigs ever our. Formtio of umers with digits. Word uildig - repeted letters o letters repeted. Permuttio of like thigs. Permuttio of Repeted thigs. Cirulr permuttio lokwise outerlokwise Distiguishle / ot distiguishle. () Comitios C r, C =, C 0 =, C r = C r, C = C y, the + y = or = y, + C r = C r- + C r. Whe ll thigs re differet. (v) Whe ll thigs re ot differet. Divisio ito groups - e.g. distit groups, idetil groups. Mied prolems o permuttio d omitios. Mthemtil idutio Usig idutio to prove vrious summtios d divisiility. OTE: Prolems o iequlities re ot required. (vi) Biomil Theorem () Sigifie of Psl s trigle. () Biomil theorem (proof usig idutio) for positive itegrl powers, i.e. ( + y ) = - C 0 + C y... Cy. Simple diret questios sed o the ove. () Biomil theorem for egtive or frtiol idies ( ) ( )! ( )( )! Whe... - Simple questios o the pplitio of the ove. - Fidig the rth term for the ove (T r). - Applyig the theorem o pproimtios e.g. (0.99) 8 = (- 0.0) 8. OTE: Algeri pproimtios re lso to e overed. (vii) Properties of Biomil Coeffiiets. C C 0 0 C C... C C 4... C C C 5... Simple prolems ivolvig pplitio of the ove. OTE: Questios o the produt of oeffiiets of (+) (+) m re eluded. 9

4 . Trigoometry (i) (ii) (iii) Agles d Ar legths Agles: Covetio of sig of gles. Mgitude of gle: Mesures of Agles; Cirulr mesure. The reltio S = r where is i rdis. Reltio etwee rdis d degree. Defiitio of trigoometri futios with the help of uit irle. Truth of the idetity si + os =. OTE: Questios o the re of setor of irle re required to e overed. Trigoometri Futios Reltioship etwee trigoometri futios. Provig simple idetities. Sigs of trigoometri futios. Domi d rge of the trigoometri futios. Trigoometri futios of ll gles. Periods of trigoometri futios. Grphs of simple trigoometri futios (oly skethes). OTE: Grphs of si, os, t, se, ose d ot re to e iluded. Compoud d multiple gles Additio d sutrtio formul: si(a B); os(a B); t(a B); t(a + B + C) et., Doule gle, triple gle, hlf gle d oe third gle formul s speil ses. Sum d differees s produts sic + sid = C D si os C D, et. Produt to sum or differee i.e. siaosb = si(a + B) + si(a B) et. 0 (iv) Trigoometri Equtios 4. Clulus Solutio of trigoometri equtios (Geerl solutio d solutio i the speified rge). () Equtios i whih oly oe futio of sigle gle is ivolved e.g. si 5 =0 () Equtios epressile i terms of oe trigoometri rtio of the ukow gle. () Equtios ivolvig multiple d su- multiple gles. (d) Lier equtios of the form os + si =, where d, 0 (i) Bsi Coepts of Reltios d Futios () Ordered pirs, sets of ordered pirs. () Crtesi Produt (Cross) of two sets, rdil umer of ross produt. Reltios s: ssoitio etwee two sets. suset of Cross Produt. () Types of Reltios: refleive, symmetri, trsitive d equivlee reltio. (d) Biry Opertio. (e) Domi, Rge d Co-domi of Reltio. (f) Futios: As speil reltios, oept of writig y is futio of s y = f(). Types: oe to oe/ my to oe, ito/oto. Domi d rge of futio. Composite futio. Iverse of futio. Clssifitio of futios. Skethes of grphs of epoetil futio, logrithmi futio, mod futio, step futio.

5 (ii) Differetil lulus () Limits otio d meig of limits. Fudmetl theorems o limits (sttemet oly). Limits of lgeri d trigoometri futios. OTE: Idetermite forms re to e itrodued while lultig limits. () Cotiuity Cotiuity of futio t poit =. Cotiuity of futio i itervl. Removle disotiuity. () Differetitio Meig d geometril iterprettio of derivtive. Coept of otiuity d differetiility of, [], et. Derivtives of simple lgeri d trigoometri futios d their formule. OTE: Differetitio usig first priiples. Derivtives of sum/differee. Derivtives of produt of futios. Derivtives of quotiets of futios. Derivtives of omposite futios.. Derivtives of omposite futios usig hi rule.. All the futios ove should e either lgeri or trigoometri i ture. (d) Applitio of derivtives Equtio of Tget d orml pproimtio. Rte mesure. Sig of derivtive. Mootooity of futio. (iii) Itegrl Clulus Idefiite itegrl Itegrtio s the iverse of differetitio. Ati-derivtives of polyomils d futios ( +), si, os, se, ose. Itegrls of the type si, si, si 4, os, os, os Coordite Geometry (i) (ii) (iii) Bsi oepts of Poits d their oordites. The stright lie Slope d grdiet of lie. Agle etwee two lies. Coditio of perpediulrity d prllelism. Vrious forms of equtio of lies. Slope iterept form. Two poit slope form. Iterept form. Perpediulr /orml form. Geerl equtio of lie. Diste of poit from lie. Diste etwee prllel lies. Equtio of lies isetig the gle etwee two lies. Defiitio of lous. Methods to fid the equtio of lous. Cirles Equtios of irle i: Stdrd form. Dimeter form. Geerl form. Prmetri form. Give the equtio of irle, to fid the etre d the rdius.

6 6. Sttistis Fidig the equtio of irle. - Give three o ollier poits. - Give other suffiiet dt tht the etre is (h, k) d it lies o lie d two poits o the irle re give. Tgets: - Tget to irle whe the slope of the tget is give: y m m Itersetio: - Cirle with lie hee to fid the legth of the hord. Fidig the equtio of irle through the itersetio of two irles i.e. S + ks = 0. OTE: Orthogol irles re ot required to e overed. Mesures of etrl tedey. Stdrd devitio - y diret method, short ut method d step devitio method. OTE: 7. Vetors Comied me d stdrd devitio.. Comied me d stdrd devitio of two groups oly re required to e overed.. Me, Medi d Mode of grouped d ugrouped dt re required to e overed. SECTIO B As direted lie segmets. Mgitude d diretio of vetor. Types: equl vetors, uit vetors, zero vetor. Positio vetor. Compoets of vetor. Vetors i two d three dimesios. iˆ, ˆ, j kˆ s uit vetors log the, y d the z es; epressig vetor i terms of the uit vetors. Opertios: Sum d Differee of vetors; slr multiplitio of vetor. Setio formul. Simple questios sed o the ove e.g. A lie joiig the mid poit of y two sides of trigle is prllel to the third side d hlf of it, ourrey of medis. 8. Co-ordite geometry i -Dimesios As etesio of -D. Diste formul. Setio d midpoit formul. Equtio of -is, y-is, z is d lies prllel to them. Equtio of y - ple, yz ple, z ple. Diretio osies, diretio rtios. Agle etwee two lies i terms of diretio osies /diretio rtios. Coditio for lies to e perpediulr/ prllel. OTE: Uderstdig of dot produt of vetors is required. 9. Sttistis SECTIO C Medi - diret d y usig the formul. Qurtiles- diret d y usig the formul. Deiles- diret d y usig the formul. Peretiles - diret d y usig the formul. Mode - grphilly, diret method d y usig the formul. Estimtio of medi/qurtiles from Ogives. OTE: The followig re lso required to e overed: The Medi, Qurtiles, Deiles d Peretiles of grouped d ugrouped dt; Mode grouped d ugrouped dt; estimtio of mode y usig grphil method. (Bimodl dt ot iluded). 0. Averge Due Dte Zero dte. Equted periods.

7 CLASS XII The syllus is divided ito three setios A, B d C. Setio A is ompulsory for ll didtes. Cdidtes will hve hoie of ttemptig questios from either Setio B or Setio C. There will e oe pper of three hours durtio of 00 mrks. Setio A (80 mrks) will osist of ie questios. Cdidtes will e required to swer Questio (ompulsory) d five out of the rest of the eight questios. Setio B/C (0 mrks) Cdidtes will e required to swer two questios out of three from either Setio B or Setio C. SECTIO A. Determits d Mtries (i) Determits Order. Miors. Coftors. Epsio. Properties of determits. Simple prolems usig properties of determits e.g. evlute Crmer's Rule et. Solvig simulteous equtios i or vriles, D D y Dz, y, z D D D Cosistey, iosistey. Depedet or idepedet. OTE: the osistey oditio for three equtios i two vriles is required to e overed. (ii) Mtries Types of mtries (m ; m, ), order; Idetity mtri, Digol mtri. Symmetri, Skew symmetri. Opertio dditio, sutrtio, multiplitio of mtri with slr, multiplitio of two mtries (the omptiility). E.g. 0 AB( sy) ut BA is ot possile. Sigulr d o-sigulr mtries. Eistee of two o-zero mtries whose produt is zero mtri. Iverse (, ) A AdjA A Mrti s Rule (i.e. usig mtries) - + y + z = d. A + y + z = d. + y + z = d. AX = B X A d B d d B X y z - Simple prolems sed o ove. OTE: The oditios for osistey of equtios i two d three vriles, usig mtries, re to e overed. Boole Alger Boole lger s lgeri struture, priiple of dulity, Boole futio. Swithig iruits, pplitio of Boole lger to swithig iruits.

8 . Cois As setio of oe. Defiitio of Foi, Diretri, Ltus Retum. PS = epl where P is poit o the ois, S is the fous, PL is the perpediulr diste of the poit from the diretri. (i) Prol (ii) Ellipse e =, y = 4, = 4y, y = -4, = -4y, (y -) = 4 ( - ), ( - ) = 4 (y - ). Rough sketh of the ove. The ltus retum; qudrts they lie i; oordites of fous d verte; d equtios of diretri d the is. Fidig equtio of Prol whe Foi d diretri re give. Simple d diret questios sed o the ove. y, e, ( e ) Cses whe > d <. Rough sketh of the ove. Mjor is, mior is; ltus retum; oordites of verties, fous d etre; d equtios of diretries d the es. Fidig equtio of ellipse whe fous d diretri re give. Simple d diret questios sed o the ove. Fol property i.e. SP + SP =. (iii) Hyperol y,,( ) e e Cses whe oeffiiet y is egtive d oeffiiet of is egtive. Rough sketh of the ove. Fol property i.e. SP - S P =. Trsverse d Cojugte es; Ltus retum; oordites of verties, foi d etre; d equtios of the diretries d the es. Geerl seod degree equtio hy y g fy 0 represets prol if h =, ellipse if h <, d hyperol if h >. Coditio tht y = m + is tget to the ois. 4. Iverse Trigoometri Futio Priipl vlues. si -, os -, t - et. d their grphs. si - = os t si - = ose ; si - + os - = d similr reltios for ot -, t -, et. 5. Clulus Additio formule si si ysi y y os os yos y y y similrly t t y t, y y Similrly, estlish formule for si -, os -, t -, t - et. usig the ove formul. Applitio of these formule. (i) Differetil Clulus Revisio of topis doe i Clss XI - mily the differetitio of produt of two futios, quotiet rule, et. Derivtives of trigoometri futios. Derivtives of epoetil futios. Derivtives of logrithmi futios. Derivtives of iverse trigoometri futios - differetitio y mes of sustitutio. Derivtives of impliit futios d hi rule for omposite futios.. 4

9 (ii) Derivtives of Prmetri futios. Differetitio of futio with respet to other futio e.g. differetitio of si with respet to. Logrithmi Differetitio - Fidig dy/d whe y =. Suessive differetitio up to d order. L'Hospitl's theorem. 0 0 form, 0 form, 0 form, form et. Rolle's Me Vlue Theorem - its geometril iterprettio. Lgrge's Me Vlue Theorem - its geometril iterprettio. Mim d miim. Itegrl Clulus Revisio of formule from Clss XI. Itegrtio of /, e. Itegrtio y simple sustitutio. Itegrls of the type f' ()[f ( )], f (). f () Itegrtio of /, e, t, ot, se, ose. Itegrtio y prts. Itegrtio y mes of sustitutio. Itegrtio usig prtil frtios, f ( ) Epressios of the form g( ) degree of f() < degree of g() E.g. A B ( )( ) whe A B C ( )( ) ( )( ) A B C 5 Whe degree of f () degree of g(), e.g.. Itegrls of the type: d d p q p q,,, d d d epressios reduile to this form. Itegrls of the form: d, d d, os si os si d, 4 d, t d,, ot d 4. Properties of defiite itegrls. Prolems sed o the followig properties of defiite itegrls re to e overed. ( ) d f f ( t) dt ( ) d f f ( ) d f ( ) d f ( ) d f ( ) d where < < f ( ) d f ( ) d 0 0 f ()() d f d () f,()() d if f f f () d 0 0 0,()() f f () f,if d is f eve futio f () d 0 0,if f is odd futio

10 Applitio of defiite itegrls - re ouded y urves, lies d oordite es is required to e overed. 6. Correltio d Regressio Defiitio d meig of orreltio d regressio oeffiiet. Coeffiiet of Correltio y Krl Perso. r If -, y - y re smll o - frtiol umers, we use - y - y - y - y If d y re smll umers, we use r y y y y Otherwise, we use ssumed mes A d B, where u = -A, v = y-b r uv - u v u u v v Rk orreltio y Sperm s (Corretio iluded). Lies of regressio of o y d y o. OTE: Stter digrms d the followig topis o regressio re required. i) The method of lest squres. ii) Lies of est fit. iii)regressio oeffiiet of o y d y o. iv) = r, 0 y y y y v) Idetifitio of regressio equtios 7. Proility Rdom eperimets d their outomes. Evets: sure evets, impossile evets, mutully elusive evets, idepedet evets d depedet evets. Defiitio of proility of evet. Lws of proility: dditio d multiplitio lws, oditiol proility (eludig Bye s theorem). 8. Comple umers Argumet d ojugte of omple umers. Sum, differee, produt d quotiet of two omple umers dditive d multiplitive iverse of omple umer. Simple lous questio o omple umer; provig d usig - z. z z ; z z z z d z z z z Trigle iequlity. Squre root of omple umer. Demoivre s theorem d its simple pplitios. Cue roots of uity: prolems. 9. Differetil Equtios,, ; pplitio Differetil equtios, order d degree. Solutio of differetil equtios. Vrile seprle. Homogeeous equtios d equtios reduile to homogeeous form. dy Lier form Py Q where P d Q re d futios of oly. Similrly for d/dy. OTE: Equtios reduile to vrile seprle type re iluded. The seod order differetil equtios re eluded. 0. Vetors SECTIO B Slr (dot) produt of vetors. Cross produt - its properties - re of trigle, ollier vetors. Slr triple produt - volume of prllelopiped, o-plrity. 6

11 Proof of Formule (Usig Vetors) Sie rule. Cosie rule Projetio formul Are of Δ = ½siC OTE: Simple geometri pplitios of the ove re required to e overed.. Co-ordite geometry i -Dimesios (i) Lies Crtesi d vetor equtios of lie through oe d two poits. Coplr d skew lies. Coditios for itersetio of two lies. Shortest diste etwee two lies. OTE: Symmetri d o-symmetri forms of lies re required to e overed. (ii) Ples. Proility Crtesi d vetor equtio of ple. Diretio rtios of the orml to the ple. Oe poit form. orml form. Iterept form. Diste of poit from ple. Agle etwee two ples, lie d ple. Equtio of ple through the itersetio of two ples i.e. - P + kp = 0. Simple questios sed o the ove. Bye s theorem; theoretil proility distriutio, proility distriutio futio; iomil distriutio its me d vrie. OTE: Theoretil proility distriutio is to e limited to iomil distriutio oly.. Disout SECTIO C True disout; ker's disout; disouted vlue; preset vlue; sh disout, ill of ehge. OTE: Bker s gi is required to e overed. 4. Auities Meig, formule for preset vlue d mout; deferred uity, pplied prolems o los, sikig fuds, sholrships. OTE: Auity due is required to e overed. 5. Lier Progrmmig Itrodutio, defiitio of relted termiology suh s ostrits, ojetive futio, optimiztio, isoprofit, isoost lies; dvtges of lier progrmmig; limittios of lier progrmmig; pplitio res of lier progrmmig; differet types of lier progrmmig (L.P.), prolems, mthemtil formultio of L.P prolems, grphil method of solutio for prolems i two vriles, fesile d ifesile regios, fesile d ifesile solutios, optimum fesile solutio. 6. Applitio of derivtives i Commere d Eoomis i the followig Cost futio, verge ost, mrgil ost, reveue futio d rek eve poit. 7. Ide umers d movig verges Prie ide or prie reltive. Simple ggregte method. Weighted ggregte method. Simple verge of prie reltives. Weighted verge of prie reltives (ost of livig ide, osumer prie ide). OTE: Uder movig verges the followig re required to e overed: Meig d purpose of the movig verges. Clultio of movig verges with the give periodiity d plottig them o grph. If the period is eve, the the etered movig verge is to e foud out d plotted. 7