SUBJECT: MATHEMATICS CLASS :XII

Transcription

1 SUBJECT: MATHEMATICS CLASS :XII KENDRIYA VIDYALAYA SANGATHAN REGIONAL OFFICE CHANDIGARH YEAR 0-0

2 INDEX Sl. No Topis Pge No.. Detil of the onepts 4. Reltions & Funtions 9. Inverse Trigonometri Funtions 5 4. Mtries & Determinnts 9 5. Continuit& Differentiilit 7 6. Applition of derivtive 7. Indefinite Integrls 9 8. Applitions of Integrtion Differentil Equtions Vetor Alger 54. Three Dimensionl Geometr 6. Liner Progrmming 69. Proilit 7 4. Answers Biliogrph 99

3 Topi wise Anlsis of Emples nd Questions NCERT TEXT BOOK Chpters Conepts Numer of Questions for revision Totl Questions From Solved Emples Questions From Eerise 0 Reltions & Funtions Inverse Trigonometri Funtions Mtries & Determinnts Continuit& Differentiilit Applition of Derivtive Indefinite Integrls Applitions of Integrtion Differentil Equtions Vetor Alger Three Dimensionl Geometr 07 9 Liner Progrmming 09 Proilit TOTAL 6 47

4 Detil of the onepts to e mstered ever hild of lss XII with eerises nd emples of NCERT Tet Book. SYMBOLS USED * : Importnt Questions, ** :Ver Importnt Questions, *** : Ver-Ver Importnt Questions S.No Topi Conepts Degree of importne Refrenes NCERT Tet Book XII Ed. 007 Reltions & Funtions Inverse Trigonometri Funtions Mtries & Determinnts 4 Continuit& Differentiilit (i).domin, Co-domin & * (Previous Knowledge) Rnge of reltion (ii).tpes of reltions *** E. Q.No- 5,9, (iii).one-one, onto & inverse of *** E. Q.No- 7,9 funtion (iv).composition of funtion * E. QNo- 7,9, (v).binr Opertions *** Emple 45 E.4 QNo- 5, (i).prinipl vlue rnh Tle ** E. QNo-, 4 (ii). Properties of Inverse *** E. QNo- 7,, 5 Trigonometri Funtions Mis E Q.No.9,0,, (i) Order, Addition, *** E. Q.No 4,6 Multiplition nd trnspose of E. Q.No 7,9,,7,8 mtries E. Q.No 0 (ii) Coftors &Adjoint of mtri (iii)inverse of mtri & pplitions (iv)to find differene etween A, dj A, ka, A.djA 4 ** E 4.4 Q.No 5 E 4.5 Q.No,,7,8 *** E 4.6 Q.No 5,6 Emple 9,0,, MisE 4,Q.No4,5,8,,5 * E 4. Q.No,4,7,8 (v) Properties of Determinnts ** E 4. Q.No,, Emple 6,8 (i).limit of funtion * (ii).continuit *** E 5. Q.No-, 6,0 (iii).differentition * E 5. Q.No- 6 E 5. Q.No- 4,7, (iv).logrithmi Differentition *** E 5.5 Q.No- 6,9,0,5 (v) Prmetri Differentition *** E 5.6 Q.No- 7,8,0, (vi). Seond order derivtives *** E 5.7 Q.No- 4,6,7

5 5 Applition of Derivtive. 6 Indefinite Integrls (vii). M. V.Th ** E 5.8 Q.No-,4 (i).rte of hnge * Emple 5E 6. Q.No- 9, (ii).inresing & deresing *** E 6.,Q.No- 6 Emple, funtions (iii).tngents & norml ** E 6.,Q.No- 5,8,,5, (iv).approimtions * E 6.4,Q.No-, (v) Mim & Minim *** E 6.5, Q.No- 8,,,5 Emple 5,6,7 (i) Integrtion sustitution * Ep 5&6 Pge0,0 (ii) Applition of trigonometri ** E 7 Pge 06, Eerise funtion in integrls 7.4Q&Q4 (iii) Integrtion of some prtiulr funtion d d,, d d,, d, (p q)d, (p q)d *** Edition Ep 8, 9, 0 Pge,Eerise 7.4 Q,4,8,9,& (iv) Integrtion using Prtil *** EditionEp & Pge 8 Frtion Ep 9,Ep 4 & 5 Pge0 (v) Integrtion Prts ** Ep 8,9&0 Pge 5 Definite Integrls (vi)some Speil Integrls *** Ep &4 Pge 9 d, d (vii) Misellneous Questions *** Solved E.4 (i) Definite integrls s limit ** Ep 5 &6 Pge, 4 of sum Q, Q5 & Q6 Eerise 7.8 () Properties of definite *** Ep Pge 4*,Ep Integrls *,4&5 pge 44 Ep 6*Ep 46 Ep 44 pge5 Eerise 7. Q7 & (i) Integrtion of modulus ** Ep 0 Pge 4,Ep 4 Pge 5

6 7 Applitions of Integrtion 8. Differentil Equtions 9. Vetor Alger funtion 5 Q5& Q6 Eerise 7. (i)are under Simple Curves * E.8. Q.,,5 (ii) Are of the region enlosed *** E. 8. Q 0, Mis.E. Q 7 etween Prol nd line (iii) Are of the region enlosed *** Emple 8, pge 69Mis.E. etween Ellipse nd line 8 (iv) Are of the region enlosed *** E. 8. Q 6 etween Cirle nd line (v) Are of the region enlosed *** E 8. Q, Mis.E.Q 5 etween Cirle nd prol (vi) Are of the region enlosed etween Two Cirles *** Emple 0, pge70e 8. Q (vii) Are of the region *** Emple 6, pge6 enlosed etween Two prols (viii) Are of tringle when *** Emple 9, pge70e 8. Q4 verties re given (i) Are of tringle when sides *** E 8. Q5,Mis.E. Q 4 re given () Misellneous Questions *** Emple 0, pge74mis.e.q 4, (i) Order nd degree of *** Q.,5,6 pg 8 differentil eqution.generl nd prtiulr ** E., pg84 solutions of differentil eqution.formtion of differentil * Q. 7,8,0 pg 9 eqution whose generl solution is given 4.Solution of differentil * Q.4,6,0 pg 96 eqution the method of seprtion of vriles 5.Homogeneous differentil ** Q.,6, pg 406 eqution of first order nd first degree Solution of differentil eqution *** Q.4,5,0,4 pg 4,44 of the tpe d/d +p=q where p nd q re funtions of And solution of differentil eqution of the tpe d/d+p=q where p nd q re funtions of (i)vetor nd slrs * Q pg 48 (ii)diretion rtio nd diretion * Q, pg 440 6

7 0 Three Dimensionl Geometr Liner Progrmmin g osines (iii)unit vetor * * E 6,8 Pg 46 (iv)position vetor of point nd olliner vetors * * Q 5 Pg 440 Q Pg440 Q 6 Pg448 (v)dot produt of two vetors ** Q6, Pg445 (vi)projetion of vetor * * * E 6 Pg 445 (vii)cross produt of two * * Q Pg458 vetors (viii)are of tringle * Q 9 Pg 454 (i)are of prllelogrm * Q 0 Pg 455 (i)diretion Rtios nd * E No Pg -466 Diretion Cosines E No 5 Pg 467 E No 4 Pg (ii)crtesin nd Vetor eqution of line in spe & onversion of one into nother form ** E No 8 Pg -470 Q N. 6, 7, - Pg 477 QN 9 Pg 478 (iii) Co-plner nd skew lines * E No 9 Pg -496 (iv)shortest distne etween two lines *** E No Pg -476 Q N. 6, 7 - Pg 478 (v)crtesin nd Vetor eqution of plne in spe & onversion of one into nother form ** E No 7 Pg -48 E No 8 Pg 484 E No 9 Pg 485 E No 7 Pg 495 (vi)angle Between (i) Two lines (ii) Two plnes (iii) Line & plne (vii)distne of point from plne (viii)distne mesures prllel to plne nd prllel to line (i)eqution of plne through the intersetion of two plnes ()Foot of perpendiulr nd imge with respet to line nd plne (i) LPP nd its Mthemtil Formultion (ii) Grphil method of solving LPP (ounded nd unounded solutions) * * ** Q N. 9, 0 - Pg 499 E No 9 Pg -47 Q N. - Pg 478 E No 6 Pg 494 Q N. - Pg 494 E No 5 Pg - 49 ** Q No 8 Pg -499 Q No 4 Pg 494 ** *** Q No 0 Pg -49 ** E. N 6 Pg 48 ** Artiles. nd.. ** Artile.. Solved Emples to 5 Q. Nos 5 to 8 E.. 7

8 (iii) Tpes of prolems () Diet Prolem *** Q. Nos, nd 9 E.. Solved Emple 9 Q. Nos nd Mis. E. () Mnufturing Prolem *** Solved Emple 8 Q. Nos,4,5,6,7 of E.. Solved Emple0 Q. Nos 4 & 0 Mis. E. () Allotion Prolem ** Solved Emple 7 Q. No 0 E.., Q. No 5 & 8 Mis. E. (d) Trnsporttion Prolem * Solved Emple Q. Nos 6 & 7 Mis. E. (e) Misellneous Prolems ** Q. No 8 E.. Proilit (i) Conditionl Proilit *** Artile. nd.. Solved Emples to 6 Q. Nos nd 5 to 5 E.. (ii)multiplition theorem on proilit ** Artile. SolvedEmples 8 & 9 Q. Nos,, 4 & 6 E.. (iii) Independent Events *** Artile.4 Solved Emples 0 to 4 Q. Nos, 6, 7, 8 nd E.. (iv) Be s theorem, prtition of smple spe nd Theorem of totl proilit (v) Rndom vriles & proilit distriution Men & vrine of rndom vriles (vi) Bernoulli,s trils nd Binomil Distriution *** Artiles.5,.5.,.5. Solved Emples 5 to, & 7,Q. Nos to E.. Q. Nos & 6 Mis. E. *** Artiles.6,.6.,.6. &.6. Solved Emples 4 to 9 Q. Nos & 4 to 5 E..4 *** Artiles.7,.7. &.7. Solved Emples & Q. Nos to E..5 8

9 TOPIC RELATIONS & FUNCTIONS SCHEMATIC DIAGRAM Topi Conepts Degree of importne Referenes NCERT Tet Book XII Ed. 007 Reltions & (i).domin, Co domin & * (Previous Knowledge) Funtions Rnge of reltion (ii).tpes of reltions *** E. Q.No- 5,9, (iii).one-one, onto & inverse *** E. Q.No- 7,9 of funtion (iv).composition of funtion * E. QNo- 7,9, (v).binr Opertions *** Emple 45 E.4 QNo- 5, SOME IMPORTANT RESULTS/CONCEPTS ** A reltion R in set A is lled (i) refleive, if (, ) R, for ever A, (ii) smmetri, if (, ) R implies tht (, ) R, for ll, A. (iii)trnsitive, if (, ) R nd (, ) R implies tht (, ) R, for ll,, A. ** Equivlene Reltion : R is equivlene if it is refleive, smmetri nd trnsitive. ** Funtion :A reltion f : A B is sid to e funtion if ever element of A is orreltedto unique element in B. * A is domin * B is odomin * For n element A, funtion f orreltes it to n element in B, whih is denoted f()nd is lled imge of under f. Agin if = f(), then is lled s pre-imge of. * Rnge = {f() A }. Rnge Codomin * The lrgest possile domin of funtion is lled domin of definition. **Composite funtion : Let two funtions e defined s f : A B nd g : B C. Then we n define funtion : A C setting () = g{f()} where A, f () B, g{f()} C. This funtion : A C is lled the omposite funtion of f nd g in tht order nd we write. = gof. A B C () f g f() g{f()} 9

10 ** Different tpe of funtions : Let f : A B e funtion. * f is one to one (injetive) mpping, if n two different elements in A is lws orrelted to different elements in B, i.e. f( ) f( )or, f( ) = f( ) = * f is mn one mpping, if t lest two elements in A suh tht their imges re sme. * f is onto mpping (sujetive), if eh element in B is hving t lest one preimge. *f is into mpping if rnge odomin. * f is ijetive mpping if it is oth one to one nd onto. ** Binr opertion : A inr opertion * on set A is funtion * : A A A. We denote *(, ) *. * A inr opertion * on A is rule tht ssoites with ever ordered pir (, ) of A A unique element *. * An opertion * on is sid to e ommuttive iff * = *, A. * An opertion * on is sid to e ssoitive iff ( * ) * = * ( * ),, A. * Given inr opertion * : A A A, n element e A, if it eists, is lled identit for the opertion *, if *e = = e *, A. * Given inr opertion * : A A A with the identit element e in A, n element A is sid to e invertile with respet to the opertion*, if there eists n element in A suh tht = e = nd is lled the inverse of nd is denoted. ASSIGNMENTS (i) Domin, Co domin & Rnge of reltion. If A = {,,,4,5}, write the reltion R suh tht + = 8,, A. Write the domin, rnge & o-domin.. Define reltion R on the set N of nturl numers R={(, ) : = +7, is nturl numer lesst hn 4 ;, N}. Write down the domin nd the rnge.. Tpes of reltions. Let R e the reltion in the set N given R = {(, ) =, > 6} Whether the reltion is refleive or not?justif our nswer.. Show tht the reltion R in the set N given R = {(, ) is divisile,, N} is refleive nd trnsitive ut not smmetri.. Let R e the reltion in the set N given R = {(,) > } Show tht the reltion is neither refleive nor smmetri ut trnsitive. 4. Let R e the reltion on R defined s (, ) R iff + > 0,R. () Show tht R is smmetri. () Show tht R is refleive. () Show tht R is not trnsitive. 5. Chek whether the reltion R is refleive, smmetri nd trnsitive. R = { (, ) = 0} on A ={,,., 4}. 0

11 I. Show tht the reltion R on A,A = { Z, 0 }, R = {(,): - is multiple of.} is n equivlene reltion..let N e the set of ll nturl numers & R e the reltion on N N defined { (, ) R (, d) iff + d = + }. Show tht R is n equivlene reltion.. Show tht the reltion R in the set A of ll polgons s: R ={(P,P ), P & P hve the sme numer of sides} is n equivlene reltion. Wht is the set of ll elements in A relted to the right tringle T with sides,4 & 5? 4. Show tht the reltion R on A,A = { Z, 0 }, R = {(,): - is multiple of.} is n equivlene reltion. 5. Let N e the set of ll nturl numers & R e the reltion on N N defined { (, ) R (,d) iff + d = + }. Show tht R is n equivlene reltion. [CBSE 00] 6. Let A = Set of ll tringles in plne nd R is defined R={(T,T ) : T,T A & T ~T } Show tht the R is equivlene reltion. Consider the right ngled s, T with size,4,5; T with size 5,,; T with side 6,8,0; Whih of the pirs re relted? (iii)one-one, onto & inverse of funtion. If f() =, then find f(/). Show tht the funtion f: RR defined f()= is neither one-one nor onto. Show tht the funtion f: NN given f()= is one-one ut not onto., if 0 4 Show tht the signum funtion f: RR given : f () 0, if 0, if 0 is neither one-one nor onto. 5 Let A = {-,0,} nd B = {0,}. Stte whether the funtion f : A B defined f() = isijetive. 6. Let f() =, -,then find f - (). Let A = {,,}, B = {4,5,6,7} nd let f = {(,4),(,5), (,6)} e funtion from A to B. Stte whether f is one-one or not. [CBSE0] 7. If f : RR defined s f() = is n invertile funtion. Find f - (). 4. Write the numer of ll one-one funtions on the set A={,, } to itself. 4. Show tht funtion f :RR defined f()= 7 for ll R is ijetive If f: RR is defined f()=. Find f -.

12 I. Show tht the funtion f: RR defined f() =. R is one- one & onto funtion. Also find the f -.. Consider funtion f :R + [-5, ) defined f() = Show tht f is invertile & f - 6 () =, where R + = (0, ).. Consider funtion f: RR given f() = 4 +. Show tht f is invertile & f - : RR with f - ()=. 4. Show tht f: RR defined f()= +4 is one-one, onto. Show tht f - ()=( 4) /. 5. Let A R {} nd B R {}. Consider the funtion f : A Bdefined f (). Show tht f is one one onto nd hene find 6. Show tht f : N N defined (iv) Composition of funtions f. [CBSE0], if is odd f () is oth one one onto., if is even [CBSE0]. If f() = e nd g() = log, > 0, find () (f + g)() () (f.g)() () f o g ( ) (d) g o f ( ).. If f() =, then show tht () f = f() () f =. Let f, g : RR e defined f()= & g() = [] where [] denotes the gretest integer funtion. Find f o g ( 5/ ) & g o f (- ).. Let f() =. Then find f(f()) 4. If = f() =, then find (fof)() i.e. f() 5 4. Let f : R R e defined s f() = 0 +7.Find the funtion g : R Rsuh tht g f ()= f g() = I R [CBSE0] 5. If f : R R e defined s f() =, then find f f(). [CBSE00] 6. Let f :RR& g : RR e defined s f() =, g() =. Find fog(). f ()

13 (v)binr Opertions. Let * e the inr opertion on N given * = LCM of &. Find *5.. Let *e the inr on N given * =HCF of {,},,N. Find 0*6.. Let * e inr opertion on the set Q of rtionl numers defined s * =. 5 Write the identit of *, if n. 4. If inr opertion * on the set of integer Z, is defined * = + Then find the vlue of * 4. LEVEL. Let A= N N & * e the inr opertion on A defined (,) * (,d) = (+, +d ) Show tht * is () Commuttive () Assoitive () Find identit for * on A, if n.. Let A = Q Q. Let * e inr opertion on A defined (,)*(,d)= (, d+). Find: (i) the identit element of A (ii) the invertile element of A.. Emine whih of the following is inr opertion (i) * = ;, N (ii) * =, Q For inr opertion hek ommuttive & ssoitive lw. LEVEL.Let A= N N & * e inr opertion on A defined (, ) (, d) = (, d) (, ),(, d) N N (i) Find (,) * (4,) (ii) Find [(,)*(4,)]*(,5) nd (,)*[(4,)* (,5)] & show the re equl (iii) Show tht * is ommuttive & ssoitive on A., if 6. Define inr opertion * on the set {0,,,,4,5} s * = 6, 6 Show tht zero in the identit for this opertion & eh element of the set is invertile with 6 eing the inverse of. [CBSE0]. Consider the inr opertions :R R Rnd o : R R R defined s = nd o =,, R. Show tht is ommuttive ut not ssoitive, o is ssoitive ut not ommuttive. [CBSE0] Questions for self evlution. Show tht the reltion R in the set A = {,,, 4, 5} given R = {(, ) : is even}, is n equivlene reltion. Show tht ll the elements of {,, 5} re relted to eh other nd ll the elements of {, 4} re relted to eh other. But no element of {,, 5} is relted to n element of {, 4}.. Show tht eh of the reltion R in the set A = { Z : 0 }, given R = {(, ) : is multiple of 4} is n equivlene reltion. Find the set of ll elements relted to.

14 . Show tht the reltion R defined in the set A of ll tringles s R = {(T, T ) : T is similr to T }, is equivlene reltion. Consider three right ngle tringles T with sides, 4, 5, T with sides 5,, nd T with sides 6, 8, 0. Whih tringles mong T, T nd T re relted? 4. If R nd R re equivlene reltions in set A, show tht R R is lso n equivlene reltion. 5. Let A = R {} nd B = R {}. Consider the funtion f : A B defined f () =. Is f one-one nd onto? Justif our nswer. 6. Consider f :R+ [ 5, ) given f () = Show tht f is invertile nd findf. 7. On R {} inr opertion * is defined s * = +. Prove tht * is ommuttive nd ssoitive. Find the identit element for *.Also prove tht ever element of R {} is invertile. 8. If A = Q Q nd * e inr opertion defined (, ) * (, d) = (, + d), for (, ), (, d) A.Then with respet to * on A (i) emine whether * is ommuttive & ssoitive (i) find the identit element in A, (ii) find the invertile elements of A. 4

15 TOPIC INVERSE TRIGONOMETRIC FUNCTIONS SCHEMATIC DIAGRAM Topi Conepts Degree of importne Referenes NCERT Tet Book XI Ed. 007 Inverse (i).prinipl vlue rnh ** E. QNo-, 4 Trigonometri Funtions Tle (ii). Properties of Inverse Trigonometri Funtions *** E. Q No- 7,, 5 Mis E Q.No. 9,0,, * Domin & Rnge of i. ii. iii. iv. v. vi. SOME IMPORTANT RESULTS/CONCEPTS the Inverse Trigonometri Funtion : Funtions sin os se tn ot ose : : : : : : Domin Rnge, /, /, 0, R, /, / 0 R, 0, / R /, / R 0, Prinipl vlue Brnh * Properties of Inverse Trigonomet ri i sin sin & sin sin ii.os os & os os iii.tn tn & tn tn iv.ot ot & ot ot v. se se & se se vi.os e os e & os e ose iii. tn i i. i ii. tn iii. os e iii. sin sin os e ot & ot sin iv os tn v se os e vi ot sin os os e se & sin os e tn Funtion ii. os ii. tn se ot & os se ot se os 5

16 5. tn 6. tn tn tn tn tn tn tn tn tn tn os if if if sin ASSIGNMENTS (i). Prinipl vlue rnh Tle Write the prinipl vlue of the following :.os.tn Write the prinipl vlue of the following :.sin 4.os π π 4π. os os sin sin [CBSE 0]. sin sin 5 7. os os 6 (ii). Properties of Inverse Trigonometri Funtions. Evlute ot[tn ot ].Prove sin sin 4 π. Find ifse os e. Write the following in simplest form : tn, 0 6

17 8 77. Prove tht sin sin tn Prove tht tn tn tn tn Prove tht tn tn tn Prove thtsin sin os π 4. [CBSE 0] [CBSE 0] I sin sin. Prove tht ot, 0, sin sin 4. Prove tht tn os 4. Solve tn tn π / 4 4. Solve tn tn tn π 5.Solve tn tn 4 8 [CBSE 0] os 6. Prove tht tn,, [CBSE 0] sin 4. Prove tht sin 5 os Questions for self evlution tn 5. Prove tht tn os,, Prove tht sin os tn Prove tht tn tn tn 5. Prove tht tn tn 4 6. Write in the simplest form os tn 7

18 7. Solve tn tn 4 8. Solve tn tn / 4 8

19 TOPIC MATRICES & DETERMINANTS SCHEMATIC DIAGRAM Topi Conepts Degree of importne Mtries & Determinnts (i) Order, Addition, Multiplition nd trnspose of mtries (ii) Coftors &Adjoint of mtri (iii)inverse of mtri & pplitions (iv)to find differene etween A, dj A, ka, A.djA (v) Properties of Determinnts Referenes NCERT Tet Book XI Ed. 007 *** E. Q.No 4,6 E. Q.No 7,9,,7,8 E. Q.No 0 ** E 4.4 Q.No 5 E 4.5 Q.No,,7,8 *** E 4.6 Q.No 5,6 Emple 9,0,, MisE 4 Q.No 4,5,8,,5 * E 4. Q.No,4,7,8 ** E 4. Q.No,, Emple 6,8 SOME IMPORTANT RESULTS/CONCEPTS A mtri is retngulr rr of m n numers rrnged in m rows nd n olumns. n n A OR A = [ ] ij m n, where i =,,., m ; j =,,.,n. m m mnmn * Row Mtri : A mtri whih hs one row is lled row mtri. A [ ] ij n * Column Mtri : A mtri whih hs one olumn is lled olumn mtri. A [ ] ij m. * Squre Mtri: A mtri in whih numer of rows re equl to numer of olumns, is lled squre mtri A [ ] ij mm * Digonl Mtri : A squre mtri is lled Digonl Mtri if ll the elements, eept the digonl elements re zero. A [ ] ij n n, where ij = 0, i j. ij 0, i = j. * Slr Mtri: A squre mtri is lled slr mtri it ll the elements, eept digonl elements re zero nd digonl elements re sme non-zero quntit. A [ ij ] n n, where ij = 0, i j. ij, i = j. * Identit or Unit Mtri : A squre mtri in whih ll the non digonl elements re zero nd digonl elements re unit is lled identit or unit mtri. 9

20 * Null Mtries : A mtries in whih ll element re zero. * Equl Mtries : Two mtries re sid to e equl if the hve sme order nd ll their orresponding elements re equl. * Trnspose of mtri : If A is the given mtri, then the mtri otined interhnging the rows nd olumns is lled the trnspose of mtri.\ * Properties of Trnspose : If A & B re mtries suh tht their sum & produt re defined, then T T (i). A A T T (ii). T T T T A B A B (iii). KA K.A where K is slr. T T T T T T (iv). AB B A (v). ABC C B A. * Smmetri Mtri : A squre mtri is sid to e smmetri if A = A T i.e. If A [ ] ij mm, then ij ji for ll i, j. Also elements of the smmetri mtri re smmetri out the min digonl * Skew smmetri Mtri : A squre mtri is sid to e skew smmetri if A T = -A. If A [ ] ij mm, then ij ji for ll i, j. *Singulr mtri:a squre mtri A of order n is sid to e singulr, if A = 0. * Non -Singulr mtri : A squre mtri A of order n is sid to e non-singulr, if A 0. *Produt of mtries: (i) If A & B re two mtries, then produt AB is defined, if Numer of olumn of A = numer of rows of B. (ii) (iii) i.e. A [ ] ij mn, B [ j k] n p then AB = AB [C ] ik mp. Produt of mtries is not ommuttive. i.e. AB BA. Produt of mtries is ssoitive. i.e A(BC) = (AB)C (iv) Produt of mtries is distriutive over ddition. *Adjoint of mtri : If A [ ] e n-squre mtri then trnspose of mtri [A ij ], where ij A ij is the oftor of A ij element of mtri A, is lled the djoint of A. T Adjoint of A = Adj. A = [A ]. ij A(Adj.A) = (Adj. A)A = A I. *Inverse of mtri :Inverse of squre mtri A eists, if A is non-singulr or squre mtri A is sid to e invertile nd A - = A Adj.A *Sstem of Liner Equtions : + + z = d. + + z = d. + + z = d. 0

21 d d A X = B X = A - B ; { A 0}. z d *Criteri of Consisten. (i) If A 0, then the sstem of equtions is sid to e onsistent & hs unique solution. (ii) If A = 0 nd (dj. A)B = 0, then the sstem of equtions is onsistent nd hs infinitel mn solutions. (iii) If A = 0 nd (dj. A)B 0, then the sstem of equtions is inonsistent nd hs no solution. * Determinnt : To ever squre mtri we n ssign numer lled determinnt If A = [ ], det. A = A =. If A =, A =. * Properties : (i) The determinnt of the squre mtri A is unhnged when its rows nd olumns re interhnged. (ii) The determinnt of squre mtri otined interhnging two rows(or two olumns) is negtive of given determinnt. (iii) If two rows or two olumns of determinnt re identil, vlue of the determinnt is zero. (iv) If ll the elements of row or olumn of squre mtri A re multiplied non-zero numer k, then determinnt of the new mtri is k times the determinnt of A. If elements of n one olumn(or row) re epressed s sum of two elements eh, then determinnt n e written s sum of two determinnts. An two or more rows(or olumn) n e dded or sutrted proportionll. If A & B re squre mtries of sme order, then AB = A B ASSIGNMENTS (i). Order, Addition, Multiplition nd trnspose of mtries:. If mtri hs 5 elements, wht re the possile orders it n hve? [CBSE 0]. Construt mtri whose elements re given ij = i j. If A = 0, B = 0, then find A B. 4. If A = 0 nd B = [ ], write the order of AB nd BA.. For the following mtries A nd B, verif (AB) T = B T A T, where A = [ ], B =, -. Give emple of mtries A & B suh tht AB = O, ut BA O, where O is zero mtri nd

22 A, B re oth non zero mtries.. If B is skew smmetri mtri, write whether the mtri (ABA T ) is Smmetri or skew smmetri. 4. If A = 0 nd I = 0, find nd so tht A + I = A I. If A = [ ], then find the vlue of A A + I. Epress the mtri A s the sum of smmetri nd skew smmetri mtri, where: A = [ ]. If A = 0, prove tht A n = [ ( ) ], n N (ii) Coftors &Adjoint of mtri. Find the o-ftor of in A =. Find the djoint of the mtri A = 0 Verif A(djA) = (dja) A = I if. A = 0. A = [ ] (iii)inverse of Mtri & Applitions. If A = 0, write A - in terms of A CBSE 0. If A is squre mtri stisfing A = I, then wht is the inverse of A?. For wht vlue of k, the mtri A = 0 is not invertile?. If A = 0, show tht A 5A 4I = 0. Hene find A -. If A, B, C re three non zero squre mtries of sme order, find the ondition on A suh tht AB = AC B = C.

23 . Find the numer of ll possile mtries A of order with eh entr 0 or nd for whih A [ ] = [ ] hs etl two distint solutions. I If A = [ ], find A - nd hene solve the following sstem of equtions: + 5z =, + 4z = - 5, + z = -. Using mtries, solve the following sstem of equtions:. + - z = z = - 4z = [CBSE 0] z = z = z = 70 [CBSE 0]. Find the produt AB, where A = [ ], B = [ ] nd use it to solve the equtions =, + + 4z = 7, + z = 7 4. Using mtries, solve the following sstem of equtions: - + = = = 5. Using elementr trnsformtions, find the inverse of the mtri [ ] (iv)to Find The Differene Between. Evlute [CBSE 0]. Wht is the vlue of, where I is identit mtri of order?. If A is non singulr mtri of order nd =, then find 4. For wht vlve of, 0 is singulr mtri?. If A is squre mtri of order suh tht = 64, find. If A is non singulr mtri of order nd = 7, then find

24 I. If A =0 nd = 5, then find.. A squre mtri A, of order, hs = 5, find (v).properties of Determinnts. Find positive vlve of if =. Evlute. Using properties of determinnts, prove the following : 4. [CBSE 0]. = ( + pz)( - )( - z) (z - ) 4. ( )( )( )( ) [CBSE 0] I. Using properties of determinnts, solve the following for :. = 0 [CBSE 0]. = 0 [CBSE 0]. = 0 [CBSE 0]. If,,, re positive nd unequl, show tht the following determinnt is negtive: = 4

25 5. 4. [CBSE 0] ) ( 7. (+) (+) (+) = ( + + ) 8. If p, q, r re not in G.P nd 0 r p p tht show 0, 0 r q q p q r q r p q p q. 9. If,, re rel numers, nd 0 Show tht either + + = 0 or = =. QUESTIONS FOR SELF EVALUTION. Using properties of determinnts, prove tht : q p z p r z r q z r q p

26 6. Using properties of determinnts, prove tht :. Using properties of determinnts, prove tht : 4..Epress A = s the sum of smmetri nd skew-smmetri mtri. 5. Let A = 4, prove mthemtil indution tht : n n 4n n A n. 6. If A = 5 7, find nd suh tht A + I = A. Hene find A. 7. Let A= 0 tn tn I nd. Prove tht os sin sin os A) (I A I. 8. Solve the following sstem of equtions : + + z = 7, + z =, =. 9. Find the produt AB, where A = B nd nd use it to solve the equtions + z = 4, z = 9, + + z =. 0. Find the mtri P stisfing the mtri eqution 5 P.

27 TOPIC 4 CONTINUITY AND DIFFRENTIABILITY SCHEMATIC DIAGRAM Topi Conepts Degree of importne Refrenes NCERT Tet Book XII Ed. 007 Continuit& Differentiilit.Limit of funtion.continuit *** E 5. Q.No-, 6,0.Differentition * E 5. Q.No- 6 E 5. Q.No- 4,7, 4.Logrithmi Differentition *** E 5.5 QNo- 6,9,0,5 5 Prmetri Differentition *** E 5.6 QNo- 7,8,0, 6. Seond order derivtives *** E 5.7 QNo- 4,6,7 7. Men Vlue Theorem ** E 5.8 QNo-,4 SOME IMPORTANT RESULTS/CONCEPTS * A funtion f is sid to e ontinuous t = if Left hnd limit = Right hnd limit = vlue of the funtion t = i.e. lim f () lim f () f () i.e. lim f ( h) lim f ( h) f (). h0 h0 * A funtion is sid to e differentile t = if Lf () Rf () i.e f ( h) f () f ( h) f () lim lim h0 h h0 h (i) d d ( n ) = n n -. (ii) d d () = (iii) d d () = 0, R (iv) d d ( ) = log, > 0,. (v) d (e ) = e. d (vi) d d (log ) =, log (vii) d d (log ) =, > 0 > 0,, 7 (iii) d d (ot ) = ose, R. (iv) d d (se ) = se tn, R. (v) d d (ose ) = ose ot, R. (vi) d (sin - ) = d -. d (vii) d (os - ) = - -. d (viii) d (tn - ) =, R d (i) d (ot - ) =, R. d () d (se - ) =,. d (i) d (ose - ) =. d (ii) d ( ) =, 0 d du (iii) (ku) = k d d d d (iv) u v du d dv d

28 (viii) d d (log ) = log (i) d d (log ) =, 0 () d d (sin ) = os, R., > 0,, 0 (v) d d (u.v) = (vi) d d u v dv du u v d d du dv v u d d v (i) d d (os ) = sin, R. (ii) d d (tn ) = se, R..Continuit LEVEL-I.Emine the ontinuit of the funtion f()= + 5 t =-.. Emine the ontinuit of the funtion f()=, R.. Show tht f()=4 is ontinuous for ll R. LEVEL-II. Give n emple of funtion whih is ontinuous t =,ut not differentile t =. k, if. For wht vlue of k,the funtion is ontinuous t =., if.find the reltionship etween nd so tht the funtion f defined : + f()= + if is ontinuous t =. if > sin,when 0 4. If f()=. Find whether f() is ontinuous t =0.,when 0 [CBSE 0] LEVEL-III os 4, 0.For wht vlue of k, the funtion f()= 8 is ontinuous t =0? k, 0 sin. If funtion f()=, for 0 is ontinuous t =0, then Find f(0). sin 8

29 sin, if os.let f() = If f() e ontinuous funtion t =, find nd. if ( sin ) if ( ) 4.For wht vlue of k,is the funtion f() =.Differentition sin os k LEVEL-I, when, when. Disuss the differentiilit of the funtion f()=(-) / t =..Differentite =tn -. If = ( )( 4) 4 5 d. Find, = os(log ). d d. Find of = tn d. d, Find. d.if =e sin, then prove tht 4.Find d d, if = t t, = d d t t. LEVEL-II d +( + )=0. d LEVEL-III 0 ontinuous t = 0? 0 d.find, if = d tn d.find = sin sin ot, 0<<. d sin sin os. If sin d os, show tht =. d os 9

30 4.Prove tht d d 4 4.Logrithmi Differentition.Differentite =log 7 (log ).. Differentite, sin(log ),with respet to..differentite = ( ).If log tn 4 LEVEL-I LEVEL-II. If. =log[ -],show tht ( d +) ++=0. d d. Find, = os(log ). d d. Find if (os) = (os) [CBSE 0] d LEVEL-III p. q. = log ( ) os pq, prove tht d, find d d d. If Show tht = * ( )+ [CBSE 0]. 4. Find d d ot when [CBSE 0] 5 Prmetri Differentition LEVEL-II.If = tn, prove tht..if = os log tn nd sin find t. 4. If = tn. /, show tht ( ) ( ) [CBSE 0] 6. Seond order derivtives LEVEL-II 0

31 d d. If = os (log ) + sin(log ), prove tht 0. d d.if =(sin - ), prove tht (- ) d d - d = d.if ( ) + ( ) = for some >0.Prove tht 7. Men Vlue Theorem d d d d LEVEL-II / is onstnt, independent.it is given tht for the funtion f()= -6 +p+q on[,], Rolle s theorem holds with =+. Find the vlues p nd q.. Verif Rolle s theorem for the funtion f() = sin, in [0, ].Find, if verified.veiflgrnge s men Vlue Theorem f() = in the intervl [,4] Questions for self evlution.for wht vlue of k is the following funtion ontinuous t =? f () ; k ; ;, if.if f() = if, ontinuous t =, find the vlues of nd.[cbse 0 Comptt.] 5 -, if. Disuss the ontinuit of f() = t = & =. 4. If f(), defined the following is ontinuous t = 0, find the vlues of,, sin( ) sin f () /, 0, 0, 0 5.If = os log tn nd sin find d t d If = log os d, find. d

32 7. If + d = tn +, find. d 9.If 8. If = log = ( ), prove tht d, find. d d = d. d 0. Find if (os) = (os) d d d.if = os (log ) + sin(log ), prove tht 0. d d.if p. q ( ) pq, prove tht d d.

33 TOPIC 5 APPLICATIONS OF DERIVATIVES SCHEMATIC DIAGRAM Topi Conepts Degree of importne Applition of Derivtive. Refrenes NCERT Tet Book XII Ed. 007.Rte of hnge * Emple 5 E 6. Q.No- 9,.Inresing & deresing *** E 6. Q.No- 6 Emple, funtions.tngents & normls ** E 6. Q.No- 5,8,,5, 4.Approimtions * E 6.4 QNo-, 5 Mim & Minim *** E 6.5Q.No- 8,,,5 Emple 5,6,7, SOME IMPORTANT RESULTS/CONCEPTS d ** Whenever one quntit vries with nother quntit, stisfing some rule = f (), then (or f ()) d d represents the rte of hnge of with respet to nd d (or f ( 0 )) represents the rte of hnge o of with respet to t = 0. ** Let I e n open intervl ontined in the domin of rel vlued funtion f. Then f is sid to e (i) inresing on I if < in I f ( ) f ( ) for ll, I. (ii) stritl inresing on I if < in I f ( ) < f ( ) for ll, I. (iii) deresing on I if < in I f ( ) f ( ) for ll, I. (iv) stritl deresing on I if < in I f ( ) > f ( ) for ll, I. ** (i) f is stritl inresing in (, ) if f () > 0 for eh (, ) (ii) f is stritl deresing in (, ) if f () < 0 for eh (, ) (iii) A funtion will e inresing (deresing) in R if it is so in ever intervl of R. ** Slope of the tngent to the urve = f () t the point ( 0, 0 ) is given d d ( ) (0,0). ** If slope of the tngent line is zero, then tn θ = 0 nd so θ = 0 whih mens the tngent line is prllel to the f 0 ** The eqution of the tngent t ( 0, 0 ) to the urve = f () is given 0 = ( ) ( 0 ). ** Slope of the norml to the urve = f () t ( 0, 0 ) is. f ( ) ** The eqution of the norml t ( 0, 0 ) to the urve = f () is given 0 = 0 f 0 ( 0 ). f ( ) 0

34 -is. In this se, the eqution of the tngent t the point (0, 0) is given = 0. ** If θ, then tn θ, whih mens the tngent line is perpendiulr to the -is, i.e., prllel to the -is. In this se, the eqution of the tngent t ( 0, 0 ) is given = 0. d ** Inrement in the funtion = f() orresponding to inrement in is given =. d ** Reltive error in =. ** Perentge error in = 00. ** Let f e funtion defined on n intervl I. Then () f is sid to hve mimum vlue in I, if there eists point in I suh tht f () f (), for ll I. The numer f () is lled the mimum vlue of f in I nd the point is lled point of mimum vlue of f in I. () f is sid to hve minimum vlue in I, if there eists point in I suh tht f () f (), for ll I. The numer f (), in this se, is lled the minimum vlue of f in I nd the point, in this se, is lled point of minimum vlue of f in I. () f is sid to hve n etreme vlue in I if there eists point in I suh tht f () is either mimum vlue or minimum vlue of f in I. The numer f (), in this se, is lled n etreme vlue of f in I nd the point is lled n etreme point. * * Asolute mim nd minim Let f e funtion defined on the intervl I nd I. Then f() for ll I. () f() is solute mimum if f() f() for ll I. () I is lled the ritil point off if f () = 0 (d) Asolute mimum or minimum vlue of ontinuous funtion f on [, ] ours t or or t ritil points off (i.e. t the points where f is zero) If,,, n re the ritil points ling in [, ], then solute mimum vlue of f = m{f(), f( ), f( ),, f( n ), f()} nd solute minimum vlue of f = min{f(), f( ), f( ),, f( n ), f()}. ** Lol mim nd minim ()A funtion f is sid to hve lol mim or simpl mimum vjue t if f( ± h) f() for suffiientl smll h ()A funtion f is sid to hve lol minim or simpl minimum vlue t = if f( ± h) f(). ** First derivtive test : A funtion f hs mimum t point = if (i) f () = 0, nd (ii) f () hnges sign from + ve to ve in the neighourhood of (points tken from left to right). However, f hs minimum t =, if (i) f () = 0, nd (ii) f () hnges sign from ve to +ve in the neighourhood of. If f () = 0 nd f () does not hnge sign, then f() hs neither mimum nor minimum nd the point is lled point of infltion. The points where f () = 0 re lled sttionr or ritil points. The sttionr points t whih the funtion ttins either mimum or minimum vlues re lled etreme points. ** Seond derivtive test 4

35 (i) funtion hs mim t if f () 0 nd f () <0 (ii) funtion hs minim t = if f () = 0 nd f () > 0..Rte of hnge ASSIGNMENTS LEVEL -I. A lloon, whih lws remins spheril, hs vrile dimeter. Find the rte of hnge of its volume with respet to..the side of squre sheet is inresing t the rte of 4 m per minute. At wht rte is the re inresing when the side is 8 m long?. The rdius of irle is inresing t the rte of 0.7 m/se. wht is the rte of inrese of its irumferene? LEVEL II. Find the point on the urve = 8 for whih the siss nd ordinte hnge t the sme rte?. A mn metre high wlks t uniform speed of 6km /h w from lmp post 6 metre high. Find the rte t whih the length of his shdow inreses. Also find the rte t whih the tip of the shdow is moving w from the lmp post.. The length of retngle is inresing t the rte of.5 m/se nd its redth is deresing t the rte of m/se. find the rte of hnge of the re of the retngle when length is m nd redth is 8 m I. A prtile moves long the urve 6 = +., Find the points on the urve t whih - oordinte is hnging 8 times s fst s the -oordinte.. Wter is leking from onil funnel t the rte of 5 m /se. If the rdius of the se of the funnel is 0 m nd ltitude is 0 m, Find the rte t whih wter level is dropping when it is 5 m from top.. From linder drum ontining petrol nd kept vertil, the petrol is leking t the rte of 0 ml/se. If the rdius of the drum is 0m nd height 50m, find the rte t whih the level of the petrol is hnging when petrol level is 0 m.inresing & deresing funtions. Show tht f() = is n inresing funtion for ll R.. Show tht the funtion + is neither inresing nor deresing on (0,). Find the intervls in whih the funtion f() = sin os, 0< < isinresing or 5

36 deresing.. Indite the intervl in whih the funtion f() = os, 0 is deresing. sin.show tht the funtion f() = is stritl deresing on ( 0, /) log. Find the intervls in whih the funtion f() = inresing or deresing. I. Find the intervl of monotonoit of the funtion f() = log, 0 4sin θ. Prove tht the funtion = θ is n inresing funtion of in [ 0, /] os θ [CBSE 0].Tngents &Normls LEVEL-I.Find the equtions of the normls to the urve = 8 whih re prllel to the line + = 4.. Find the point on the urve = where the slope of the tngent is equl to the -oordinte of the point.. At wht points on the irle = 0, the tngent is prllel to is? LEVEL-II. Find the eqution of the norml to the urve = t the point ( m, m ). For the urve = + + 8, find ll the points t whih the tngent psses through the origin.. Find the eqution of the normls to the urve = whih re prllel to the line = 0 4. Show tht the eqution of tngent t (, ) to the prol =( + ). [CBSE 0Comptt.] LEVEL- III.Find the eqution of the tngent line to the urve = 5 whih is prllel to the line 4 + =0. Show tht the urve + = 0 nd + =0 ut orthogonll t the point (0,0) 6

37 . Find the ondition for the urves nd = to interset orthogonll. 4.Approimtions LEVEL-I Q. Evlute 5. Q. Use differentils to pproimte the ue root of 66 Q. Evlute Q.4 Evlute [CBSE 0] LEVEL-II. If the rdius of sphere is mesured s 9 m with n error of 0.0 m, then find the pproimte error in lulting its surfe re 5 Mim & Minim. Find the mimum nd minimum vlue of the funtion f() = sin. Show tht the funtion f() = hs neither mimum vlue nor minimum vlue. Find two positive numers whose sum is 4 nd whose produt is mimum. Prove tht the re of right-ngled tringle of given hpotenuse is mimum when the tringle is isoseles..a piee of wire 8(units) long is ut into two piees. One piee is ent into the shpe of irle nd other into the shpe of squre. How should the wire e ut so tht the omined re of the two figures is s smll s possile.. A window is in the form of retngle surmounted semiirulr opening. The totl perimeter of the window is 0 m. Find the dimensions of the window to dmit mimum light through the whole opening. I.Find the re of the gretest isoseles tringle tht n e insried in given ellipse hving its verte oinident with one etremit of mjor is..an open o with squre se is to e mde out of given quntit of rd ord of re squre units. Show tht the mimum volume of the o is 6 7 ui units.[cbse 0 Comptt.]

38 .A window is in the shpe of retngle surmounted n equilterl tringle. If the perimeter of the window is m, find the dimensions of the retngle tht will produe the lrgest re of the window. [CBSE 0] Questions for self evlution.snd is pouring from pipe t the rte of m /s. The flling snd forms one on the ground in suh w tht the height of the one is lws one-sith of the rdius of the se. How fst is the height of the snd one inresing when the height is 4 m?. The two equl sides of n isoseles tringle with fied se re deresing t the rte of m per seond. How fst is the re deresing when the two equl sides re equl to the se?. Find the intervls in whih the following funtion is stritl inresing or deresing: f() = Find the intervls in whih the following funtion is stritl inresing or deresing: f() = sin + os, 0 5. For the urve = 4 5, find ll the points t whih the tngent psses through the origin. 6. Find the eqution of the tngent line to the urve = +7 whih is () prllel to the line + 9 = 0 () perpendiulr to the line 5 5 =. 7. Prove tht the urves = nd = k ut t right ngles if 8k =. 8. Using differentils, find the pproimte vlue of eh of the following up to ples of deiml : (i) 6 (ii) Prove tht the volume of the lrgest one tht n e insried in sphere of rdius R is of the 7 volume of the sphere. 0. An open topped o is to e onstruted removing equl squres from eh orner of metre 8 metre retngulr sheet of luminium nd folding up the sides. Find the volume of the lrgest suh o. 8

39 TOPIC 6 INDEFINITE & DEFINITE INTEGRALS SCHEMATIC DIAGRAM Topis Conept Degree of Importne Referenes Tet ook of NCERT, Vol. II 007 Edition Indefinite (i) Integrtion sustitution * Ep 5&6 Pge0,0 Integrls (ii) ) Applition of trigonometri ** E 7 Pge 06, Eerise 7.4Q&Q4 funtion in integrls (iii) Integrtion of some prtiulr funtion *** Ep 8, 9, 0 Pge, Eerise 7.4 Q,4,8,9,& d d,, d d,, d, (p q)d, (p q)d Definite Integrls (iv) Integrtion using Prtil *** Ep & Pge 8 Frtion Ep 9,Ep 4 & 5 Pge0 (v) Integrtion Prts ** Ep 8,9&0 Pge 5 (vi)some Speil Integrls *** Ep &4 Pge 9 d, d (vii) Misellneous Questions *** Solved E.4 (i) Definite Integrls sed upon tpes of indefinite integrls * Eerise 7 Pge 6, Q,,4,5,9,,6 Eerise 7.9 (ii) Definite integrls s limit of sum ** Ep 5 &6 Pge, 4 Q, Q5 & Q6 Eerise 7.8 (iii) Properties of definite Integrls *** Ep Pge 4*,Ep *,4&5 pge 44 Ep 6***Ep 46 Ep 44 pge5 Eerise 7. Q7 & (iv) Integrtion of modulus funtion ** Ep 0 Pge 4,Ep 4 Pge 5 Q5& Q6 Eerise 7. 9

40 SOME IMPORTANT RESULTS/CONCEPTS n n * d n.d * n n * d * * d e d e * * d log * sin d os * sin d os * os d sin * se d tn * os e d ot * se.tn d se * os e.ot d os e * tn d log os log se * ot d = log sin + C * se d log se tn C =log tn C 4 * ose d log ose - ot = log ose + ot + C = log tn + C d * log C, if > d * log C, if > d * log C, if > C * d sin = - os - C` d * log C d * log C * d log * d log * d sin C * f () f()...f n () d 40 =f ()d f ()d... * f ()d f ()d C * u.vd u. v.d v.d * f () d du. d d = F() F(), where F() = f() d f n () d * Generl Properties of Definite Integrls. * f () d = f (t) d * f () d = - f () d * f () d * * 0 = f() d = f( f() d = f( 0 f() d + f() d + - ) - ) * f() d = f()d, 0 0 d d if f() is n even funtion of if f() is n odd funtion of C C.

41 d * tn C, ot C` * f() d = f()d, if f( - ) f() if f( - ) -f() (i) Integrtion sustitution se. (log ) e d.. d tn. d sin.os. Assignments mtn d d 6 I tn. d se os. e d (ii) Applition of trigonometri funtion in integrls sin e. d. d sin.os. sin. d. os. d. os.os.os. d. se 4 sin 4.tn. d. d sin I. os 5. d. sin.os. d (iii) Integrtion using stndrd results. 4 d 9. d 0 d. 9. d 4 os. d sin 4sin 5 4. d 7 6

42 . 4. d 4 d I. d (iv) Integrtion using Prtil Frtion. d ( )( ) 8. d ( )( ) 8. d ( )( 4) (v) Integrtion Prts..se. d. [CBSE 0] d 5 6. d. ( )( )( ) d ( ) ( ). d ( ) I d. sin sin.sin.d..sin 4. os.d. d ( ) ( ). d. log. d. e (tn logse) d 5. se. d.d.sin. d. log I e ( ) os d. ( ) d sin 4. e os.d (vi) Some Speil Integrls. 4. d 5. e.os. d. 4. d d. 4. d I log. ( log ) d 4

43 . ( ). d (vii) Misellneous Questions. d os d 4. sin 8os sin os. d os sin 4. d 4 d. tn. ( 5) d d. d. sin 4sin 5os sin se 5. d d sin os 5se 4 tn I 5. 4 d Definite Integrls 4. 4 d 6. tn. d (i) Definite Integrls sed upon tpes of indefinite integrls d 4 / 5. sin.os. d 0. e d. d 0 (ii) Definite integrls s limit of sum. Evlute ( ) d s the limit of sum Evlute ( ) d definite integrl s the limit of sum. 0 4

44 . Evlute ( ) d s the limit of sum.. Evlute ( ) d s the limit of sum. 0. Evlute ( )d s the limit of sum. 4. Evlute e d s the limit of sum. I (iii) Properties of definite Integrls / tn. d tn 0 π/. d sin os 0 4. d 4 sin. d os 0 / 4 sin. d 4 4 sin os 0 tn. d se.os e 0 / d 4. [CBSE 0] tn / 6 I / / 4 sin. d [CBSE 0]. os log sin d. log tn d [CBSE 0] (iv) Integrtion of modulus funtion I 5 /. ( 4) d. d. sin os d / Questions for self evlution ( )d. Evlute. Evlute 8 44 ( ).d 5

45 d. Evlute os 4. d 4. Evlute sin os sin os 5. Evlute d sin 4os / 5 7. Evlute sin.os. d 0 / 9. Evlute log sin d 0.sin 6. Evlute d / 8. Evlute sin d 4 0. Evlute d 45

46 TOPIC 7 APPLICATIONS OF INTEGRATION SCHEMATIC DIAGRAM Topi Conepts Degree of Importne Referene NCERT Tet BookEdition 007 Applitions of (i)are under Simple Curves * E.8. Q.,,5 Integrtion (ii) Are of the region enlosed *** E. 8. Q 0, Mis.E.Q 7 etween Prol nd line (iii) Are of the region enlosed etween Ellipse nd line *** Emple 8, pge 69 Mis.E. 8 (iv) Are of the region enlosed *** E. 8. Q 6 etweencirle nd line (v) Are of the region enlosed *** E 8. Q, Mis.E.Q 5 etween Cirle nd prol (vi) Are of the region enlosed etween Two Cirles *** Emple 0, pge70 E 8. Q (vii) Are of the region enlosed *** Emple 6, pge68 etween Two prols (viii) Are of tringle when verties re given *** Emple 9, pge70 E 8. Q4 (i) Are of tringle when sides *** E 8. Q5,Mis.E. Q 4 re given () Misellneous Questions *** Emple 0, pge74 Mis.E.Q 4, SOME IMPORTANT RESULTS/CONCEPTS ** Are of the region PQRSP = da = d = f () d. ** The re A of the region ounded the urve = g (), -is nd d the lines =, = d is given A= d d = g () d 46

47 ASSIGNMENTS (i) Are under Simple Curves. Sketh the region of the ellipse nd find its re, using integrtion, Sketh the region {(, ) : = 6} nd find its re, using integrtion. (ii) Are of the region enlosed etween Prol nd line. Find the re of the region inluded etween the prol = nd the line + =.. Find the re of the region ounded = 4, =, = 4 nd the -is in the first qudrnt. I. Find the re of the region :(, ):,,0 (iii) Are of the region enlosed etween Ellipse nd line. Find the re of smller region ounded the ellipse nd the stright line (iv) Are of the region enlosed etween Cirle nd line. Find the re of the region in the first qudrnt enlosed the -is, the line = nd the irle + =. I. Find the re of the region :(, ): (v) Are of the region enlosed etween Cirle nd prol I. Drw the rough sketh of the region {(, ): 6, + 6} n find the re enlosed the region using the method of integrtion.. Find the re ling ove the -is nd inluded etween the irle + = 8 nd the prol = 4. (vi) Are of the region enlosed etween Two Cirles I. Find the re ounded the urves + = 4 nd ( + ) + = 4 using integrtion. (vii) Are of the region enlosed etween Two prols. Drw the rough sketh nd find the re of the region ounded two prols 47

48 4 = 9 nd = 6 using method of integrtion. (viii) Are of tringle when verties re given I. Using integrtion ompute the re of the region ounded the tringle whose verties re (, ), (, 4), nd (5, ).. Using integrtion ompute the re of the region ounded the tringle whose verties re (, ), (0, 5), nd (, ). (i) Are of tringle when sides re given I. Using integrtion find the re of the region ounded the tringle whose sides re = +, = +, = 4.. Using integrtion ompute the re of the region ounded the lines + =, =, nd + = 7. () Misellneous Questions I. Find the re of the region ounded the urves = nd = +.. Find the re ounded the urve = nd =.. Drw rough sketh of the urve = sin nd = os s vries from = 0 to = nd find the re of the region enlosed them nd -is 4. Sketh the grph of =.Evlute d.wht does this vlue represent on the grph. 5. Find the re ounded the urves = 6 nd =. 6. Sketh the grph of = nd evlute the re under the urve = ove -is nd etween = 6 to = 0. [CBSE 0] Questions for self evlution. Find the re ounded the urve = 4 nd the line = 4.. Find the re ounded the prol = nd =.. Find the re of the region :(,): 0, 0,0 4. Find the re of the smller region ounded the ellipse nd the line Find the re of the region :(,):, 6. Find the re ling ove the -is nd inluded etween the irle + = 8 nd the prol = Find the re ounded the urves + = 4 nd ( + ) + = 4 using integrtion. 48

49 8. Using integrtion ompute the re of the region ounded the tringle whose verties re (, ), (, 4), nd (5, ). 9. Using integrtion ompute the re of the region ounded the lines + = 4, = 6, nd + 5 = 0., 0. Sketh the grph of : f ()., 4 Evlute f ()d. Wht does the vlue of this integrl represent on the grph? 0 49

50 TOPIC 8 DIFFERENTIAL EQUATIONS SCHEMATIC DIAGRAM (ii).generl nd prtiulr solutions of differentil eqution (iii).formtion of differentil eqution whose generl solution is given (iv).solution of differentil eqution the method of seprtion of vriles (vi).homogeneous differentil eqution of first order nd first degree (vii)solution of differentil eqution of the tpe d/d +p=q where p nd q re funtions of And solution of differentil eqution of the tpe d/d+p=q where p nd q re funtions of ** E., pg84 * Q. 7,8,0 pg 9 * Q.4,6,0 pg 96 ** Q.,6, pg 406 *** Q.4,5,0,4 pg 4,44 SOME IMPORTANT RESULTS/CONCEPTS ** Order of Differentil Eqution : Order of the heighest order derivtive of the given differentil eqution is lled theorder of the differentil eqution. ** Degree of the Differentil Eqution :Heighest power of of ll the derivtives re of the given differentil eqution is lled d f, ** Homogeneou s Differentil Eqution :, where f, &f d f (, ) funtion of sme degree. ** Liner Differentil Eqution : i. ii. d d p q, wherep&q e thefuntion of Solution of theeqution is :.e d pq, wherep &q e thefuntion of or onstnt. d Solution of theeqution is:.e p d p d e e or onstnt. p d p d 50 the heighest order derivtive when powers.q d, where e.q d, where e p d the degree of the differentil p d equtin (, ) e the homogeneou s is Integrting Ftor (I.F.) is Integrting Ftor (I.F.)

51 ASSIGNMENTS. Order nd degree of differentil eqution. Write the order nd degree of the following differentil equtions d d (i) 0 d d. Generl nd prtiulr solutions of differentil eqution. Show tht e d is the solution of e d. Formtion of differentil eqution. Otin the differentil eqution eliminting nd from the eqution = e (os + sin) I. Find the differentil eqution of the fmil of irles ( - )² - ( - )² = r². Otin the differentil eqution representing the fmil of prol hving verte t the origin nd is long the positive diretion of -is 4. Solution of differentil eqution the method of seprtion of vriles d d. Solve. Solve e os given tht (0)=0. d d d. Solve tn d 5.Homogeneous differentil eqution of first order nd first degree. Solve ( )d ( ) d I Show tht the given differentil eqution is homogenous nd solve it. d. ( ). d log( ) d d 0 d 5

52 .Solve d d d 4.Solve d ( )d 0 5.Solve d d ( ) d d d CBSE0 6.Solve 7. Solve d ( ) d 0 CBSE0 8.Solve d ( ) d 0 6. Liner Differentil Equtions d.find the integrting ftor of the differentil d d d.solve tn sin. Solve e ( ) d d d. Solve log d I d. Solve os( ).Solve e d ( e ) d d d. Solve ( ) d 4. Solve d 4 d ( ) 5. Solve the differentil eqution d ;given tht when =,= d Questions for self evlution d d d. Write the order nd degree of the differentil eqution sin 0 d d d. Form the differentil eqution representing the fmil of ellipses hving foi on is nd entre t origin.. Solve the differentil eqution : (tn )d ( ) d, given tht = 0 when = Solve the differentil eqution :d d = d d 5. Solve the differentil eqution : log log. d 6. Solve the differentil eqution : d + ( + ) d.= 0, () = 5

53 7. Show tht the differentil eqution.e d e d 0 is prtiulr solution given tht (0) =. 8. Find the prtiulr solution of differentil eqution homogeneous ndfind its d ot d ot, given tht 0. 5

54 TOPIC 9 VECTOR ALGEBRA SCHEMATIC DIAGRAM Topi Conept Degree of importne Vetor lger (i)vetor nd slrs * Q pg48 (ii)diretion rtio nd diretion * Q, pg 440 osines (iii)unit vetor * * E 6,8 Pg 46 Refrene NCERT Tet Book Edition 007 (iv)position vetor of point nd * * Q 5 Pg 440, Q Pg440, Q 6 olliner vetors Pg448 (v)dot produt of two vetors ** Q6, Pg445 (vi)projetion of vetor * * * E 6 Pg 445 (vii)cross produt of two vetors * * Q Pg458 (viii)are of tringle * Q 9 Pg 454 (i)are of prllelogrm * Q 0 Pg 455 * Position vetor of point A, * If A(,,z ) nd point B * If î ĵ zkˆ ; SOME IMPORTANT RESULTS/CONCEPTS,z OA î ĵ zkˆ,,z then AB î ĵ z z z kˆ *Unit vetor prllel to * Slr Produt (dot produt)etween two vetors :. os ; is ngle etween the vetors. * os * If î ĵ kˆ nd î ĵ kˆ then. 54