COMPLEX NUMBER & QUADRATIC EQUATION

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1 MCQ COMPLEX NUMBER & QUADRATIC EQUATION Syllus : Comple numers s ordered prs of rels, Representton of comple numers n the form + nd ther representton n plne, Argnd dgrm, lger of comple numers, modulus nd rgument (or mpltude) of comple numer, squre root of comple numer, trngle nequlty, Qudrtc equton n rel nd comple numer system nd ther solutons. Relton etween roots nd co-effcent, nture of roots, formton of qudrtc equtons wth gven roots. Ensten Clsses, Unt No. 0, 0, Vrdhmn Rng Rod Plz, Vks Pur Etn., Outer Rng Rod New Delh 0 08, Ph. : 96905, 857

2 MCQ C O N C E P T S (Comple Numer) C The comple numer system Comple numer s denoted y z.e. z = +, where s clled s rel prt of z (Re z) nd s clled s mgnry prt of z (Im z). Here =, lso =, = ; 4 = etc. Prctce Prolems :. If n s nturl numer then the vlue of n + n + + n + + n + s 0. The vlue of ( 00 + ) ( 99 + )...( + ) wll e 0 [Answers : () () ] C Algerc Opertons on Comple Numer :. Addton ( + ) + (c + d) = + + c + d = ( + c) + ( + d). Sutrcton ( + ) (c + d) = + c d = ( c) + ( d). Multplcton ( + ) (c + d) = c + d + c + d = (c d) + (d + c) c d 4. Dvson. c d c d c d c d c d c d c d 5. Inequltes n comple numers re not defned. 6. In rel numers f + = 0 then = 0 = however n comple numers, z + z = 0 does not mply z = z = Equlty In Comple Numer : If z = z Re(z ) = Re(z ) nd I m (z ) = I m (z ) Prctce Prolems : 4. The vlue of 4 s equl to m. If then the lest ntegrl vlue of m s ( ) ( )y. If, then the rel vlue of nd y re gven y =, y = =, y = =, y = =, y = [Answers : () d () () ] C Modulus of Comple Numer : If z = +, then t s modulus s denoted nd defned y z orgn. Propertes of modulus. Infct z s the dstnce of z from () z z = z z () z z z (provde z z 0) Ensten Clsses, Unt No. 0, 0, Vrdhmn Rng Rod Plz, Vks Pur Etn., Outer Rng Rod New Delh 0 08, Ph. : 96905, 857

3 () z + z z + z (v) z z z z MCQ (Equlty n () nd (v) holds f nd only f orgn, z nd z re collner wth z nd z on the sme sde of orgn). C4 Representton of Comple Numer : Crtesn Form (Geometrc Representton) : Every comple numer z = + y cn e represented y pont on the Crtesn plne known s comple plne (Argnd dgrm) y the ordered pr (, y) Length OP = z = y nd = y tn s clled the rgument or mpltude. If s the rgument of comple numer then n + ; n I wll lso e the rgument of tht comple numer. The unque vlue of such tht < s clled the prncpl vlue of the rgument. Unless otherwse stted, mp z mples prncpl vlue of the rgument. The rgument of z =,, +,, = tn qudrnt. Propertes of Argument of Comple Numer : () rg (z z ) = rg (z ) + rg (z ) + m for some nteger m. () rg (z /z ) = rg (z ) rg (z ) + m for some nteger m. () rg (z ) = rg (z) + m for some nteger m. (v) rg (z) = 0 z s rel, for ny comple numer z 0 y, ccordng s z = + y les n I, II, III or IVth (v) rg (z) = ± / z s purely mgnry, for ny comple numer z 0 (v) rg (z z ) = ngle of the lne segment jonng the pont (z ) nd pont (z ) Trgnometrc/Polr Representton : z = r (cos + sn ) where z = r; rg z = ; z = r(cos sn ) cos + sn s lso wrtten s CS or e. Euler s Representton : z re ; z r;rg z ;z re e e e e Also cos nd sn re known s Euler s denttes. Prctce Prolems :. The prncple vlue of the rgument of the comple numer s 0 none of these. If z = 4 nd rg z = 5 then z equls to 6 Ensten Clsses, Unt No. 0, 0, Vrdhmn Rng Rod Plz, Vks Pur Etn., Outer Rng Rod New Delh 0 08, Ph. : 96905, 857

4 MCQ 4. If r = r r then... s 0 4. The mpltude or rgument of ( )( ) ( ) wll e [Answers : () c () c () c (4) c] C5 Conjugte of comple Numer Conjugte of comple numer z = + s denoted nd defned y Propertes of conjugte z. () z = z () zz z () z z ) (z ) (z ) ( z (z) (v) ( z z ) (z ) (z ) (v) ( zz ) (z) (z ) (v) (z 0) z (z ) (v) z z (z z )(z z ) z z zz zz (v) ( z ) z () rg( z) rg(z) 0 Prctce Prolems :. The modulus of comple numer ( )( ) wll e ( ) 0 ½ [Answers : () ] C6 Demovere s Theorem : If n s ny nteger then () (cos + sn ) n = cos n + sn n () (cos + sn ) (cos + sn ) (cos + sn ) (cos + sn )... (cos n + sn n ) = cos ( n ) + sn ( n ) C7 Cue Root of Unty : () The cue root of unty re,, () If s one of the mgnry cue roots of unty then + + = 0. In generl + r + r = 0; where r I ut s not the multple of. () In polr form the cue roots of unty re : (v) 4 4 cos0 sn 0;cos sn,cos sn The three cue roots of unty when plotted on the rgnd plne consttute the vrtes of n equlterl trngle. (v) The followng fctorston should e rememered : Ensten Clsses, Unt No. 0, 0, Vrdhmn Rng Rod Plz, Vks Pur Etn., Outer Rng Rod New Delh 0 08, Ph. : 96905, 857

5 (,, c R nd s the cue root of unty) = ( ) ( ) ( ) ; + + = ( ) ( ) + = ( + ) ( + ) ( + ) ; + + = ( ) ( ) + + c c = ( + + c) ( + + c) ( + + c) Prctce Prolems :. If s cue root of unty then ( + ) ( + ) s equl to MCQ 5 0. If n s postve nteger not multple of then + n + n = none of these 0 none of these. If s comple cue root of unty then ( ) ( ) ( 4 ) ( 8 ) equl to If s comple cue root of unty, then ( + ) ( + ) ( + 4 )... to n fctors 0 n c c 5. The vlue of c c wll e 0 [Answers : () d () c) () d (4) (5) ] C8 nth Roots of Unty : If,,,... n re the n, nth root of unty then : () They re n G.P. wth common rto e (/n) () p p p p = 0, f p s not n ntegrl multple of n n = n f p s n ntegrl multple of n () ( ) ( )... ( n ) = n ( + ) ( + )... ( + n ) = 0 f n s even nd f n s odd. (v) n = or ccordng s n s odd or even. Prctce Prolems :. If,,,... n re the n, nth roots of unty then ( ) ( )...( n ) equls 0 n n. If s ny nth root of unty then S = up to n terms s equl to n n n n [Answers : () c () ] C9 Geometrcl Propertes : () () Dstnce Formul : If z nd z re ffes of the two ponts P nd Q respectvely then dstnce etween P nd Q s gven y z z Secton Formul : If z nd z re ffes of the two ponts P nd Q respectvely nd pont C devdes the lne jonng P nd Q nternlly n the rto m : n then ff z to C s gven y mz nz z m n mz nz. If C devdes PQ n the rto m : n eternlly then z m n Ensten Clsses, Unt No. 0, 0, Vrdhmn Rng Rod Plz, Vks Pur Etn., Outer Rng Rod New Delh 0 08, Ph. : 96905, 857

6 () (v) (v) MCQ 6 Condton of collnerty : If,, c re three rel numers such tht z + z + cz = 0; where + + c = 0 nd,, c re not ll smultneoulsy zero, then the comple numers z, z nd z re collner. If the vertces A, B, C of trngle represents the comple numers z, z, z respectvely, then centrod of the trngle s z z z The equton of crcle hvng centre z 0 nd rdus s z z 0 = z z (v) If k where k 0 or then locus of z s crcle. z z (v) (v) If z, z, z re vertces of n equlterl trngle wth z 0 ts crcumcentre, then z + z + z s equl to z 0 The ponts z, z, z, z 4 n the comple plne re the vertces of prllelogrm tken n order f nd only f z + z = z + z 4 Prctce Prolems :. Length of the lne segment jonng the ponts ( ) nd ( + ) s z 5. The comple numers z = + y whch stsfy the equton le on z 5 the -s the strght lne y = 5 crcle pssng through the orgn none of these. The locus represented y z = z + s A crcle of rdus An ellpse wth foc t (, 0) nd (0, ) A strght lne through the orgn A crcle on the lne jonng (, 0), (0, ) s dmeter 4. The locus of the ponts representng the comple numer z for whch z = z z + 5 = 0 s crcle wth centre t the orgn strght lne pssng through the orgn the sngle pont (0, ) none of the ove [Answers : () c () () c (4) c] Ensten Clsses, Unt No. 0, 0, Vrdhmn Rng Rod Plz, Vks Pur Etn., Outer Rng Rod New Delh 0 08, Ph. : 96905, 857

7 MCQ 7 CONCEPTS (Qudrtc Equtons) C Qudrtc Equton : A qudrtc equton s ny equton equvlent to one of the form + + c = 0 hs only two roots. Here 0 nd,, c re rel numers (,, c R). Roots of the qudrtc equton 4c Here 4c = D s known s dscrmnnt of the equton (). 4c Sum of roots :, Product of roots : c, Dfference of roots : 4c D The qudrtc equton whose roots re nd s ( ) ( ) = 0 Prctce Prolems :. If, re the roots of the equton + + = 0 ( < 0), then s greter thn 0 none of these. If the rto of the roots of + + c = 0 e equl to the rto of the roots of + + c = 0, then, c, c re n A.P. G.P. H.P. None. The vlue of for whch one root of the qudrtc equton ( 5 + ) + ( ) + = 0 s twce s lrge s the other, s / / / / 4. Let, e the roots of the equton + p = 0 nd let, e the roots of the equton + q = 0. If the numers,,, (n order) form n ncresng G.P. then p =, q = 6 p =, q = p = 4, q = 6 p = 4, q = [Answers : () d () () d (4) ] C Nture of roots of one qudrtc equton :. D = 0, then equton () hs rel nd equl roots.. D > 0, then equton () hs rel nd dstnct roots.. If D < 0, then equton () hs mgnry roots. 4. If,, c re rel nd D < 0 then roots re l + m nd l m. 5. If,, c re rtonl nd D s perfect squre then roots re rtonl. 6. If,, c re rtonl nd D s not perfect squre then roots re rrtonl. The roots re l + m nd l m. Ensten Clsses, Unt No. 0, 0, Vrdhmn Rng Rod Plz, Vks Pur Etn., Outer Rng Rod New Delh 0 08, Ph. : 96905, 857

8 MCQ 8 Prctce Prolems :. If ( c) + (c ) y + c ( ) y s perfect squre, then the qunttes,, c re n A.P. G.P. H.P. none of these. If, re odd ntegers, then the roots of the equton + ( + ) + = 0, 0 re Rtonl Irrtonl Non-rel none of these. If,, c re ll postve nd n H.P., then the roots of + + c = 0 re Rel Imgnry Rtonl Equl [Answers : () c () () ] C Condton for common roots : Let the equtons re + + c = 0 nd + + c = 0 Here, 0, 0, 0. One root s common : (c c ) = ( c c ) ( ) Both roots re common : c c Prctce Prolems :. If,, c re n G.P., then the equton + + c = 0 nd d + e + f = 0 hve common root f d/, e/, f/c re n A.P. G.P. H.P. none of these. The equtons + + = nd + = 0 hve roots n common. Then + must e equl to 0 none [Answers : () () c] C4 Importnt Ponts :. If > 0 nd > 0 (oth roots re postve) then + > 0 nd > 0.. If < 0 nd < 0 then + < 0 nd > 0.. If roots re of opposte sgn then < 0. Prctce Prolems :. If the equton + (k + ) + 9k 5 = 0 hs only negtve roots, then : k 0 k 0 k 6 k 6. The set of vlues of p for whch the roots of the equton + + (p )p = 0 re of opposte, s (, 0) (0, ) (, ) (0, ) [Answers : () c () ] Ensten Clsses, Unt No. 0, 0, Vrdhmn Rng Rod Plz, Vks Pur Etn., Outer Rng Rod New Delh 0 08, Ph. : 96905, 857

9 MCQ 9 INITIAL STEP EXERCISE. The vlue of determnnt where = s If,,,..., n re nth roots of unty. The vlue ( ) ( ) ( )...( n ) s n 0 n n. The centre of squre ABCD s t z = 0. The ff of the verte A s z. Then the ff of the centrod of the trngle ABC s z (cos ± sn ) z (cos sn ) z (cos sn ) z (cos sn ) 4. If p, q re two equl numers lyng etween 0 nd such tht z = p +, z = + q nd z = 0 form n equlterl trngle, then (p, q) = ( +, + ) (, ) ( + 5, 5) none of these 5. If ( + )( + ) ( n + n ) = A + B, then n tn s : B A 6. The equton z = z hs B tn A B tn A tn A B no soluton two solutons four solutons n nfnte numer of solutons 7. If for comple numers z nd z, rg (z z ) = 0 then z z s equl to z + z z z z z 0 8. If, nd re the cue roots of p, p < 0, then for ny, y nd z, the vlue of y z y z,, re, none of these 9. If s n mgnry cue root of unty, then the vlue of s 0 0. Let z nd z e n th roots of unty whch sutend rght ngle t the orgn. Then n must e of the form 4k + 4k + 4k + 4k. If the comple numers z, z, z re n A.P., then they le on crcle prol lne ellpse. If nd re the roots of the equton + + = 0 nd 4 nd 4 re the roots of p + q = 0 the roots of 4 + p = 0 re lwys oth non-rel oth postve oth negtve postve nd negtve. If R, the lest vlue of the epresson 6 5 s none of these Ensten Clsses, Unt No. 0, 0, Vrdhmn Rng Rod Plz, Vks Pur Etn., Outer Rng Rod New Delh 0 08, Ph. : 96905, 857

10 4. If R, then 4 cn tke ll vlues f (0, ) [0, ] [, ] none of the ove 5. The roots of the equton ( ) 5 = re ( ) 5, wher ±, ± ± 4, ± 4 ±, ± 5 ± 6, ± 0 6. If nd e the roots of + p + = 0 nd, the roots of + q + = 0, then the vlue of ( ) ( ) ( + ) ( + ) s equl to p q q p p q MCQ 0 7. The numer of rel roots of equton ( ) + ( ) + ( ) = 0 s 0 8. The root common to the equton ( ) + ( c) + (c ) = 0 nd ( )( c) + ( ) (c ) + (c )( ) = 0 s / 0 9. If,, c re postve rel numers, then the numer of rel roots of the equton + + c = 0 s 4 0 none of these FINAL STEP EXERCISE. The soluton of the equton ( + ) = 0 form n A.P. G.P. H.P. none of these. The vlue of for whch the equton ( ) + = 0 hs roots elongng to (0, ) s > > 5 5. If z nd z re two dstnct ponts n n Argnd plne. If z = z, then the z z s pont on z z lne segment [, ] of the rel s the mgnry s unt crcle z = the lne rgz = tn z z z 4. If log, then the locus of z s z = 5 z < 5 z > 5 none of these 5. Let A 0 A A A A 4 A 5 e regulr hegon nscred n crcle of unt rdus. Then the product of the lengths of the lne segments A 0 A, A 0 A nd A 0 A 4 s 4 6. If z s comple numer hvng lest solute vlue nd z + =, then z = ( /) ( ) ( /) ( + ) ( + /) ( ) ( + /) ( + ) 7. Let,, c e non-zero rel numer such tht 0 0 ( cos 8 )( c)d 8 ( cos )( c)d, then the qudrtc equton + + c = 0 hs no root n (0, ) t lest one root n (0, ) two roots n (0, ) two mgnry roots Ensten Clsses, Unt No. 0, 0, Vrdhmn Rng Rod Plz, Vks Pur Etn., Outer Rng Rod New Delh 0 08, Ph. : 96905, 857

11 ( )( c) 8. For rel vlues of, the epresson wll ssume ll rel vlues provded c c c c 9. The rto of the roots of the equton + + c = 0 s sme s the rto of the roots of the equton p + q + r = 0. If D nd D re the dscrmnnts of + + c = 0 nd p + q + r = 0 respectvely, then D : D = p c r q none of these 0. The vlue of for whch ectly one root of the equton e e + e = 0, les etween nd re gven y 5 ln < < ln ln 4 0 ln none of the ove. The vlue of c for whch = 4 7, where nd re the roots of c = 0 s Root(s) of the equton = 0 elongng to the domn of defnton of the functon f() = log ( ) s/re 5, 5 5,. If the equtons + + c = 0 nd = 0 hve common root, where,, c re the lengths of the sdes of ABC, then sn A + sn B + sn C s equl to / MCQ 4. Let [] denote the gretest nteger n. Then n [0, ], the numer of solutons of the equton + [] = 0 s If the roots of the equton ( ) ( c) + ( c) ( ) + ( ) ( ) = 0 re equl, then + + c = c = 0 + c = 0 none of the ove 6. If s root of the equton = 0, then s equl to 6 s greter thn 6 s less thn 6 cn not e estmted 7. If Z nd the equton ( ) ( 0) + = 0 hs ntegrl roots, then the vlue of re 0, 8, 0, 8 none of these 8. The vlue of for whch ( ) + ( ) + s postve for ny re or 9. Gven tht for ll rel, the epresson 4 4 les etween nd. The vlues etween whch the epresson les re nd nd 0 nd 0 nd If, e the roots of + + c = 0 nd +, + e those of A + B + C = 0, then the vlue of ( c)/(b AC) s A 0 A Ensten Clsses, Unt No. 0, 0, Vrdhmn Rng Rod Plz, Vks Pur Etn., Outer Rng Rod New Delh 0 08, Ph. : 96905, 857

12 . If the product of the roots of the equton k + e log k = 0 s 7, then the roots of the equton re rel for k equl to 4. If, re the roots of the equton + p + q = 0 nd 4, 4 re the roots of r + s = 0, then the equton 4q + q r = 0 hs lwys (p, q, r, s re rel numers) two rel roots two negtve roots two postve roots one postve nd one negtve roots. If p, q {,,, 4}, the numer of equtons of the form p + q + = 0 hvng rel roots s MCQ 4. The numer of roots of the qudrtc equton 8 sec 6 sec + = 0 s nfnte 0 5. If A, G, H e respectvely, the A.M, G.M. nd H.M. of three postve numers,, c; then the equton whose roots re these numers s gven y A + G ( ) = 0 A + (G /H) G = 0 + A + (G /H) G = 0 A (G /H) + G = 0 6. If nd ( 0) re the roots of the equton + + = 0, then the lest vlue of + + ( R) s : 9/4 9/4 /4 /4 ANSWERS (INITIAL STEP EXERCISE). c. c. d c 6. c d. c. d. c c d ANSWERS (FINAL STEP EXERCISE) c. 4. c 5. c c. c. d 4. c 7. c 8. d d Ensten Clsses, Unt No. 0, 0, Vrdhmn Rng Rod Plz, Vks Pur Etn., Outer Rng Rod New Delh 0 08, Ph. : 96905, 857

13 MCQ AIEEE ANALYSIS [00] 6 0. If 4 y then =, y = = 0, y = = 0, y = 0 =, y =. If nd = 5, = 5, then the equton hvng / nd / s ts roots s = = = 0 9 = 0 6. If,, re the cue roots of unty, then n n s equl to n n 0 7. If, then n n = n, where n s ny postve nteger = 4n +, where n s ny postve nteger = n +, where n s ny postve nteger = 4n, where n s ny postve nteger 8. If z nd re two non-zero comple numers such tht z =, nd Arg (z) Arg () =, then equl to z s 77. The numer of rel roots of 9 s zero 4 4. If ( ) s cuc root of unty, then equls zero 5. If s n mgnry cue root of unty then ( + ) 7 equls AIEEE ANALYSIS [00] 9. Let z nd z e two roots of the equton z + z + = 0, z eng comple. Further, ssume tht the orgn, z nd z form n equlterl trngle. Then = = = 4 = 0. The vlue of for whch one root of the qudrtc equton ( 5 + ) + ( ) + = 0 s twce s lrge s the other, s / / / /. The numer of rel solutons of the equton + = 0 s 4. If the sum of the roots of the qudrtc equton + + c = 0 s equl to the sum of the squres of ther recprocls, then c, nd c Geometrc Progresson Hrmonc Progresson re n Arthmetc-Geometrc Progresson Arthmetc Progresson Ensten Clsses, Unt No. 0, 0, Vrdhmn Rng Rod Plz, Vks Pur Etn., Outer Rng Rod New Delh 0 08, Ph. : 96905, 857

14 MCQ 4 AIEEE ANALYSIS [004/005]. Let z, w e comple numers such tht z w 0 nd rg (zw) =. The rg z equls /4 / /4 5/4 [004] 4. If z = y nd z / = p + q, then y p q (p q ) equl to [004] 5. If z = z +, then z les on crcle the mgnry s the rel s n ellpse [004] 6. If ( p) s root of qudrtc equton + p + ( p) = 0 then ts roots re 0,, 0,, [004] 7. If one root of the equton + p + = 0 s 4, whle the equton + p + q = 0 hs equl roots, then the vlue of q s 49/4 4 [004] 8. If + + 6c = 0, the t lest one root of the equton + + c = 0 les n the ntervl (, ) (, ) (0, ) (, ) [004] 9. If the cue root of unty re,, then the roots of the equton ( ) + 8 = 0, re,,, +,, +, +,, [005] 0. If z nd z re two non-zero comple numers such tht z + z = z + z, then rg z rg z s equl to s [005] 0 z. If nd =, then z les on z [005] crcle n ellpse prol strght lne. In trngle PQR, R =. P Q tn nd tn re the roots of + + c = 0, 0 then c = + = + c = + c = c [005]. The vlue of for whch the sum of the squres of the roots of the equton ( ) = 0 ssume the lest vlue s 0 [005] 4. If the roots of the equton + c = 0 e two consecutve ntegers, then 4c equls [005] 5. If oth the roots of the qudrtc equton k + k + k 5 = 0 re less thn 5, then k les n the ntervl (6, ) (5, 6] [4, 5] (, 4) [005] 6. If the equton n n + n n = 0 0, n, hs postve root =, then the equton n n n + (n ) n n = 0 hs postve root, whch s smller thn greter thn equl to greter thn or equl to [005] If Ensten Clsses, Unt No. 0, 0, Vrdhmn Rng Rod Plz, Vks Pur Etn., Outer Rng Rod New Delh 0 08, Ph. : 96905, 857

15 MCQ 5 AIEEE ANALYSIS [006] 7. The numer of vlues of n the ntervl [0, ] stsfyng the equton sn + 5sn = 0 s All the vlues of m for whch oth roots of the equton m + m = 0 re greter thn ut less thn 4, le n the ntervl < m < 4 < m < 0 m > < m < 9. If the roots of the qudrtc equton + p + q = 0 re tn 0 0 nd tn 5 0, respectvely then the vlue of + q p s 0. If s rel, the mmum vlue of 7/7 / s 9 7. If z + z + = 0, where z s comple numer, then the vlue of z z s z z z z... z z 0 k k 0. The vlue of sn cos s k AIEEE ANALYSIS [007]. If the dfference etween the roots of the equton + + = 0 s less thn 5, then the set of possle vlues of s (, ) (, ) (, ) (, ) 4. If z + 4, then the mmum vlue of z + s ANSWERS AIEEE ANALYSIS. c d 6. d 7. d 8. c d d c 8. c 9. d 0. d. d.. 4. c 5. d d 9. c 0.. c.. c 4. Ensten Clsses, Unt No. 0, 0, Vrdhmn Rng Rod Plz, Vks Pur Etn., Outer Rng Rod New Delh 0 08, Ph. : 96905, 857

16 MCQ 6 TEST YOURSELF. The regon of the rgnd plne defned y z + z + 4 s nteror of n ellpse eteror of crcle nteror nd oundry of n ellpse eteror nd oundry of crcle. If z s comple numer lyng n the fourth kz qudrnt of the rgnd plne nd k for ll vlues of k (k ) then rnge of rg (z) s, 0 8, 0 4, 0 6 none of these. If the comple numers z, z, z, re the vertces of prllelogrm ABCD, then the fourth verte s 7. The numer of rel roots of the equton sn (e ) = s 0 nfntely mny 8. If, re the roots of P( + ) C = 0 then vlue of ( + ) ( + ) s C 4 + C + C C 9. If 4 s root of p + q = 0 where p, q R, p q then vlue s p q / / 5/ none of these 0. If,, c re n AP nd one root of the equton + + c = 0 s then the other root s 4 4 (z + z ) (z + z 4 z z 4 ) (z + z + z ) z + z z 4. The trngle wth vertces z, z nd ( )z + z s n sosceles trngle rght ngled trngle n sosceles rght ngled trngle n equlterl trngle z 5. If z = + y nd, then = mples tht z n the comple plne z les on the mgnry s z les on the rel s z les on the unt crcle none of these 6. If, re the roots of + p + q = 0 nd, re the roots of + r + s = 0, then vlue of ( ) ( ) ( ) ( ) s (r p) (q s) (r p) + (q s) (r p) + (q s) rp(r p) (q s) none of these. c. c. d 4. c 5. ANSWERS 6. d d c Ensten Clsses, Unt No. 0, 0, Vrdhmn Rng Rod Plz, Vks Pur Etn., Outer Rng Rod New Delh 0 08, Ph. : 96905, 857

INTRODUCTION TO COMPLEX NUMBERS

INTRODUCTION TO COMPLEX NUMBERS The numers -4, -3, -, -1, 0, 1,, 3, 4 represent the negtve nd postve rel numers termed ntegers. As one frst lerns n mddle school they cn e thought of s unt dstnce spced