BRILLIANT PUBLIC SCHOOL, SITAMARHI (Affiliated up to +2 level to C.B.S.E., New Delhi)

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1 BRILLIANT PUBLIC SCHOOL, SITAMARHI (Affilited up to level to C.B.S.E., New Delhi) Clss-XII IIT-JEE Advced Mthemtics Study Pckge Sessio: -5 Office: Rjoptti, Dumr Rod, Sitmrhi (Bihr), Pi-8 Ph.66-5, Moile:966758, 9969 Wesite: E-mil:

2 STUDY PACKAGE Trget: IIT-JEE (Advced) SUBJECT: MATHEMATICS-XII Chpter Pges Eercises Iverse Trigoometric 7 5 Fuctios Determits d Mtrices 5 Cotiuity & Differetiility 8 Applictios of Derivtive Itegrtio Are Uder Curves 5 7 Differetil Equtios 8

3 STUDY PACKAGE Trget: IIT-JEE (Advced) SUBJECT: MATHEMATICS TOPIC: 8 XII M. Iverse Trigoometric Fuctios Ide:. Key Cocepts. Eercise I to II. Aswer Key. Assertio d Resos 5. Yrs. Que. from IIT-JEE 6. Yrs. Que. from AIEEE

4 . Pricipl Vlues & Domis of Iverse Trigoometric/Circulr Fuctios: Fuctio Domi Rge (i) y si where y (ii) y cos where y (iii) y t where R < y < (iv) y cosec where or y, y (v) y sec where or y ; y (vi) y cot where R < y < NOTE: () st qudrt is commo to the rge of ll the iverse fuctios. () rd qudrt is ot used i iverse fuctios. (c) th qudrt is used i the clockwise directio i.e. y. (d) No iverse fuctio is periodic. (See the grphs o pge 7) Solved Emple # Solutio Fid the vlue of t Let y t t cos t 6 y Self prctice prolems: 6 cos t As. t Fid the vlue of the followigs :.. () si si As. () cosec [sec ( ) cot ( )] As. Solved Emple # Fid domi of si ( ) Solutio. Let y si ( ) For y to e defied ( ) [, ] Self prctice prolems: Fid the domi of followigs : () y sec ( )

5 () y cos (5) y t ( ) Aswers () (, ] [, ] [, ) () R (5) (, ] [, ). Properties of Iverse Trigoometric Fuctios: Property - (A) (i) si (si ), (ii) cos (cos ), (iii) t (t ), R (iv) cot (cot ), R (v) sec (sec ),, (vi) cosec (cosec ),, These fuctios re equl to idetity fuctio i their whole domi which my or my ot e R.(See the grphs o pge 8) Solved Emple # Fid the vlue of cosec cot cot Solutio. Let y cosec cot cot cot (cot ), R cot cot from equtio (i), we get y cosec y As. Self prctice prolems:. Fid the vlue of ech of the followig : (6) cos si si 6...(i) (7) si cos cos Aswers (6) (7) ot defied Property - (B) (i) si (si ), (ii) cos (cos ) ; (iii) t (t ) ; < < (iv) cot (cot ) ; < < (v) sec (sec ) ;, These re equl to idetity fuctio for short itervl of oly. (See the grphs o pge 9-) Solved Emple # Solutio. Fid the vlue of t t Let y t t Note t (t ) if, (vi) cosec (cosec ) ;,

6 , t t, grph of y t (t ) is s : from the grph we c see tht if < <, the y t (t ) c e writte s y y t t y solved Emple # 5 Fid the vlue of si (si7) Solutio. Let y si (si 7) Note : si (si 7) 7 s 7 5 < 7 < grph of y si (si ) is s :, 5 From the grph we c see tht if the y si (si ) c e writte s : y si (si 7) 7 Similrly if we hve to fid si (si( 5)) the < 5 <

7 from the grph of si (si ), we c sy tht si (si( 5)) ( 5) 5 Self prctice prolems: (8) Fid the vlue of cos (cos ) (9) 5 Fid si (si θ), cos (cosθ), t (tθ ), cot (cotθ) for θ, As. (8) (9) si - (siθ) θ ; cos (cos θ ) θ ; t (t θ) θ ; cot (cot θ) θ Property - (C) (i) s i () si, (ii) t () t, R (iii) cos () cos, (iv) cot () cot, R The fuctios si, t d cosec re odd fuctios d rest re either eve or odd. Solved Emple # 6 Solutio. Let Fid the vlue of cos {si( 5)} y cos {si( 5)} cos ( si 5) cos ( ) cos, cos (si 5) cos cos 5 < 5 < grph of cos (cos ) is s :...(i) from the grph we c see tht if the y cos (cos) c e writte s y from the grph cos cos from equtio (i), we get 5 y 5 y 5 Self prctice prolems: As. Fid the vlue of the followig 7 () cos ( cos ) () t t 8 () t cot Aswers. () () 8 5 ()

8 Property - (D) (i) c o s e c si ;, (ii) sec cos ;, (iii) cot t t ; > ; < Solved Emple # 7 Fid the vlue of t cot Solutio Let y t cot cot ( ) cot, R equtio (i) c e writte s y t cot y t cot cot t if >...(i) y t t y Self prctice prolems: Fid the vlue of the followigs () sec Aswers. () Property - (E) cos () cosec () si (i) si cos, (ii) t cot, R (iii) cosec sec, Solved Emple # 8 Fid the vlue of si (cos si ) whe 5 Solutio. Let y si [cos si ] si cos, y si cos cos si cos cos (cos ) 5 y cos cos...(i) 5 cos (cos ) if [, ] 6

9 [, ] 5 cos cos 5 5 y 5. Self prctice prolems: Solve the followig equtios from equtio (i), we get (5) 5 t cot (6) si cos Aswers. (5) (6) Property - (F) (i) si (cos ) cos (si ), (ii) t (cot ) cot (t ), R, (iii) cosec (sec ) sec (cosec ) Solved Emple # 9 Fid the vlue of si t. Solutio. Let y si t Note : To fid y we use si(si ), For this we covert t i si Let θ t, >...(i) tθ d θ, si θ 5 si (si θ) si 5 θ, si (si θ) θ...(ii) equtio (ii) c e writte s : θ si 5 θ t from equtio (i), we get y si si 5 y 5 Solved Emple # Fid the vlue of t cos Solutio. Let y t cos (i) 7 t si 5

10 Let 5 5 cos θ θ, d cos θ equtio (i) ecomes θ y t...(ii) θ t cos θ cos θ ( 5) θ t ± 5...(iii) θ, θ, t θ > from equtio (iii), we get θ t 5 from equtio (ii), we get 5 y As. Solved Emple # Fid the vlue of cos (cos si ) whe 5 Solutio. Let y cos [cos si ] Aliter : Let si cos, y coscos cos cos cos si (cos ) 5 y si cos 5 si (cos ), si cos 5 5 from equtio (i), we get y (i) cos θ cos θ d θ, 5 5 siθ 5 si (si θ) si 5...(ii) 8 θ, si (si θ) θ

11 equtio (ii) c e writte s θ si θ cos 5 5 cos si 5 5 Now equtio (i) c e writte s y si si 5...(iii) [, ] si si 5 5 from equtio (iii), we get y Self prctice prolems: 5 5 Fid the vlue of the followigs : (7) t cosec (9) si cot Aswers : (7) 5 (8) (8) sec cot 6 () t t (9) 5 5. Idetities of Additio d Sustrctio: A. () 5 (i) si si y si y y,, y & ( y ) 7 7 si y y,, y & y > Note tht: y si si y y > < si si y < (ii) cos cos y cos y y,, y (iii) t t y t y, >, y > & y < y t y, >, y > & y > y, >, y > & y Note tht : y < < t t y < ;y > < t t y < B. (i) si si y si y y,, y (ii) cos cos y cos y y,, y, y 9

12 y (iii) t t y t y,, y Note: For < d y < these idetities c e used with the help of properties (C) i.e. chge d y to d y which re positive. Solved Emple # Solutio. 5 8 Show tht si si si >, > d si si si 5 7 Solved Emple # Evlute: Solutio. cos si t > si si Let z cos si t 5 6 si 5 cos 5 z cos cos 5 z cos cos 5 >, > d < t. 6 6 t 6 cos cos cos 5 5 equtio (i) c e writte s 6 6 z cos t z si t si t 65 6 from equtio (ii), we get 6 6 z t t (i) 6 69 cos 65...(ii) z As. Solved Emple #

13 Solutio. Evlute t 9 t >, > d 9 > 5 9 t 9 t t t ( ) Self prctice prolems: 5 t 9 t. 5 6 () Evlute si si si 5 65 () If t t 5 cot λ the fid λ () Prove tht cos Solve the followig equtios () t () t () 6 7 cot cos 6 5 (5) si si Aswers. () C. (i) si (ii) cos ( ) 9 () λ 9 si si ( si ) cos cos if if if if if () 6 (5) > < < (iii) t (iv) si (v) cos (See the grphs o pge ) t t ( t ) t t ( t ) t t if if if if if if if if < < < > > < Solved Emple # 5 Defie y cos ( ) i terms of cos d lso drw its grph. Solutio. Let y cos ( ) Note Domi : [, ] d rge : [, ]

14 Let cos θ θ [, ] d cos θ y cos ( cos θ cos θ ) y cos (cos θ)...(i) Fig.: Grph of cos (cos ) θ [, ] θ [, ] to defie y cos (cos θ), we cosider the grph of cos (cos ) i the itervl [, ]. Now, from the ove grph we c see tht (i) if θ cos (cos θ) θ from equtio (i), we get y θ if θ θ y θ if θ y cos if (ii) if < θ cos (cos θ) θ from equtio (i), we get y θ if < θ y θ if < θ y cos if < (iii) < θ cos (cos θ) θ from equtio (i), we get y θ if < θ y θ if < θ y cos if < from (i), (ii) & (iii), we get cos y cos ( ) cos cos Grph : For y cos ( ) domi : [, ] rge : [, ] (i) if, y cos. ; ; ; < < dy d ( ) /...(i) dy < if d, decresig if, gi if we differetite equtio (i) w.r.t., we get

15 d d y ( ) / d y < if, d cocvity dowwrds if, (ii) if <, y cos. (iii) dy d icresig if, d y () if, the < d dy > if d, d y d d cocvity dowwrds if, d y () if, the > d cocvity upwrds if, dy d y Similrly if < the < d d >. d the grph of y cos ( ) is s ) / ( Self prctice prolems: (6) Defie y si ( ) i terms of si d lso drw its grph. (7) Defie y t Aswers i terms of t d lso drw its grph. (6) y si ( ) si si si ; ; ; < < grph of y si ( )

16 (7) y t t t t ; ; ; < < < < < < D. Fig.: Grph of y t y z yz If t t y t z t y yz z if, >, y >, z > & (y yz z) < NOTE: (i) If t t y t z the y z yz (ii) If t t y t z the y yz z (iii) t t t (iv) t t t

17 Iverse Trigoometric Fuctios Some Useful Grphs. (i) y si,, y y, (ii) y cos,, y [, ] y O O (iii) y t, R, y,, y (iv) y cot, R, y (, ) y O O (v) y sec,, y, U, (vi) y cosec,, y O y, U, y O 5

18 Prt - (A) (i) y si (si ) cos (cos ), [, ], y [, ]; y is periodic y y )5º O (ii) y t (t - ) cot (cot - ), R, y R; y is periodic y )5º O y (iii) y cosec (cosec ) sec (sec ),, y ; y is periodic y y O y 6

19 Prt -(B) (i) y si (si ), R, y,, is periodic with period y y ( ) )5º O y y y y (ii) y cos (cos ), R, y [, ], is periodic with period y y y y y O (iii) y t (t ), R ( ), Ι, y, is periodic with period y y y y O y y (iv) y sec (sec ), y is periodic with period ; R ( ), Ι, y, U, y y y y y O 7

20 (v) y cot (cot ), y is periodic with period ; R {, Ι}, y,, (vi) y cosec (cosec ), y is periodic with period ; R {, Ι}, y, {} 8

21 Prt - (C) (i) g r p h o f y s i (ii) grph of y cos ( ) Note : I this grph it is dvisle ot to check its derivility just y the ispectio of the grph ecuse it is difficult to judge from the grph tht t there is shpr corer or ot. (iii) grph of y t (iv) grph of y si (v) grph of y cos 9

22 KEY CONCEPTS (INVERSE TRIGONOMETRY FUNCTION) GENERAL DEFINITION(S):. si, cos, t etc. deote gles or rel umers whose sie is, whose cosie is d whose tget is, provided tht the swers give re umericlly smllest ville. These re lso writte s rc si, rc cos etc. If there re two gles oe positive & the other egtive hvig sme umericl vlue, the positive gle should e tke.. PRINCIPAL VALUES AND DOMAINS OF INVERSE CIRCULAR FUNCTIONS : (i) y si where ; y d si y. (ii) y cos where ; y d cos y. (iii) y t where R ; < < d t y. (iv) y cosec where or ; y, y d cosec y. (v) y sec where or ; y ; y (vi) y cot where R, < y < d cot y. NOTE THAT : () st qudrt is commo to ll the iverse fuctios. () rd qudrt is ot used i iverse fuctios. d sec y. (c) th qudrt is used i the CLOCKWISE DIRECTION i.e. y.. PROPERTIES OF INVERSE CIRCULAR FUNCTIONS : P (i) si (si ), (ii) cos (cos ), (iii) t (t ), R (iv) si (si ), (v) cos (cos ) ; (vi) t (t ) ; < < P (i) cosec si ;, (ii) sec cos (iii) cot t t ;, ; > ; < P (i) si () si, (ii) t () t, R (iii) cos () cos, (iv) cot () cot, R P (i) si cos (ii) t cot R (iii) cosec sec P5 t t y t y y where >, y > & y <

23 t y y where >, y > & y > t t y t y y where >, y > P6 (i) si si y si y y where, y & ( y ) Note tht : y si si y (ii) si si y si y y where, y & y > Note tht : y > < si si y < (iii) si si y si [ y y ] where >, y > (iv) cos cos y cos [ ] y y where, y P7 If t t y t z t y z y z if, >, y >, z > & y yz z < y y z z Note : (i) If t t y t z the y z yz (ii) If t t y t z the y yz z P8 t si Note very crefully tht : cos t si t if t if > ( t ) if < cos t if t if < t REMEMBER THAT : t t ( t ) if if if < < > (i) si si y si z y z (ii) cos cos y cos z y z (iii) t t t d t t t INVERSE TRIGONOMETRIC FUNCTIONS SOME USEFUL GRAPHS. y si,, y,. y cos,, y [, ]

24 . y t, R, y,. y cot, R, y (, ) 5. y sec,, y,, 6. y cosec,, y,, 7. () y si (si ), R, y,, 7.() y si (si ), Periodic with period [, ], y [, ], y is periodic

25 8. () y cos (cos ), R, y [, ], periodic with period 8. () y cos (cos ), [, ], y [, ], y is periodic 9. () y t (t ), R, y R, y is periodic 9. () y t (t ), R ( ) I periodic with period, y,,. () y cot (cot ),. () y cot (cot ), R { }, y (, ), periodic with R, y R, y is periodic. () y cosec (cosec ),. () y cosec (cosec ), ε R {, ε I }, y y is periodic with period,,, y, y is periodic. () y sec (sec ),. () y sec (sec ), y is periodic with period ; ; y ], y is periodic R ( ) I y,,

26 Q. Fid the followig (i) tcos t EXERCISE (iv) t t (v) cos t Q. Fid the followig : (ii) si si (iii) cos cos 7 6 (i) si si (ii) cos cos 6 (v) sicos Q. Prove tht: 5 () cos cot 6 6 cos 7 5 (vi) t si (vi) t si α t t α 5 cos α 5 cot (iii) t t (iv) cos cos where < α < () cos cos si (c) rc cos rc cos 6 6 (d) Solve the iequlity: (rc sec ) 6(rc sec ) 8 > Q. Fid the domi of defiitio the followig fuctios. ( Red the symols [*] d {*} s gretest itegers d frctiol prt fuctios respectively.) (i) f() rc cos (iii) f () si log ( ) (iv) f() si cos ( { }) log ( ) 5 (ii) cos(si ) si, where {} is the frctiol prt of. 6 5 (v) f () cos log ( ) si ( log ) (vi) f () log ( log 7 ( 5 )) cos 9 si si ( ) (vii) f() e t ( [ ] ) l (viii) f() si(cos ) l ( cos cos ) e cos si si Q.5 Fid the domi d rge of the followig fuctios. (Red the symols [*] d {*} s gretest itegers d frctiol prt fuctios respectively.) (i) f () cot ( ²) (ii) f () sec (log t log t ) 6 5

27 (iii) f() cos (iv) f () t log ( 5 8 ) 5 ( ). Q.6 Fid the solutio set of the equtio, cos si Q.7 Prove tht: () si cos (si ) cos si (cos ), () t (cosec t t cot ) t ( ) (c) t m m pq t t MN p q M N where M mp q, N p mq, q N < ; < d < m p M (d) t (t t y t z) cot (cot cot y cot z) Q.8 Fid the simplest vlue of, rc cos rc cos, Q.9 If cos cos y α the prove tht.y α y cos si α. Q. If rc si rc siy rc siz the prove tht : (, y, z > ) () y y z z yz () y z y z ( y y z z ), Q. If > > c > the fid the vlue of : cot c cot c c cot. c Q. Solve the followig equtios / system of equtios: () si si () t t t (c) t () t () t () t () (d) si 5 cos (e) cos t (f) si si y & cos cos y (g) t cos cos ( >, > ). Q. Let l e the lie y d l e the lie y 8. L is the lie formed y reflectig l cross the lie y d L is the lie formed y reflectig l cross the -is. If θ is the cute gle etwee L d L such tht t θ, where d re coprime the fid ( ). Q. Let y si (si 8) t (t ) cos (cos ) sec (sec 9) cot (cot 6) cosec (cosec 7). If y simplifies to the fid ( ). Q.5 Show tht : si si cos cos t t cot cot

28 Q.6 Let α si 6, β cos d γ t 8, fid (α β γ) d hece prove tht (i) cot α α cot, (ii) t α tβ Q.7 Prove tht : si cot t cos si cosec cot t where (, ] Q.8 If si si y < for ll, y R the prove tht si (t. ty),. y Q.9 Fid ll the positive itegrl solutios of, t cos y si. Q. Let f () cot ( α α) e fuctio defied R, the fid the complete set of rel vlues of α for which f () is oto. EXERCISE Q. Prove tht: () t cos t cos () cos cos cosy cos cosy t y t. t (c) t. t cos cos cos Q. If y t prove tht ² si y. Q. If u cot cosθ t cosθ the prove tht si u t θ. Q. If α rc t & β rc si vlue of α β will e if >. for < <, the prove tht α β, wht the Q.5 If, the epress the fuctio f () si ( ) cos ( ) i the form of cos, where d re rtiol umers. Q.6 Fid the sum of the series: () si si... si... 6 ( ) () t t 9... t... (c) cot 7 cot cot cot... to terms. (d) t t t t (e) t t 8 t 8 t... Q.7 Solve the followig () cot cot (² ) cot ( ) 6 to terms.

29 () sec sec sec sec ;,. (c) t t t 6 β Q.8 Epress cosec t α β α sec t α β s itegrl polyomil i α & β. Q.9 Fid the itegrl vlues of K for which the system of equtios ; K rccos ( rcsi y) ( rcsi y). ( rccos ) 6 possesses solutios & fid those solutios. Q. If the vlue of (k )k(k )(k ) Lim cos is equl to, fid the vlue of k. k k(k ) k Q. If X cosec. t. cos. cot. sec. si & Y sec cot si t cosec cos ; where. Fid the reltio etwee X & Y. Epress them i terms of. Q. Fid ll vlues of k for which there is trigle whose gles hve mesure t, t k, d t k. Q. Prove tht the equtio,(si ) (cos ) α hs o roots for α < d α > 7 8 Q. Solve the followig iequlities : () rc cot 5 rc cot 6 > () rc si > rc cos (c) t (rc si ) > Q.5 Solve the followig system of iequtios rc t 8rc t < & rc cot rc cot > Q.6 Cosider the two equtios i ; cos si (i) si (ii) cos y y T h e s e t s X, X [, ] ; Y, Y I {} re such tht X : the solutio set of equtio (i) X : the solutio set of equtio (ii) Y : the set of ll itegrl vlues of y for which equtio (i) possess solutio Y : the set of ll itegrl vlues of y for which equtio (ii) possess solutio Let : C e the correspodece : X Y such tht C y for X, y Y & (, y) stisfy (i). C e the correspodece : X Y such tht C y for X, y Y & (, y) stisfy (ii). Stte with resos if C & C re fuctios? If yes, stte whether they re ijjective or ito? cos ( si( )) Q.7 Give the fuctios f() e, g() cosec cos & the fuctio h() f() defied oly for those vlues of, which re commo to the domis of the fuctios f() & g(). Clculte the rge of the fuctio h(). Q.8 () If the fuctios f() si their domi & rge. () & g() cos re ideticl fuctios, the compute If the fuctios f() si ( ) & g() si re equl fuctios, the compute the mimum rge of. 7

30 Q.9 Show tht the roots r, s, d t of the cuic ( )( 7), re rel d positive. Also compute the vlue of t (r) t (s) t (t). Q. Solve for : si si <. EXERCISE Q. The umer of rel solutios of t ( ) si is : (A) zero (B) oe (C) two (D) ifiite [JEE '99, (out of )] Q. Usig the pricipl vlues, epress the followig s sigle gle : Q. Solve, si c t t 5 si 65 5 Q. Solve the equtio: cos ( ) cos ( 6 ). [ REE '99, 6 ] si c si, where c, c. [REE (Mis), out of ] Q.5 If si... cos [ REE (Mis), out of ] 6... for < < the equls to [JEE (screeig)] (A) / (B) (C) / (D) Q.6 Prove tht cos t si cot [JEE (mis) 5] Q.7 Domi of f () si () 6 is (A), (B), (C), (D), [JEE (Screeig) ] Q.8 If si( cot ( ) ) cos(t ), the (A) (B) (C) (D) 9 [JEE (Screeig)] 8

31 Q. (i), (ii), (iii) 5, (iv) 6, (v) 5, (vi) 7 6 INVERSE TRIGONOMETRY EXERCISE Q. (d) (, sec ) [, ) Q. (i) / (ii) {, } (iii) < < (iv) (/, /), (v) (/, ] Q. (i), (ii), (iii), (iv), (v) 5, (vi) α (vi) {7/, 5/9} (vii) (, ) {,, } (viii) { 6, I} Q5. (i) D : ε R R : [/, ) Q 6. (ii) D:, I ; R :, (iii) D : R R :, (iv) D : R R :,, Q. () 7 Q 8. Q. () (c),, (e) or (f), y (g) Q. 57 Q. 5 Q 9. ; y & ; y 7 Q. EXERCISE (d) ± 7 Q. Q5. 6 cos 9 9, so 6, Q 6. () () (c) rc cot 5 (d) rc t ( ) rc t (e) Q 7. () ² or () (c) Q 8. (α β ) (α β) Q 9. K ; cos, & cos, Q. 7 Q. X Y Q. k Q. () (cot, ) (, cot ) () F H G, O Q P (c),, Q5. t, cot Q6. C is ijective fuctio, C is my to my correspodece, hece it is ot fuctio Q7. [e /6, e ] Q 8.() D : [, ], R : [, /] () (c) D : [, ], R : [, ] Q.9 Q. (, ) EXERCISE Q. C Q. Q. {,, } Q. Q.5 B Q.7 D Q.8 A 9

32 EXERCISE (Iv. Trigoo.) Prt : (A) Oly oe correct optio. If cos λ cos µ cos v the λµ µv vλ is equl to (A) (B) (C) (D). Rge of f() si t sec is (A), (B), (C), (D) oe of these. The solutio of the equtio si t si 6 is (A) (B) (C) (D) oe of these. The vlue of si [cos{cos (cos) si (si )}], where, is (A) (B) (C) (D) 5. The set of vlues of k for which k si (si ) > for ll rel is (A) { } (B) (, ) (C) R (D) oe of these 6. si (cos(si )) cos (si (cos )) is equl to (A) (B) (C) (D) 7. cos. cos cos holds for (A) (B) R (C) (D) 8. t t, where >, >, >, is equl to (A) t (C) t (B) t (D) t 9. The set of vlues of for which the formul si si ( ) is true, is (A) (, ) (B) [, ] (C), (D),. The set of vlues of for which si ( 5) cos ( 5) hs t lest oe solutio is (A) (, (C) R ] [, ) (B) (, ) (, ) (D) oe of these. All possile vlues of p d q for which cos p cos p cos q holds, is (A) p, q (B) q >, p (C) p, q (D) oe of these. If [cot ] [cos ], where [.] deotes the gretest iteger fuctio, the complete set of vlues of is (A) (cos, ] (B) (cot, cos ) (C) (cot, ] (D) oe of these. The complete solutio set of the iequlity [cot ] 6 [cot ] 9, where [.] deotes gretest iteger fuctio, is (A) (, cot ] (B) [cot, cot ] (C) [cot, ) (D) oe of these. t cos t cos, is equl to (A) (B) (C) (D) siθ 5. If si, the t θ is equl to 5 cosθ (A) / (B) (C) (D)

33 u 6. If u cot t α t t α, the t (A) is equl to t α (B) cot α (C) t α (D) cot α si si 7. The vlue of cot, < <, is: si si (A) (B) (C) (D) 8. The umer of solutio(s) of the equtio, si cos ( ) si ( ), is/re (A) (B) (C) (D) more th 9. The umer of solutios of the equtio t t t is (A) (B) (C) (D). If t t t.... t. t ( ) θ, the θ is equl to (A) (B) (C) (D). If cot >, N, the the mimum vlue of is: 6 (A) (B) 5 (C) 9 (D) oe of these. The umer of rel solutios of (, y) where, y si, y cos (cos ),, is: (A) (B) (C) (D). The vlue of cos cos is equl to 8 (A) / (B) / (C) /6 (D) / Prt : (B) My hve more th oe optios correct. α, β d γ re three gles give y α t ( ), β si si d γ cos. The (A) α > β (B) β > γ (C) α < γ (D) α > γ 5. cos t the (A) 5 (B) 5 (C) si (cos ) 5 (D) t (cos ) 6. For the equtio t ( t ) t (t t ), which of the followig is ivlid? (A) (B) (C) (D), 7. The sum t is equl to: (A) t t (B) t (C) / (D) sec ( ) 8. If the umericl vlue of t (cos (/5) t (/)) is / the (A) (B) (C) (D) 9. If α stisfies the iequtio >, the vlue eists for (A) si α (B) cos α (C) sec α (D) cosec α. If f () cos cos the: (A) f (B) f cos 5 (C) f (D) f cos m

34 . Fid the vlue of the followig : (i) si si EXERCISE 8 (ii) t cos t (iii) si cos si. Solve the equtio : cot t. Solve the equtio : t t. Solve the followig equtios : (i) t t, ( > ) (ii) t t t 5. Fid the vlue of t si cos y y, if > y >. 6. If si ( t ) d y si t the fid the reltio etwee d y. 7. If rc si rc siy rc siz the prove tht:(, y, z > ) (i) y y z z yz (ii) y z y z ( y y z z ) 8. Solve the followig equtios : (i) sec sec sec sec ;,. (ii) si si si (iii) Solve for, if (t ) (cot ) 9. If α t & β si >? 5 8 for < <, the prove tht α β. Wht the vlue of α β will e if. If X cosec t cos cot sec si & Y sec cot si t cosec cos ; where. Fid the reltio etwee X & Y. Epress them i terms of ''.. Solve the followig iequlities: (i) cos > cos (ii) si > cos (iii) t > cot. (iv) si (si 5) >. (v) t (rc si ) > (vi) rccot 5 rccot 6 > (vii) t t. Fid the sum of ech of the followig series : (i) cot cos cot... cot. (ii) t t... t Prove tht the equtio, (si ) (cos ) α hs o roots for α <.. (i) Fid ll positive itegrl solutios of the equtio, t cot y t. (ii) If 'k' e positive iteger, the show tht the equtio: t t y t k hs o ozero itegrl solutio.

35 FUNCTIONS (ASSERTION AND REASON) Some questios (Assertio Reso type) re give elow. Ech questio cotis Sttemet (Assertio) d Sttemet (Reso). Ech questio hs choices (A), (B), (C) d (D) out of which ONLY ONE is correct. So select the correct choice : Choices re : (A) Sttemet is True, Sttemet is True; Sttemet is correct epltio for Sttemet. (B) Sttemet is True, Sttemet is True; Sttemet is NOT correct epltio for Sttemet. (C) Sttemet is True, Sttemet is Flse. (D) Sttemet is Flse, Sttemet is True.. Let f() cos si. Sttemet : f() is ot periodic fuctio. Sttemet : L.C.M. of rtiol d irrtiol does ot eist. Sttemet : If f() d the equtio f() f () is stisfied y every rel vlue of, the R d. Sttemet : If f() d the equtio f() f () is stisfied y every rel vlue of, the d R.. Sttemets-: If f() d F(), the F() f() lwys Sttemets-: At, F() is ot defied.. Sttemet : If f(),,, the the grph of the fuctio y f (f(f()), > is stright Sttemet : f(f()))) lie 5. Let f( ) f( ) d f( ) f( ) Sttemet : f() is periodic with period 6 Sttemet : 6 is ot ecessrily fudmetl period of f() {} 6. Sttemet : Period of the fuctio f() si e does ot eist Sttemet : LCM of rtiol d irrtiol does ot eist 7. Sttemet : Domi of f() is (, ) Sttemet : > for R 8. Sttemet : Rge of f() is [, ] Sttemet : f() is icresig for d decresig for. 9. Let, R, d let f(). Sttemet : f is oe oe fuctio. Sttemet : Rge of f is R {}. Sttemet : si cos () is o periodic fuctio. Sttemet : Lest commo multiple of the periods of si d cos () is irrtiol umer.. Sttemet : The grph of f() is symmetricl out the lie, the, f( ) f( ). Sttemet : eve fuctios re symmetric out the y-is. cos is ()!!!. Sttemet : Period of f() si ( ) Sttemet : Period of cos si is.. Sttemet : Numer of solutios of t( t ) cos equls Sttemet :?. Sttemet : Grph of eve fuctio is symmetricl out y is Sttemet : If f() cos hs ()ve solutio the totl umer of solutio of the ove equtio is. (whe f() is cotiuous eve fuctio).

36 5. If f is polyomil fuctio stisfyig f().f(y) f() f(y) f(y), y R Sttemet-: f() 5 which implies f(5) 6 Sttemet-: If f() is polyomil of degree '' stisfyig f() f(/) f(). f(/), the f() 6. Sttemet-: The rge of the fuctio si - cos - t - is [/, /] Sttemet-: si -, cos - re defied for d t - is defied for ll ''. where is rtiol 7. A fuctio f() is defied s f() where is irrtiol Sttemet- : f() is discotiuous t ll R Sttemet- : I the eighourhood of y rtiol umer there re irrtiol umers d i the vicity of y irrtiol umer there re rtiol umers. 8. Let f() si ( ) cos( ) Sttemet- : f() is periodic fuctio Sttemet-: LCM of two irrtiol umers of two similr kid eists. 9. Sttemets-: The domi of the fuctio f() cos - t - si - is [-, ] Sttemets-: si -, cos - re defied for d t - is defied for ll.. Sttemet- : The period of f() si cos [] cos si [] is / Sttemets-: The period of [] is, where [ ] deotes gretest iteger fuctio.. Sttemets-: If the fuctio f : R R e such tht f() [], where [ ] deotes the gretest iteger less th or equl to, the f - () is equls to [] Sttemets-: Fuctio f is ivertile iff is oe-oe d oto.. Sttemets- : Period of f() si {} t [] were, [ ] & { } deote we G.I.F. & frctiol prt respectively is. Sttemets-: A fuctio f() is sid to e periodic if there eist positive umer T idepedet of such tht f(t ) f(). The smllest such positive vlue of T is clled the period or fudmetl period.. Sttemets-: f() is oe-oe fuctio Sttemets-: is mootoiclly decresig fuctio d every decresig fuctio is oe-oe.. Sttemets-: f() si ( si - cos ) is periodic with fudmetl period / Sttemets-: Whe two or more th two fuctios re give i sutrctio or multiplictio form we tke the L.C.M. of fudmetl periods of ll the fuctios to fid the period. 5. Sttemets-: e l hs oe solutio. Sttemets-: If f() f() f () hve solutio o y. 6. Sttemets-: F() si. G() - H() F(X) G(), is periodic fuctio. Sttemets-: If F() is o-periodic fuctio & g() is o-periodic fuctio the h() f() ± g() will e periodic fuctio., 7. Sttemets-: f () is odd fuctio., < Sttemets-: If y f() is odd fuctio d lies i the domi of f() the f() ; Q 8. Sttemets-: f () is oe to oe d o-mootoic fuctio. C ; Q Sttemets-: Every oe to oe fuctio is mootoic. 5

37 9. Sttemet : Let f : [, ] [5, 6] [, ] [5, 6] defied s, [, ] f () 7, [5, 6] equtio f() f () hs two solutios. Sttemets-: f() f () hs solutios oly o y lie.. Sttemets-: The fuctio p q r s (ps qr ) cot tti the vlue p/r. Sttemets-: The domi of the fuctio g(y) q sy ry p is ll rel ecept /c.. Sttemets-: The period of f() si [] cos [] cos si [] is / Sttemets-: The period of [] is.. Sttemets-: If f is eve fuctio, g is odd fuctio the g (g ) is odd fuctio. the the Sttemets-: If f( ) f() for every of its domi, the f() is clled odd fuctio d if f( ) f() for every of its domi, the f() is clled eve fuctio.. Sttemets-: f : A B d g : B C re two fuctio the (gof) f og. Sttemets-: f : A B d g : B C re ijectios the f & g re lso ijectios.. Sttemets-: The domi of the fuctio f () log si is ( ), N. Sttemets-: Epressio uder eve root should e 5. Sttemets-: The fuctio f : R R give f () log ( ) >, is ivertile. Sttemets-: f is my oe ito. 6. Sttemets-: φ() si (cos ), is oe-oe fuctio. Sttemets-: φ'(), 7. Sttemets-: For the equtio k ( k) k R {} ectly oe root lie i (, ). Sttemets-: If f(k ) f(k ) < (f() is polyomil) the ectly oe root of f() lie i (k, k ). 8. Sttemets-: Domi of f () si is {, } Sttemets-: whe > d whe <. 9. Sttemets-: Rge of f() ( ) is [, ) Sttemets-: If fuctio f() is defied R d for if f() d f() is eve fuctio th rge of f() f() is [, ].. Sttemets-: Period of {}. Sttemets-: Period of []. Sttemets-: Domi of f φ. If f() [] Sttemets-: [] R. Sttemets-: The domi of the fuctio si cos t is [, ] Sttemets-: si, cos re defied for d t is defied for ll ANSWER KEY. A. D. A. C 5. A 6. A 7. A 8. C 9. B. C. A. C. B. A 5. A 6. A 7. A 8. A 9. A. A. D 6 5

38 . A. A. A 5. D 6. C 7. D 8. C 9. C. A. A. A. D. A 5. C 6. A 7. C 8. A 9. A. A. A. A SOLUTIONS. f(f()) f () f(f(f())) f (f ()) 5. f( ) f( )... () f( ) f( )... () i () f( ) f()... () i () f( ) f(8 ) f()... () () d () f( ) f(8 )... (5) Use i (5), we get f() f(6 ) f() is periodic with period 6 Oviously 6 is ot ecessry the fudmetl period. As. C As. A 6. L.C.M. of {, } does ot eist (A) is the correct optio. 7. () Clerly oth re true d sttemet II is correct epltio of Sttemet I. 8. (c) f () f() is icresig for d decresig for. 9. Suppose >. Sttemet II is true s i its cotiuous prt. Also f () lim f () d ( ), which is lwys egtive d hece mootoic lim f (). Moreover lim f () d lim f (). Hece rge of f is R {}. F is oviously oe oe s f( ) f( ). However sttemet II is ot correct resoig for sttemet I Hece () is the correct swer.. Sttemet I is true, s period of si d cos re d respectively whose L.C.M does ot eist. Oviously sttemet II is flse Hece (c) is the correct swer.. Grph of f() is symmetric out the lie if f(- ) f() i.e. if f( ) f( ) Grph of y f() is symmetric out, if f( ) f( ). Hece () is the correct swer. 7 6

39 !. Period of si ( ) ( )! Period of cos ( )!! Period of f() L.C.M of ( )! Ad ()! (!) Now, f() cops si si f() is periodic fuctio with period. Hece (c) is the correct swer.. t( t ), sice t t Oviously cos d meets t ectly two poits (B) is the correct optio.. (A)Sice cos is lso eve fuctio. Therefore solutio of cos f() is lwys sym. lso out y is. 9. () Both A d R re oviously correct.. () f() [] f( ) ([] ) [] So, period of [] is. Let f() si ( []) f si si ( [] ) si ( []) So, period is /. f() f() f is ot oe-oe f - () is ot defied As. (D). Clerly t [] R d period of si {}. As. (A). f() ( ) ( ) f () < ( ) ( ) So f() is mootoiclly decresig & every mootoic fuctio is oe-oe. So is correct.. f() si ( si - cos ) is periodic with period / ecuse f(/ ) si (/ ) ( si (/ ) - cos (/ ) ) si ( ) ( cos - si ) -si ( cos - si ) si ( si - cos ) Sometimes f( r) f() where r is less th the L.C.M. of periods of ll the fuctio, ut ccordig to defiitio of periodicity, period must e lest d positive, so r is the fudmetl period. So f is correct. 7. (D) If f() is odd fuctio, the f() f() D f 8. (C) For oe to oe fuctio if f( ) f( ) for ll, D f > ut f ( ) < f () d > f(5) > f() f() is oe-to-oe ut o-mootoic 8 7

40 9. (C), d, () there re two solutios d they do ot lie o y.. If we tke y p q the q s does ot eist if y p/r r s r p Thus sttemet- is correct d follows from sttemet- (A). f() si( [] f( /) si si ( [] ] si ( [].) i.e., period is /. f() [] f( ) ([] ) [] i.e., period is. (A). (A) Let h() f () g() h( ) f ( ) f () f () h() g( ) g( ) g() h() f g is odd fuctio.. (D) Assertio : f : A B, g : B C re two fuctios the (gof) f og (sice fuctios eed ot posses iverses. Reso : Bijective fuctios re ivertiles.. (A) for f() to e rel log (si ) si º si ( ), N. 5. (C) f is ijective sice y (, y R) log { } log { y y } f() f(y) f is oto ecuse ( ) log y. Sice {} [] { } [ ] [] [] [] Period of []. f() [] [] [] [] > It is imposile or [] So the domi of f is φ ecuse reso [] y y. As (A) As. (A) 9 8

41 STUDY PACKAGE Trget: IIT-JEE (Advced) SUBJECT: MATHEMATICS TOPIC: 9 XII M. Determits d Mtrices Ide:. Key Cocepts. Eercise I to X. Aswer Key. Assertio d Resos 5. Yrs. Que. from IIT-JEE 6. Yrs. Que. from AIEEE

42 . D e fi i t i o : Let us cosider the equtios y, y y we epress this elimit s The symol Determit is clled the determit of order two. DET. & MATRICES/Pge : of 5 Its vlue is give y: D. E p sio of D etermi t : c The symol c is clled the determit of order three. c Its vlue c e foud s: c D c c c c c c c D c c c... & so o. I this mer we c epd determit i 6 wys usig elemets of ; R, R, R or C, C, C.. M i o r s : The mior of give elemet of determit is the determit of the elemets which remi fter deletig the row & the colum i which the give elemet stds. For emple, the mior of i c c c c is & the mior of c is. c c Hece determit of order two will hve miors & determit of order three will hve 9 miors.. Cofctor: Cofctor of the elemet ij is C ij () ij. M ij ; Where i & j deotes the row & colum i which the prticulr elemet lies. Note tht the vlue of determit of order three i terms of Mior & Cofctor c e writte s: D M M M OR D C C C & so o. 5. Trspose of Det ermit: The trspose of determit is determit otied fter iterchgig the rows & colums. c T D c D c c c c 6. Sy mm et ric, Skew-Sym metric, Asy mm et ric D et ermi t s: (i) A determit is symmetric if it is ideticl to its trspose. Its i th row is ideticl to its i th (ii) colum i.e. ij ji for ll vlues of ' A determit is skew-symmetric if i it ' d is ideticl ' j ' to its trspose hvig sig of ech elemet iverted i.e. ij ji for ll vlues of ' i ' d ' j '. A skew-symmetric determit hs ll elemets zero i its pricipl digol. (iii) A determit is symmetric if it is either symmetric or skew-symmetric. 7. Prop erties of Det ermits: (i) The vlue of determit remis ultered, if the rows & colums re iter chged, c i.e. D c D c c c c (ii) If y two rows (or colums) of determit e iterchged, the vlue of determit is chged i sig oly. e.g. c c Let D c & D c The D D. c c NOTE : A skew-symmetric detemit of odd order hs vlue zero. OR

43 DET. & MATRICES/Pge : of 5 (iii) If determit hs ll the elemets zero i y row or colum the its vlue is zero, i.e. D c c. (iv) If determit hs y two rows (or colums) ideticl, the its vlue is zero, i.e. D c c c. (v) If ll the elemets of y row (or colum) e multiplied y the sme umer, the the determit is multiplied y tht umer, i.e. D c c c d D c c Kc K K The D KD (vi) If ech elemet of y row (or colum) c e epressed s sum of two terms the the determit c e epressed s the sum of two determits, i.e. c c z y c c c c c z c y (vii) The vlue of determit is ot ltered y ddig to the elemets of y row (or colum) costt multiple of the correspodig elemets of y other row (or colum), i.e. D c c c d D c c c mc c m m. The D D. Emple : Simplify c c c Solutio. Let R R R R c c c c c ( c) c c Apply C C C, C C C ( c) c c c ( c) (( c) ( ) (c ) ) ( c) ( c c c c ) ( c) ( c c c ) c c Emple : Simplify c c c c Solutio. Give deteremit is equl to c c c c c c c c c c Apply C C C, C C C c c c c

44 DET. & MATRICES/Pge : of 5 ( ) ( c) c c c c c ( ) ( c) [ c c C c c c c] ( ) ( c) [c( c c) ( c c)] ( ) ( c) (c ) ( c c) Use of fctor theorem. USE OF FACTOR THEOREM TO FIND THE VALUE OF DETERMINANT If y puttig the vlue of determit vishes the ( ) is fctor of the determit. Emple : Prove tht c c c c ( ) ( c) (c ) ( c c) y usig fctor theorem. Solutio. Let D c c c c Hece ( ) is fctor of determit Similrly, let c, D c, D Hece, ( ) ( c) (c ) is fctor of determit. But the give determit is of fifth order so c c c c ( ) ( c) (c ) (λ ( c ) µ ( c c)) Sice this is idetity so i order to fid the vlues of λ d µ. Let,, c () (λ µ) (λ µ)....(i) Let,, c ( ) ( ) (5λ µ) 5λ µ...(ii) from (i) d (ii) λ d µ Hece c c c c ( ) ( c) (c ) ( c c). Self Prctice Prolems. Fid the vlue of c c c c. As.. Simplify c c c c c c. As.. Prove tht c c c c c ( c).. Show tht c c c ( ) ( c) (c ) y usig fctor theorem M ultip lic t io Of Two Determi ts: M ultip lic t io Of Two Determi ts: M ultip lic t io Of Two Determi ts: M ultip lic t io Of Two Determi ts: m m m m m m l l l l l l c c c m m m l l l c m c m m c c m c m m c c m c m m c l l l l l l l l l

45 DET. & MATRICES/Pge : 5 of 5 We hve multiplied here rows y rows ut we c lso multiply rows y colums, colums y rows d colums y colums. If ij is deteremit of order, the the vlue of the determit A ij. This is lso kow s power cofctor formul. Emple : Fid the vlue of d prove tht it is equl to 6 8. Solutio. ) ( Emple : Prove tht y y y y y y y y y Solutio. Give determit c e splitted ito product of two determits i.e. y y y y y y y y y c c c y y y Emple : Prove tht ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ( ) ( ) ( ) ( ) ( ) ( ). Solutio. ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ( ) ( ) ( ) ( ) ( ) ( ) Emple : Prove tht R) cos(c Q) cos(c P) cos(c R) cos(b Q) cos(b P) cos(b R) cos(a Q) cos(a P) cos(a Solutio. R) cos(c Q) cos(c P) cos(c R) cos(b Q) cos(b P) cos(b R) cos(a Q) cos(a P) cos(a sicsir cosccosr sicsiq cosccosq sicsip cosccosp sib sir cosbcosr sib siq cosbcosq sib sip cosbcosp sia sir cosacosr sia siq cosacosq sia sip cosacosp sic cosc sib cosb sia cosa sir siq sip cosr cosq cosp. 5

46 DET. & MATRICES/Pge : 6 of 5 Self Prctice Prolems. Fid the vlue of c c c c c As. (c c ). If A, B, C re rel umers the fid the vlue of C) cos(b C) cos(a B) cos(c B) cos(a A) cos(c A) cos(b. As Su mm tio of D et erm i ts Su mm tio of D et erm i ts Su mm tio of D et erm i ts Su mm tio of D et erm i ts Let (r) h(r) g(r) f(r) where,,,,, re costts idepedet of r, the r r) ( r r r h(r) g(r) (r) f Here fuctio of r c e the elemets of oly oe row or colum. Noe of the elemets other the tht row or colum should e depedet o r. If more th oe colum or row hve elemets depedet o r the first epd the determit d the fid the summtio. Emple : Evlute r y cos r r C r θ Solutio : r r D y cos ) (r r r r C r r θ y cos θ Emple : D r C C C r r r evlute r D r Solutio : r r D r C C C r r r C... C C C... C C C... C C C C C 6

47 ( ) r Emple : If r r r, fid r r r Solutio. O epsio of determiet, we get D r (r ) ( r) 7 r r 8r r ( ) Self Prctice Prolem. r 6 Evlute D r (r ) y r (r ) z. Iteg rtio of determ it r As. DET. & MATRICES/Pge : 7 of 5 f() g() h() Let () c c where,, c,,, c re costts idepedet of. Hece ( ) d f() d g() d h() d c c Note : If more th oe row or oe colum re fuctio of the first epd the determit d the itegrte it. cos Emple : If f() cos, the fid / f() d cos Solutio. Here f() cos ( cos ) cos cos cos cos Emple : If / so cos d α 6 si / β γ, the fid () d Solutio. ( ) d α 6 d β d γ d 6. Differe tit io of D et ermi t: f () f () f () Let () g () h () g h α () () g h β () () γ α 6 6 β γ 7

48 f () f () f () the () g() g() g() h () h () h () Emple : If f() Solutio. f () Emple : 6 6 f() f() f() g () g () g () h () h () h (), the fid the vlue of f (). f () g() h () f () g() h () f () g() h () f () f (). Let α e repeted root of qudrtic equtio f() d A(), B() d C() e polyomil of degree, d 5 respectively, the show tht A() B() C() A( α) A ( α) Solutio. Let g() B( α) B ( α) A() A( α) A ( α) A () C( α) C ( α) B() B( α) B ( α) B () g () A( α) B( α) A ( α) B ( α) Sice g(α) g (α) C() C( α) C ( α) C () C( α) C ( α) divisile y f(). g() ( α) h() i.e. α is the repeted root of g() d h() is y polyomil epressio of degree. Also f() hve repeted root α. So g() is divisile y f(). Emple : Prove tht F depeds oly o, d F DET. & MATRICES/Pge : 8 of 5 Solutio : d simplify F. df d Hece F is idepedet of. Emple : Solutio : df df Similrly. d d Hece F is idepedet of d lso. So F is depedet oly o,, Put,, F ( ) ( ) ( ). e si If cos l( ) Put i e si cos l( ) A B C..., the fid the vlue of A d B. A B C... 8

49 A A. Differetitig the give determit w.r.t, we get e cos cos l( ) Put, we get B A, B Self Prctice Prolem si si B C... e. If c d. Fid (i) d As. [ ] (ii) c d As. [ 5]. (iii) As. [ ] Crmer's s Rule: System of Lier Eq utios (i) Two Vriles () () Cosistet Equtios: Defiite & uique solutio. [ itersectig lies ] Icosistet Equtio: No solutio. [ Prllel lie ] (c) Depedet equtio: Ifiite solutios. [ Ideticl lies ] Let y c & y c the: c c Give equtios re icosistet & (ii) c Give equtios re depedet c Three Vriles DET. & MATRICES/Pge : 9 of 5 Let, y c z d... (I) y c z d... (II) y c z d... (III) The, D, Y D, Z D. D D D (iii) () () (c) c d c Where D c d ; D c ; D d c & D d c d c d c d Cosistecy of system of Equtios If D d ltest oe of D, D, D, the the give system of equtios re cosistet d hve uique o trivil solutio. If D & D D D, the the give system of equtios re cosistet d hve trivil solutio oly. If D D D D, the the give system of equtios hve either ifiite solutios or o solutio. d c d (d) (e) (Refer Emple & Self Prctice Prolem with*) If D ut tlest oe of D, D, D is ot zero the the equtios re icosistet d hve o solutio. If give system of lier equtios hve Oly Zero Solutio for ll its vriles the the give equtios re sid to hve TRIVIAL SOLUTION. (iv) Three equtio i two vriles : If d y re ot zero, the coditio for y c ; y c & c y c to e cosistet i d y is 9 c c.

50 Emple: Solutio. Let D Fid the ture of solutio for the give system of equtios. y z y z y 5z 5 pply C C C, C C C D Now, D 5 D 5 DET. & MATRICES/Pge : of 5 *Emple : C C C D R R R, R R R D 5 D But D Hece o solutio Solve the followig system of equtios y z y z y z Solutio. D D, D, D Let z t y t y t Sice oth the lies re prllel hece o vlue of d y Hece there is o solutio of the give equtio. *Emple : Solve the followig system of equtios y z y z y z 6 Solutio. D D, D, D All the cofctors of D, D, D d D re ll zeros, hece the system will hve ifiite solutios. Let z t, y t t t where t, t R. Emple : Cosider the followig system of equtios y z 6 y z y λz µ Fid vlues of λ d µ if such tht sets of equtio hve (i) uique solutio (ii) ifiite solutio (iii) o solutio Solutio. y z 6 y z y λz µ D λ Here for λ secod d third rows re ideticl hece D for λ.

51 D D 6 µ 6 µ λ λ 6 D µ If λ the D D D for µ (i) For uique solutio D i.e. λ (ii) For ifiite solutios D λ D D D µ. (iii) For o solutio D λ Atlest oe of D, D or D is o zero µ. Self Prctice Prolems *. Solve the followig system of equtios y z y z y 5z As. t y t z t where t R. Solve the followig system of equtios y z y z y z As., y, z. Solve: ( c) (y z) c, (c ) (z ) y c, ( ) ( y) cz where c. As. c c, y c c, z c. Let y ; 5y 6, 6y 5y 8 y t, if the system of equtios i d y re cosistet the fid the vlue of t. As. t 7. Ap plic tio of Det ermit s: Followig emples of short hd writig lrge epressios re: (i) Are of trigle whose vertices re ( r, y r ); r,, is: y (ii) D y y If D the the three poits re collier. Equtio of stright lie pssig through (, y ) & (, y ) is (iii) The lies: y c... () y c... () y c... () c (iv) re cocurret if, c. y y y c Coditio for the cosistecy of three simulteous lier equtios i vriles. ² hy y² g fy c represets pir of stright lies if: h g c fgh f² g² ch² h g f f c DET. & MATRICES/Pge : of 5

52 Mtrices Ay rectgulr rrgemet of umers (rel or comple) (or of rel vlued or comple vlued epressios) is clled mtri. If mtri hs m rows d colums the the order of mtri is sid to e m y (deoted s m ). The geerl m mtri is... j j A i i i... ij... i m m m... mj... m where ij deote the elemet of i th row & j th colum. The ove mtri is usully deoted s [ ij ] m. Note : (i) The elemets,,,... re clled s digol elemets. Their sum is clled s trce of A deoted s T r (A) (ii) Cpitl letters of Eglish lphets re used to deote mtri.. Bsic Defiitios (i) Row mtri : A mtri hvig oly oe row is clled s row mtri (or row vector). Geerl form of row mtri is A [,,,..., ] (ii) Colum mtri : A mtri hvig oly oe colum is clled s colum mtri. (or colum vector) Colum mtri is i the form A... m (iii) Squ re m tri : A m tri i whi c h um er of rows & colums re equl is clled squre mtri. Geerl form of squre mtri is... A which we deote s A [ ij ]. (iv) Zero mtri : A [ ij ] m is clled zero mtri, if ij i & j. (v) Upper trigulr mtri : A [ ij ] m is sid to e upper trigulr, if ij for i > j (i.e., ll the elemets elow the digol elemets re zero). (vi) Lower trigulr mtri : A [ ij ] m is sid to e lower trigulr mtri, if ij for i < j. (i.e., ll the elemets ove the digol elemets re zero.) (vii) Digol mtri : A squre mtri [ ij ] is sid to e digol mtri if ij for i j. (i.e., ll the elemets of the squre mtri other th digol elemets re zero) (viii) Note : Digol mtri of order is deoted s Dig (,,... ). Sclr mtri :Sclr mtri is digol mtri i which ll the digol elemets re sme A [ ij ] is sclr mtri, if (i) ij for i j d (ii) ij k for i j. (i) Uit mtri (Idetity mtri) : Uit mtri is digol mtri i which ll the digol elemets re uity. Uit mtri of order '' is deoted y Ι (or Ι). i.e. A [ ij ] is uit mtri whe ij for i j & ii eg. Ι, Ι. () Comprle mtrices : Two mtrices A & B re sid to e comprle, if they hve the sme order (i.e., umer of rows of A & B re sme d lso the umer of colums). (i) Equlity of mtrices : Two mtrices A d B re sid to e equl if they re comprle d ll the correspodig elemets re equl. Let A [ ij ] m & B [ ij ] p q A B iff (i) m p, q (ii) ij ij i & j. (ii) Multiplictio of mtri y sclr : Let λ e sclr (rel or comple umer) & A [ ij ] m e mtri. Thus the product λa is defied s λa [ ij ] m where ij λ ij i & j. Note : If A is sclr mtri, the A λι, where λ is the digol elemet. (iii) Additio of mtrices : Let A d B e two mtrices of sme order (i.e. comprle mtrices). DET. & MATRICES/Pge : of 5

53 (iv) The A B is defied to e. A B [ ij ] m [ ij ] m. [c ij ] m where c ij ij ij i & j. Sustrctio of mtrices : Let A & B e two mtrices of sme order. The A B is defied s A B where B is ( ) B. (v) Properties of dditio & sclr multiplictio : Cosider ll mtrices of order m, whose elemets re from set F (F deote Q, R or C). Let M m (F) deote the set of ll such mtrices. The () A M m (F) & B M m (F) A B M m (F) (vi) () A B B A (c) (A B) C A (B C) (d) O [o] m is the dditive idetity. (e) For every A M m (F), A is the dditive iverse. (f) λ (A B) λa λb (g) λa Aλ (h) (λ λ ) A λ A λ A Multiplictio of mtrices : Let A d B e two mtrices such tht the umer of colums of A is sme s umer of rows of B. i.e., A [ ij ] m p & B [ ij ] p. The AB [c ij ] m where c ij colum vector of B. p ik kj k, which is the dot product of i th row vector of A d j th DET. & MATRICES/Pge : of 5 Note - : The product AB is defied iff umer of colums of A equls umer of rows of B. A is clled s premultiplier & B is clled s post multiplier. AB is defied / BA is defied. Note - : I geerl AB BA, eve whe oth the products re defied. Note - : A (BC) (AB) C, wheever it is defied. (vii) Properties of mtri multiplictio : Cosider ll squre mtrices of order ''. Let M (F) deote the set of ll squre mtrices of order. (where F is Q, R or C). The () A, B M (F) AB M (F) () I geerl AB BA (c) (AB) C A(BC) (d) Ι, the idetity mtri of order, is the multiplictive idetity. AΙ A Ι A A M (F) (e) For every o sigulr mtri A (i.e., A ) of M (F) there eist uique (prticulr) mtri B M (F) so tht AB Ι BA. I this cse we sy tht A & B re multiplictive iverse of oe other. I ottios, we write B A or A B. (f) If λ is sclr (λa) B λ(ab) A(λB). (g) A(B C) AB AC A, B, C M (F) (h) (A B) C AC BC A, B, C M (F). Note : (i) Let A [ ij ] m. The AΙ A & Ι m A A, where Ι & Ι m re idetity mtrices of order & m respectively. (ii) For squre mtri A, A deotes AA, A deotes AAA etc. Solved Emple # Solutio. Let A siθ / cosθ / / cosθ & B cosθ tθ cosθ siθ cosθ. Fid θ so tht A B. By defiitio A & B re equl if they hve the sme order d ll the correspodig elemets re equl. Thus we hve si θ, cosθ & t θ θ ( ). Solved Emple # f() is qudrtic epressio such tht f() f() for three uequl umers,, c. Fid f(). c c f( ) c Solutio. The give mtri equtio implies f() f() f( ) f() f() f( ) c f() cf() f( ) c f() f() f( ) for three uequl umers,, c...(i)

54 (i) is idetity f(), f() & f( ) &. f() ( ) & f(). Self Prctice Prolems : cos θ siθ. If A(θ), vrify tht A(α) A(β) A(α β). siθ cos θ Hece show tht i this cse A(α). A(β) A(β). A(α). 6. Let A, B d C [ ]. 5 The which of the products ABC, ACB, BAC, BCA, CAB, CBA re defied. Clculte the product whichever is defied. As. oly CAB is defied. CAB [5 ]. Trspose of Mtri DET. & MATRICES/Pge : of 5 Let A [ ij ] m. The the trspose of A is deoted y A ( or A T ) d is defied s A [ ij ] m where ij ji i & j. i.e. A is otied y rewritig ll the rows of A s colums (or y rewritig ll the colums of A s rows). (i) For y mtri A [ ij ] m, (A ) A (ii) Let λ e sclr & A e mtri. The (λa) λa (iii) (A B) A B & (A B) A B for two comprle mtrices A d B. (iv) (A ± A ±... ± A ) A ± A ±... ± A, where A i re comprle. (v) Let A [ ij ] m p & B [ ij ] p, the (AB) B A (vi) (A A...A ) A. A...A. A, provided the product is defied. (vii) Symmetric & skew symmetric mtri : A squre mtri A is sid to e symmetric if A A i.e. Let A [ ij ]. A is symmetric iff ij ji i & j. A squre mtri A is sid to e skew symmetric if A A i.e. Let A [ ij ]. A is skew symmetric iff ij ji i & j. h g e.g. A h f is symmetric mtri. g f c o y B o z is skew symmetric mtri. y z Note- I skew symmetric mtri ll the digol elemets re zero. ( ii ii ii ) Note- For y squre mtri A, A A is symmetric & A A is skew symmetric. Note- Every squre mtri c e uiquly epressed s sum of two squre mtrices of which oe is symmetric d other is skew symmetric. A B C, where B (A A ) & C (A A ). Solved Emple # Show tht BAB is symmetric or skew symmetric ccordig s A is symmetric or skew symmetric (where B is y squre mtri whose order is sme s tht of A). Solutio. Cse - Ι A is symmetric A A (BAB ) (B ) A B BAB BAB is symmetric. Cse - ΙΙ A is skew symmetric (BAB ) (B ) A B A A B ( A) B Self Prctice Prolems : (BAB ) BAB is skew symmetric. For y squre mtri A, show tht A A & AA re symmetric mtrices.. If A & B re symmetric mtrices of sme order, th show tht AB BA is symmetric d AB BA is skew symmetric.. Sumtri, Miors, Cofctors & Determit of Mtri (i) of Sumtri : Let A e give mtri. The mtri otied y deletig some rows or colums A is clled s sumtri of A. eg. c d A y z w p q r s The c c d z,, p r p q s y z re ll sumtrices of A. p q r

55 (ii) Determit of squre mtri : Let A [] e mtri. Determit A is defied s A. e.g. A [ ] A Let A, the A is defied s d c. c d 5 e.g. A, A (iii) Miors & Cofctors : Let A [ ij ] e squre mtri. The mior of elemet ij, deoted y M ij is defied s the determit of the sumtri otied y deletig i th row & j th colum of A. Cofctor of elemet ij, deoted y C ij (or A ij ) is defied s C ij ( ) i j M ij. e.g. A c d M d C M c, C c M, C M C c e.g. A p q r y z q r M qz yr C y z. M y y, C (y ) y etc. DET. & MATRICES/Pge : 5 of 5 (iv) Determit of y order : Let A [ ij ] e squre mtri ( > ). Determit of A is defied s the sum of products of elemets of y oe row (or y oe colum) with correspodig cofctors. e.g. A A C C C (usig first row). A C C C (usig secod colum).. (v) Some properties of determit () A A for y squre mtri A. () If two rows re ideticl (or two colums re ideticl) the A. (c) (d) (e) Let λ e sclr. Th λ A is otied y multiplyig y oe row (or y oe colum) of A y λ Note : λa λ A, whe A [ ij ]. Let A [ ij ]. The sum of the products of elemets of y row with correspodig cofctors of y other row is zero. (Similrly the sum of the products of elemets of y colum with correspodig cofctors of y other colum is zero). If A d B re two squre mtrices of sme order, the AB A B. Note : As A A, we hve A B AB (row - row method) A B A B (colum - colum method) A B A B (colum - row method) (vi) (vii) Sigulr & o sigulr mtri : A squre mtri A is sid to e sigulr or o sigulr ccordig s A is zero or o zero respectively. Cofctor mtri & djoit mtri :Let A [ ij ] e squre mtri. The mtri otied y replcig ech elemet of A y correspodig cofctor is clled s cofctor mtri of A, deoted s cofctor A. The trspose of cofctor mtri of A is clled s djoit of A, deoted s dj A. 5

56 i.e. if A [ ij ] the cofctor A [c ij ] whe c ij is the cofctor of ij i & j. Adj A [d ij ] where d ij c ji i & j. (viii) Properties of cofctor A d dj A: () A. dj A A Ι (dj A) A where A [ ij ]. () dj A A, where is order of A. I prticulr, for mtri, dj A A (c) If A is symmetric mtri, the dj A re lso symmetric mtrices. (d) If A is sigulr, the dj A is lso sigulr. (i) Remrks : Iverse of mtri (reciprocl mtri) : Let A e o sigulr mtri. The the mtri dj A is the multiplictive iverse of A (we cll it iverse of A) d is deoted y A A. We hve A (dj A) A Ι (dj A) A A dja Ι A dja A, for A is o sigulr A A dj A. A DET. & MATRICES/Pge : 6 of 5. The ecessry d sufficiet coditio for eistece of iverse of A is tht A is o sigulr.. A is lwys o sigulr.. If A di (,,..., ) where ii i, the A dig (,,..., ).. (A ) (A ) for y o sigulr mtri A. Also dj (A ) (dj A). 5. (A ) A if A is o sigulr. 6. Let k e o zero sclr & A e o sigulr mtri. The (ka) k A. 7. A A for A. 8. Let A e osigulr mtri. The AB AC B C & BA CA B C. 9. A is o-sigulr d symmetric A is symmetric.. I geerl AB does ot imply A or B. But if A is o sigulr d AB, the B. Similrly B is o sigulr d AB A. Therefore, AB either oth re sigulr or oe of them is. Solved Emple # For skew symmetric mtri A, show tht dj A is symmetric mtri. Solutio. A c c dj A (cof A) c c c c c c c cof A c c c which is symmetric. Solved Emple # 5 For two osigulr mtrices A & B, show tht dj (AB) (dj B) (dj A) Solutio. We hve (AB) (dj (AB)) AB Ι A B Ι A (AB)(dj (AB)) A B A B dj (AB) B dj A ( A B B dj (AB) B B dj A dj (AB) (djb) (dj A) A dj A) 6