Post on 02-Jun-2018
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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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1 BACHILLERATO - Matemticas - Unidad 7 - Nmeros com !e"os
#$%1 7 (15 Im osi*!e reso!4er!a en e! con"+ntoR2 1806 11 = Z ( k k Z 60306
3601806 11 ++ ==
( ) i seni Z
21
23
3030cos11 301 +=+== ( ) i seni Z =+== 9090cos11 902
( ) i seni Z 21
23
150150cos11 1503 +=+== ( ) i seni Z 21
23
210210cos11 2104 =+==
( ) i seni Z =+== 270270cos11 2705 ( ) i seni Z 21
23
330330cos11 3306 =+==
#$%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
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1 BACHILLERATO - Matemticas - Unidad 7 - Nmeros com !e"os
#$%175 (23 a) ( ) ( ) =+++ i126184 i1814 + *) ( ) ( ) =++ i12486 i162
c) =
+
+ i
73
61
141
1 i4225
1413 d) ( ) ( ) =++ i352103 ( ( i235310 +
e) =
+
+ i
53
1452
21 i5
675
107 + ,) i96
#$%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
#$%175 (27 a) ( ) =+=++=+ 1211211 222 iiii i2( ) =+=+++=+ iiiiii 331131311 32233 i22 +( ) =++=++++=+ 1464114161411 4322344 iiiiiii 4
*) ( ) =+=++=+ 91243322232 2222 iiii i125 +( ) =+=+++=+ iiiiii 27543683323323232 3322233 i946 +
01
8/11/2019 Unidad 7 - Nmeros Complejos - Problemas resueltos
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1 BACHILLERATO - Matemticas - Unidad 7 - Nmeros com !e"os( ) =++++=+ 4433222344 3324326324232 iiiii
=++= 812162169616 ii i120119 c) ( ) 1211211 222 =+= iiii 6 i2
( ) =+=+= iiiiii 331131311 32233 i22 ( ) =++=+++= 1464114161411 4322344 iiiiiii 4
d) ( ) ( ) ( ) ==++=+ 1442222 222 iiii i43 ( ) ( ) ( ) ( ) =++=+++=+ iiiiii 6128232322 32233 i112 +( ) ( ) ( ) ( ) ( ) =++++=+ 4322344 24.262422 iiiii
=++= 18243216 ii i247
#$%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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1 BACHILLERATO - Matemticas - Unidad 7 - Nmeros com !e"os
#$%171 (37 =
+
=
22
23
21
z 1 ( ) == 3...arg tg arc z 120
( ) ( ) == + 12033601010120 11 1201 z#$%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 $irada< s+cesi4amente
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1 BACHILLERATO - Matemticas - Unidad 7 - Nmeros com !e"os
=
++=
21
21
21
23
21
23
21
21
4 i i++ 1313
#$%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
#$%173 (52 a) == +8
36008 1 k z k 451 =1 z 1 =2 z 451 = ( )i+122 =3 z 901 = i
D01
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1 BACHILLERATO - Matemticas - Unidad 7 - Nmeros com !e"os
=4 z 1351 = ( )i+122 =5 z 1801 = 1 =6 z 2251 =
( )i122
=7 z 2701 = i =8 z 3151 = ( )i122
*)
i z == 1211 2
=1 z i+1
=2 z i1
c) == 3 1803 11 z =+3
3601801 k k 120601 + =1 z 601 i
23
21 +=
=2 z 1801 1= =3 z 3001 i23
21 =
d) ( ) ( ) =+ 02728 323 z z ( )( ) 0271 33 = z z ( == +3
3600331 2727 k z
= k 1203 03 3= 1203 6 i2
3323 + 2403 i
333
23 =
e) === +6
360906690
6 222 k i zk 6015
6 2 + ( 1561 2= z ( 7562 2= z( 13563 2= z ( 19564 2= z ( 25565 2= z ( 31566 2= z
,) ( ) = 3154 2515 i z == +4
3603158 50 k z ( ) k 90'45788 50 +
( '457881 50= z ( '4516882 50= z ( '4525883 50= z ( '4534884 50= z
#$%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
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1 BACHILLERATO - Matemticas - Unidad 7 - Nmeros com !e"os
=
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
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1 BACHILLERATO - Matemticas - Unidad 7 - Nmeros com !e"os
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
#$%17 (&7 a) ( )( )( ) =+=
+
2
22
2222
22
2i
iiii
i3
221 i
1 01
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1 BACHILLERATO - Matemticas - Unidad 7 - Nmeros com !e"os
*) ( )( )( )( ) =+=
++=
=
++
11214
11168
16869
2
2 iiiii
ii
iiii
i 7
#$%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
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