CLASS XII. AdjA A. x X. a etc. B d

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1 CLASS XII The syllus is divided into three setions A B nd C Setion A is ompulsory for ll ndidtes Cndidtes will hve hoie of ttempting questions from either Setion B or Setion C There will e one pper of three hours durtion of 00 mrks Setion A (80 mrks) will onsist of nine questions Cndidtes will e required to nswer Question (ompulsory) nd five out of the rest of the eight questions Setion B/C (0 mrks) Cndidtes will e required to nswer two questions out of three from either Setion B or Setion C SECTIO A Determinnts nd Mtries (i) Determinnts Order Minors Coftors Epnsion Properties of determinnts Simple prolems using properties of determinnts eg evlute Crmer's Rule et Solving simultneous equtions in or vriles D D y Dz y z D D D Consisteny inonsisteny Dependent or independent OTE: the onsisteny ondition for three equtions in two vriles is required to e overed (ii) Mtries Types of mtries (m n; m n ) order; Identity mtri Digonl mtri Symmetri Skew symmetri Opertion ddition sutrtion multiplition of mtri with slr multiplition of two mtries (the omptiility) Eg 0 AB( sy) ut BA is not possile Singulr nd non-singulr mtries Eistene of two non-zero mtries whose produt is zero mtri Inverse ( ) A AdjA A Mrtin s Rule (ie using 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 on ove OTE: The onditions for onsisteny of equtions in two nd three vriles using mtries re to e overed Boolen Alger Boolen lger s n lgeri struture priniple of dulity Boolen funtion Swithing iruits pplition of Boolen lger to swithing iruits 5

2 Conis As setion of one Definition of Foi Diretri Ltus Retum PS = epl where P is point on the onis S is the fous PL is the perpendiulr distne of the point 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; qudrnts they lie in; oordintes of fous nd verte; nd equtions of diretri nd the is Finding eqution of Prol when Foi nd diretri re given Simple nd diret questions sed on the ove y e ( e ) Cses when > nd < Rough sketh of the ove Mjor is minor is; ltus retum; oordintes of verties fous nd entre; nd equtions of diretries nd the es Finding eqution of ellipse when fous nd diretri re given Simple nd diret questions sed on the ove Fol property ie SP + SP = (iii) Hyperol y ( ) e e Cses when oeffiient y is negtive nd oeffiient of is negtive Rough sketh of the ove Fol property ie SP - S P = Trnsverse nd Conjugte es; Ltus retum; oordintes of verties foi nd entre; nd equtions of the diretries nd the es Generl seond degree eqution hy y g fy 0 represents prol if h = ellipse if h < nd hyperol if h > Condition tht y = m + is tngent to the onis 4 Inverse Trigonometri Funtion Prinipl vlues sin - os - tn - et nd their grphs sin - = os tn sin - = ose ; sin - + os - = nd similr reltions for ot - tn - et 5 Clulus Addition formule sin sin ysin y y os os yos y y y similrly t n tn y tn y y Similrly estlish formule for sin - os - tn - tn - et using the ove formul Applition of these formule (i) Differentil Clulus Revision of topis done in Clss XI - minly the differentition of produt of two funtions quotient rule et Derivtives of trigonometri funtions Derivtives of eponentil funtions Derivtives of logrithmi funtions Derivtives of inverse trigonometri funtions - differentition y mens of sustitution Derivtives of impliit funtions nd hin rule for omposite funtions 6

3 (ii) Derivtives of Prmetri funtions Differentition of funtion with respet to nother funtion eg differentition of sin with respet to Logrithmi Differentition - Finding dy/d when y = Suessive differentition up to nd order L'Hospitl's theorem 0 0 form 0 form 0 form form et Rolle's Men Vlue Theorem - its geometril interprettion Lgrnge's Men Vlue Theorem - its geometril interprettion Mim nd minim Integrl Clulus Revision of formule from Clss XI Integrtion of / e Integrtion y simple sustitution Integrls of the type f' ()[f ( )] n f () f () Integrtion of / e tn ot se ose Integrtion y prts Integrtion y mens of sustitution Integrtion using prtil frtions f ( ) Epressions of the form g( ) degree of f() < degree of g() Eg A B ( )( ) when A B C ( )( ) ( )( ) A B C 7 When degree of f () degree of g() eg Integrls of the type: d d p q p q d d nd epressions reduile to this form Integrls of the form: d d d os sin os sin d 4 d tn d ot d 4 Properties of definite integrls Prolems sed on the following properties of definite integrls 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 n f even funtion f () d 0 0if f is n odd funtion

4 Applition of definite integrls - re ounded y urves lines nd oordinte es is required to e overed 6 Correltion nd Regression Definition nd mening of orreltion nd regression oeffiient Coeffiient of Correltion y Krl Person r If - y - y re smll non - frtionl numers we use - y - y - y - y If nd y re smll numers we use r y y y y Otherwise we use ssumed mens A nd B where u = -A v = y-b r uv - u v u u v v Rnk orreltion y Spermn s (Corretion inluded) Lines of regression of on y nd y on OTE: Stter digrms nd the following topis on regression re required i) The method of lest squres ii) Lines of est fit iii)regression oeffiient of on y nd y on iv) = r 0 y y y y v) Identifition of regression equtions 7 Proility Rndom eperiments nd their outomes Events: sure events impossile events mutully elusive events independent events nd dependent events Definition of proility of n event Lws of proility: ddition nd multiplition lws onditionl proility (eluding Bye s theorem) 8 Comple umers Argument nd onjugte of omple numers Sum differene produt nd quotient of two omple numers dditive nd multiplitive inverse of omple numer Simple lous question on omple numer; proving nd using - z z z ; z z z z nd z z z z Tringle inequlity Squre root of omple numer Demoivre s theorem nd its simple pplitions Cue roots of unity: prolems 9 Differentil Equtions ; pplition Differentil equtions order nd degree Solution of differentil equtions Vrile seprle Homogeneous equtions nd equtions reduile to homogeneous form dy Liner form Py Q where P nd Q re d funtions of only Similrly for d/dy OTE: Equtions reduile to vrile seprle type re inluded The seond order differentil equtions re eluded 0 Vetors SECTIO B Slr (dot) produt of vetors Cross produt - its properties - re of tringle olliner vetors Slr triple produt - volume of prllelopiped o-plnrity 8

5 Proof of Formule (Using Vetors) Sine rule Cosine rule Projetion formul Are of Δ = ½sinC OTE: Simple geometri pplitions of the ove re required to e overed Co-ordinte geometry in -Dimensions (i) Lines Crtesin nd vetor equtions of line through one nd two points Coplnr nd skew lines Conditions for intersetion of two lines Shortest distne etween two lines OTE: Symmetri nd non-symmetri forms of lines re required to e overed (ii) Plnes Proility Crtesin nd vetor eqution of plne Diretion rtios of the norml to the plne One point form orml form Interept form Distne of point from plne Angle etween two plnes line nd plne Eqution of plne through the intersetion of two plnes ie - P + kp = 0 Simple questions sed on the ove Bye s theorem; theoretil proility distriution proility distriution funtion; inomil distriution its men nd vrine OTE: Theoretil proility distriution is to e limited to inomil distriution only Disount SECTIO C True disount; nker's disount; disounted vlue; present vlue; sh disount ill of ehnge OTE: Bnker s gin is required to e overed 4 Annuities Mening formule for present vlue nd mount; deferred nnuity pplied prolems on lons sinking funds sholrships OTE: Annuity due is required to e overed 5 Liner Progrmming Introdution definition of relted terminology suh s onstrints ojetive funtion optimiztion isoprofit isoost lines; dvntges of liner progrmming; limittions of liner progrmming; pplition res of liner progrmming; different types of liner progrmming (LP) prolems mthemtil formultion of LP prolems grphil method of solution for prolems in two vriles fesile nd infesile regions fesile nd infesile solutions optimum fesile solution 6 Applition of derivtives in Commere nd Eonomis in the following Cost funtion verge ost mrginl ost revenue funtion nd rek even point 7 Inde numers nd moving verges Prie inde or prie reltive Simple ggregte method Weighted ggregte method Simple verge of prie reltives Weighted verge of prie reltives (ost of living inde onsumer prie inde) OTE: Under moving verges the following re required to e overed: Mening nd purpose of the moving verges Clultion of moving verges with the given periodiity nd plotting them on grph If the period is even then the entered moving verge is to e found out nd plotted 9