For all Engineering Entrance Examinations held across India. Mathematics

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2 For ll Egieerig Etrce Exmitios held cross Idi. JEE Mi Mthemtics Sliet Fetures Exhustive coverge of MCQs subtopic wise. 95 MCQs icludig questios from vrious competitive exms. Precise theory for every topic. Net, lbelled d uthetic digrms. Hits provided wherever relevt. Additiol iformtio relevt to the cocepts. Simple d lucid lguge. Self evlutive i ture. Prited t: Repro Idi Ltd., Mumbi No prt of this book my be reproduced or trsmitted i y form or by y mes, C.D. ROM/Audio Video Cssettes or electroic, mechicl icludig photocopyig; recordig or by y iformtio storge d retrievl system without permissio i writig from the Publisher. TEID : 7

3 Prefce Mthemtics is the study of qutity, structure, spce d chge. It is oe of the oldest cdemic disciplie tht hs led towrds hum progress. Its root lies i m s fscitio with umbers. Mths ot oly dds gret vlue towrds progressive society but lso cotributes immesely towrds other scieces like Physics d Chemistry. Iterdiscipliry reserch i the bove metioed fields hs led to moumetl cotributios towrds progress i techology. Trget s Mths Vol. II hs bee compiled ccordig to the otified syllbus for JEE (Mi), which i tur hs bee frmed fter reviewig vrious tiol syllbi. Trget s Mths Vol. II comprises of comprehesive coverge of theoreticl cocepts d multiple choice questios. I the developmet of ech chpter we hve esured the iclusio of shortcuts d uique poits represeted s Importt Note for the beefit of studets. The flow of cotet d MCQs hs bee pled keepig i mid the weightge give to topic s per the JEE (Mi). MCQs i ech chpter re mix of questios bsed o theory d umericls d their level of difficulty is t pr with tht of vrious egieerig competitive exmitios. This editio of Mths Vol. II hs bee coceptulized with bsolute focus o the ssistce studets would require to swer tricky questios d would give them edge over the competitio. Lstly, I m grteful to the publishers of this book for their persistet efforts, commitmet to qulity d their uedig support to brig out this book, without which it would hve bee difficult for me to prter with studets o this jourey towrds their success. Yours fithfully, Author All the best to ll Aspirts! No. Topic Nme Pge No. Limits, Cotiuity d Differetibility Itegrl Clculus 0 4 Differetil Equtios Vector Algebr Three Dimesiol Geometry 68 7 Sttistics d Probbility Mthemticl Resoig 786

4 TARGET Publictios Physics (Vol. II) 0 Syllbus For JEE (Mi). Determits.. Determits of order two d three, properties d evlutio of determits.. Are of trigle usig determits.. Test of cosistecy d solutio of simulteous lier equtios i two or three vribles. Mtrices.. Mtrices of order two d three, Algebr of mtrices d Types of mtrices.. Adjoit d Evlutio of iverse of squre mtrix usig determits d elemetry trsformtios.. Test of cosistecy d solutio of simulteous lier equtios i two or three vribles

5 Mths (Vol. II) TARGET Publictios. Determits. Determit of order two d three: i. Determit of order two: The rrgemet of four umbers,, b, b i rows d colums eclosed betwee two verticl b brs s is determit of order two. b b The vlue of the determit b is defied s b b. (7) ( ) ii. Determit of order three: The rrgemet of ie umbers, b, c,, b, c,, b, c i rows d colums eclosed b c betwee two verticl brs s b c is determit of order three. b c Here, the elemets i the horizotl lie re sid to form row. The three rows re deoted by R, R, R respectively. Similrly, the elemets i the verticl lie re sid to form colum. The three colums re deoted by C, C, C respectively. b c The vlue of the determit b c is give by b c b c b c b c b + c (b c b c ) b ( c c ) + c ( b b ) c b + ( 9) (6 ) + (9 ) Importt Notes A determit of order c be expded log y row or colum. If ech elemet of row or colum of determit is zero, the its vlue is zero.. Miors d Cofctors: Let Here, ij deotes the elemet of the determit i i th row d j th colum.

6 TARGET Publictios Mths (Vol. II) i. Mior of elemet: The mior of elemet ij is defied s the vlue of the determit obtied by elimitig the i th row d j th colum of. It is deoted by M ij. If, the M mior of M mior of M mior of Similrly, we c fid the miors of other elemets. Fid the mior of i the determit Solutio: Mior of 0 5 (0)(7) (5)(6) ii. Cofctor of elemet: The cofctor of elemet ij i is equl to ( ) i + j M ij, where M ij is the mior of ij. It is deoted by C ij or A ij. Thus, C ij ( ) i + j M ij If, the C ( ) + M M C ( ) + M M C ( ) + M M Similrly, we c fid the cofctors of other elemets. Fid the cofctor of i the determit Solutio: Cofctor of ( ) (8 5) Importt Note The sum of the products of the elemets of y row (or colum) with the cofctors of correspodig elemets of y other row (or colum) is zero.

7 Mths (Vol. II) TARGET Publictios. Properties of determits: i. The vlue of determit is uchged, if its rows d colums re iterchged. b c Thus, if D b c, the the vlue of D b b b is D. b c c c c 4 ii. iii. iv. Let D By iterchgig rows d colums, we get D 5 8 By expdig the determits D d D, we get the vlue of ech determit equl to. Iterchgig of y two rows (or colums) will chge the sig of the vlue of the determit. 8 5 Let D 5 8. The, D 6 Let D be the determit obtied by iterchgig secod d third row of D. The, 8 5 D D D If y two rows (or colums) of determit re ideticl, the its vlue is zero.. b c 0.[ R R ] b [ C C ] 47 If ll the elemets of y row (or colum) re multiplied by umber k, the the vlue of ew determit so obtied is k times the vlue of the origil determit Let D 5 8. The, D Let D be the determit obtied by multiplyig the third row of D by k. The, D 5 8 8(5k) 5( 8k) + 6( k) k k k k kd

8 TARGET Publictios Mths (Vol. II) v. If ech elemet of y row (or colum) of determit is the sum of two terms, the the determit c be expressed s the sum of two determits. + x b+ y c + z b c x y z. b c b c + b c b c b c b c b + p c b c p c b. b + q c b c + q c b + r c b c r c vi. If costt multiple of ll elemets of y row (or colum) is dded to the correspodig elemets of y other row (or colum), the the vlue of the ew determit so obtied remis uchged Let D 5 8. Let D be the determit obtied by multiplyig the elemets of the first row of D by k d ddig these elemets to the correspodig elemets of the third row. The, D k + 5k + 6k By solvig, we get D D 4. Product or Multiplictio of two determits: b c Let the two determits of third order be b c b c, β γ β γ β γ d be their product. Rule: Tke the st row of d multiply it successively with st, d d rd rows of. The three expressios thus obtied will be elemets of st row of. I similr mer the elemets of d d rd row of re obtied. b c β γ b c β γ b c β γ + bβ + cγ + bβ + cγ + bβ + cγ + bβ + cγ + bβ + cγ + bβ + cγ + b β + c γ + b β + c γ + b β + c γ This is row by row multiplictio rule for fidig the product of two determits. We c lso multiply rows by colums or colums by rows or colums by colums. Fid the vlue of 4 Solutio:

9 Mths (Vol. II) TARGET Publictios 5. Are of trigle usig determits: 6 Are of trigle whose vertices re (x, y ), (x, y ) d (x, y ) is equl to Sice, re cot be egtive, therefore we lwys tke the bsolute vlue of the bove determit for the re. If the vertices of trigle re (, ), ( 5, 7) d (, 4), the fid its re. Solutio: Let A (, ), B ( 5, 7) d C (, 4) re of ABC sq.uits Importt Notes x y If poits (x, y ), (x, y ) d (x, y ) re collier, the x y x y 0. x y The equtio of the lie joiig poits (x, y ) d (x, y ) is x y 0. x y 6. System of lier equtios: A system of lier equtios i ukows x, y, z is of the form x + b y + c z d x + b y + c z d x + b y + c z d i. If d, d d d re ll zero, the system is clled homogeeous d o homogeeous if t lest oe d i is o-zero. ii. A system of lier equtios my hve uique solutio, or my solutios, or o solutio t ll. If it hs solutio (whether uique or ot) the system is sid to be cosistet. If it hs o solutio, it is clled icosistet system. 7. Solutio of o-homogeeous system of lier equtios: i. The solutio of the system of lier equtios x + b y c x + b y c D is give by x D, y D D, where D b b, D c b c b d D c c provided tht D 0. Coditios for cosistecy:. If D 0, the the give system of equtios is cosistet d hs uique solutio give by D x D, y D D. b. If D 0 d D D 0, the the give system of equtios is cosistet d hs ifiitely my solutios. x x x y y y c. If D 0 d oe of D d D is o-zero, the the give system of equtios is icosistet..

10 TARGET Publictios ii. The solutio of the system of lier equtios Mths (Vol. II) x + b y + c z d x + b y + c z d x + b y + c z d D is give by x D, y D D d z D D, where D b c b c b c, D d b c d b c d b c l, D d c d c d c d D b d b d b d provided tht D 0. Coditios for cosistecy:. If D 0, the the give system of equtios is cosistet d hs uique solutio give by D x D, y D D d z D D. b. If D 0 d D D D 0, the the give system of equtios is either cosistet with ifiitely my solutios or hs o solutio. c. If D 0 d t lest oe of the determits D, D d D is o-zero, the the give system of equtios is icosistet. egs. i. Usig Crmer s Rule, solve the followig lier equtios: x y 5, x y Solutio: We hve, D D d D 5 4 by crmer s rule, D x D 7 D d y D 4 7 x d y. ii. Verify whether the system of equtios: x y z, y z, x 5y is cosistet or icosistet. Solutio: We hve, D 0 0, D Sice, D 0 d D 0 Hece, the give system of equtios is icosistet. 7

11 Mths (Vol. II) TARGET Publictios 8. Solutio of homogeeous system of lier equtios: If x + b y + c z 0 x + b y + c z 0 x + b y + c z 0 is homogeeous system of equtios, such tht b c D b c 0, the x y z 0 is the oly solutio d it is kow s the trivil solutio. b c 8 If D 0, the the system is cosistet with ifiitely my solutios. Solve the followig system of homogeeous equtios: x 4y + 5z 0 x + y z 0 x + y + z Solutio: D Mtrices the give system of equtios hs oly the trivil solutio i.e., x y z 0.. Mtrix: A rectgulr rrgemet of m umbers (rel or complex) i m rows d colums is clled mtrix. This rrgemet is eclosed by [ ] or ( ). Geerlly mtrices re represeted by cpitl letters A, B, C, etc. d its elemets re represeted by smll letters, b, c, etc.. Order of mtrix: If mtrix A hs m rows d colums, the A is of order m or simple m mtrix (red s m by mtrix). A mtrix A of order m is usully writte s... j j... A i i i... ij... i m m m... mj... m or A [ ij ] m, where i,,. m j,,. Here, ij deotes the elemet of the mtrix A i i th row d j th colum. A mtrix of order m cotis m elemets. Every row of such mtrix cotis elemets d every colum cotis m elemets. Order of the mtrix is Types of mtrices: i. Row mtrix: A mtrix hvig oly oe row is clled row-mtrix or row vector. Thus, A [ ij ] m is row mtrix, if m.

12 TARGET Publictios ii. iii. iv. [ ],[ 5 ] re row mtrices of order, respectively. Colum mtrix: A mtrix hvig oly oe colum is clled colum mtrix or colum vector. Thus, A [ ij ] m is colum mtrix, if [, ] re colum mtrices of order, respectively. 5 Mths (Vol. II) Rectgulr mtrix: A mtrix A [ ij ] m is clled rectgulr mtrix, if umber of rows is ot equl to umber of colums (m ). [ ], re rectgulr mtrices of order, respectively. 4 Squre mtrix: A mtrix A [ ij ] m is clled squre mtrix, if umber of rows is equl to umber of colums (m ). 5, 4 re squre mtrices of order, respectively. 4 5 v. Null mtrix or Zero mtrix: A mtrix whose ll elemets re zero is clled ull mtrix or zero mtrix. It is deoted by O. Thus, A [ ij ] m is zero mtrix, if ij 0 i d j 0 0 [0], 0 0 re zero mtrices of order, respectively. vi. Digol mtrix: A squre mtrix i which ll its o-digol elemets re zero is clled digol mtrix. Thus, squre mtrix A [ ij ] is digol mtrix, if ij 0 i j , 0 0 re digol mtrices of order d respectively. 0 0 Importt Notes A digol mtrix of order hvig d, d,., d s digol elemets is deoted by dig [d, d,, d ] is digol mtrix d is deoted by dig [, 4, 7] Number of zeros i digol mtrix of order is. 9

13 Mths (Vol. II) TARGET Publictios vii. Sclr mtrix: A squre mtrix A [ ij ] is clled sclr mtrix, if ll its o-digol elemets re zero d digol elemets re sme. Thus, squre mtrix A [ ij ] is sclr mtrix, if ij 0, i j, where λ is costt. λ, i j , 0 0 re sclr mtrices of order d respectively. 0 0 Importt Note A sclr mtrix is lwys digol mtrix. viii. Uit mtrix or Idetity mtrix: A squre mtrix A [ ij ] is clled idetity or uit mtrix, if ll its o-digol elemets re zero d digol elemets re oe. 0, i j Thus, squre mtrix A [ ij ] is uit mtrix, if ij, i j A uit mtrix is deoted by I , 0 0 re idetity mtrices of order d respectively. 0 0 Importt Note Every uit mtrix is digol s well s sclr mtrix. ix. Trigulr mtrix: A squre mtrix is sid to be trigulr mtrix if ech elemet bove or below the digol is zero.. Upper trigulr mtrix: A squre mtrix A [ ij ] is clled upper trigulr mtrix, if every elemet below the digol is zero. Thus, squre mtrix A [ ij ] is upper trigulr mtrix, if ij 0 i > j is upper trigulr mtrix b. Lower trigulr mtrix: A squre mtrix A [ ij ] is clled lower trigulr mtrix, if every elemet bove the digol is zero. Thus, squre mtrix A [ ij ] is lower trigulr mtrix, if ij 0 i < j is lower trigulr mtrix. 4

14 TARGET Publictios Mths (Vol. II) Importt Notes ( ) Miimum umber of zeros i trigulr mtrix of order is. A digol mtrix is upper s well s lower trigulr mtrix. x. Sigulr mtrix: A squre mtrix A is clled sigulr mtrix, if A 0. If A, the A 0 A is sigulr mtrix. xi. No-sigulr mtrix: A squre mtrix A is clled o-sigulr mtrix, if A 0. 5 If A 4, the A A is o-sigulr mtrix. 4. Trce of mtrix: The sum of ll digol elemets of squre mtrix A is clled the trce of mtrix A. It is deoted by tr(a). Thus, tr(a) If A i ii , the tr(a) Properties of trce of mtrix: Let A [ ij ] m d B [b ij ] m d λ be sclr. The, i. tr(a ± B) tr(a) ± tr(b) ii. iii. tr(λa) λ tr(a) tr(ab) tr(ba) iv. tr(a) tr(a T ) v. tr(i ) vi. tr(o) 0 vii. tr(ab) tr(a). tr(b)

15 Mths (Vol. II) TARGET Publictios 5. Submtrix: A mtrix which is obtied from give mtrix by deletig y umber of rows or colums or both is clled submtrix of the give mtrix. 5 5 is submtrix of the mtrix Equlity of mtrices: Two mtrices A [ ij ] d B [b ij ] re sid to be equl, if i. they re of the sme order ii. their correspodig elemets re equl (i.e., ij b ij i, j) egs. b. If A c d d B 4 re equl mtrices, the, b, c d d 4 b. C 4 d D 5 re ot equl mtrices becuse their orders re ot sme. 7. Algebr of mtrices: i. Additio of mtrices: Let A [ ij ] m d B [b ij ] m be two mtrices. The their sum (deoted by A + B) is defied to be mtrix [c ij ] m, where c ij ij + b ij for i m, j. If A d B 8 4, the A + B Similrly, their subtrctio A B is defied s A B [ ij b ij ] m i, j. Importt Note Mtrix dditio d subtrctio c be possible oly whe mtrices re of the sme order. Properties of mtrix dditio: If A, B d C re three mtrices of sme order, the. A + B B + A (Commuttive lw) b. (A + B) + C A + (B + C) (Associtive lw) c. A + O O + A A, where O is zero mtrix of the sme order s A. d. A + ( A) O ( A) + A, where ( A) is obtied by chgig the sig of every elemet of A which is dditive iverse of the mtrix. e. A+ B A+ C B C B+ A C+ A (Ccelltio lw)

16 TARGET Publictios ii. Mths (Vol. II) Multiplictio of mtrices: Let A d B be y two mtrices, the their product AB will be defied oly whe umber of colums i A is equl to the umber of rows i B. If A [ ik ] m d B [b kj ] p, the their product AB is of order m p d is defied s (AB) ij k b ik kj i b j + i b j +. + i b j (i th row of A) (j th colum of B) Fid AB, if A d B Solutio: Here, umber of colums of A umber of rows of B. AB is defied s mtrix AB Properties of mtrix multiplictio: If A, B d C re three mtrices such tht their product is defied, the. AB BA (geerlly ot commuttive) b. (AB)C A(BC) (Associtive lw) c. AI IA A, where A is squre mtrix d I is idetity mtrix of sme order. d. A (B + C) AB + AC (A + B)C AC + BC (Distributive lw) e. AB AC B C (Ccelltio lw is ot pplicble) f. If AB 0, the it does ot imply tht A 0 or B 0 Importt Notes Multiplictio of two digol mtrices is digol mtrix. Multiplictio of two sclr mtrices is sclr mtrix. If A d B re squre mtrices of the sme order, the i. (A + B) A + AB + BA + B ii. (A B) A AB BA + B iii. (A+B) (A B) A AB + BA B iv. A ( B) ( A) B (AB) (A + B) A + AB + B uless AB BA. Sclr multiplictio of mtrices: Let A [ ij ] m be mtrix d λ be umber (sclr), the the mtrix obtied by multiplyig every elemet of A by λ is clled the sclr multiple of A by λ. It is deoted by λa. Thus, if A [ ij ] m, the λa Aλ [λ ij ] m i, j

17 Mths (Vol. II) TARGET Publictios Properties of sclr multiplictio: If A, B re two mtrices of sme order d, β re y umbers, the. (A ± B) A ± B. ( ± β)a A ± βa. (βa) β(a) (βa) 4. ( )A (A) ( A) 5. O.A O 6..O O b. Positive itegrl powers of mtrices: The positive itegrl powers of mtrix A re defied oly whe A is squre mtrix. The, we defie A A d A + A.A, where N From this defiitio, A A.A, A A. A A.A.A For y positive itegers m d,. A m A A m +. (A m ) A m (A ) m. I I, I m I 4. A 0 I, where A is squre mtrix of order 8. Trspose of mtrix: A mtrix obtied from the mtrix A by iterchgig its rows d colums is clled the trspose of A. It is deoted by A t or A T or A. Thus, if the order of A is m, the the order of A T is m. 4 If A, the AT 4 Properties of trspose of mtrix: If A d B re two mtrices, the i. (A T ) T A ii. (ka) T ka T, where k is sclr iii.. (A + B) T A T + B T b. (A B) T A T B T A d B beig of the sme order iv. (AB) T B T A T, A d B beig coformble for the product AB 4 v. (A ) T (A T ), N vi.. Trce A T Trce A b. Trce AA T 0

18 TARGET Publictios 9. Symmetric mtrix: A squre mtrix A [ ij ] is clled symmetric mtrix, if A A T or ij ji i, j. 4 If A 4 5, the A T A A T A is symmetric mtrix. Importt Notes A uit mtrix is lwys symmetric mtrix. ( + ) Mximum umber of differet elemets i symmetric mtrix of order is. 0. Skew-symmetric mtrix: A squre mtrix A [ ij ] is clled skew-symmetric mtrix, if A A T or ij ji i j j. 0 If A 0, the A T 0 0 A T 0 0 A A T A is skew-symmetric mtrix. Importt Notes All digol elemets of skew-symmetric mtrix re lwys zero. Trce of skew-symmetric mtrix is lwys zero. Mths (Vol. II) Properties of symmetric d skew-symmetric mtrices: i. If A is squre mtrix, the A + A T, AA T,A T A re symmetric mtrices d A A T is skew-symmetric mtrix. ii. iii. iv. The mtrix B T AB is symmetric or skew-symmetric ccordig s A is symmetric or skew-symmetric. If A is skew-symmetric mtrix, the. A is symmetric mtrix for N. b. A + is skew-symmetric mtrix for N. If A d B re symmetric mtrices, the. A ± B, AB + BA re symmetric mtrices b. AB BA is skew- symmetric mtrix c. AB is symmetric mtrix iff AB BA. v. If A d B re skew-symmetric mtrices, the. A ± B, AB BA re skew-symmetric mtrices. b. AB + BA is symmetric mtrix. vi. A squre mtrix A c be expressed s the sum of symmetric d skew-symmetric mtrix s A ( A A T + ) + ( A A T ) 5

19 Mths (Vol. II) TARGET Publictios. Orthogol mtrix: A squre mtrix A is clled orthogol mtrix, if AA T A T A I cosθ si θ If A si θ cosθ, the A T cosθ si θ si θ cosθ A.A T cosθ si θ cosθ si θ si θ cosθ si θ cosθ 0 0 I Similrly, A T A I A is orthogol mtrix. Importt Note A uit mtrix is lwys orthogol mtrix. Properties of orthogol mtrix: i. If A is orthogol mtrix, the A T d A re lso orthogol mtrices. ii. If A d B re two orthogol mtrices, the AB d BA re lso orthogol mtrices.. Idempotet mtrix: A squre mtrix A is clled idempotet mtrix, if A A. If A, the A A.A A A is idempotet mtrix. Importt Note A uit mtrix is lwys idempotet mtrix.. Nilpotet mtrix: A squre mtrix A is sid to be ilpotet mtrix of idex p, if p is the lest positive iteger such tht A p O. 4 8 If A 4, the A A.A O A is ilpotet mtrix of idex. Importt Note Determit of every ilpotet mtrix is zero. 6

20 TARGET Publictios 4. Ivolutory mtrix: A squre mtrix A is sid to be ivolutory mtrix, if A I. If A A A.A , the b b b I 0 0 A is ivolutory mtrix. Every uit mtrix is ivolutory. Importt Note Mths (Vol. II) 5. Cojugte of mtrix: The mtrix obtied from give mtrix A by replcig ech etry cotiig complex umbers with its complex cojugte is clled cojugte of A. It is deoted by A. If A + i i + 4i 4 5i 5 6i 6 7i +, the A i 7 i + i 4i 4 5i 5 6i 6 7i i 7 Trspose cojugte of mtrix: The trspose of the cojugte of mtrix A is clled trspose cojugte of A. It is deoted by A θ. If A + i i + 4i 4 5i 5 6i 6 7i +, the A θ i 7 i 4+ 5i 8 i 5 6i 7 8i + 4i 6+ 7i 7 6. Hermiti mtrix: A squre mtrix A [ ij ] is sid to be hermiti mtrix, if A A θ or ij ji i, j. If A A A θ + 4i 4i 5, the Aθ A is hermiti mtrix. + 4i 4i 5 Importt Note Determit of hermiti mtrix is purely rel. 7

21 Mths (Vol. II) TARGET Publictios 7. Skew-Hermiti mtrix: A squre mtrix A [ ij ] is sid to be skew-hermiti mtrix if A A θ or ij ji i, j i i If A i 0, the A θ i + i + i 0 i i i 0 A A θ A is skew-hermiti mtrix. 8. Adjoit of squre mtrix: The djoit of squre mtrix A [ ij ] is the trspose of the mtrix of cofctors of elemets of A. It is deoted by dj A. Let A [ ij ] be squre mtrix d C ij be the cofctor of ij i A. The, dj A [C ij ] T T C C C C C C If A, the dj A C C C C C C C C C C C C If A, fid dj A. 4 Solutio: Here, C ( ) + 4 6, C ( ) + 4, C ( ) +, C ( ) + 4, C ( ) + 4 5, C ( ) +, C ( ) + 5, C ( ) + 4, C ( ) + 6 dj A T Properties of djoit mtrix: If A d B re squre mtrices of order such tht A 0 d B 0, the i. A(dj A) A I (dj A)A ii. dj A A 8 iii. dj (dj A) A A

22 TARGET Publictios Mths (Vol. II) iv. dj (dj A) A v. dj (A T ) (dj A) T ( ) vi. dj (AB) (dj B) (dj A) vii. dj (A m ) (dj A) m, m N viii. dj (ka) k (dj A), k R ix. dj (I ) I x. dj (O) O Importt Notes Adjoit of digol mtrix is digol mtrix. Adjoit of trigulr mtrix is trigulr mtrix. Adjoit of sigulr mtrix is sigulr mtrix. Adjoit of symmetric mtrix is symmetric mtrix. 9. Iverse of mtrix: Let A be -rowed squre mtrix. The, if there exists squre mtrix B of the sme order such tht AB I BA, mtrix B is clled the iverse of mtrix A. It is deoted by A. Thus, AA I A A A squre mtrix A hs iverse iff A is o-sigulr i.e., A exists iff A 0. Iverse by djoit method: The iverse of o-sigulr squre mtrix A is give by A If A 4, fid A. Solutio: A A exists Here, C ( ) + (), C ( ) + ( 4) 4, C ( ) + ( ), C ( ) + () T 4 dj A 4 A dja A 8 4 Importt Notes (dja), if A 0. A Mtrix A is ivertible if A exists. The iverse of squre mtrix, if exists, it is uique. A ilpotet mtrix is lwys o - ivertible. 9

23 Mths (Vol. II) TARGET Publictios Properties of iverse mtrix: If A d B re ivertible mtrices of the sme order, the i. (A ) A ii. (A T ) (A ) T iii. (AB) B A iv. (A ) (A ), N v. dj (A ) (dja) vi. A A Importt Notes Iverse of digol mtrix is digol mtrix. Iverse of trigulr mtrix is trigulr mtrix. Iverse of sclr mtrix is sclr mtrix. Iverse of symmetric mtrix is symmetric mtrix. 0. Elemetry trsformtios: The elemetry trsformtios re the opertios performed o rows (or colums) of mtrix. i. Iterchgig y two rows (or colums). It is deoted by R i R j (C i C j ). ii. iii. Multiplyig the elemets of y row (or colum) by o-zero sclr. It is deoted by R i kr i (C i kc i ) Multiplyig the elemets of y row (or colum) by o-zero sclr k d ddig them to correspodig elemets of other row (or colum). It is deoted by R i + kr j (C i + kc j ). Iverse of o-sigulr squre mtrix by elemetry trsformtios: Let A be o-sigulr squre mtrix of order. 0 To fid A by elemetry row (or colum) trsformtios: i. Cosider, AA I ii. Perform suitble elemetry row (or colum) trsformtios o mtrix A, so s to covert it ito idetity mtrix of order. iii. The sme row (or colum) trsformtios should be performed o the R.H.S. i.e. o I. Let, I gets coverted ito mtrix B. iv. Thus, AA I reduces to IA B i.e. A B. If A A. Solutio: A 4 0 A exists. Cosider, AA I 0 A 0

24 TARGET Publictios Mths (Vol. II) Applyig R R + R, we get A 0 Applyig R R + R, we get 0 A Applyig R R + R, we get 0 0 A A. Rk of mtrix: A positive iteger r is sid to be the rk of o zero mtrix A, if i. there exists t lest oe mior i A of order r which is ot zero d ii. every mior of order (r + ) or more is zero. It is deoted by ρ(a) r. Fid the rk of mtrix A Solutio: A rk of A < rk of A. () () + () 0 Importt Note The rk of the ull mtrix is ot defied d the rk of every o-ull mtrix is greter th or equl to. Properties of rk of mtrix: i. If I is uit mtrix of order, the ρ(i ). ii. If A is o-sigulr mtrix, the ρ(a). iii. The rk of sigulr squre mtrix of order cot be. iv. Elemetry opertios do ot chge the rk of mtrix. v. If A T is trspose of A, the ρ(a T ) ρ(a). vi. If A is m mtrix, the r(a) mi (m, ).

25 Mths (Vol. II) TARGET Publictios. Echelo form of mtrix: A o-zero mtrix A is sid to be i echelo form if either A is the ull mtrix or A stisfies the followig coditios: i. Every o-zero row i A precedes every zero row. ii. The umber of zeros before the first o-zero elemet i row is less th the umber of such zeros i the ext row. The umber of o-zero rows of mtrix give i the echelo form is its rk. 0 5 The mtrix 0 0 is i the echelo form becuse it hs two o-zero rows, so the rk is.. System of simulteous lier equtios: Cosider the followig system of m lier equtios i ukows s give below: x + x +. + x b x + x +. + x b m x + m x +. + m x b m This system of equtios c be writte i mtrix form s AX B, x b where A, X x b d B m m m m x b m m The m mtrix A is clled the coefficiet mtrix of the system of lier equtios. i. Solutio of o-homogeeous system of lier equtios:. Mtrix method: If AX B, the X A B gives uique solutio, provided A is o-sigulr. But if A is sigulr mtrix i.e., if A 0, the the system of equtio AX B my be cosistet with ifiitely my solutios or it my be icosistet. b. Rk method: Rk method for solutio of o-homogeeous system AX B. Write dow A, B. Write the ugmeted mtrix [A : B]. Reduce the ugmeted mtrix to echelo form by usig elemetry row opertios. 4. Fid the umber of o-zero rows i A d [A : B] to fid the rks of A d [A : B] respectively. 5. If ρ(a) ρ(a : B), the the system is icosistet. 6. If ρ(a) ρ(a : B) umber of ukows, the the system hs uique solutio. If ρ(a) ρ(a : B) < umber of ukows, the the system hs ifiite umber of solutios.

26 TARGET Publictios Mths (Vol. II) c. Criterio of cosistecy: Let AX B be system of -lier equtios i ukows.. If A 0, the the system is cosistet d hs the uique solutio give by X A B.. If A 0 d (dj A) B O, the the give system of equtios is cosistet d hs ifiitely my solutios.. If A 0 d (dj A) B O, the the give system of equtios is icosistet. egs.. Solve the followig system of equtios by mtrix method: x 4y 5 d 4x + y Solutio: The give system of equtios c be writte i the mtrix form s AX B, 4 where A 4, X x 5 d B y Now, A 0 The give system of equtios hs uique solutio give by X A B. Here, C, C 4, C 4, C T 4 dj A A dja 4 A 4 X A B x y x d y b. For wht vlue of λ, the system of equtios x + y + z 6, x + y + z 0, x + y + λz is icosistet? x 6 Solutio: The give system of equtios c be writte s y 0 λ z Applyig R R R, R R R, we get x 6 0 y 4 0 λ z 6 Applyig R R R, we get x 6 0 y λ z For λ, rk of mtrix A is d tht of the ugmeted mtrix is. So, the system is icosistet.

27 Mths (Vol. II) TARGET Publictios c. Solve the system of equtios x y 5, 4x y 7. Solutio: The give system of equtios c be writte i the mtrix form s AX B, where A 4, X x 5, B y 7 Now, A 0 A is sigulr. Either the give system of equtios hs o solutio or ifiite umber of solutios. Here, C, C 4, C, C dj A (dj A) B 4 T Hece, the give system of equtios is icosistet. 4. Solutio of homogeeous system of lier equtios: i. Mtrix method: Let AX O be homogeeous system of -lier equtios with -ukows. If A is o-sigulr mtrix, the the system of equtios hs uique solutio X O i.e., x x. x 0. This solutio is kow s trivil solutio. A system AX O of homogeeous lier equtios i ukows, hs o-trivil solutio iff the coefficiet mtrix A is sigulr. ii. Rk method: I cse of homogeeous system of lier equtios, the rk of the ugmeted mtrix is lwys sme s tht of the coefficiet mtrix. So, homogeeous system of lier equtios is lwys cosistet. If r(a) umber of vribles, the AX O hs uique solutio X 0 i.e., x x. x 0 If r(a) r < ( umber of vribles), the the system of equtios hs ifiitely my solutios. 5. Properties of determit of mtrix: i. If A d B re squre mtrices of the sme order, the AB A B. ii. If A is squre mtrix of order, the A A T. iii. iv. If A is squre mtrix of order, the ka k A. If A d B re squre mtrices of sme order, the AB BA. v. If A is skew-symmetric mtrix of odd order, the A 0. 4 vi. A A, N

28 TARGET Publictios Formule Mths (Vol. II). Determits. Determit of order two d three: i. If A, the det A ii. If A det A, the Miors d Cofctors: i. Mior of elemet: If, the M mior of M mior of M mior of d so o. ii. Cofctor of elemet: If, the C ( ) + M M C ( ) + M M C ( ) + M M d so o.. Properties of determits: i. The vlue of determit is uchged, if its rows d colums re iterchged. ii. Iterchgig of y two rows (or colums) will chge the sig of the vlue of the determit. iii. If y two rows (or colums) of determit re ideticl, the its vlue is zero. iv. If ll the elemets of y row (or colum) re multiplied by umber k, the the vlue of ew determit so obtied is k times the vlue of the origil determit. v. If ech elemet of y row (or colum) of determit is the sum of two terms, the the determit c be expressed s the sum of two determits. vi. If costt multiple of ll elemets of y row (or colum) is dded to the correspodig elemets of y other row (or colum), the the vlue of the ew determit so obtied remis uchged. 5

29 Mths (Vol. II) 4. Product of two determits: b c Let the two determits of third order be b c b c β γ b c b c β γ b c β γ + bβ + c γ + bβ + c γ + bβ + c γ + b β + c γ + b β + c γ + b β + c γ, + b β + c γ + b β + c γ + b β + c γ β γ β γ β γ TARGET Publictios d be their product. 5. Are of trigle: Are of trigle whose vertices re (x, y ), (x, y ), (x, y ) is give by Whe the re of the trigle is zero, the the poits re collier. x x x y y y. 6. Solutio of o-homogeeous system of lier equtios: i. The solutio of the system of lier equtios x + b y c x + b y c D is give by x D, y D D, where D b b, D c b c b d D c c provided tht D 0. Coditios for cosistecy:. If D 0, the the give system of equtios is cosistet d hs uique solutio give by D x D, y D D. b. If D 0 d D D 0, the the give system of equtios is cosistet d hs ifiitely my solutios. c. If D 0 d oe of D d D is o-zero, the the give system of equtios is icosistet. ii. The solutio of the system of lier equtios x + b y + c z d x + b y + c z d x + b y + c z d D is give by x D, y D D d z D D, b c d b cl where D b c, D d b c, D b c d b c d c d c d c d D b d b d b d 6 provided tht D 0.

30 TARGET Publictios Mths (Vol. II) Coditios for cosistecy:. If D 0, the the give system of equtios is cosistet d hs uique solutio give by D x D, y D D d z D D. b. If D 0 d D D D 0, the the give system of equtios is either cosistet with ifiitely my solutios or hs o solutio. c. If D 0 d t lest oe of the determits D, D d D is o-zero, the the give system of equtios is icosistet. 7. Solutio of homogeeous system of lier equtios: If x + b y + c z 0 x + b y + c z 0 x + b y + c z 0 is homogeeous system of equtios, such tht b c D b c 0, the x y z 0 is the oly solutio d it is kow s the trivil solutio. b c If D 0, the the system is cosistet with ifiitely my solutios.. Mtrices. Mtrix: A rectgulr rrgemet of m umbers (rel or complex) i m rows d colums is clled mtrix.. Types of mtrices: i. Row mtrix: A mtrix hvig oly oe row is clled row-mtrix or row vector. ii. iii. iv. Colum mtrix: A mtrix hvig oly oe colum is clled colum mtrix or colum vector. Rectgulr mtrix: A mtrix A [ ij ] m is clled rectgulr mtrix, if umber of rows is ot equl to umber of colums (m ). Squre mtrix: A mtrix A [ ij ] m is clled squre mtrix, if umber of rows is equl to umber of colums (m ). v. Null mtrix or zero mtrix: A mtrix whose ll elemets re zero is clled ull mtrix or zero mtrix. vi. vii. Digol mtrix: A squre mtrix A [ ij ] is digol mtrix, if ij 0 i j. Sclr mtrix: A squre mtrix A [ ij ] is clled sclr mtrix, if ll its o-digol elemets re zero d digol elemets re sme. viii. Uit mtrix or Idetity mtrix: A squre mtrix A [ ij ] is clled idetity or uit mtrix, if ll its o-digol elemets re zero d digol elemets re oe. 7

31 Mths (Vol. II) TARGET Publictios ix. Trigulr mtrix: A squre mtrix is sid to be trigulr mtrix if ech elemet bove or below the digol is zero.. A squre mtrix A [ ij ] is clled upper trigulr mtrix, if ij 0 i > j b. A squre mtrix A [ ij ] is clled lower trigulr mtrix, if ij 0 i < j x. Sigulr mtrix: A squre mtrix A is clled sigulr mtrix, if A 0. xi. No-sigulr mtrix: A squre mtrix A is clled o-sigulr mtrix, if A 0.. Trce of mtrix: The sum of ll digol elemets of squre mtrix A is clled the trce of mtrix A. It is deoted by tr(a). Thus, tr(a) i ii Submtrix: A mtrix which is obtied from give mtrix by deletig y umber of rows or colums or both is clled submtrix of the give mtrix. 5. Equlity of mtrices: Two mtrices A [ ij ] d B [b ij ] re sid to be equl, if i. they re of the sme order ii. their correspodig elemets re equl. 6. Algebr of mtrices: i. Additio of mtrices: Let A [ ij ] m d B [b ij ] m be two mtrices. The their sum (deoted by A + B) is defied to be mtrix [c ij ] m, where c ij ij + b ij for i m, j. Similrly, their subtrctio A B is defied s A B [ ij b ij ] m i, j. ii. Multiplictio of mtrices: Let A d B be y two mtrices, the their product AB will be defied oly whe umber of colums i A is equl to the umber of rows i B. If A [ ik ] m d B [b kj ] p, the their product AB is of order m p d is defied s (AB) ij ikbkj i b j + i b j +. + i b j k 7. Trspose of mtrix: A mtrix obtied from the mtrix A by iterchgig its rows d colums is clled the trspose of A. Thus, if the order of A is m, the the order of A T is m. 8. Symmetric mtrix: A squre mtrix A [ ij ] is clled symmetric mtrix, if A A T or ij ji i, j. 9. Skew symmetric mtrix: A squre mtrix A [ ij ] is clled skew-symmetric mtrix, if A A T or ij ji i j j. 0. Orthogol mtrix: A squre mtrix A is clled orthogol mtrix, if AA T A T A I. Idempotet mtrix: A squre mtrix A is clled idempotet mtrix, if A A. 8

32 TARGET Publictios Mths (Vol. II). Nilpotet mtrix: A squre mtrix A is sid to be ilpotet mtrix of idex p, if p is the lest positive iteger such tht A p O.. Ivolutory mtrix: A squre mtrix A is sid to be ivolutory mtrix, if A I. 4. Cojugte of mtrix: The mtrix obtied from give mtrix A by replcig ech etry cotiig complex umbers with its complex cojugte is clled cojugte of A.. 5. Hermiti mtrix: A squre mtrix A [ ij ] is sid to be hermiti mtrix, if A A θ or ij ji i, j. 6. Skew hermiti mtrix: A squre mtrix A [ ij ] is sid to be skew-hermiti mtrix if A A θ or ij ji i, j 7. Adjoit of mtrix: The djoit of squre mtrix A [ ij ] is the trspose of the mtrix of cofctors of elemets of A. Let A [ ij ] be squre mtrix d C ij be the cofctor of ij i A. The, dj A [C ij ] T T C C C C C C If A, the dj A C C C C C C C C C C C C 8. Iverse of mtrix: Let A be -rowed squre mtrix. The, if there exists squre mtrix B of the sme order such tht AB I BA, mtrix B is clled the iverse of mtrix A. A squre mtrix A hs iverse iff A is o-sigulr i.e., A exists iff A 0. The iverse of o-sigulr squre mtrix A is give by A (dja), if A 0. A 9. i. Solutio of o-homogeeous system of lier equtios: If AX B, the X A B gives uique solutio, provided A is o-sigulr. But if A is sigulr mtrix i.e., if A 0, the the system of equtio AX B my be cosistet with ifiitely my solutios or it my be icosistet. ii. Criterio of cosistecy: Let AX B be system of -lier equtios i ukows.. If A 0, the the system is cosistet d hs the uique solutio give by X A B. b. If A 0 d (dj A) B 0, the the give system of equtios is cosistet d hs ifiitely my solutios. c. If A 0 d (dj A) B 0, the the give system of equtios is icosistet. Solutio of homogeeous system of lier equtios: Let AX O be homogeeous system of -lier equtios with -ukows. If A is o-sigulr mtrix, the the system of equtios hs uique solutio X O i.e., x x. x 0. This solutio is kow s trivil solutio. A system AX O of homogeeous lier equtios i ukows, hs o-trivil solutio iff the coefficiet mtrix A is sigulr. 9

33 Mths (Vol. II) TARGET Publictios Shortcuts. b b c c ( b) (b c) (c ). b c b c b b c c ( b) (b c) (c ) ( + b + c). b c b c c b bc b c ( + b + c) ( + b + c b bc c) ( + b + c) [( b) + (b c) + (c ) ] 4. bc bc b c bc c b bc b b b c c c bc ( b) (b c) (c ) 5. If b c b c b c d A B C A B C A B C, where A, B, C re co-fctors of, b, c, etc. the. 6. If squre mtrix A is orthogol i.e., if A T A I, the det A is or. 7. If A is ivolutory mtrix, the (I + A) d (I A) re idempotet d (I + A). (I A) If squre mtrix A is uitry i.e., if A θ A I, the det A. 9. i. If A is squre mtrix of order, the dj A A. ii. If A is squre mtrix of order, the dj A A. 0. If A c b d, the A ( d bc ) d b c, (d bc 0). If A b 0, the A 0 0 c b c d A b 0 0 c. If A m I for some positive iteger m, where A is squre mtrix, the A is ivertible d A A m. 0

34 TARGET Publictios Multiple Choice Questios. Determits.. Determits of order two d three, properties d evlutio of determits 4 6. The vlue of the determit is 4 (A) 75 (B) 5 (C) 0 (D) (A) 0 (B) 87 (C) 54 (D) 54 bc. b c [RPET 00] c b (A) 0 (B) bc (C) bc (D) Noe of these b 4. c [MP PET 99] b c (A) + + b + c (B) + b + c (C) + + b c (D) + b + c 5. If x 0, the x 4 5 [Krtk CET 994] (A) 5 (B) 5 (C) 5 (D) Wht is vlue of x, if 5 x? 6 (A) 4 (B) 8 (C) 5 (D) 9 Mths (Vol. II) k 7. If k 0, the the vlue of k is [IIT 979] (A) (B) 0 (C) (D) Noe of these 8. The vlue of x, if x 0 x 0 x 0, is equl to [Pb. CET 00] (A) ± 6 (B) ± (C) ± (D), 9. If 0 b d 0 c d, the is equl to [RPET 984] (A) c (B) bd (C) (b ) (d c) (D) Noe of these 0. If. 4 x x (A) 4 (B) (C) 6 (D) 7 log 5 log 4 log8 log49 8, the x log log log 4 log 4 [RPET 996] [Tmildu (Egg.) 00] (A) 7 (B) 0 (C) (D) 7. The miors of 4 d 9 d the co-fctors of 4 d 9 i determit re respectively [J & K 005] (A) 4, ; 4, (B) 4, ; 4, (C) 4, ; 4, (D) 4, ; 4,. If A 5 6 4, the cofctors of the 4 7 elemets of d row re [RPET 00] (A) 9,, (B) 9,, (C) 9, 7, (D) 9,,

35 4. Mths (Vol. II) m b m b m b [RPET 989] (A) 0 (B) m (C) m b (D) mb 5. The vlue of (A) 5 (B) 0 (C) 5 (D) 5 9 x y z x y z 6. If p q r, the p 4q r equls b c b c [RPET 999] (A) (B) 4 (C) (D) Noe of these b c 6 b c 7. If m p k, the m p x y z x y z [Tmildu (Egg.) 00] (A) k 6 (B) k (C) k (D) 6k b c k kb kc 8. If x y z, the kx ky kz p q r kp kq kr [RPET 986] (A) (B) k (C) k (D) k 5 π 9. loge e 5 5 log 0 5 e 0. 0 is (A) π (B) e (C) (D) 0 / bc /b b c /c c b [RPET 990, 99] (A) bc (B) /bc (C) b + bc + c (D) 0. The vlue of the determit.. TARGET Publictios equl to [Roorkee 99] (A) 4 (B) 0 (C) (D) 4 si x cos x cos x si x 0 (A) 0 (B) cos x 0si x (C) si x 0cos x (D) 0six is [EAMCET 994] [MP PET 996] 5 8 (A) 0 (B) 9 (C) 96 (D) The vlue of [Krtk CET 00] (A) (B) 4 (C) 0 (D) 5. The determit b b c c x y y z z x is equl to p q q r r p (A) 0 (B) (C) (D) oe of these If A 0 d B 6 0, the [Tmildu (Egg.) 00] (A) B 4A (B) B 4A (C) B A (D) B 6A 7. x 4 y+ z y 4 z+ x z 4 x + y [Krtk CET 99] (A) 4 (B) x + y + z (C) xyz (D) 0

36 TARGET Publictios b c 8. If 6, b, c stisfy b c 0, the bc 4 b [EAMCET 000] (A) + b + c (B) 0 (C) b (D) b + bc 9. If ω is complex cube root of uity, the the ω ω determit 0 (A) 0 (B) (C) (D) Noe of these 0. If ω is complex cube root of uity, the ω ω / 0 (A) 0 (B) (C) ω (D) ω b+ c. The vlue of the determit b c+ is c + b [MP PET 99; Krtk CET 994; Pb. CET 004] (A) + b + c (B) ( + b + c) (C) 0 (D) + + b + c. + x [RPET 996] + y (A) (B) 0 (C) x (D) xy 0 x 6. The roots of the equtio x re 0 9 x [Pb. CET 00; Krtk CET 994] (A) 0,, (B) 0,, (C) 0,, 6 (D) 0, 9, 6 4. The roots of the determit (i x) x m m m 0 re b x b [EAMCET 99] (A) x, b (B) x, b (C) x, b (D) x, b Mths (Vol. II) The roots of the equtio 5 0 re x 5x [IIT 987; MP PET 00] (A), (B), (C), (D), x If x + 5 0, the x x + 4 [MP PET 99] (A), 9 (B), 9 (C), 9 (D), 9 x + 7. If x + 0, the x is x + [Kerl (Egg.) 00] (A) 0, 6 (B) 0, 6 (C) 6 (D) 6 x 8. Solutio of the equtio p+ p+ p+ x 0 x+ x+ re [AMU 00] (A) x, (B) x, (C) x, p, (D) x,, p 9. A root of the equtio x 6 6 x 0 is 6 x [Roorkee 99; RPET 00; J & K 005] (A) 6 (B) (C) 0 (D) Noe of these 40. The roots of the equtio + x + x 0 re + x [MP PET 989; Roorkee 998] (A) 0, (B) 0, 0, (C) 0, 0, 0, (D) Noe of these

37 4 Mths (Vol. II) 4. The roots of the equtio x x 0 re x [Krtk CET 99] (A), (B), (C), (D), 4. Oe of the roots of the give equtio x + b c b x + c 0 is c x + b [MP PET 988, 00; RPET 996] (A) ( + b) (B) (b + c) (C) (D) ( + b + c) 4. If b c, the vlue of x which stisfies the 0 x x b equtio x+ 0 x c 0 is x+ b x+ c 0 [EAMCET 988; Krtk CET 99; MNR 980; MP PET 988, 99, 00; DCE 00] (A) x 0 (B) x (C) x b (D) x c x If x 8 0, the the vlues x 8 of x re [RPET 997] (A) 0, (B), (C), (D), 45. If 9 is root of the equtio 46. x 7 x 7 6 x 0, the the other two roots re [IIT 98; MNR 99; MP PET 995; DCE 997; UPSEAT 00] (A), 7 (B), 7 (C), 7 (D), 7 C C C C C C (A) 0 (B) (C) (D) oe of these 47. The vlue of the determit TARGET Publictios x x z z (A) (x y) (y z) (z x) (B) (x y) (y z) (z x) (C) (x y) (y z) (z x) (x + y + z) (D) oe of these 48. The vlue of the determit (A) (0!!) (B) (0!!) (C) (0!!!) (D) (!!!) y y is 0!!!!!! is!! 4! [Oriss JEE 00] 49. The vlue of the determit b + c c+ + b is b + c c+ b + b c [RPET 986] (A) bc (B) + b + c (C) b + bc + c (D) Noe of these 50. bc b b c c c b [IIT 988; MP PET 990, 9; RPET 00] (A) 0 (B) + b + c bc (C) bc (D) ( + b + c) 5. The determit is ot equl to 6 (A) (C) (B) (D) [MP PET 988]

38 TARGET Publictios 5. The vlue of bc c b b + c c+ + b [Krtk CET 004] (A) (B) 0 (C) ( b) (b c) (c ) (D) ( + b) (b + c) (c + ) b c b b c b c c c b [RPET 990, 95] (A) ( + b + c) (B) ( + b + c) (C) ( + b + c) (b + bc + c) (D) Noe of these b + c b c+ b c b + b c c [MP PET 990] (A) + b + c bc (B) bc b c (C) + b + c b b c c (D) ( + b + c) ( + b + c + b + bc + c) + b + b + b + b + b + 4b + 4b + 5b + 6b [IIT 986; MNR 985; MP PET 998; Pb. CET 00] (A) + b + c bc (B) b (C) + 5b (D) 0 bc b b c c c b (A) 0 (B) ( b) (b c) (c ) (C) + b + c bc (D) Noe of these b + c b c + b c c + b is [RPET 988] [IIT 980] (A) bc (B) 4bc (C) 4 b c (D) b c 58. Mths (Vol. II) + x + y + z [RPET 99; Kerl (Egg.) 00] (A) xyz x y z (B) xyz (C) (D) + x + y + z x + y + z 59. If + b + c 0 such tht + + b λ, the the vlue of λ is + c [RPET 000] (A) 0 (B) bc (C) bc (D) Noe of these 60. If ω is cube root of uity, the x + ω ω ω x +ω ω x +ω [MNR 990; MP PET 999] (A) x + (B) x + ω (C) x + ω (D) x y+ z x y 6. If z+ x z x k(x + y + z) (x z), the k x+ y y z (A) xyz (B) (C) xyz (D) x y z 6. The vlue of 6. If is equl to [RPET 989] (A) 0 (B) 679 (C) 779 (D) 000 x + x x+ x x + x x x x + x+ x x Ax, the the vlue of A is [IIT 98] (A) (B) 4 (C) (D) 4 5

39 6 Mths (Vol. II) 64. If, b, c re positive itegers, the the + x b c determit b b + x bc is c bc c + x divisible by (A) x (B) x (C) + b + c (D) Noe of these p If D p p 5 9,the p D + D + D + D 4 + D 5 [Kurukshetr CEE 998] (A) 0 (B) 5 (C) 65 (D) Noe of these b c bc b c c b [EAMCET 99; UPSEAT 999] (A) 0 (B) (C) (D) bc 67. The vlue of If + b + b + b + b + b + b is equl to [Kerl (Egg.) 00] (A) 9 ( + b) (B) 9b ( + b) (C) ( + b) (D) b ( + b) ( x + x) ( x x) ( x + x) ( x x) ( x + x) ( x x) b b b b c c c c [UPSEAT 00; AMU 005] (A) 0 (B) bc (C) b c (D) Noe of these + si θ si θ si θ cos θ + cos θ cos θ 0, the 4si 4θ 4si 4θ + 4si 4θ si 4θ is equl to [Oriss JEE 005] (A) (B) (C) (D) 70. TARGET Publictios x+ x+ x+ 4 x+ x+ 5 x+ 8 x+ 7 x+ 0 x+ 4 [MNR 985; UPSEAT 000] (A) (B) (C) x (D) Noe of these 7. If + b + b+ c 4 + b 5 + 4b + c, b + 9b + 6c where i, b ω, c ω, the is equl to (A) i (B) ω (C) ω (D) i x x+ 7. If f(x) x x( x ) ( x+ ) x, x( x ) x( x )( x ) ( x+ ) x( x ) the f(00) is equl to [IIT 999, DCE 005] (A) 0 (B) (C) 00 (D) The vlue of the determit k k + kb k + b kc k + c is (A) k ( + b) (b + c) (c + ) (B) k ( + b + c ) (C) k ( b) (b c) (c ) (D) k ( + b c) (b + c ) (c + b) 74. If D D + b b+ c c+ b + c c+ + b c+ + b b+ c b c b c, the c b d (A) D D (B) D D (C) D D (D) D D 75. If p, b q, c 0 d the p b c p+ q+ b c 0, b r p p + q q b + r r c [EAMCET 00] (A) (B) (C) (D) 0

40 TARGET Publictios 76. If, b, c re uequl, wht is the coditio tht the vlue of the followig determit is zero + b b b + c c c + [IIT 985; DCE 999] (A) + bc 0 (B) + b + c + 0 (C) ( b) (b c) (c ) 0 (D) Noe of these 77. Let, b, c be positive d ot ll equl, the b c vlue of the determit b c, is c b [DCE 006] (A) positive (B) egtive (C) zero (D) oe of these 78. If b + bc + c 0 d the oe of the vlue of x is (A) ( + b + c ) b c (B) ( + + ) x c b c b x b c x [AMU 000] (C) ( + b + c ) (D) Noe of these 79. If pλ 4 + qλ + rλ + sλ + t λ + λ λ λ+ λ+ λ λ 4, the vlue of t is λ λ+ 4 λ (A) 6 (B) 8 (C) 7 (D) If + b + c d + x (+ b ) x + c x f(x) ( ) ( + ) + b ( + c ) ( + ) x ( + b ) x + c x [IIT 98] x x x, the f(x) is polyomil of degree [AIEEE 005] (A) (B) (C) 0 (D) 0, Mths (Vol. II) 8. The determit cos β cos γ cos β cos γ β cos γ cos β γ ( ) ( ) ( ) ( ) ( ) ( ) equl to (A) cos cos β cos γ (B) cos + cos β + cos γ (C) (D) 0 8. The vlue of the determit cos si si cos is cos si (A) (B) (C) (D) ( +β) ( +β) idepedet of idepedet of β idepedet of d β oe of these x x x does ot x x x 8. cos( ) cos( + ) cos( + ) si( ) si( + ) si( + ) deped [RPET 000] (A) o x (B) o (C) o both x d (D) oe of these 84. If A, B, C be the gles of trigle, the cosc cosb cosc cosa cos B cos A is [Krtk CET 00] (A) (B) 0 (C) cos A cos B cos C (D) cos A + cos B cos C 85. The vlue of the determit ( β) cos cos cos( β) cosβ is cos cosβ [UPSEAT 00] (A) + β (B) β (C) (D) 0 7 is

41 8 Mths (Vol. II) 86. If A (A) (B) (C) (D) ( θ+) ( θ+) ( θ+β) ( θ+β) ( θ+γ) ( θ+γ) si cos si cos, the si cos [Oriss JEE 00] A 0 for ll θ A is odd fuctio of θ A 0 for θ + β + γ A is idepedet of θ 87. If x is positive iteger, the x! ( x+ )! ( x+ )! ( x+! ) ( x+! ) ( x+! ) is equl to x+! x+! x+ 4! ( ) ( ) ( ) (A) x! (x + )! (B) x! (x + )! (x + )! (C) x! (x + )! (D) (x + )! (x + )! (x + )! 88. Let, b, c be such tht b ( + c) 0. If + + b+ c b b+ b + b c+ c c c+ ( ) + ( ) + b c the the vlue of is (A) (B) (C) (D) 89. If f(x) 90. If zero y eve iteger y odd iteger y iteger [DCE 009] ( ) 0, [AIEEE 009] x x 8 x 8 x 5 x 50 4x 500, the f().f() + f().f(5) + f(5).f() [Kerl (Egg.) 005] (A) f() (B) f() (C) f() + f() (D) f() + f(5) ( + ) b c ( + ) b c b ( + ) c c b k bc(+b+c), the the vlue of k is [Tmildu (Egg.) 00] (A) (B) (C) (D) TARGET Publictios 9. If, b, c re o-zero complex umbers stisfyig + b + c 0 d b + c b c b c + bc k b c, the k is c bc + b equl to [AIEEE 0] (A) (B) (C) 4 (D).. Are of trigle usig determits 9. If the vertices of trigle re A(5, 4), B(, 4) d C(, 6), the its re i sq. uits is (A) 70 sq. uits (B) 8 sq. uits (C) 0 sq. uits (D) 5 sq. uits 9. If the poits (x, ), (5, ), (8, 8) re collier, the x is equl to (A) (B) (C) (D) 94. If the re of trigle is 4 squre uits whose vertices re (k, 0), (4, 0), (0, ), the the vlues of k re (A) 0, 8 (B) 4, (C), 4 (D), If the poits (, ), (λ, ) d (0, 4) re collier, the the vlue of λ is (A) 4 7 (B) (C) 0 7 (D) If the poits (, 0), (0, b) d (, ) re collier, the + is equl to b (A) (B) (C) (D) b 97. The equtio of the lie joiig the poits A (, ) d B (, 6) is (A) y x (B) x y 0 (C) y x (D) x + y If the poits (, b ), (, b ) d ( +, b + b ) re collier, the (A) b b (B) b b (C) b b (D) b + b Poits (k, k), ( k +, k) d ( 4 k, 6 k) re collier, if k is equl to (A), (B), (C), (D),

42 TARGET Publictios 00. If (x, y ), (x, y ) d (x, y ) re vertices of equilterl trigle whose ech side is equl to x y, the x y is x y (A) 4 (B) 4 (C) (D) A trigle hs its three sides equl to, b d c. If the co-ordites of its vertices re (x, y ), x y (x, y ) d (x, y ), the x y is x y (A) ( + b + c)(b + c )(c + b)( + b c) (B) ( b + c) (b + c ) ( + b c) (C) ( b c)(b + c )(c b)( + b c) (D) ( + b c)(b + c )(c + b)( b + c).. Test of cosistecy d solutio of simulteous lier equtios i two or three vribles 0. If x + y 5z 7, x + y + z 6, x 4y + z, the x (A) (B) (C) (D) Noe of these 0. The umber of solutios of the system of equtios x + y z 7, x y + z, x + 4y z 5 is [EAMCET 00] (A) (B) (C) (D) The vlue of λ for which the system of equtios x y z, x y + z 4, x + y + λz 4 hs o solutio is [IIT Screeig 004] (A) (B) (C) (D) Mths (Vol. II) 05. The system of lier equtios x + y + z, x + y z, x + y + kz 4 hs uique solutio, if [EAMCET 994; DCE 000] (A) k 0 (B) < k < (C) < k < (D) k Let, b, c be positive rel umbers. The followig system of equtios x y z + b c, x y z + b c, x y z + + b c hs (A) o solutio (B) uique solutio (C) ifiitely my solutios (D) oe of these 07. If the system of equtios x + y z, (k + )z, (k + ) x + z 0 is icosistet, the the vlue of k is [Roorkee 000] (A) (B) (C) 0 (D) 08. The system of equtios x+ y+ z x+ y+ z x+ y+ z hs o solutio, if is [AIEEE 005] (A) ot equl to (B) (C) (D) either or 09. The system of equtios x x + x, x x + x 6 d x + x + x 8 hs [AMU 00] (A) o solutio (B) exctly oe solutio (C) ifiite solutios (D) oe of these 0. If x+ b y+ c z 0, x+ b y + c z 0, b c x+ by + cz 0 d b c 0, the b c the give system hs [Roorkee 990] (A) oe trivil d oe o-trivil solutio (B) o solutio (C) oe solutio (D) ifiite solutios 9

43 40 Mths (Vol. II). The followig system of equtios x y + z 0, λx 4y + 5z 0, x + y z 0 hs solutio other th x y z 0 for λ equl to [MP PET 990] (A) (B) (C) (D) 5. x + ky z 0, x ky z 0 d x y + z 0 hs o-zero solutio for k [IIT 988] (A) (B) 0 (C) (D). The umber of solutios of the equtios x + y z 0, x y z 0, x y + z 0 is [MP PET 99] (A) 0 (B) (C) (D) Ifiite 4. If x + y z 0, x y z 0, x y + z 0 hs o zero solutio, the [MP PET 990] (A) (B) 0 (C) (D) 5. The umber of solutios of the equtios x + 4y z 0, x 4y z 0, x y + z 0 is [MP PET 99] (A) 0 (B) (C) (D) Ifiite 6. The vlue of for which the system of equtios x + ( + ) y + ( + ) z 0, x + ( + ) y + ( + ) z 0, x + y + z 0, hs o-zero solutio is [Pb. CET 000] (A) (B) 0 (C) (D) Noe of these 7. The vlue of k for which the system of equtios x + ky + z 0, x + ky z 0, x + y 4z 0 hs o-trivil solutio is (A) 5 (B) (C) 6 (D) 8. If the system of equtios x ky z 0, kx y z 0 d x + y z 0 hs o zero solutio, the the possible vlues of k re [IIT Screeig 000] (A), (B), (C) 0, (D), TARGET Publictios 9. Set of equtios + b c 0, b + c 0 d 5b + 4c is cosistet for equl to [Oriss JEE 004] (A) (B) 0 (C) (D) 0. If the system of equtios x y + z 0, λx 4y + 5z 0, x + y + z 0 hve o-trivil solutio, the λ [EAMCET 99] (A) 5 (B) 5 (C) 9 (D) 9. For wht vlue of λ, the system of equtios x + y + z 6, x + y + z 0, x + y + λz is icosistet? [AIEEE 00] (A) λ (B) λ (C) λ (D) λ. If the system of equtios x + y 0, z + y 0 d x + z 0 hs ifiite solutios, the the vlue of is [IIT Screeig 00] (A) (B) (C) 0 (D) No rel vlues. Vlue of λ for which the homogeeous system of equtios x + y z 0, x y + z 0, 7x + λy z 0 hs o-trivil solutios is (A) (B) (C) (D) The system of equtios λ x+ y + z 0, x+λ y + z 0, x y +λ z 0, will hve o-zero solutio if rel vlues of λ re give by [IIT 984] (A) 0 (B) (C) (D) 5. Let the homogeeous system of lier equtios px + y + z 0, x + qy + z 0, x + y + rz 0, where p, q, r, hve o-zero solutio, the the vlue of p + q + r is (A) (B) 0 (C) (D) 6. If f(x) x + bx + c is qudrtic fuctio such tht f() 8, f() d f( ) 6, the f(0) is equl to (A) 0 (B) 6 (C) 8 (D)