complex numbers.edudept.np.gov.lk/learning_materials/passpapers...Β Β· title: microsoft word -...
TRANSCRIPT
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SVM Complex Number 1
rpf;fnyz;fs; (Complex Numbers) rpf;fnyz;
a, b vd;gd nka; vz;fshf π + ππ vd;gJ rpf;fnyz; vd tiuaWf;fg;gLk;. π = ββ1 MFk;. rpf;fnyz;fs; nghJthf π ,dhy; Fwpf;fg;gLk;.
Fwpg;G : ,q;F a vd;gJ rpf;fnyz;zpd; nka;g;gFjp vdTk; b vd;gJ rpf;fnyz;zpd;
fw;gidg;gFjp vdTk; miof;fg;gLk;.
π = π π (π) vdTk; π = πΌπ (π) vdTk; Fwpf;fg;gLk;
ππ = βπ ,
ππ = ππ. π = βπ,
ππ = ππππ = (βπ) . (βπ) = π
ππ = ππ. π = π
ππ = ππ. π = π. π = ππ = βπ
ππ = ππ. π = βππ
ππ = ππ. ππ = π
πππ = π
πππ π = π
rpf;fnyz; xd;wpd; cld;Gzup (Conjugate Complex Number) π = π + ππ vd;w rpf;fnyz;zpd; cld; Gzupr; rpf;fnyz;> οΏ½Μ οΏ½ = π β ππ vd
tiuaWf;fg;gLk;.
rpf;fnyz;fspd; $l;ly;
π = π + ππ
π = π + ππ
π + π = (π + ππ) + (π + ππ )
= (π + π) + π(π + π )
rpf;fnyz;fspd; fopj;jy;
π β π = (π + ππ) β (π + ππ )
= (π β π) + π(π β π )
rpf;fnyz;fspd; ngUf;fy;
π π = (π + ππ)(π + ππ )
= π(π + ππ ) + ππ(π + ππ )
= ππ + πππ + πππ + π ππ
= ππ + π ππ + πππ + πππ
= (ππ β ππ) + π(ππ + ππ)
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SVM Complex Number 2
rpf;fnyz;fspd; tFj;jy; π1
π2 =
π+ππ
π+ππ
=(π ππ)(π ππ )
(π ππ )(π ππ ) =
π(π ππ ) ππ(π ππ )
(π ππ )(π ππ )
= π(π ππ ) ππ(π ππ )
(π ππ )(π ππ ) =
ππ ππππ πππ πππ
ππ πππ π
= ππ ππ π(ππ ππ )
ππ π π
rpf;fnyz;zpd; gz;Gfs;
1. π + ππ = π vdpd; π = π πππ π = π MFk;
epWty;: π + ππ = π
π = βππ ,lg;gf;fj;jpy; nka;naz;Zk; tyg;gf;fj;jpy; fw;gidnaz;Zk;
cs;sd. ,it rkdhf ,Uf;f KbahJ.
MfNt ,it jdpj;jdp
Β€r;rpakhf ,Ug;gpd;
kl;LNk rhj;jpakhFk;.
π₯ = 0 πππ π¦ = 0 MFk;
2. π + ππ = π + ππ vdpd; π = π , π = π MFk;
epWty;: π + ππ = π + ππ
π + ππ β (π + ππ) = π
(π β π) + π(π β π) = π
(π β π) = π πππ (π β π) = π
(π = π) πππ (π = π)
cjhuzk;:
π + ππ = π βΊ π₯ = 0 πππ π¦ = 0
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SVM Complex Number 3
1. π + ππ = π + ππ Ij; jPHf;f
π β π + π(π β π) = π
π β π = π πππ (π β π)=0
π = π, π = π
2. ππ β πππ = (π β π)π Ij; jPHf;f
ππ β πππ = π + ππ β ππ
ππ β πππ = π β ππ
(ππ β π) + π(π β ππ) = π
ππ β π = π , π β ππ = π
π = ππ
= π , π =π
π
3. (1 + π)(2 + 3π) vd;gij π₯ + ππ¦ vd;w tbtpy; jUf
(1 + π)(2 + 3π)
= 1(2 + 3π) + π(2 + 3π)
= (2 + 3π) + 2π + 3π
= (2 + 3π ) + 3π + 2π
= (2 β 3) + 5π
= (β1) + 5π
4. (1 + π) vd;gij π₯ + ππ¦ vd;w tbtpy; jUf
(1 + π)
= 1 + π + 2π
= 1 β 1 + 2π
= 2π
= 0 + 2π
5. (1 + π) vd;gij π₯ + ππ¦ vd;w tbtpy; jUf
= (1 + π)
= [(1 + π) ]
= (2π)
= 2 . π
= 32. π π
= 32. (1). π
= 32π
6. vd;gij π₯ + ππ¦ vd;w tbtpy; jUf
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SVM Complex Number 4
=( )( )
( ) ( )
=
= = π
gpd; tUtdtw;iw π₯ + ππ¦ vd;w tbtpy; jUf
1. (2 + 5π )(5 β 8π)
2. (1 β 4π)
3.
4. ( )( )
Njw;wq;fs;
π + π = οΏ½Μ οΏ½ + οΏ½Μ οΏ½
π β π = οΏ½Μ οΏ½ β οΏ½Μ οΏ½
π π = οΏ½Μ οΏ½ οΏ½Μ οΏ½
=π1
π2
Z1Z2 =Z1Z2
π , π ,U rpf;fnyz;fshf π οΏ½Μ οΏ½ , π π vd;gd cld; GzupfshFk;.
π = π + ππ, π = π + ππ vd vLj;J NkYs;s Njw;wq;fs; epWt KbAk;.
rpf;fnyz;zpd; KidT tbtk;.
π = π₯ + ππ¦ vd;f.
= π₯ + π¦ π +
π₯ + π¦ = π > 0, = πΆππ π, = ππππ vdpd;
π = π[πΆππ π + πππππ] vd;gJ π ,d; KidT tbtk; MFk;.
π₯ = π πΆππ π, π¦ = π ππππ
π vd;gJ rpf;fnyz; π,d; kl;L (Modulus) vdg;gLk;. |π| = π
π vd;gJ rpf;fnyz; π ,d; tPr;rk; vdg;gLk;. πππ(π) = π
βπ < π β€ π vdpd; π jiyik tPr;rk; vdg;gLk;. ,J Arg(Z) = π vdf;Fwpf;fg;gLk;
π = π(πΆππ π + πππππ) vdpd;
=( )
= (πΆππ π β ππ πππ)
= [πΆππ (βπ) + ππππ(βπ)]
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SVM Complex Number 5
/ = =| |
πππ = (βπ) = βπππ(π)
οΏ½Μ οΏ½ = π(πΆππ π β π ππππ)
π = πΆππ (βπ) + π πππ(βπ)
/ |οΏ½Μ οΏ½| = |π| = π
πππ(οΏ½Μ οΏ½) = (βπ) = βπππ(π§)
|π π | = |π π |
=| |
| |
πππ(π π ) = πππ(π ) + πππ(π )
πππ π π = πππ(π ) β πππ(π )
π = π₯ + ππ¦ ,d;
π₯ = π π(π) = π ,d; nka;g;gFjp
π¦ = πΌπ(π) = π ,d; fw;gidg;gFjp
rpf;fnyz;fspd; rkdpyp
|π + π | β€ |π | + |π |
|π β π | β₯ |π | β |π |
Argand Diagram (Mfz; tupg;glk;)
|π| = ππ = π
πππ(π) = π
Fwpg;G ;: P vd;w Gs;spahy; Z vd;w rpf;fnyz; Fwpf;fg;gLk;
cjhuzk;
1. β3 + π I KidT tbtpy; je;J kl;L tPriyf; fhz;f
β3 + π = 2β
+ π
= 2 cos + π sin
β3 + π ,d; kl;L 2 MFk;
β3 + π ,d; tPry; MFk;
2. β β3 β π I KidT tbtpy; je;J kl;L tPriyf; fhz;f
ββ3 β π = 2 ββ
β π
P(π)
0
π
π nka; mr;R
fw;gid mr;R
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SVM Complex Number 6
= 2 ββ
+ π β
= 2 cos β + π sin β
ββ3 β π ,d; kl;L 2 MFk;
ββ3 β π ,d; tPry; β MFk;
3. -1 I KidT tbtpy; je;J kl;L tPriyf; fhz;f
β1 = (β1 + 0π)
= 1(β1 + 0π)
= 1(cos π + π sin π)
β1 ,d; kl;L 1 MFk;
β1 ,d; tPry; (π) MFk
4. π I KidT tbtpy; je;J kl;L tPriyf; fhz;f
π = 0 + 1π
= 1(0 + 1π)
= 1 cos + sin
π ,d; kl;L 1 MFk;
π ,d; tPry; MFk;
5. I KidT tbtpy; je;J kl;L tPriyf; fhz;f
=
=
= βπ
= 0 β π
= 0 + (β1)π
= [0 + (β1)π]
= 1 cos β + πsin β
,d; kl;L 1 MFk;
,d; tPry; β MFk;
gpd;tUk; rpf;fnyz;fspd; kl;L> tPr;rj;jpid jUf
1. i1 2. i31 3. i1 4. i31 2. gpd;tUtdtw;wpy; iba tbtpy; jUf
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SVM Complex Number 7
i
i
2
1.1
i
i
25
43.2
2
1
1.3
i
i
2
3
3
2.4
i
i
βπ + ππ ,d; tHf;f %yq;fisf; fhz;f β3 + 4π ,d; tHf;f %yk; (π₯ + ππ¦ ) vd; nfhs;f
ββ3 + 4π = π₯ + ππ¦
β ββ3 + 4π = (π₯ + ππ¦)
β β3 + 4π = π₯ + π π¦ + 2ππ₯π¦ β β3 + 4π = π₯ β 1π¦ + 2ππ₯π¦ β π₯ β 1π¦ = β3 πππ 2π₯π¦ = 4 β π₯ β π¦ = β3 πππ π₯π¦ = 2
β π₯ β = β3
β π₯ β 4 = β3π₯ β π₯ + 3π₯ β 4 = 0 β (π₯ + 4)(π₯ β 1) = 0 β π₯ + 4 β 0 β΄ π₯ β 1 = 0 β π₯ = Β±1 β΄ π¦ = Β±2
β΄ ββ3 + 4π = 1 + 2π ππ β 1 β 2π 1 ,d; fd%yq;fisf; fhzy;
x3/11 vdf; nfhs;f
π₯ = 1 π₯ β 1 = 0 (π₯ β 1)(π₯ + π₯ + 1) = 0
1x or 012 xx
2
3
2
1
2
31
2
31
2
411
ix
i
x
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SVM Complex Number 8
,jd;xU %yk; vdpy; ,jd; kw;wa %ykhf mikAk;
,2
3
2
1 i
2
3
2
1 i
3/2sin3/2cos12
3
2
1
i
i
3/2sin3/2cos1 i
π = + β π
= β β β π
= 2
3
2
1 i
= ; π₯ β 1 = 0 %yq;fshf 2,,1 vd;gd mikAk;. ;
mj;Jld; 13 0101 22
1. -1 ,d; fd%yq;fisf; fhz;f 2. 5 + 12π ,d; tHf;f %yq;fisf; fhz;f 3. 2π ,d; tHf;f %yq;fisf; fhz;f
mikg;G 1
Z jug;gbd; (-Z) I mikj;jy;
Z = π₯ + ππ¦ vd;f
-Z=βπ₯ β ππ¦
Z I P tiff;Fwpf;fpd;wJ
PM = MQ MFkhW PM MdJ Q tiu ePl;lg;gl;Ls;sJ
P (x,y)
Q (-x,-y)
Q MdJ βZ I tiff;Fwpf;fpd;wJ
mikg;G 2
Z1 , Z2 jug;gbd; Z1 + Z2 I Fwpf;Fk; Gs;spia Mfd; tupg;glj;jpy; Fwpj;jy;
π (A)
(C)
M
P
Q
M
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SVM Complex Number 9
π = π + ππ
π = π + ππ
π + π = (π + ππ) + (π + ππ )
= (π + π) + π(π + π )
epWty;:
A (a,b)
B (c,d)
A,B vd;gd KiwNa Z1 , Z2 I
Fwpf;Fk;
OA , OB vd;gtw;iw may; Xuq;fshff;
nfhz;l ,izfuk; OACB I tiuf
M ,
C [π + π , π + π]
C MdJ π + π If; Fwpf;Fk;
mikg;G 3
Z1 , Z2 jug;gbd; Z1 - Z2 I Fwpf;Fk; Gs;spia Mfd; tupg;glj;jpy; Fwpj;jy;
π = π + ππ
π = π + ππ
A (a,b)
B (c,d) vd;f
A,B vd;gd KiwNa Z1 , Z2 I
Fwpf;Fk;
BO = OD MFkhW BO it D tiu
ePl;Lf
D = (-c,-d ) MFk;
vdNt D MdJ - Z2 If;Fwpf;Fk;
AODC vd;w ,izfuj;ij tiuf
MfNt Gs;sp C MdJ Z1 - Z2 If;Fwpf;Fk;
Fwpg;G:
| Z β Z | = OC = AB vd;gij mtjhdpf;f
Arg (ππ β π π) = πΆ vd;gijAk; mtjhdpf;f
O
B
A
C
D
πΌ
C(Z1 - Z2 )
O
B(Z2)
A(Z1)
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SVM Complex Number 10
mikg;G 4
Z1 , Z2 jug;gbd; Z1 . Z2 I Fwpf;Fk; Gs;spia Mfd; tupg;glj;jpy; Fwpj;jy;
π = π (πΆππ π + πππππ)
π = π (πΆππ πΌ + πππππΌ)
Z1 . Z2 = π (πΆππ π + πππππ) π (πΆππ πΌ + πππππΌ)
= π π [cos π cos πΌ + π sin π sin πΌ + π(sin π cos πΌ + cos π sin πΌ)]
= π π [cos π cos πΌ β sin π sin πΌ + π(sin π cos πΌ + cos π sin πΌ)]
= π π [cos(π + πΌ) + π sin(π + πΌ)]
Arg(Z1 . Z2 ) = π + πΌ
|π π | = π π
Fwpg;G : 1. Arg(Z1 . Z2 ) = π + πΌ
= Arg (Z1) + Arg (Z2)
2. |π π | = |π | |π |
A,B vd;gd KiwNa Z1 , Z2 I
Fwpf;Fk;
π΄ππ(π ) = π
π΄ππ(π ) = πΌ
|π | = ππ΄ = π
|π | = ππ΅ = π
OM = 1 myF MFkhW nka;ar;rpy; M vd;w Gs;spiaf; Fwpf;f
OB cld; ,lQ;Ropahf π Nfhzj;jpy; l vd;w Nfhl;il tiuf
β‘πππ΄ = β‘ππ΅πΆ MFkhW l vd;w Nfhl;by; C vd;w Gs;spiaf; Fwpf;f
β‘πππ΄ = π
β‘πππ΅ = πΌ
βOMA, βOBC ,ay;nghj;jit
β΄ = βΉπ2
=π1
βΉ ππΆ = π π
= |π π |
mj;Jld; β‘πππΆ = π + πΌ
= Arg (Z1 . Z2 )
MfNt Gs;sp C MdJ Z1 . Z2 vd;w rpf;fnyz;izf;Fwpf;Fk;
mikg;G 5
O
A
B
M
C
l
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SVM Complex Number 11
Z1 , Z2 jug;gbd; ππ
ππ I Fwpf;Fk; Gs;spia Mfd; tupg;glj;jpy; Fwpj;jy;
π = π (πΆππ π + πππππ)
π = π (πΆππ πΌ + πππππΌ) ππ
ππ=
( )
( )
= (πΆππ π + πππππ)((πΆππ πΌ β πππππΌ)
=
[cos π cos πΌ β π sin π sin πΌ ] + π[sin π cos πΌ β cos π sin πΌ]
=
[cos(π β πΌ) + π sin(π β πΌ)]
ππ
ππ =
Arg ππ
ππ = π β πΌ
Fwpg;G
1.ππ
ππ =
ππ
ππ
2. Arg ππ
ππ = Arg (Z1) - Arg (Z2)
π΄ππ(π ) = π
π΄ππ(π ) = πΌ
|π | = ππ΄ = π
|π | = ππ΅ = π
OM = 1 myF MFkhW nka;ar;rpy; M vd;w Gs;spiaf; Fwpf;f
OA cld; tyQ;Ropahf πΌ Nfhzj;jpy; l vd;w Nfhl;il tiuf
β‘ππ΅π = β‘ππ΄πΆ MFkhW l vd;w Nfhl;by; C vd;w Gs;spiaf; Fwpf;f
β‘πππ΄ = π
β‘πππ΅ = πΌ
βOMB, βOCA ,ay;nghj;jit
β΄ = βΉπ2
=π1
βΉ ππΆ =
=
mj;Jld; β‘πππΆ = π β πΌ
O
B
A
M
C
l
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SVM Complex Number 12
= Arg
MfNt Gs;sp C MdJ vd;w rpf;fnyz;izf;Fwpf;Fk;
mikg;G 6
Z jug;gbd; I mikj;jy;
π = π(πΆππ π + πππππ)
= π(πΆππ π+πππππ)
= (πΆππ π β πππππ)
= [πΆππ (βπ) + ππππ()]
MfNt =
= | |
Arg = (βπ)
Mfd; tupg;glj;jpy; A vd;w Gs;sp Z If; Fwpf;Fk;. OM = 1 myF MFkhW nka;ar;rpy; M If;Fwpf;f. β‘πππ΄ = π
nka;ar;Rld; tyQ;Ropahf π Nfhzj;jpy; l vd;w Nfhl;il tiuf. β‘ππ΄π = β‘πππΆ MFkhW l vd;w Nfhl;by; C vd;w Gs;spiaf; Fwpf;f
βOMA, βOCM ,ay;nghj;jit
β΄ = βΉπ
=1
βΉ OC =
=
β‘πππΆ = βπ
= Arg
MfNt C vd;w Gs;sp vd;w rpf;fnyz;izf;Fwpf;Fk;
mikg;G 5
Z jug;gbd; (ππ) I mikj;jy; ,q;F π xU nka;naz;.
A
M C
l
O
A
M C
l
O
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SVM Complex Number 13
π = π(πΆππ π + πππππ) ππ = ππ(πΆππ π + πππππ) β΄ |ππ| = ππ = π|π| mj;Jld; π΄ππ(ππ) = π = π΄ππ(π) ππ΄ = |π| ππ΅ = π. ππ΄ = π. |π|
MfNt B MdJ ππ If; Fwpf;Fk;
xOf;F Loci
1. |π| = 1 vdpd; π ,d; xOf;F 2. |π| β€ 1 vdpd; π ,d; xOf;F
3. |π| < 1 vdpd; ; π ,d; xOf;F
1 π
0
0 π
1
A
B
Z
O π
π 1
Z
0
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SVM Complex Number 14
4. πππ(π) = vdpd; π ,d; xOf;F 5. 0 β€ πππ(π) β€ vdpd; π ,d; xOf;F
6. 0 β€ πππ(π) <π
6 vdpd;; π ,d; xOf;F
7. |π| = 2 MfTk; β€ πππ(π) β€ MfTk; ,Ug;gpd ; π ,d; xOf;F.
AππΆ =
BππΆ =
π ,d; xOf;F AB vd;w tpy;.
8. |π| β€ π πππ 0 β€ πππ(π) < vdpd;; π ,d; xOf;F ahJ?
π ,d; xOf;F epow;gLj;jpagFjp MFk;
;.
9. |π β 1| = 1 vdpd; π ,d; xOf;F
π
π
π π
π
π
π
π
π π6
π
3
π
π
A
B
πΆ C
π
π π
π
π
4
π
Z
1
1
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SVM Complex Number 15
π ,d; xOf;F (1,0) I ikakhfTk; 1 myF MiuahfTk; cila tl;lkhFk;.
10. πππ(π β 1) = vdpd; π ,d; xOf;F
11. πππ(π β 1) β πππ(π§ + 1) = vdpd;; π ,d; xOf;F
π ,d; xOf;F tl;ltpy; MFk;.
12. |π| = 2 πππ πππ(π β 1) = vdpd; ; π If; fhz;f.
13.
32arg
Z vdpd; |π|,d; ,opTg; ngWkhdk; ahJ?
|π| = ππ = 2πππ
= β3
14. π π π β = 0 vdpd;; π ,d; xOf;F ahJ?
π = π(πΆππ π + π ππππ)
π = 1 + β3π π
2 β3
1 2
π
π
π
π
-1 0 1
π
π
π
π
1
π > 0 π > 0
1
P
π
π
-π 0
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SVM Complex Number 16
οΏ½Μ οΏ½ = π(πΆππ π β π ππππ)
= (πΆππ π β π ππππ)
π π π β = ππΆππ π β πΆππ π = 0
β πΆππ π π β = 0
β πΆππ π = 0
π β 0 β πΆππ π(π β 1) = 0
πΆππ π = 0 ππ π = 1
π = Β± ππ |π| = 1
xOf;F tl;lk; my;yJ cw;gj;jp jtpHe;j fw;gid mr;rhFk;
Some Important Results
2
zzz
zz nka;ahdJ
zz nka;ahdJ
z nka; vdpd;; 00 ZImandorZarg
z fw;gid vdpd;; 0Re2
arg ZandZ
Njw;wk; : π , π ,π vd;gd Mfd; tupg;glj;jpy; KiwNa π , π , π Mfpa
Gs;spfis Fwpf;fpd;wd. β‘π π π = πΌ , π , π , π vd;gd ,lQ;RopahfNth md;wp
tyQ;RopahfNth miktjw;Nfw;g = (cos πΌ Β± sin πΌ) MFk;
epWty;:
π , π , π vd;gd ,lQ;Ropahfnjupapd;
|π β π | = π π
Arg (π β π ) = π½
β΄ π β π = π π (cos π½ + π sin π½)
|π β π | = π π
Arg (π β π ) = π
β΄ π β π = π π (cos π + π sin π)
π½ π
π
π
πΌ
π
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SVM Complex Number 17
vdNt = ( )
( )
= ([cos(π β π½) + π sin(π β π½)])
= (cos πΌ + π sin πΌ)
π , π , π vd;gd tyQ;Ropahf njupapd;
= (cos πΌ β π sin πΌ)
π , π , π vd;gd ,lQ;RopahfNth md;wp tyQ;RopahfNth miktjw;Nfw;g =
(cos πΌ Β± sin πΌ) MFk;
cjhuzk;
ππ΄π΅ XH rkgf;f Kf;Nfhzp A vd;w Gs;spia 1 + β3π vd;w rpf;fnyz; Fwpg;gpd; B
If;Fwpf;Fk; rpf;fnyz;izf;fhz;f
B I Z vd;w rpf;fnyz; Fwpf;Fnkd;f
β= cos Β± π sin
π = 1 + β3π cos Β± π sin ( OA=OB )
= 1 + β3π Β±β
π
= β1 + β3π or 2
B
60
60 O
A
B
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SVM Complex Number 18
gapw;rp
1. 3 + π , β2 β 2π , β1 + 2π vd;w rpf;fnyz;izf;Fwpf;Fk; Gs;spfs; XH ,Urkgf;f Kf;Nfhzpiaf;Fwpf;Fk; vdf; fhl;Lf
gpd;tUk; xOf;Ffis Mfd; thpg; glj;;jpy; Fwpj;J tpghpf;f
21.6
31.5
21.4
2.3
2.2
2.1
Z
Z
Z
Z
Z
Z
iZiZ
iZ
iZ
iZ
iZ
232.11
22.10
11.9
121.8
1.7
2
arg.15
arg.146
1arg.13
3arg.12
iZ
iZ
Z
Z
01
Re.18
21arg1arg.17
3
232arg.16
zZ
ZZ
iZ
iZiZ
ZZ
ZZ
arg4
arg.22
62arg2arg21
3arg4arg.20
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SVM Complex Number 19
01
Re.26
02
Re25
04
1Im24
0Im.23 2
zZ
z
z
zZ
Z
3
32arg0.27
iZ
ffhz; ijf;ngWkhdj; opTg;, d;, vdpd; ZZ3
3arg.28
ffhz; ijf;ngWkhdj; opTg;, caHT d;, ;vdpd; 1233.29 ZiZ
Lffhl; vdf; gidahdJfw; Lfz; ijf;ngWkhdj;d;,6
32
530
i
i
ffhz; iwf;gtw;vd; rk;tPr; Lkl; d;,4
3
1
1.31
i
i
2121 ,1
2
1
1ZZ
iZ
i
iZ
I Mfd; thpg;glj;jpy; Fwpj;J
21 ZZ I mikf;f
128
tan
vd ca;j;jwpf
32. Z1= )31(2
1i Z2=i Mf cs;s rpf;fy;nyz;zpd; kl;ilAk; tPriyAk; jzpf>
rpf;fnyz;z; Z1,Z2,Z1+Z2 vd;gtw;iw Mfd; thpg;glj;jpy; Fwpf;f. ,jpypUe;J
12
5tan
,d; ngWkhdj;ij Jzpf.
33 . gpd;tUtdtw;why; jug;gLk; gpuNjrj;ij epow;Wf
6
arg6
,5.1 ZZ
3
2arg
3.2
Z
-
SVM Complex Number 20
34. gpd;tUtdtw;wpy; Z If; fhz;f
4
arg4.1
ZandZ
01Re52.2 ZandiZ
21121.3 ZandZiZ
24
arg.4 ZandZ
6
21arg
61arg.5
ZandZ
35. gpd;tUtdtw;wpy; Z If; Fwpj;Jf; fhl;Lf.
33
1arg0.1 iZandZ
6
51arg423.2
ZandiZ
4
3arg4,1.3
ZandZZ
36. , vd;gd ,U rpf;fnyz;fshapd; gpd;tUtdtw;iw epWTf.
1) Re2 2) Re2 3)
22 ImRe
4) 2 ImRe
5)
37. Lffhl; vdf; d;vdp; ZZZZZZ 212121 ,2argarg
38. 2
argarg212121
ZZZZZZ vdpd; Lffhl; vdf;
39. ZZZZZZ
i
2
12121
, d;vdp; Lffhl; vdf;
40. xUrkgf;fKf;Nfhzpapd; cr;rpfs; Z1,Z2,Z3 MYk; ikag;Nghyp Z0 MYk; Fwpf;fg;gbd;
1) 0111
ZZZZZZ 133221
vd epWTf.
2) ZZZZ2
0
2
3
2
2
2
13 vd epWTf
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SVM Complex Number 21
41. 2
2
2
1
2
21
2
2122 ZZZZZZ vd epWTf
42. 11 Z vdpd; gpd;tUtdtw;iw epWTf
1) ZZ arg21arg
2) 22
arg
Z
Z
3) ZiZ argtan
2
2
43 (a) Gs;sp P Mfd; tupg;glj;jpy; rpf;fy; vz; z If; Fwpf;fpd;wJ. rpf;fy; vz; Z2 I
Gs;sp Q Fwpg;gpd;> Q I Nfj;jpu fzpj Kiwahy; tpgupf;f. (fzpg;Gf;fs; mtrpakpy;iy)
1. P MdJ (1,0) I ikakhfTk; 1 I MiuahfTk; nfhz;l tl;lj;jpy; fplf;fpd;wJ. vdpd;>
(i) ZZZ 2
vdTk;
(ii) ZZArgZArgZArg _
3
21 22
vdTk; fhl;Lf.
45. 4/
1
Z
iZArg
MFkhW Z ,d; xOf;iff; fhz;f.
46. 5Z , ck; 6/6/ ArgZ ck; MFkhWs;s gpuNjrj;ij Mfd; tupg;glj;jpy; epow;wpf; fhl;Lf.
47. 21, ZZ vd;gd vitNaDk; ,U rpf;fnyz;fnsdf; nfhs;Nthk;. Mfd; tupg;glj;jpy;
rpf;fnyz; 21 ZZ I tifFwpf;Fk; Gs;spia mikf;f.
2121 ZZZZ Mf ,Uf;Fk; re;jHg;gj;ij vLj;Jf; fhl;Lk; tupg;glj;ij
tiuf.
nghJthf 2121 ZZZZ Mf ,Ug;gJ Vnddf; Nfj;jpufzpj
Kiwapy; tpsf;Ff.
iZ 5121 MfTk; 52 Z MfTk; ,Ug;gpd;> 21 ZZ ,d; kpfg; ngupa
ngWkhdj;jijf; fhz;f.
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SVM Complex Number 22
21 ZZ mjd; kpfg; ngupa ngWkhdj;ijf; nfhz;Lk;
2arg2
z
MfTk; ,Ug;gpd;> 2Z I iqp tbtj;jpy; vLj;Jiuf;f.
48. Mfd; tupg;glj;jpy; A, B, C, D vd;Dk; Gs;spfs; KiwNa 4321 ,,, zzzz vd;Dk; rpf;fnyz;fis tifFwpf;fpd;wd.
AB Ak; CD Ak; nrq;Fj;jhf ,ilntl;Lnkdpd;> mg;NghJ
43
21
zz
zz mwf; fw;gidahdnjdf; fhl;Lf.
49. z vd;gJ rpf;fnyz; 312
1i vdf; nfhs;Nthk;.
22 3,2
zz
vd;Dk; rpf;fnyz;fs; xt;nthd;wpdJk; kl;bidAk; tPriyAk;
fhz;f.
Xu; Mfd; tpupg;glj;jpNy O MdJ cw;gj;jpiaAk; A MdJ rpf;fnyz;
22z IAk; B MdJ rpf;fnyz; 23
z IAk; tifFwpf;fpd;wd.
O tpw;Fk; B apw;Fk; Clhfr; nry;Yk; Nfhl;bd; kPJ z I tifFwpf;fk; Gs;sp fplf;fpd;wjh?
OACB Xu; ,izfukhf ,Uf;FkhW Gs;sp C njupe;njLf;fg;gl;Ls;sJ. C
apdhy; tifFwpf;fg;gLk; rpf;fnyz; p + iq itj; njf;fhl;bd; tbtj;jpy; Jzpf.
OACB apd; %iytpl;lq;fspd; ePsq;fisf; fhz;f.
50. rpf;fnyz; iw 3 I 0sin0cos ir vd;Dk; tbtj;jpy; vLj;Jiuf;f; ,q;Nf r > 0. mNjhL 200 Mf ,Uf;FkhW 0 Miuadpy; cs;sJ.
5432 ,,, wwww Mfpatw;iw Nkw;Fwpj;j tbtpy; ngWf. 306 z
MfTk; 65
arg6
z
MfTk; ,Uf;FkhW rpf;fnyz;fs; z I
Mfd; tupg;glj;jpy; tiff;Fwpf;fk; Gs;spfisf; nfhz;l gpuNjrk; R ,y; fplf;fpd;wdtw;iwj; Jzpf.
rpf;fnyz; 5,.......,2,1nwn I tifFwpf;Fk; Gs;spfspilNa vit gpuNjrk; R ,y; fplf;fpd;wdntdj; Jzpf.
51. rpf;fnyz; z MdJ 0, yiyxz ,dhy; jug;gLfpd;wJ. Mfd; tupg;glj;jpy; z, 2iz, z + 2iz Mfpatw;iw Nenuhj;j Gs;spfs; KiwNa A, B, CMfpa
Gs;spfisf; Fwpj;J>
AOB , tan
AOC Mfpatw;iwj; Jzpf.
-
SVM Complex Number 23
a. C MdJ fw;gidar;rpd; kPJ fple;jhy;> x ,w;Fk; y apw;FkpilNa cs;s njhlHGilikiag; ngWf.
b. y = 2x vdpd; rpf;fnyz; 2z I tifFwpf;Fk; Gs;spahdJ NfhL OC kPJ fplf;fpd;wnjdf; fhl;Lf.
c. |π| β€ 4MfTk; tan β€ arg (π) β€ tan 2MfTk; ,Uf;Fk; rpf;fnyz; z
I tifFwpf;Fk; Gs;spfisf; nfhz;l gpuNjrj;ij NtnwhU tupg;glj;jpy;
epow;Wf. epow;wpa gpuNjrj;jpd; gug;gsitf; fhz;f.
52. (a) azarg vdpd; z ,d; xOf;if tptupf;f; ,q;F a ck; 0
ck; MFk;.
61arg
z
vdTk;
3
21arg
z
vdTk; jug;gl;Ls;sJ.
(b) rpf;fnyz ; i
i
32
5
I i1 vd;Dk; tbtj;jpy; vLj;Jiuf;fyhnkdf; fhl;Lf. ,q;F nka;ahdJ. ngWkhdj;ijf; $Wf.
,jpypUe;J
6
32
5
i
i
fw;gidahdnjdf; fhl;b mjd; ngWkhdj;ijj; Jzpf 53.
π1 =β3
2+ π
1
2, π2 = β
1
2+ π
β3
2 vd;Dk; rpf;fnyz;fs; Xu; Mfz; tupg;glj;jpy;
KiwNa A, B vd;Dk; Gs;spfspdhy; tiff;Fwpf;fg;gLfpd;wd. π΄ππ π , π΄ππ π Mfpatw;iwf; fhz;f.
OACB vd;gJ Mfz; tupg;glj;jpy; xU rJunkdj; jug;gl;bUg;gpd;> C apdhy; tiff;Fwpf;fg;gLk; rpf;fnyz;zpd; kl;ilAk; tPriyAk; fhz;f. ,q;F O
MdJ cw;gj;jpahFk;.
54. (i) π β + π β β€ 2 vd;Dk; epge;jidf;F cl;gl;L |π β 3|,d; kpfr; rpwpa
ngWkhdj;ijAk; kpfg;ngupa ngWkhdj;ijAk; fhz;f.
55. π΄ππ(π β 1) = vd;Dk; epge;jidf;F cl;gl;L Z ,d; kpfr; rpw;a ngWkhdj;ijf;
fhz;f.
56. (a) β1 + β3π , vd;gtw;wpd; kl;L> tPriyf; fhz;f.
(b) 7 + 24π ,d; th;f;f %yq;fis π + ππ vd;w tbtpy; fhz;f.
(c) |π§ + 2 β π| = β5, πππ(π§ + 2) = Β± MFkhWs;s π§ ,d; xOf;Ffis Mfz;
tupg;glj;jpy; Fwpj;J π§ ,dhy; jug;gLk; nghJ rpf;fnyz;fis π + ππ vd;w
tbtpy; jUf.
57. (a) ππ π vdpd; + MdJ Neh;nka; vdf; fhl;Lf.
NkYk; + MdJ Neu;nka; vd ca;j;jwpf.
-
SVM Complex Number 24
(b) Mfz; tupg;glj;jpy; ,Uf;Fk; P1, P2 vd;Dk; Gs;spfs; KiwNa Z1, Z2 I
tif Fwpg;gpd; (π , π ) I tif Fwpf;Fk; Gs;sp P I ngWtjw;fhf Nfj;jpufzpj mikg;ig epWtYld; jUf.
π = , π =β
vdpd; π , π , π Mfpa Gs;spfis mfz; tupg;glj;jpy;
Fwpf;Ff. ,jpypUe;J π‘ππ = 2 β β3 vd ca;j;jwpf. π‘ππ = 3.732
vdTk; fhl;Lf.
58. i2
3
2
1
vdpd; )(arg|,| vd;gtw;iwf; fhz;f.
Mfd; thpg;glj;jpy; ,Uf;Fk; Gs;sp P MdJ rpf;fy; vz; Z I tif
Fwpf;fpd;wJ. Z tif Fwpf;Fk; Gs;sp Qit ngWtjw;fhd Nfj;jpufzpj
mikg;ig epWtYld; jUf. ,jpypUe;J 2Z tif Fwpf;Fk; Gs;sp RI
ca;j;jwpf. P,Q,R vd;gd rk gf;f Kf;Nfhzpapd; cr;rpfs; vdf; fhl;Lf.
NkYk; ,k; Kf;Nfhzpapd; gug;G 2||4
33Z vdTk; fhl;Lf.
59. iz 5121 MfTk; MfTk; ,Ug;gpd;> 21 zz ,d; kpfg;nghpa ngWkhdj;ijf;
fhz;f.
21 zz mjd; kpfg;nghpa ngWkhdj;ijf; nfhz;Lk;
2arg2z MfTk;
,Ug;gpd; 2z I iqp tbtj;jpy; vLj;Jiuf;f.
60. = cos(π β πΌ) + sin(π β πΌ) vdf; fhl;Lf
π = β1 + π , π = 1 + β3π vdpd; If; fhz;f
π , π vd;gtw;wpd; kl;L tPry; vd;gtw;iwf; fhz;f
cos = β6 β β2 vdf; fhl;Lf
61. 0 β€ πΌπ(π) <β
, |π β 2| β€ 1 MFkhW Mfd; tupg;glj;jpy; Z If;Fwpf;Fk;
gpuNjrj;ijf; fhz;f
π΄ππ(π) caHthFk; NghJ Z If; fhz;f
62. |π + π| = |π β π| MFkhW rpf;fnyz; Z ,d; xOf;iff; fhz;f a nka;naz;.
|π + 2π | = |π β 2π | MFkhW ,U rpf;fnyz; vdpd; nka;ahdJ
vdf; fhl;Lf
52 z
-
SVM Complex Number 25
π΄ππ(π ) ~ π΄ππ(π ) = vdf;fhl;Lf
π + 2π , π β 2π vd;gtw;iw A , B vd;w Gs;spfs; Fwpg;gpd; OA,OB
nrq;Fj;jd;W vdpd; Aππ΅ =tan vdf; fhl;Lf
OA β₯ OB vdpd; k ,d; ngWkhdq;fisf; fhz;f
63. π , π vd;gd A, B, vd;w Gs;spfisf;Fwpf;fpd;wd. π΄ π π΅ = vdpd;
π + π = 0 vdf; fhl;Lf
64. Z vd;w rpf;fnyz; A vd;w Gs;spiaf;Fwpf;Fk; B MdJ iZ , CMdJ Z+iZ
IAk;Fwpg;gpd; OC = β2 |π| vdTk; |π + ππ| = |π β ππ| vdTk; fhl;Lf.
65. .|π β π| < vdpd;
< |π| < vdf; fhl;Lf
66. |π β π| < MfTk;
< |π| < MfTk; cs;s gpuNjrj;ij
epow;Wf
-
SVM Complex Number
tpilfs;
1.. 2.
3. 4.
.
5. 6.
2 Z
-2 2
2 Z
-2 2
2 Z
-2 2
Z
3
1 O
Z
2
-1 O
-
SVM Complex Number 27
7. 8. 9. 10. 11.
Z
1
-1
O
-1
Z
2
2 O
1
-2 3
-2
1
Z
1
-i
O
Z
1 2
O 1
-
SVM Complex Number 28
12. 13. 14.
15. 16. 17. 18.
π
π
π
π
π
π
π
π
π
π
ππ
π
π
β¦ β¦ β¦ β¦
π
βπ
π
1 Z
-1 1
π
βπ
-
SVM Complex Number 29
20. 21. 22. 23. 24. 25.
π
π π
π
π π
π
π
π
π -2
π
π
π i
βπ
π
π
2 Z
-2 2
π
π π
π
-
SVM Complex Number
26. 27.
28 . β 29. 7 ,3 30. -8i 31. 1 , 32. 1 ,
33 i.
ii.
1
π
π
π
βπ
βπ ----------------------------------β¦β¦β¦..
π
3
π
3
π
3
π
-
SVM Complex Number 31
34. 1. Z = 2β2(1 + π) 2. Z = 1+3i , 1-3i 3. Z= -i , 2+i
4. Z =β2(1 β π)
5. Z = 1 + β3π
35 i ii.
-i -1
π
3
Z
-3
2
-1
π
6
Z
Z
π
4
-
SVM Complex Number 32
iii. 45.
46. |π + π | = 18 , π = β + π
49. |2π | = 2 , = 3 , π΄ππ(2π ) = , π΄ππ(3/π ) = β
Mk;. β β β π %iy tpl;lq;fs; β7 , β19
50. π€ = 2 cos + π sin w2 = 4 cos + π sin , w3 = 8 cos + π sin
w4 = 16 cos + π sin , w5 =32 cos + π sin
R ,Ys;sit w3 , w4
51. π΄ π π΅ = , tan π΄ π πΆ = 2 , a) x=2y c) R = 8 tan
52.
π
4
1
-i
-
SVM Complex Number 33
Z = ;. + β π ,π = 1 , β 8π fw;gidahdJ 53.
π΄ππ(π ) = , π΄ππ(π ) =
C ,dhy; Fwpf;fg;gLk; rpf;fnyz; π + π ,d; kl;L β2 MFk;.
π΄ππ(π + π ) =
54. |π β 3| = β3 β 2 , |π β 3| = β7 + 2 55. |π| = 1
56. a. π = β1 + β3π = 2 β +β
π
= 2 cos + π sin β |π| = 2 , π΄ππ(π) =
Z= = π β |π| = 1 , π΄ππ(π) =
b. 4 + 3π , 4 β 3π c.
Ξ± a
Ξ±
a
1
-2
π
π
β5
π = -2+ β5 + 1 π = -2- β5 β 1