Unidad 7 - Números Complejos - Problemas resueltos

8/11/2019 Unidad 7 - Nmeros Complejos - Problemas resueltos

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1 BACHILLERATO - Matemticas - Unidad 7 - Nmeros com !e"os

#$%1&' (I a) Raciona! *) Nat+ra! c) Entero d) Irraciona! e) Entero ,) Irraciona!

#$%1&' (II a) ( ) =+= 22 21u 5 ( ) =

= 2.1

2..arg tg arctg arcu '33116

*) ( ) ( ) =+= 22 11v 2 ==

= 1..1

1..arg tg arctg arcv 225

c) ( ) =+= 22 12w 5 ( ) =

= 21.21

.arg, tg arctg arcw '26333

#$%1&' (III =

=

=23

521

53

25

32

cos5 senv

235

25

#$%1 1 (1 z z z z a) i3 i+3 i+ 3 ( ) =+ 22 13 10*) i22 i22 + i22 + ( ) ( ) =+ 22 22 6c) 5 5 5 5d) i3 i3 i3 3

#$%1 1 (2 En e! !ano com !e"o de !os nmeros enteros( ) Z E! con"+nto edido es +nacircunferencia .+ centro es e! ori$en de( )O coordenadas.+ radio i$+a! a dos( )2

#$%1 / (3 a) ( ) =++ i2735 i935 +*) ( [ ] =+ i2735 i535 ++c) ( ) ( ) ( ) =++=+++=++ iiiiiii 237024234970252375 22

=+= 214048370324 iii ( ( i37048140324 +

d) ( )( )( ) =+=

+=

+ 24323

2323

23

23

1i

iii

ii 7

23 i+

e) ( ) ( ) ( ) ( ) =++=+ iii 35142510755752 i4915 +,) ( ) =++=++ 2141037352375 iiiii ( ) ( )i10373514 +

#$%1 / (4 ( ) ( ) iinm 622253 ++ 253 =m 622 =+ n 37=m 4=n

#$%1 / (5 a) ( )( )( )( ) ( ) ( ) =+=+

+22

2

2

5410822254

iiii

iii

i56

513

*) ( ) ( ) ( ) ( ) =+===

121

121

21

21

21

21

2

42328677

iiiiiii

1

c) ( ) ( ) ===+ iii 338338413384 11 id) ( ) ( ) ( ) ( ) ( ) =++ 54322345 2215211021102151 iiiii

=++= iii 32808040101 i3841 +

e) ( ) ( ) ( ) ( ) ( ) =+=+++=+

iiiiii 27927

827

333338

273

23 32233

3

i27

101


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3/16


#$%1 7 (15 Im osi*!e reso!4er!a en e! con"+ntoR2 1806 11 = Z ( k k Z 60306

3601806 11 ++ ==

( ) i seni Z

21

23

3030cos11 301 +=+== ( ) i seni Z =+== 9090cos11 902

( ) i seni Z 21

23

150150cos11 1503 +=+== ( ) i seni Z 21

23

210210cos11 2104 =+==

( ) i seni Z =+== 270270cos11 2705 ( ) i seni Z 21

23

330330cos11 3306 =+==

#$%175 (16 a) i75 *) i43 +c) i3d) i2e) i21 ,) i37 +

#$%175 (17 i A 33 +i B

iC + 3i D 1

i E 3i F 5

#$%175 (18 a) i72 + *) i425 c) i113 + d) 7 e) i5 ,) i32

#$%175 (19 a) i23 + *) i57 c) i231 d) i5 e) i5121 + ,) i3

#$%175 (20 a) Z i Z iiZ Z Z Z Z =+= 22 11 2

*) Z i Z C+adrante 3 Z i Z C+adrante iZ Z C+adrante / iZ Z C+adrante 1

#$%175 (21 a) ( ) ( ) =+ i3622 i3 *) ( ) ( ) =+ i2436 i23 + c) =

++

i

21

1223

i23

21 + d) ( ) ( ) =++ i5213 ( ) i331 +

e) ( ) =++ i512 i62 + ,) ( ) =+ i3107 i33 +

#$%175 (22 a) ( ) ( ) =++ i6322 i94 *) ( ) ( ) =+++ i2416 i67 + c) =

+

i71

21

31

3 i149

310 d) ( ) =++ i5213 ( i713 +

e) ( ) =

++ i

51

273 i5

1110 + ,) ( ) =+ i233 i53

/01


4/16


#$%175 (23 a) ( ) ( ) =+++ i126184 i1814 + *) ( ) ( ) =++ i12486 i162

c) =

+

+ i

73

61

141

1 i4225

1413 d) ( ) ( ) =++ i352103 ( ( i235310 +

e) =

+

+ i

53

1452

21 i5

675

107 + ,) i96

#$%175 (24 a) ( )( ) =++

=++

= 25452

525252

521 i

iii

i i295

292

+

*) ( )( ) =+=+

=+ 94937

373737

371 i

iii

i i583

587

c) ( )( ) =++=+

+= 41624

242424

241 i

iii

i i101

51 +

d) ( )( ) ==

=777

1 iii

ii i7

1

e) =+

=

+

+=

+

+=

3613

2

1

3

1

2

1

3

12

1

3

1

2

1

3

1

2

1

3

12

1

3

1

21

31

122

ii

ii

i

i

i1318

1312 +

,) ( )( ) =++=

++=

3232

3232

32

32

1 i

ii

i

ii

53

52 +

#$%175 (25 a) ( )( )( )( )( ) ( )=

++=

+

364126184

62626232 i

iiii

i203

2011

*)( )( )

( )( )( ) ( )=+

+=+++

4141286

21212146 i

iiii

i58

514 +

c) =+=+

+

=

+

i

i

ii

ii

586744111

581444115

491

91 6

1

7

3

14

11

71

31

71

31

7

1

3

1

2

13

i116231

116515 +

d) ( ( )( )( )( ( =

+++=

+++

251352103

51515123iiii

i26

35226

310 ++

e)( )

=+=+

++

+

=

+

+

ii

ii

ii

1226573

12265103

251

49

53

1452

21

51

751

7

51

723i

1226365

122655 +

,)( )

( )( ) =+

=+

4996

2323233 i

iiii

i139

136

#$%175 (26 a) ( ) iii = 194 i *) ( ) ==+ 112534i 1 c) ==+ 338144 1 ii i d) ==

== + 4335423 1

111

ii

iii i

#$%175 (27 a) ( ) =+=++=+ 1211211 222 iiii i2( ) =+=+++=+ iiiiii 331131311 32233 i22 +( ) =++=++++=+ 1464114161411 4322344 iiiiiii 4

*) ( ) =+=++=+ 91243322232 2222 iiii i125 +( ) =+=+++=+ iiiiii 27543683323323232 3322233 i946 +

01


5/16

1 BACHILLERATO - Matemticas - Unidad 7 - Nmeros com !e"os( ) =++++=+ 4433222344 3324326324232 iiiii

=++= 812162169616 ii i120119 c) ( ) 1211211 222 =+= iiii 6 i2

( ) =+=+= iiiiii 331131311 32233 i22 ( ) =++=+++= 1464114161411 4322344 iiiiiii 4

d) ( ) ( ) ( ) ==++=+ 1442222 222 iiii i43 ( ) ( ) ( ) ( ) =++=+++=+ iiiiii 6128232322 32233 i112 +( ) ( ) ( ) ( ) ( ) =++++=+ 4322344 24.262422 iiiii

=++= 18243216 ii i247

#$%175 (28 a) ( )( )( ) =++=+

=+++

11628

4442

421 2 i

iiii

iii

i178

172 +

*) ===+=+ 22 4451

51

ii

iiii

i i4

c) ( ) =++=+++=+=

+ 1331131311 322333

12 iiiiiiii i22 +

d) ( )( ) ( )

( )( )

=+

+=+

=+

=

++

2

2

22

2222

11

2125

11

12515

132 ii

iii

ii

iii

i2

25

#$%175 (29 a) ( ) ( ) == + ii 901904 1 i *) ( ) == + 2862864 1 ii 1 c) ( ) ( ) ( ) ( ) === + 433334 1 iiii i d) ( ) === + 218184 1 iiii i e) ( ) == 374235976837423597684 1i 1 ,) ( ) == + 33241 ii i

#$%175 (30 a) + yi x z 322 =+ y x z =+ 222 3 y x 922 =+ y x

#ertenecen a circ+n,erencia Centro( )00 Radio3 *) = 9 z z ( )( ) =+ 9 yi x yi x

= 9222 i y x 222 3=+ y x Es !a misma circ+n,erencia anterior Centro( )00 Radio3

c) + yi x z 5= x

Recta ara!e!a a! e"e O8 A*scisa a! ori$en5 d) + yi x z 3= y Recta ara!e!a a! e"e O9 Ordenada a! ori$en3 e) =

+1

yi x yi x +=+ yi x yi x

= 02 x 0= x Es e! e"e coordenado O8

,) =+ i yi x yi x =+ 2 yi xi yi x ( ) ( ) = i yi x 11 y x = Es !a *isectri: de c+adrantes 1 /

&01


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#$%171 (37 =

+

=

22

23

21

z 1 ( ) == 3...arg tg arc z 120

( ) ( ) == + 12033601010120 11 1201 z#$%171 (38 a) i31 +

*) ( ) ( ) =+=+ iii 31131 90 i+ 3 E! inicia! $irado '5 c) ( ) ( ) ( ) =+=+ iii 313 270 i31 + E! ori$ina!< am*os $iros 6 +no de / 5

#$%171 (39 ( ) =+= 22 838 z 16 ( ) == 31...arg tg arc z 210 == ++ k k 72425

5

36021055210 161616 ( 425 16 ( 1145 16 ( 1865 16 ( ) 2565 16 ( ) 3305 16

( ) ( ) 120236044210 1616 += 6 ( ) =

+=+ iisen

23

21

16120120cos16 44 i33276832768 +

#$%171 (40 ( ) ( )

( ) ( ) =+=++=+

=+ iiii

iiiii

111

22

11

12

1( ) 452

( ) ( ) ( ) == ++ k k 90'151184 3604584

45 222 ( ) '15118 2 ( ) '151018 2 ( ) '151918 2

( ) '152818 2

#$%171 (41 a) =

+ 30272

123

27 i ( = +3

360303330 2727 k 103 1303 2503

*) 60823

21

8 =

+ i ( k k 12020

3

360603360

288+

+ == 202 1402 2602

c) =

+ 4522

222

2 i ( = +3

360453345 22 k ( ) 153 2 ( 1353 2 ( ) 2553 2

d) ( ) ( = 452515 i ( ) ( ) ( ) == ++ k k 1201563

3604563

45 505025

( ) 156 50 ( ) 3456 50 ( ) 1056 50 ( ) 2256 50 #$%171 (42 E! n$+!o centra!< de cada !ado de! oct=$ono< es de 458360 =

Conocida +na ra>: ? 458 )< !as dems son @sta $irada< s+cesi4amente


8/16


=

++=

21

21

21

23

21

23

21

21

4 i i++ 1313

#$%171 (46 a) ( ) ( ) =+ 01 xi x ( ) ( ) =+ 011 x x 012 =+ x *) ( ) ( ) =+ 011 i xi x ( ) = 01 22 i x 0222 =+ x x c) ( )( ) =+ 02323 i xi x ( ) ( ) = 023 22 i x 01362 =+ x x

d) i+12 45 i12 315 er *) 01362

=+ x x#$%171 (47 a) i x = 51 i x = 52 ( )( ) =+ 055 i xi x 052 =+ x

*) i x = 21 i x += 22 ( )( ) =+ 022 i xi x 0542 =+ x x c) i x 311 = i x 312 += ( )( ) =+++ 03131 i xi x 01022 =++ x x d) i x

31

21

1 += i x 31

21

2 = ( )( ) = 021 x x x x 036132 =+ x x

#$%171 (48 a) ( ) ( )( ) 04416 22 =+ i xi xi x i x 41 = i x 42 = *) ( ) ( ) ( ) 072492 222 =+ i x x i x 721 += i x 722 =

c)

( ) ( ) ( ) 043163 22

=+++ i x x i x 431 += i x 432 =

d) 043

21

169

21

222

=

++

+ i x x i x

43

21

1 += i x 43

21

2 =

#$%173 (49 a) ( )( ) 0422 2 =++ x x x 21 = x 312 i x += 313 i x = *) = 2703 1 x k k x 12090

3360270

3270 111 ++ === 901 1= x 2102 1= x 3303 1= x

c) ( )( ) =+ 0544 2 x x x ( )( )( ) 0224 =+ i xi x x 41 = x i x += 22 i x = 23

d) =+++ 010167 23

x x x ( ) ( ) =+++ 01061 2

x x x ( )( )( ) 0331 =++++ i xi x x 11 = x i x += 32 i x = 33

#$%173 (50 a) = 4 1804 33 ( ( k k 904544

3601804 33 ++ =

( 4541 3= x ( )i+ 122

34 ( 13542 3= x ( )i+ 122

34

( 22543 3= x ( ) ( )i 122

34 ( 31544 3= x ( )i 122

34

*) 01616 k k

x 9043600

4

216 == +

21 = x i x 22 = 23 = x i x 24 = c) ( ) ( ) ( ) =++ 0522 x xi xi x ( )( )( )( ) 02121 =++ i xi xi xi x

i x =1 i x =2 i x 213 += i x 214 = d) ( )( )( ) =++ 05411 2 x x x x ( )( )( )( ) 0212111 =++ i xi x x x

11 = x 12 = x i x += 23 i x = 24

#$%173 (51 No +eden ser !astres imaginarias E! nmero de ra>ces com !e"as tiene +e ser ar

Con coe,icientes rea!es< si a +na ra>: com !e"a< tam*i@n a !a con"+$a

#$%173 (52 a) == +8

36008 1 k z k 451 =1 z 1 =2 z 451 = ( )i+122 =3 z 901 = i

D01


9/16


=4 z 1351 = ( )i+122 =5 z 1801 = 1 =6 z 2251 =

( )i122

=7 z 2701 = i =8 z 3151 = ( )i122

*)

i z == 1211 2

=1 z i+1

=2 z i1

c) == 3 1803 11 z =+3

3601801 k k 120601 + =1 z 601 i

23

21 +=

=2 z 1801 1= =3 z 3001 i23

21 =

d) ( ) ( ) =+ 02728 323 z z ( )( ) 0271 33 = z z ( == +3

3600331 2727 k z

= k 1203 03 3= 1203 6 i2

3323 + 2403 i

333

23 =

e) === +6

360906690

6 222 k i zk 6015

6 2 + ( 1561 2= z ( 7562 2= z( 13563 2= z ( 19564 2= z ( 25565 2= z ( 31566 2= z

,) ( ) = 3154 2515 i z == +4

3603158 50 k z ( ) k 90'45788 50 +

( '457881 50= z ( '4516882 50= z ( '4525883 50= z ( '4534884 50= z

#$%173 (53 ( ) ( ) ( ) ( ) =++ 102827242 234 iiii + 102814284 ii 0

( ) ( ) ( ) ( ) =++ 102827242 234 iiii

= ( ) ( ) ( ) ( ) ++ 108162821448247 iiii 0 #or tanto< am*os com !e"os2i i2 ces sern !os nmeros con"+$ados2i i+2

#$%173 (54 ( ) ( ) ( )=++=i

iiii x

2

242222 2 ( ) =+i

ii2

2222

222 i

i x = 21 i x =2

#$%173 (55 ( ) ( )( )( ) +

+++=+

+ i x x

x x

i xi xi xi

1

2

1

12222 +

=++

1

2

1

1222 x

x x x =+ 212 x x 3= x

#$%173 (56 a) +=+ i

ii x

211

( ) ( ) +=++=+ iii x 31221 3= x

*) ( ) ( )( ) ( )

++=+++ i x x

iiii x

2

1

2

1

11

1 =+ 0

21 x 1= x

c) ( ) ( )( ) ( ) ++=+

++ i x xiiii x

2

1

2

1

11

1 = 0

21 x 1= x

#$%173 (57 ( )( )( )( )

=++

iiiia

222

iaa5

25

12 ++ =

+

+ 2

52

512

22 aa 3=a

#$%173 (58 904 sr [ ] I 4= sr [ ] II k 36090 +=+

( ) =

4

13

sr

[ ] III 413 = sr [ ] IV n36003 +=

'01


10/16


11/16


12/16


=

212

1

2

1

Z Z

=

22

1

2

1

Z Z

2

1

2

1

Z Z

Z Z =

e) Z = Z ( ) ( ) 222 Z Z Z === 2 Z Z Z =

,) Z

===

1111

Z Z

= 11 Z = Z ( )

= 1

1 Z

( ) 11 = Z Z

#$%17/ (74 ( ) = 03 A i Z A 03 += ( ) +=+= iii Z & 3003' ( )30' = A ( ) = 41 B i Z B 41 += ( ) =+= iii Z B 441' ( )14' = B ( ) = 50C i Z C 50 = ( ) +== iii Z C 0550' ( )05' =C

#$%17/ (75 Miem*ro1 ( ) ( ) ==+=+++ 0111 22322 nnn iiiiiiii 0 Miem*ro2 ( ) ( ) ==++=+++ +++ 0

111

11

13232

23

32 nnn iii

iiii

i 0

#or tanto< am*os miem*ros son i$+a!es#$%17/ (76 ( ) aia Z ++ 1 ( ) ( ) ( )( ) ( ) =++++++= 32233 13131 aiaiaaiaa Z

( ) ( )iaaaaa 362132 233 +++++= =++ 0362 23 aaa ( ) 0362 2 =++ aaa 0=a 1= Z

+=2

33a i Z 2

332

31 +++= =2

33a i Z 2

332

31 +=

#$%17/ (77 a) Z 1

( )( ) =+=

++=

++=

1cos

coscos

coscoscos

22

isenaaa sena

isenaaisenaaisenaa

isenaaisenaa +cos

*) Z 1

( )( ) =++

=+

=+= a senaisenaa

isenaaisenaaisenaa

isenaa 22coscos

coscoscos

cos1

isenaa +cos

c) = 45a 451 = Z ( k k Z 120153

3604533 11 ++ == 151 1051

2251

k k Z 7295

3601555 11 ++ == 91 631 1351 2071 2791

#$%17/ (78 Trin$+!o e +i!tero ori$ina! ABC ( )02 a A 3aa B 3aaC Las res ecti4as medianas< a!t+ras< *isectrices son AOA


13/16


14/16


c) =+=2

22

sen sen) 2 sen sen + ( )274,1 cm)

#$%17 (&3 a) ( ) ( )12211531

31

=++== AB A% ( ) ( ) += 1221 % ( )11 = %

( )1231

== AB %* ( ) ( ) += 1211* ( )03=*

( ) ( ) ( ) =++=+= 1421

013121

21 * % +

=

21

2 +

*) ,B,*,% ,A +++ ( ) ( ) ( ) ( ) =+++= 23314153 ( )140

=

= 3

21

22,+

27

0 ,+ ,B,*,% ,A =+++ 4

c) Iterando !a a !icaci=n de! a artadoa)< se +ede escri*ir s+cesi4amente2( ( ( ( ( =+=+=++++++ ,*,% ,*,% ,B,% ,*,A,B,*,% ,A 222

,+ ,+ == 422 ,+ ,B,*,% ,A =+++ 4

#$%17 (&4 a) ( ) ( ) ( ) ( ) ==+== 15253125132V $ 17 *) =+= 22 32$ 13 =+= 22 51V 26

c) ( ) ==

=338

17

2613

17cos

V $

V $ V $ "12'37157,. V $ "ng

d) = 0- $ ( ) ( ) === 038342432 mmm 38=m

#$%17 (&5 ( ) :r ( ) ( ) + 03121 ( )r A = 21r m 2=#ar perpendicum #er endic+!ar a( )r or A 2 ( ) = 121 x y 012 = y x ( )t

Corte de( )t ( ) : s52= x

59= y @rtice / 6

59

52

( ) :r ( ) ( ) + 03023 ( )r B#er endic+!ar a( )r or : B ( ) = 320 x y 62 = x y ( )u

Corte de( )u ( ) : s58= x

514= y @rtice 6

514

58

#$%17 (&6 a) = AC +edi! % . =

+

+

250

276

25

213 =

= 56705

AC m 51=mediatriz m

: +ediatriz = 213

51

25 x y 0195 =+ y x

*) ( ) ( ) =+= 260567 22 AC Radio c>rc+!o 6226

Circ+n,erencia24

2625

213

22

=

+

y x

c) Corte de circ+n,erencia mediatri: anteriores2 ( ) ( ) =+ 2652132 22 y x ( )[ ] ( ) =+ 2652135192 22 y y 0652 =+ y y

= 31 y 41 = x ( )34= B = 22 y 92 = x ( )29= D

#$%17 (&7 a) ( )( )( ) =+=

+

2

22

2222

22

2i

iiii

i3

221 i

1 01


15/16


*) ( )( )( )( ) =+=

++=

=

++

11214

11168

16869

2

2 iiiii

ii

iiii

i 7

#$%177 ('1 =+ 2cos

3coscos x

x x =

+

2

cos

2

3cos

2

3cos2

x

x x x x ( ) 2

coscos2cos2 = x

x x

= 222cos x = 452 x '3022= x '3022= x '30337= x

#$%177 ('2 ( ) ABCD 6== CD AB 8== BC AD 150== B D 30== C A ( ) ACD += 150cos68268 222 AC 312252 += AC ( )cm53,13

( ) ABD += 30cos86286 222 BD 312252 = BD ( )cm11,4

#$%177 ('3 Centro c>rc+!o 6C segund!Crece %O% "ng /2'. = M imoO) 6 C+ando ' %% sea tan$ente a! c>rc+!o 6 90'. =C F% "ng

( ) C F% ' =+= 22 FC 4 2' =C% ( ) == 2142'CF% sen ='"ngCF% 30 ( ) FO) ( ) = 'CF% tg OF O) ( ) =++= 30222 tg O) m"x 32 ( )cm46.3 Con = .m"xO) = 60'. %O% "ng == s,iemp! /2

60 ( ) s30

#$%177 ('4 Bisectri: = x y =1sec triz bim == '' BB AA mm 1

( ) ' AA ( ) = 112 x y 03 =+ y x Corte con : x y =

=

23

23

' A

( ) ' BB ( ) = 315 x y 08 =+ y x Corte con : x y = ( )44' = B

Lon$it+d =

=

+

= 22

34

2

34

2

34''

22

B A2

25

( ) == 2513 AB ( )32 =

=

23

423

4'' B A

25

25

=

=

+

+

+

=26

5

2513

25

2

5

2

532

2

53

2

52

cos22

22

26

5cos.arc= '1911

#$%177 ('5 Kistancia de( ) y x a rectas2 +

+=+

+2222

12

22

21

12 y x y x ( )2212 +=+ y x y x

E! !+$ar $eom@trico escon(unto derectas 2( ) I 033 =+ y x ( ) II 013 =++ y x

.on !as *isectrices de !os n$+!os ,ormados or !as rectas( )r ( ) s Adems son er endic+!ares< or *isectrices or +e I II mm /1=

#$%177 ('6 a) 'ip.rb!#a = 1916

22 y x 134 2

2

2

2

= y x

)emie/es 4=a 3=b += 22 34c 5=c Centr! ( )00

1&01


16/16

Unidad 7 - Números Complejos - Problemas resueltos

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