( ) ( ) A number of the form x+iy, where x & y are integers and i = 1 is called a complex number.

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Caitlin Morgan
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1 A umber of the form y, where & y are tegers ad s called a comple umber. Dfferet Forms )Cartesa Form y )Polar Form ( cos s ) r or r cs )Epoetal Form r e Demover s Theorem If s ay teger the cos s cos s If s fracto the oe of the value of cos s s cos s

2 ) The value of ) - ) ) ) As (), teger ay For ) ( Q

3 ) ) ) ) e e e e y ) Epoetal form of s ta ta ta e r e y y r As()

4 )Fd the smallest ve teger for whch ( ) ( ) ) ) ) ) Noe of these usg we get ( ) As ()

5 ) Fd the modulus ad Ampltude, of ( ) ) (0, ) ) (,0) ) ) Noe of these. As amp( ) 0 ( 0) modulus As )

6 5) Ampltude of ) ) ) 6 ) Noe of these W. K ( ). T Amp [ ] ta or Amp 6 Amp Amp [ ( ) ] Amp Amp ( ) As()

7 of Ampltude 6) ) ) 6 ) ) 6 () 6 6 As Amp Amp Amp Amp Amp

8 7) Ampltude of O s ) 0 ) ) 6 ) Noe of these , y 0 Amp ( ) ta y ta 0 0 r 0 ca have ay value Amp of '0' ot defed As ()

9 s 5 - cos 5 s )The Ampltude of 8 0 ) 5 ) 5 ) 5 ) [ ] [ ] s cos 0 s 0 cos 0 s cos s s 0 s 0 cos 0 s Q

10 s cos 0 0 t sof the form r 0 s 0 [ cos s ] As( )

11 s cos s 9) Imagary part of ta ) cos ) ta ) ta ) ec s cos cos cos s cos s cos

12 s cos s cos cos As () ta cos s cos cos

13 0) Whch of the followg are correct for ay two comple umbers ad? ) ) ) ) arg ( ) ( arg ) ( arg ) By the propertes of comple umbers ), ) & ) are ot correct. As ()

14 )The modulus of s ) ) ) ) / / As ()

15 - )If fd ) 6 ) 6 ) -6 ) 6 ( ) ( ) ( ) ( ) Q 6 6 As ()

16 ) The value of s cos ) )- ) ) - s cos s (cos s ) ( ) As ()

17 )If the cuberootsof uty are,, the the ) rootsof ),,, the equato, ( -) 8 0 are ), )Noe of ( -) - - ( ), -, (,, ) -, usg, - r, these 8 r, r, r As ()

18 - 5 5 ) If thecomple umber y whch satsfy 5 leso the ) - as ) y-as ) both the aes ) Noe Cosder - 5 y - 5 ( y 5) ( y 5) - 5 ( y 5) & z 5 y 5 ( y 5)

19 ( y 5) ( y 5) ( y 5) Smplfyg weget, 0y 0 y 0 As()

20 6) The modulusad ampltudeof 8 are respectvely ) 56ad ( ) )56ad cs 8 8 cs )56ad 8 cs ) ad cs As ()

21 7)The pots represetg the comple umbers 7 9, - 7 ad - form a ) Rght agled tragle ) Isosceles tragle ) Equlateral tragle ) Noe of these A ( 7,9) B (, 7) C (,) AB ( 7) ( 7 9) ( ) ( 6) BC ( 7)

22 6 ( 0) AC AB BC AC As ()

23 8) If ad are two o - zero comple umber such that ad arg - arg the (z) ) ) - ) ) - Note arg arg z arg arg arg Qarg z arg As ()

24 d c b a y If ) 9 y the ) ) ) ) d c b a d c b a d a b a b a d a b a y b a y ) ( squarg d c b a d c b a d c b a y d c b a y As d c b a y d c b a y y y

25 0) The value of 6 k s k 7 cos k 7 ) - ) 0 ) ) k k sol cos s k th cs sum of 7 root of k k cs 0 Q k 0 7 k s ( uty ecept ) k 7 cs 0 0 As ()

26 s to the cs If..... ) ) - ) - ) )...,, cs cs cs ,, cs cs to cs cs cs cs cs cs

27 s s of GP a r wth a cs cs, r As ()

28 ( ) are ) The real ad magary part of loge ) log ) log log e e e ( ) ad ad e loge ) log 6 log loge log e ad ) log log log e e e ad 6 ( Q log e ) e As ()

29 ) If s a comple umber such that - the ) Real Part of s the same as ts magary part. ) s ay comple umber. ) s purely magary ) s purely real. sol let z y z y gve z z y y 0 z s purely magary As ()

30 the form y, we get cos - s ( cos s ) ( cos s ) ( cos - s ) )If we epress 9 ) cos - s )cos9 s9 )cos s ) cos9 - s9 ( ) 5 cs - cs cs( 8 ) cs( 0 ) ) ( cs(- )) cs( 6 ) cs7 9 (cs cs cs cs ( 8 0 ) cs( 8 ) cs( 8 ) ( 6 7 ) cs ( 9 ) As() 5

31 5) If s for whch a cube root of ( ) ( ) uty fd theleast ve valueof ) ) ) ) ( ) ( ) ( ) ( ) ( ) As()

32 s a c b b a c cs If c b a c b a,the thevalueof 6) c b a c b a ) 0 ) ) ) - c b a c b a Q a c b b a c a c b b a c As()

33 7) Let & be th roots of uty whch subted a rght agle at the org. The must be of the form ) k )k ) k 5) k - Recall of Argad dagram of a comple umber square roots, cube roots, fouth roots z - o z - Square roots of uty Cube roots of uty Fourth roots of uty It s obvous that must be a multple of As()

34 8) The smallest possble s 8 s 8 cos 8 cos 8 tegral value of '' such that to be purely magarythe ) ) ) 8 )

35 8 cos 8 s Put magary purely 0 s cos 8 s 8 cos 8 s 8 cos 8 cos 8 s s get we whe As ()

36 9 ) If,, are three comple umbers A.P. the they le o ) a Crcle ) a straght le ) a parabola ) a ellpse Sce,, are A.P. s md pot of the le jog,, are le o a straght le. & As ()

37 0) Multplcato of a comple umber by correspods to ) Clock wse rotato of the le jog to org Argad dagram through a agle of )At clockwse rotato of the le jog to org Argad dagram through a agle of )Rotato of the le jog to org Argad dagram through a agle.) No rotato - As () z

38 ? root of fourth a s followg the of )Whch 6 ) ) ) ) cs cs cs cs Requred cs cs form Polar cs As

39 s log of ampltude prcpal The ) 0 ) log ta ) log ta ) ) e log ta log ta log log log logm log log log log log log e m e e e e e Q As()

40 the If, cos ) s ) ) cos s ) cos ) the If s cos cos ± Note take cos s s cos s cos s cos As ()

41 ) If ( )( )( )... ( ) y the ( )? ) y ) y ) y ) Noe of these... y... y. 5 squarg B. S. weget, ( ) y As() y

42 les - umber thecomple of heargumet 5) T quadrat Secod ) quadrat Frst ) quadrat Fourth ) Thrd quadrat ), As ()

43 6) Two of the three cube roots of a comple umber are cos0 0 s0 0 ad cos50 0 s50 0 the thrd root s ) ) ) cs0 0 ) Noe thrd root s cs 0 0 As ()

44 7) The real part of log( ) s ) log ) ½ log 5 ) log 5 ) oe of these 5 ta e log5 log ta log5 log log5 5 log log 5 ta ta part real e e e e e As()

45 e e 8) The Real part of s e e cos cos ) ) e ) e cos ) oeof the above e cos s real part e cos cos(s) e e e cos s e cos e s e cos [ cos(s) s(s) ] As ()

46 9) If ( cos s ) ( cos s )...( cos s ) the thevalueof s )m ) m ) m ) m ( ) ( ) ( ) cs cs cs... cs cs cs (... ) (... )

47 cs 0 & s cos I m m m m, As ()

48 0 ) Modulus ad Ampltude of cos s ) cos, ) s, ) cos, ) s, cos s cos cos modulus cos cos ( cos s ) cos Amp. of As

49 m m ) If m, m s a teger the ) ) ) - ) m m Note m m m m ' ' For ay teger As ()

50 000 ) The value of ) - ) - ) - ) ,, 0 As ()

51 ) If, the, are the cube roots of uty, 8 s equal to ) ) ) ) 8 ( ) ( ) ( ) 7 As ()

52 ) The comple umber z whch satsfes the equato z les o z a) a crcle y b) The -as c) The y-as d) The le y

53 z z z z z z As () as o the les 0 0 get, we smplfyg ) ( ) ( z y y y y z z

54 5) Fd modulus ad Ampltude of s cos ) s, ) cos, ) s, ) Noe of ths

55 cos let s ( ) s( ) cosα sα s α sα Modulus sα Substutgα α ( α ) s( α ) cos( α ) [ s ( α ) cos ( α ) ] { cos( α ) s( α )} compargwth Polar weget form r ( cos s ) Ampltude ( α ) As

56 6)The value of ) ) m ( a b) ( a b) s equal to b a b a m m ( ) a b a b cos ta ) cos ta m m m ( ) a b a b cos ta ) cos ta m m m m b a b a Remember Stadard result As()

57 7) Fd the cojugate of ) 9 ) 9 ) ) smplfg z we get 9 0, Noe of ths As()

58 of Ampltude ) 8 ) 6 ) 8 ) ) 6 8 () As Amp

59 9) If ad ame the locus of - ) Crcle )straght le )ellpse )parabola. Note Dstace of from of the le jog the pots Dstace of from ad..hece les o the bsector Requred locus s a straght le. whch s the perpedcular bsector of the le jog (0,0) ad (0,-/) As()

60 50) The comple umbers, -, form a tragle whch s ) Rght agled ) Isosceles ) Equlateral ) Isosceles Rght agled (, 0 ), B (, 0 ), C ( 0, ) Gve A C Usg dstace formula AB BC CA ABC s equlateral As

61 5) The locus of the pot satsfyg the codto amp - s )Straght le passg through org. ) Crcle ) Parabola ) Straght le ot passg through org

62 Amp y y y y y y y y y crcle a s whch y y y y y y y y Amp 0 ta ta As()

63 5) If ( ) the value of 0. ) -0 ) ) )0 z 0 ( ) ( 0 z Q ) As )

64 5)If & s a multple of are two th roots of uty the arg. ) ) ) ) Noe of these Polar cs form of ( r ) cs cs0 r ( r ) cs ( usg Demovres theorem)

65 arg. wherer &s tegersbet.0& a multpleof s r s r s cs r cs As

66 6 5) If cos, the ) cos6 ) cos )cos )Noe cos cs or cs 6 6 cs ( ) 6 cs ( cs ) cs cs 6 ( ) cos s cos s cos As() 6

67 55) If ad ( where ), the Re s ) 0 ) ) ) - 0 ( ) ( Q ) As ()

68 56) The least ve teger for whch ) ) ) ) of multple a be must specto By As ()

69 57 ) If ad s a ve t eger, the ) ) ) ) 0 As ()

70 58) If s a teger, the s ),,, ), ), ),,, ad so o.,,, As

71 6 ) 59 the If ) 79 ) -79 ) 6 ) Q Q As ()

72 60)The cotued product of the cube roots of s ) ) ) ) cotued product of cube roots of s. As ()

Log1 Contest Round 2 Theta Complex Numbers. 4 points each. 5 points each

Log1 Contest Round 2 Theta Complex Numbers. 4 points each. 5 points each 01 Log1 Cotest Roud Theta Complex Numbers 1 Wrte a b Wrte a b form: 1 5 form: 1 5 4 pots each Wrte a b form: 65 4 4 Evaluate: 65 5 Determe f the followg statemet s always, sometmes, or ever true (you may