دليل رياضيات سادس علمي

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11

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* .1 = i ,ib ayp 4 = 2x ,xp 4 = 2y) (k ,h v2x

2ay +

2

2bb > a ,1 =

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1

)1 - 1(

. : 11(1)i 5 = i 4+1 = i = 0 + i5)i 4 n+1 = i 4 n( )+1 = i = 0 + i,nN= 1 + 2i + i 2 1 + 2i + i 2 1 + 2i + i

2 1 + 2i + i 2

2(

i5 =i(4)+1 =i =0+i

2)i6 =i(4)(1)+2 =i2 =-1=-1+0i3)i124 =i4(31)+0 =i0 =1+0i4)i999 =i4(249)+3 =i3 =i2.i =-i =0-i5)i4n+1 =i4(n)+1 =i =0+i ,n N6)(2+3i)2 +(12+2i)

=(4+12i+9i2)+(12+2i)=(-5+12i)+(12+2i)=7+14i7)(10+3i) (0+6i)

=(10+3i)(6i)=60i+18i2 =60i-18=-18+60i8)(1+i)4 (1-i)4

= (1+i)2( )2 (1-i)2( )2(1+2i+i2)2 (1-2i+i2)2 =(2i)2 (-2i)2=-4+4=0=0+0i

(1+i)4 (1+i)4

= (1+i)2 (1-i)2 (1+i)2 +(1-i)2

= 1+2i+i2 (1-2i+i2) 1+2i+i2 +1-2i+i2

= 1 +2i+ i2 1 +2i- i2 1+2i +12 + 2i+i2

i5 =i(4)+1 =i =0+i


=(4+12i+9i2)+(12+2i)=(-5+12i)+(12+2i)=7+14i7)(10+3i) (0+6i)

=(10+3i)(6i)=60i+18i2 =60i-18=-18+60i8)(1+i)4 (1-i)4

= (1+i)2( )2 (1-i)2( )2(1+2i+i2)2 (1-2i+i2)2 =(2i)2 (-2i)2=-4+4=0=0+0i

(1+i)4 (1+i)4

= (1+i)2 (1-i)2 (1+i)2 +(1-i)2

= 1+2i+i2 (1-2i+i2) 1+2i+i2 +1-2i+i2

= 1 +2i+ i2 1 +2i- i2 1+2i +12 + 2i+i2

i5 =i(4)+1 =i =0+i


=(4+12i+9i2)+(12+2i)=(-5+12i)+(12+2i)=7+14i7)(10+3i) (0+6i)

=(10+3i)(6i)=60i+18i2 =60i-18=-18+60i8)(1+i)4 (1-i)4

= (1+i)2( )2 (1-i)2( )2(1+2i+i2)2 (1-2i+i2)2 =(2i)2 (-2i)2=-4+4=0=0+0i

(1+i)4 (1+i)4

= (1+i)2 (1-i)2 (1+i)2 +(1-i)2

= 1+2i+i2 (1-2i+i2) 1+2i+i2 +1-2i+i2

= 1 +2i+ i2 1 +2i- i2 1+2i +12 + 2i+i2

i5 =i(4)+1 =i =0+i


=(4+12i+9i2)+(12+2i)=(-5+12i)+(12+2i)=7+14i7)(10+3i) (0+6i)

=(10+3i)(6i)=60i+18i2 =60i-18=-18+60i8)(1+i)4 (1-i)4

= (1+i)2( )2 (1-i)2( )2(1+2i+i2)2 (1-2i+i2)2 =(2i)2 (-2i)2=-4+4=0=0+0i

(1+i)4 (1+i)4

= (1+i)2 (1-i)2 (1+i)2 +(1-i)2

= 1+2i+i2 (1-2i+i2) 1+2i+i2 +1-2i+i2

= 1 +2i+ i2 1 +2i- i2 1+2i +12 + 2i+i2

((

i5 =i(4)+1 =i =0+i


=(4+12i+9i2)+(12+2i)=(-5+12i)+(12+2i)=7+14i7)(10+3i) (0+6i)

=(10+3i)(6i)=60i+18i2 =60i-18=-18+60i8)(1+i)4 (1-i)4

= (1+i)2( )2 (1-i)2( )2(1+2i+i2)2 (1-2i+i2)2 =(2i)2 (-2i)2=-4+4=0=0+0i

(1+i)4 (1+i)4

= (1+i)2 (1-i)2 (1+i)2 +(1-i)2

= 1+2i+i2 (1-2i+i2) 1+2i+i2 +1-2i+i2

= 1 +2i+ i2 1 +2i- i2 1+2i +12 + 2i+i2

5(1)i 5 = i 4+1 = i = 0 + i5)i 4 n+1 = i 4 n( )+1 = i = 0 + i,nN= 1 + 2i + i 2 1 + 2i + i 2 1 + 2i + i

2 1 + 2i + i 2 6(

i5 =i(4)+1 =i =0+i


=(4+12i+9i2)+(12+2i)=(-5+12i)+(12+2i)=7+14i7)(10+3i) (0+6i)

=(10+3i)(6i)=60i+18i2 =60i-18=-18+60i8)(1+i)4 (1-i)4

= (1+i)2( )2 (1-i)2( )2(1+2i+i2)2 (1-2i+i2)2 =(2i)2 (-2i)2=-4+4=0=0+0i

(1+i)4 (1+i)4

= (1+i)2 (1-i)2 (1+i)2 +(1-i)2

= 1+2i+i2 (1-2i+i2) 1+2i+i2 +1-2i+i2

= 1 +2i+ i2 1 +2i- i2 1+2i +12 + 2i+i2

i5 =i(4)+1 =i =0+i


=(4+12i+9i2)+(12+2i)=(-5+12i)+(12+2i)=7+14i7)(10+3i) (0+6i)

=(10+3i)(6i)=60i+18i2 =60i-18=-18+60i8)(1+i)4 (1-i)4

= (1+i)2( )2 (1-i)2( )2(1+2i+i2)2 (1-2i+i2)2 =(2i)2 (-2i)2=-4+4=0=0+0i

(1+i)4 (1+i)4

= (1+i)2 (1-i)2 (1+i)2 +(1-i)2

= 1+2i+i2 (1-2i+i2) 1+2i+i2 +1-2i+i2

= 1 +2i+ i2 1 +2i- i2 1+2i +12 + 2i+i2

7(

i5 =i(4)+1 =i =0+i


=(4+12i+9i2)+(12+2i)=(-5+12i)+(12+2i)=7+14i7)(10+3i) (0+6i)

=(10+3i)(6i)=60i+18i2 =60i-18=-18+60i8)(1+i)4 (1-i)4

= (1+i)2( )2 (1-i)2( )2(1+2i+i2)2 (1-2i+i2)2 =(2i)2 (-2i)2=-4+4=0=0+0i

(1+i)4 (1+i)4

= (1+i)2 (1-i)2 (1+i)2 +(1-i)2

= 1+2i+i2 (1-2i+i2) 1+2i+i2 +1-2i+i2

= 1 +2i+ i2 1 +2i- i2 1+2i +12 + 2i+i2

i5 =i(4)+1 =i =0+i


=(4+12i+9i2)+(12+2i)=(-5+12i)+(12+2i)=7+14i7)(10+3i) (0+6i)

=(10+3i)(6i)=60i+18i2 =60i-18=-18+60i8)(1+i)4 (1-i)4

= (1+i)2( )2 (1-i)2( )2(1+2i+i2)2 (1-2i+i2)2 =(2i)2 (-2i)2=-4+4=0=0+0i

(1+i)4 (1+i)4

= (1+i)2 (1-i)2 (1+i)2 +(1-i)2

= 1+2i+i2 (1-2i+i2) 1+2i+i2 +1-2i+i2

= 1 +2i+ i2 1 +2i- i2 1+2i +12 + 2i+i2

8(

i5 =i(4)+1 =i =0+i


=(4+12i+9i2)+(12+2i)=(-5+12i)+(12+2i)=7+14i7)(10+3i) (0+6i)

=(10+3i)(6i)=60i+18i2 =60i-18=-18+60i8)(1+i)4 (1-i)4

= (1+i)2( )2 (1-i)2( )2(1+2i+i2)2 (1-2i+i2)2 =(2i)2 (-2i)2=-4+4=0=0+0i

(1+i)4 (1+i)4

= (1+i)2 (1-i)2 (1+i)2 +(1-i)2

= 1+2i+i2 (1-2i+i2) 1+2i+i2 +1-2i+i2

= 1 +2i+ i2 1 +2i- i2 1+2i +12 + 2i+i2

i5 =i(4)+1 =i =0+i


=(4+12i+9i2)+(12+2i)=(-5+12i)+(12+2i)=7+14i7)(10+3i) (0+6i)

=(10+3i)(6i)=60i+18i2 =60i-18=-18+60i8)(1+i)4 (1-i)4

= (1+i)2( )2 (1-i)2( )2(1+2i+i2)2 (1-2i+i2)2 =(2i)2 (-2i)2=-4+4=0=0+0i

(1+i)4 (1+i)4

= (1+i)2 (1-i)2 (1+i)2 +(1-i)2

= 1+2i+i2 (1-2i+i2) 1+2i+i2 +1-2i+i2

= 1 +2i+ i2 1 +2i- i2 1+2i +12 + 2i+i2

i5 =i(4)+1 =i =0+i


=(4+12i+9i2)+(12+2i)=(-5+12i)+(12+2i)=7+14i7)(10+3i) (0+6i)

=(10+3i)(6i)=60i+18i2 =60i-18=-18+60i8)(1+i)4 (1-i)4

= (1+i)2( )2 (1-i)2( )2(1+2i+i2)2 (1-2i+i2)2 =(2i)2 (-2i)2=-4+4=0=0+0i

(1+i)4 (1+i)4

= (1+i)2 (1-i)2 (1+i)2 +(1-i)2

= 1+2i+i2 (1-2i+i2) 1+2i+i2 +1-2i+i2

= 1 +2i+ i2 1 +2i- i2 1+2i +12 + 2i+i2

i5 =i(4)+1 =i =0+i


=(4+12i+9i2)+(12+2i)=(-5+12i)+(12+2i)=7+14i7)(10+3i) (0+6i)

=(10+3i)(6i)=60i+18i2 =60i-18=-18+60i8)(1+i)4 (1-i)4

= (1+i)2( )2 (1-i)2( )2(1+2i+i2)2 (1-2i+i2)2 =(2i)2 (-2i)2=-4+4=0=0+0i

(1+i)4 (1+i)4

= (1+i)2 (1-i)2 (1+i)2 +(1-i)2

= 1+2i+i2 (1-2i+i2) 1+2i+i2 +1-2i+i2

= 1 +2i+ i2 1 +2i- i2 1+2i +12 + 2i+i2

15

= (4i)(0) = 0 = 0+0i= 1+2i+i 2 -1+2i-i 2 1+2i+i 2 +1-2i+i 2 9(

9) 12+ii

= 12+ii

.-i-i

= 12+ii

.ii= 12i+i

2

i2= 12i-1

-1= -12i+1=1-12i

10)3+4i3-4i

= 3+4i3-4i

= 3+4i3-4i

= 9+12i+12i+16i2

9+16= 9+24i-16

25

= -7+24i25

= -725

+ 2425

i

11) i2+3i

= i2+3i

.2-3i2-3i

= 2i-3i2

13= 2i-3

13= 313

+ 213

i

12) 3+i1+i

3

3+i1+i

.1-i1-i

3

= 3-3i+i-i2

2

3

= 4-2i2

3

= 42 22i

3

=(2-i)3 =(2-i)2(2-i)

=(4-4i+i2) (2-i)=(3-4i) (2-i)=6-3i-8i+4i2

= 2-11i

13)2+3i1-i

1+4i4+i

= 2+8i+3i+12i2

4+i-4i-i2

= -10+11i5-3i

= -10+11i5-3i

.5+3i5-3i

= -50-30i+55i+33i2

25+9

= 83+25i24

= -8324

+ 25i24

9) 12+ii

= 12+ii

.-i-i

= 12+ii

.ii= 12i+i

2

i2= 12i-1

-1= -12i+1=1-12i

10)3+4i3-4i

= 3+4i3-4i

= 3+4i3-4i

= 9+12i+12i+16i2

9+16= 9+24i-16

25

= -7+24i25

= -725

+ 2425

i

11) i2+3i

= i2+3i

.2-3i2-3i

= 2i-3i2

13= 2i-3

13= 313

+ 213

i

12) 3+i1+i

3

3+i1+i

.1-i1-i

3

= 3-3i+i-i2

2

3

= 4-2i2

3

= 42 22i

3

=(2-i)3 =(2-i)2(2-i)

=(4-4i+i2) (2-i)=(3-4i) (2-i)=6-3i-8i+4i2

= 2-11i

13)2+3i1-i

1+4i4+i

= 2+8i+3i+12i2

4+i-4i-i2

= -10+11i5-3i

= -10+11i5-3i

.5+3i5-3i

= -50-30i+55i+33i2

25+9

= 83+25i24

= -8324

+ 25i24

10(

9) 12+ii

= 12+ii

.-i-i

= 12+ii

.ii= 12i+i

2

i2= 12i-1

-1= -12i+1=1-12i

10)3+4i3-4i

= 3+4i3-4i

= 3+4i3-4i

= 9+12i+12i+16i2

9+16= 9+24i-16

25

= -7+24i25

= -725

+ 2425

i

11) i2+3i

= i2+3i

.2-3i2-3i

= 2i-3i2

13= 2i-3

13= 313

+ 213

i

12) 3+i1+i

3

3+i1+i

.1-i1-i

3

= 3-3i+i-i2

2

3

= 4-2i2

3

= 42 22i

3

=(2-i)3 =(2-i)2(2-i)

=(4-4i+i2) (2-i)=(3-4i) (2-i)=6-3i-8i+4i2

= 2-11i

13)2+3i1-i

1+4i4+i

= 2+8i+3i+12i2

4+i-4i-i2

= -10+11i5-3i

= -10+11i5-3i

.5+3i5-3i

= -50-30i+55i+33i2

25+9

= 83+25i24

= -8324

+ 25i24

. 3+ 4i3+ 4i

9) 12+ii

= 12+ii

.-i-i

= 12+ii

.ii= 12i+i

2

i2= 12i-1

-1= -12i+1=1-12i

10)3+4i3-4i

= 3+4i3-4i

= 3+4i3-4i

= 9+12i+12i+16i2

9+16= 9+24i-16

25

= -7+24i25

= -725

+ 2425

i

11) i2+3i

= i2+3i

.2-3i2-3i

= 2i-3i2

13= 2i-3

13= 313

+ 213

i

12) 3+i1+i

3

3+i1+i

.1-i1-i

3

= 3-3i+i-i2

2

3

= 4-2i2

3

= 42 22i

3

=(2-i)3 =(2-i)2(2-i)

=(4-4i+i2) (2-i)=(3-4i) (2-i)=6-3i-8i+4i2

= 2-11i

13)2+3i1-i

1+4i4+i

= 2+8i+3i+12i2

4+i-4i-i2

= -10+11i5-3i

= -10+11i5-3i

.5+3i5-3i

= -50-30i+55i+33i2

25+9

= 83+25i24

= -8324

+ 25i24

11(

9) 12+ii

= 12+ii

.-i-i

= 12+ii

.ii= 12i+i

2

i2= 12i-1

-1= -12i+1=1-12i

10)3+4i3-4i

= 3+4i3-4i

= 3+4i3-4i

= 9+12i+12i+16i2

9+16= 9+24i-16

25

= -7+24i25

= -725

+ 2425

i

11) i2+3i

= i2+3i

.2-3i2-3i

= 2i-3i2

13= 2i-3

13= 313

+ 213

i

12) 3+i1+i

3

3+i1+i

.1-i1-i

3

= 3-3i+i-i2

2

3

= 4-2i2

3

= 42 22i

3

=(2-i)3 =(2-i)2(2-i)

=(4-4i+i2) (2-i)=(3-4i) (2-i)=6-3i-8i+4i2

= 2-11i

13)2+3i1-i

1+4i4+i

= 2+8i+3i+12i2

4+i-4i-i2

= -10+11i5-3i

= -10+11i5-3i

.5+3i5-3i

= -50-30i+55i+33i2

25+9

= 83+25i24

= -8324

+ 25i24

9) 12+ii

= 12+ii

.-i-i

= 12+ii

.ii= 12i+i

2

i2= 12i-1

-1= -12i+1=1-12i

10)3+4i3-4i

= 3+4i3-4i

= 3+4i3-4i

= 9+12i+12i+16i2

9+16= 9+24i-16

25

= -7+24i25

= -725

+ 2425

i

11) i2+3i

= i2+3i

.2-3i2-3i

= 2i-3i2

13= 2i-3

13= 313

+ 213

i

12) 3+i1+i

3

3+i1+i

.1-i1-i

3

= 3-3i+i-i2

2

3

= 4-2i2

3

= 42 22i

3

=(2-i)3 =(2-i)2(2-i)

=(4-4i+i2) (2-i)=(3-4i) (2-i)=6-3i-8i+4i2

= 2-11i

13)2+3i1-i

1+4i4+i

= 2+8i+3i+12i2

4+i-4i-i2

= -10+11i5-3i

= -10+11i5-3i

.5+3i5-3i

= -50-30i+55i+33i2

25+9

= 83+25i24

= -8324

+ 25i24

9) 12+ii

= 12+ii

.-i-i

= 12+ii

.ii= 12i+i

2

i2= 12i-1

-1= -12i+1=1-12i

10)3+4i3-4i

= 3+4i3-4i

= 3+4i3-4i

= 9+12i+12i+16i2

9+16= 9+24i-16

25

= -7+24i25

= -725

+ 2425

i

11) i2+3i

= i2+3i

.2-3i2-3i

= 2i-3i2

13= 2i-3

13= 313

+ 213

i

12) 3+i1+i

3

3+i1+i

.1-i1-i

3

= 3-3i+i-i2

2

3

= 4-2i2

3

= 42 22i

3

=(2-i)3 =(2-i)2(2-i)

=(4-4i+i2) (2-i)=(3-4i) (2-i)=6-3i-8i+4i2

= 2-11i

13)2+3i1-i

1+4i4+i

= 2+8i+3i+12i2

4+i-4i-i2

= -10+11i5-3i

= -10+11i5-3i

.5+3i5-3i

= -50-30i+55i+33i2

25+9

= 83+25i24

= -8324

+ 25i24

9) 12+ii

= 12+ii

.-i-i

= 12+ii

.ii= 12i+i

2

i2= 12i-1

-1= -12i+1=1-12i

10)3+4i3-4i

= 3+4i3-4i

= 3+4i3-4i

= 9+12i+12i+16i2

9+16= 9+24i-16

25

= -7+24i25

= -725

+ 2425

i

11) i2+3i

= i2+3i

.2-3i2-3i

= 2i-3i2

13= 2i-3

13= 313

+ 213

i

12) 3+i1+i

3

3+i1+i

.1-i1-i

3

= 3-3i+i-i2

2

3

= 4-2i2

3

= 42 22i

3

=(2-i)3 =(2-i)2(2-i)

=(4-4i+i2) (2-i)=(3-4i) (2-i)=6-3i-8i+4i2

= 2-11i

13)2+3i1-i

1+4i4+i

= 2+8i+3i+12i2

4+i-4i-i2

= -10+11i5-3i

= -10+11i5-3i

.5+3i5-3i

= -50-30i+55i+33i2

25+9

= 83+25i24

= -8324

+ 25i24

9) 12+ii

= 12+ii

.-i-i

= 12+ii

.ii= 12i+i

2

i2= 12i-1

-1= -12i+1=1-12i

10)3+4i3-4i

= 3+4i3-4i

= 3+4i3-4i

= 9+12i+12i+16i2

9+16= 9+24i-16

25

= -7+24i25

= -725

+ 2425

i

11) i2+3i

= i2+3i

.2-3i2-3i

= 2i-3i2

13= 2i-3

13= 313

+ 213

i

12) 3+i1+i

3

3+i1+i

.1-i1-i

3

= 3-3i+i-i2

2

3

= 4-2i2

3

= 42 22i

3

=(2-i)3 =(2-i)2(2-i)

=(4-4i+i2) (2-i)=(3-4i) (2-i)=6-3i-8i+4i2

= 2-11i

13)2+3i1-i

1+4i4+i

= 2+8i+3i+12i2

4+i-4i-i2

= -10+11i5-3i

= -10+11i5-3i

.5+3i5-3i

= -50-30i+55i+33i2

25+9

= 83+25i24

= -8324

+ 25i24

9) 12+ii

= 12+ii

.-i-i

= 12+ii

.ii= 12i+i

2

i2= 12i-1

-1= -12i+1=1-12i

10)3+4i3-4i

= 3+4i3-4i

= 3+4i3-4i

= 9+12i+12i+16i2

9+16= 9+24i-16

25

= -7+24i25

= -725

+ 2425

i

11) i2+3i

= i2+3i

.2-3i2-3i

= 2i-3i2

13= 2i-3

13= 313

+ 213

i

12) 3+i1+i

3

3+i1+i

.1-i1-i

3

= 3-3i+i-i2

2

3

= 4-2i2

3

= 42 22i

3

=(2-i)3 =(2-i)2(2-i)

=(4-4i+i2) (2-i)=(3-4i) (2-i)=6-3i-8i+4i2

= 2-11i

13)2+3i1-i

1+4i4+i

= 2+8i+3i+12i2

4+i-4i-i2

= -10+11i5-3i

= -10+11i5-3i

.5+3i5-3i

= -50-30i+55i+33i2

25+9

= 83+25i24

= -8324

+ 25i24

9) 12+ii

= 12+ii

.-i-i

= 12+ii

.ii= 12i+i

2

i2= 12i-1

-1= -12i+1=1-12i

10)3+4i3-4i

= 3+4i3-4i

= 3+4i3-4i

= 9+12i+12i+16i2

9+16= 9+24i-16

25

= -7+24i25

= -725

+ 2425

i

11) i2+3i

= i2+3i

.2-3i2-3i

= 2i-3i2

13= 2i-3

13= 313

+ 213

i

12) 3+i1+i

3

3+i1+i

.1-i1-i

3

= 3-3i+i-i2

2

3

= 4-2i2

3

= 42 22i

3

=(2-i)3 =(2-i)2(2-i)

=(4-4i+i2) (2-i)=(3-4i) (2-i)=6-3i-8i+4i2

= 2-11i

13)2+3i1-i

1+4i4+i

= 2+8i+3i+12i2

4+i-4i-i2

= -10+11i5-3i

= -10+11i5-3i

.5+3i5-3i

= -50-30i+55i+33i2

25+9

= 83+25i24

= -8324

+ 25i24

9) 12+ii

= 12+ii

.-i-i

= 12+ii

.ii= 12i+i

2

i2= 12i-1

-1= -12i+1=1-12i

10)3+4i3-4i

= 3+4i3-4i

= 3+4i3-4i

= 9+12i+12i+16i2

9+16= 9+24i-16

25

= -7+24i25

= -725

+ 2425

i

11) i2+3i

= i2+3i

.2-3i2-3i

= 2i-3i2

13= 2i-3

13= 313

+ 213

i

12) 3+i1+i

3

3+i1+i

.1-i1-i

3

= 3-3i+i-i2

2

3

= 4-2i2

3

= 42 22i

3

=(2-i)3 =(2-i)2(2-i)

=(4-4i+i2) (2-i)=(3-4i) (2-i)=6-3i-8i+4i2

= 2-11i

13)2+3i1-i

1+4i4+i

= 2+8i+3i+12i2

4+i-4i-i2

= -10+11i5-3i

= -10+11i5-3i

.5+3i5-3i

= -50-30i+55i+33i2

25+9

= 83+25i24

= -8324

+ 25i24

12(

9) 12+ii

= 12+ii

.-i-i

= 12+ii

.ii= 12i+i

2

i2= 12i-1

-1= -12i+1=1-12i

10)3+4i3-4i

= 3+4i3-4i

= 3+4i3-4i

= 9+12i+12i+16i2

9+16= 9+24i-16

25

= -7+24i25

= -725

+ 2425

i

11) i2+3i

= i2+3i

.2-3i2-3i

= 2i-3i2

13= 2i-3

13= 313

+ 213

i

12) 3+i1+i

3

3+i1+i

.1-i1-i

3

= 3-3i+i-i2

2

3

= 4-2i2

3

= 42 22i

3

=(2-i)3 =(2-i)2(2-i)

=(4-4i+i2) (2-i)=(3-4i) (2-i)=6-3i-8i+4i2

= 2-11i

13)2+3i1-i

1+4i4+i

= 2+8i+3i+12i2

4+i-4i-i2

= -10+11i5-3i

= -10+11i5-3i

.5+3i5-3i

= -50-30i+55i+33i2

25+9

= 83+25i24

= -8324

+ 25i24

9) 12+ii

= 12+ii

.-i-i

= 12+ii

.ii= 12i+i

2

i2= 12i-1

-1= -12i+1=1-12i

10)3+4i3-4i

= 3+4i3-4i

= 3+4i3-4i

= 9+12i+12i+16i2

9+16= 9+24i-16

25

= -7+24i25

= -725

+ 2425

i

11) i2+3i

= i2+3i

.2-3i2-3i

= 2i-3i2

13= 2i-3

13= 313

+ 213

i

12) 3+i1+i

3

3+i1+i

.1-i1-i

3

= 3-3i+i-i2

2

3

= 4-2i2

3

= 42 22i

3

=(2-i)3 =(2-i)2(2-i)

=(4-4i+i2) (2-i)=(3-4i) (2-i)=6-3i-8i+4i2

= 2-11i

13)2+3i1-i

1+4i4+i

= 2+8i+3i+12i2

4+i-4i-i2

= -10+11i5-3i

= -10+11i5-3i

.5+3i5-3i

= -50-30i+55i+33i2

25+9

= 83+25i24

= -8324

+ 25i24

16

1(

2+3i1- i

1+ 4i4+ i

= (2+3i1- i

.1+ i1+ i

) (1+ 4i4+ i

.4 - i4 - i

)

= (2+ 2i +3i +3i2

2) (4 - i +16i - 4i

2

17)

= (-1+5i2

) (8+15i17

) = -8 -15i + 40i +75i2

34

= -83+ 25i34

= -8334

+ 25i34

2+3i1- i

1+ 4i4+ i

= (2+3i1- i

.1+ i1+ i

) (1+ 4i4+ i

.4 - i4 - i

)

= (2+ 2i +3i +3i2

2) (4 - i +16i - 4i

2

17)

= (-1+5i2

) (8+15i17

) = -8 -15i + 40i +75i2

34

= -83+ 25i34

= -8334

+ 25i34

2+3i1- i

1+ 4i4+ i

= (2+8i +3i +12i2

4+ i 4i + i2

= 10+11i5+3i

5+3i5+3i

= 5030i +55i +33i2

25+9

= 83+ 25i34

= 8334

+ 2534

i

.

= 2+ 8i + 3i +12i2

4 + i 4i i 2

2+3i1- i

1+ 4i4+ i

= (2+8i +3i +12i2

4+ i 4i + i2

= 10+11i5+3i

5+3i5+3i

= 5030i +55i +33i2

25+9

= 83+ 25i34

= 8334

+ 2534

i

2+3i1- i

1+ 4i4+ i

= (2+8i +3i +12i2

4+ i 4i + i2

= 10+11i5+3i

5+3i5+3i

= 5030i +55i +33i2

25+9

= 83+ 25i34

= 8334

+ 2534

i

.

17

1(

-83+ 25i34

= -8334

+ 25i34

14)(1+ i)3 + (1- i)3

= (1+ i)2 (1+ i)+ (1- i)2(1- i)= (1+ 2i + i2) (1+ i)+ (1- 2i + i2) (1- i)= 2i(1+ i)+ (-2i)(1- i)= 2 i + 2i2 - 2 i + 2i2

4i2 = -4= 4 +0i(1+ i)3 + (1 i)3

(1+ i)+ (1- i) (1+ i)2 (1+ i)(1- i)+ (1- i)2

1+ i +1- i 1+ 2i + i2 (2)+ 1 - 2i + i 2

= 2(+2 i - 2 - 2 i)

-83+ 25i34

= -8334

+ 25i34

14)(1+ i)3 + (1- i)3


4i2 = -4= 4 +0i(1+ i)3 + (1 i)3

(1+ i)+ (1- i) (1+ i)2 (1+ i)(1- i)+ (1- i)2

1+ i +1- i 1+ 2i + i2 (2)+ 1 - 2i + i 2

= 2(+2 i - 2 - 2 i)

-83+ 25i34

= -8334

+ 25i34

14)(1+ i)3 + (1- i)3


4i2 = -4= 4 +0i(1+ i)3 + (1 i)3

(1+ i)+ (1- i) (1+ i)2 (1+ i)(1- i)+ (1- i)2

1+ i +1- i 1+ 2i + i2 (2)+ 1 - 2i + i 2

= 2(+2 i - 2 - 2 i)

-83+ 25i34

= -8334

+ 25i34

14)(1+ i)3 + (1- i)3


4i2 = -4= 4 +0i(1+ i)3 + (1 i)3

(1+ i)+ (1- i) (1+ i)2 (1+ i)(1- i)+ (1- i)2

1+ i +1- i 1+ 2i + i2 (2)+ 1 - 2i + i 2

= 2(+2 i - 2 - 2 i)

) (

-83+ 25i34

= -8334

+ 25i34

14)(1+ i)3 + (1- i)3


4i2 = -4= 4 +0i(1+ i)3 + (1 i)3

(1+ i)+ (1- i) (1+ i)2 (1+ i)(1- i)+ (1- i)2

1+ i +1- i 1+ 2i + i2 (2)+ 1 - 2i + i 2

= 2(+2 i - 2 - 2 i)

-83+ 25i34

= -8334

+ 25i34

14)(1+ i)3 + (1- i)3


4i2 = -4= 4 +0i(1+ i)3 + (1 i)3

(1+ i)+ (1- i) (1+ i)2 (1+ i)(1- i)+ (1- i)2

1+ i +1- i 1+ 2i + i2 (2)+ 1 - 2i + i 2

= 2(+2 i - 2 - 2 i)

-83+ 25i34

= -8334

+ 25i34

14)(1+ i)3 + (1- i)3


4i2 = -4= 4 +0i(1+ i)3 + (1 i)3

(1+ i)+ (1- i) (1+ i)2 (1+ i)(1- i)+ (1- i)2

1+ i +1- i 1+ 2i + i2 (2)+ 1 - 2i + i 2

= 2(+2 i - 2 - 2 i)

-83+ 25i34

= -8334

+ 25i34

14)(1+ i)3 + (1- i)3


4i2 = -4= 4 +0i(1+ i)3 + (1 i)3

(1+ i)+ (1- i) (1+ i)2 (1+ i)(1- i)+ (1- i)2

1+ i +1- i 1+ 2i + i2 (2)+ 1 - 2i + i 2

= 2(+2 i - 2 - 2 i) = 2 2( ) = 4 = 4 +0i

: x,y :2

a(

a)y +5i = (2x + i)(x + 2i)y +5i = 2x 2 + 4xi + xi + 2i2

y +5i = 2x 2 +5xi - 2y = 2x 2 - 2...(1)

5x = 5 x = 55=1 x =1...(2)

y = (2)(1) 2y = 2 - 2 = 0 y = 0


y +5i = 2x 2 +5xi - 2y = 2x 2 - 2...(1)

5x = 5 x = 55=1 x =1...(2)

y = (2)(1) 2y = 2 - 2 = 0 y = 0


y +5i = 2x 2 +5xi - 2y = 2x 2 - 2...(1)

5x = 5 x = 55=1 x =1...(2)

y = (2)(1) 2y = 2 - 2 = 0 y = 0

)1( )2(

18

b( b)8i = (x + 2i)(y + 2i)+1

-1+8i = (x + 2i)(y + 2i)-1+8i = xy + 2xi + 2yi + 4i2

-1+8i = xy + 2xi + 2yi - 4-1= xy - 4...(1)8 = 2x + 2y 2

4 = x + yy = 4 - x...(2) -1= x(4 - x) - 4-1= 4x - x 2 - 4

)1( )2(

) 2(

) 2(

x 2 - 4x -1+ 4 = 0 x 2 - 4x +3 = 0 (x - 3)(x 1) = 0x 3 = 0 x = 3 y = 4 3 =1x 1= 0 x =1 y = 4 1= 3

x 2 - 4x -1+ 4 = 0 x 2 - 4x +3 = 0 (x - 3)(x 1) = 0x 3 = 0 x = 3 y = 4 3 =1x 1= 0 x =1 y = 4 1= 3

c(

C )1- i1+ i

+ (x + yi) = (1+ 2i)2

1- i1+ i

.1- i1- i

+ x + yi =1+ 4i + 4i2

1- i - i + i2

2+ x + yi = 3+ 4i

-2i2+ x + yi = 3+ 4i

-i + x + yi = 3+ 4i x = -3-1+ y = 4 y = 4+1 y = 5

C )1- i1+ i

+ (x + yi) = (1+ 2i)2

1- i1+ i

.1- i1- i

+ x + yi =1+ 4i + 4i2

1- i - i + i2

2+ x + yi = 3+ 4i

-2i2+ x + yi = 3+ 4i

-i + x + yi = 3+ 4i x = -3-1+ y = 4 y = 4+1 y = 5

19

d(

d)2 - i1+ i

x + 3 - i2+ i

y = 1i

2 - i1+ i

.i - i1 i

x +

3 - i2+ i

.2 - i2 - i

y =

1

i.

ii

2 - 2i - i + i2

2

x + 6 - 3i - 2i + i

2

5

y = -i

1

2 3

2i

x +

5

5 5i

5

y = -i

1

2x -

3

2xi + y - yi = -i

1

2x + y = 0...(1)

-3

2x - y = -1...(2)

d)2 - i1+ i

x + 3 - i2+ i

y = 1i

2 - i1+ i

.i - i1 i

x +

3 - i2+ i

.2 - i2 - i

y =

1

i.

ii

2 - 2i - i + i2

2

x + 6 - 3i - 2i + i

2

5

y = -i

1

2 3

2i

x +

5

5 5i

5

y = -i

1

2x -

3

2xi + y - yi = -i

1

2x + y = 0...(1)

-3

2x - y = -1...(2)

d)2 - i1+ i

x + 3 - i2+ i

y = 1i

2 - i1+ i

.i - i1 i

x +

3 - i2+ i

.2 - i2 - i

y =

1

i.

ii

2 - 2i - i + i2

2

x + 6 - 3i - 2i + i

2

5

y = -i

1

2 3

2i

x +

5

5 5i

5

y = -i

1

2x -

3

2xi + y - yi = -i

1

2x + y = 0...(1)

-3

2x - y = -1...(2)

1 - 32

x = -1 22

x = 1 x =1

1

2+ y = 0 y = -1

2 )1( x =1

20

: :a(

a)

1

(2 - i)2-

1

(2+ i)2= 8

25i

LHS

= 14 - 4i + i2

-1

4+ 4i + i21

3 - 4i.3+ 4i3+ 4i

13+ 4i

.3 - 4i3 - 4i

= 3+ 4i9+16

3 - 4i9+16

= 3+ 4i - 3+ 4i25

= 8i25

=

1

3 - 4i-

1

3+ 4i

= (3+ 4i) - (3 - 4i)(3 - 4i)(3+ 4i)

= 3+ 4i - 3+ 4i25

= 8i25

RHS

LHS=

= RHS

= 14 - 4i + i2

-1

4+ 4i + i21

3 - 4i.3+ 4i3+ 4i

13+ 4i

.3 - 4i3 - 4i

= 3+ 4i9+16

3 - 4i9+16

= 3+ 4i - 3+ 4i25

= 8i25

=

1

3 - 4i-

1

3+ 4i

= (3+ 4i) - (3 - 4i)(3 - 4i)(3+ 4i)

= 3+ 4i - 3+ 4i25

= 8i25

21

b(

b)(1- i)2

1+ i+ (1+ i)

2

1- i= -2

= 1- 2i + i2

1+ i+ 1+ 2i + i

2

1- i

= -2i1+ i

+ 2i1- i

b)(1- i)2

1+ i+ (1+ i)

2

1- i= -2

= 1- 2i + i2

1+ i+ 1+ 2i + i

2

1- i

= -2i1+ i

+ 2i1- i

LHS

= -2i1+ i

.1- i1 i

+ 2i1- i

.1+ i1+ i

= -2i + 2i2

2+ 2i + 2i

2

2= -2 i - 2+ 2 i - 2

2

= -4

2= 2 =RHS

c(

c)(1- i)(1- i2)(1- i3 ) = 4= (1- i)(1+1)(1- i2.i)= (1- i)(2)(1+ i)= (2)(1- i)(1+ i)= 2.2= 4 =

c)(1- i)(1- i2)(1- i3 ) = 4= (1- i)(1+1)(1- i2.i)= (1- i)(2)(1+ i)= (2)(1- i)(1+ i)= 2.2= 4 =

LHS

= 2( ) 2( )= 4 = RHS

22

: 58152192 ib+ a b,a .

(a

4 +52 = 92)a)i2 +5()i2 - 5( = 2i4 - 52 = 52+ 4 = 92

4 +52 = 92)a)i2 +5()i2 - 5( = 2i4 - 52 =52+ 4 = 92

4 +52 = 92)a)i2 +5()i2 - 5( = 2i4 - 52 =52+ 4 = 92

i52 - 4 = )i5+2()i5 - 2( = 2

(b

4 +121 = 521i4 121 = ) (i2 +11 ) (i2 11 = 121+ 4 = 521i121 4 = ) (i11+ 2 ) (i11 2 =

4 +121 = 521i4 121 = ) (i2 +11 ) (i2 11 = 121+ 4 = 521i121 4 = ) (i11+ 2 ) (i11 2 =

4 +121 = 521i4 121 = ) (i2 +11 ) (i2 11 = 121+ 4 = 521i121 4 = ) (i11+ 2 ) (i11 2 =

001+ 52 = 5212 i001 52 =

) (i2 +1 ) (i2 1 52 = ) (2 i4 1 52 =

4 +121 = 521i4 121 = ) (i2 +11 ) (i2 11 = 121+ 4 = 521i121 4 = ) (i11+ 2 ) (i11 2 =

4 +121 = 521i4 121 = ) (i2 +11 ) (i2 11 = 121+ 4 = 521i121 4 = ) (i11+ 2 ) (i11 2 =

001+ 52 = 5212 i001 52 =

) (i2 +1 ) (i2 1 52 = ) (2 i4 1 52 =

4 +121 = 521i4 121 = ) (i2 +11 ) (i2 11 = 121+ 4 = 521i121 4 = ) (i11+ 2 ) (i11 2 = (c

61+52 =14)c 2i61- 52 =

)i4 +5()i4 - 5( =

2

52 +61=14

2i52 - 61=

)i5+4()i5 - 4( =

(d

18+4 = 58

2i18 - 4 =

)i9+2()i9 - 2( =

4 +18 = 58

2i4 -18 =

)i2 +9()i2 - 9( =

63 + 94 = 582 i63 94 = ) (i6 + 7 ) (i6 7 =

2 i94 63 = 58

) (i7+6 ) (i7 6 =

2

.

i+3i-2

6 , iy+x

5: xy

6

iy + xi - 3 =

i +2 :

)i +2(6 = )iy + x()i - 3(

i6+21 = iy + xi - 3

i6+21 = iy + xi - 3

.i +3 i +3

i6+ i81+ i21+63 = iy + x

2

01

i03+03 = iy + x01

03 = iy + x01

03 +01

i

i3+3 = iy + x3 = x 3 = y

i + 3i 2

6 =iy x

25

)1 - 2(

:1 a(

a)Z 2 = -12Z = -12 Z = 12i Z = 2 3i

2 3 i{ }

a)Z 2 = -12Z = -12 Z = 12i Z = 2 3i

2 3 i{ }

a)Z 2 = -12Z = -12 Z = 12i Z = 2 3i

2 3 i{ }

a)Z2 = -12

Z2 +12 = 0 Z 2 12i2 = 0 (Z - 12i)(Z + 12i) = 0

Z 12 i = 0

Z = 12i = 2 3i Z + 12i = 0 Z = 12i Z = 2 3i

b(

b)Z 2 3Z +3+ i = 0a =1,b = -3,c = 3+ i

Z = -b b2 - (4)(a)(c)2a

= Z = (-3) 9 - (4)(1)(3+ i)(2)(1)

= 3 9 -12 - 4i2

= 3 -3 - 4i2

b)Z 2 3Z +3+ i = 0a =1,b = -3,c = 3+ i

Z = -b b2 - (4)(a)(c)2a

= Z = (-3) 9 - (4)(1)(3+ i)(2)(1)

= 3 9 -12 - 4i2

= 3 -3 - 4i2

b)Z 2 3Z +3+ i = 0a =1,b = -3,c = 3+ i

Z = -b b2 - (4)(a)(c)2a

= Z = (-3) 9 - (4)(1)(3+ i)(2)(1)

= 3 9 -12 - 4i2

= 3 -3 - 4i2

b)Z 2 3Z +3+ i = 0a =1,b = -3,c = 3+ i

Z = -b b2 - (4)(a)(c)2a

= Z = (-3) 9 - (4)(1)(3+ i)(2)(1)

= 3 9 -12 - 4i2

= 3 -3 - 4i2

x 2 - (-2

x)2 = -3

b)Z 2 3Z +3+ i = 0a =1,b = -3,c = 3+ i

Z = -b b2 - (4)(a)(c)2a

= Z = (-3) 9 - (4)(1)(3+ i)(2)(1)

= 3 9 -12 - 4i2

= 3 -3 - 4i2

b)Z 2 3Z +3+ i = 0a =1,b = -3,c = 3+ i

Z = -b b2 - (4)(a)(c)2a

= Z = (-3) 9 - (4)(1)(3+ i)(2)(1)

= 3 9 -12 - 4i2

= 3 -3 - 4i2

62

i4 - 3- = )iy + x(

)iy + x(

i4 - 3- = 2

i4 - 3- = 2i2 y + yx2 + 2 x )1(...3- = 2 y - 2 x

4- = y 4- = yx2x2

2- = y x

)2(...

2-( - 2 x

x3- = 2)

(2) (1)

4 2x 2x

3 =

2x3 = 4 4xx

4 2x 0 22x

3 =

2x3 = 4 4x

0 = )1- 2 x()4 + 2 x( 0 = 4 - 2 x3+ 4 x R x, 4- = 2 xx

1m = x 1= 2

2 = y

0 = )1- 2 x()4 + 2 x( 0 = 4 - 2 x3+ 4 x R x, 4- = 2 x

0 = )1- 2 x()4 + 2 x( 0 = 4 - 2 x3+ 4 xR x, 4- = 2 x

(2)

)i2 +1-(,)i2 -1(

i4 - 3- 3 = z

2

i4-3-

72

= Z

)i2 -1( +32

4 =22

2i - 2 = i

)i2 -1( - 32

2 =22 +

2i +1= i

} {)1-2(,)i+1(} {)1-2(,)i+1( (c

Z2)c 0 = 31+ Z5 - 2

31= c,5- = b,2 = a

)31()2()4( - 52 5 = Z

)2()2(401- 52 5 =

4

97- 5 = Z

45 =

4i97

4 } {)i - 2( ,)i +1(

54i 97

4

} {)i - 2( ,)i +1(

28

d(d)Z 2 + 2Z + i(2 i) = 0Z 2 + 2Z + 2i i 2 = 0Z 2 + 2Z + 2i +1= 0

Z = 2 4 (4)(1)(1+ 2i)(2)(1)

= 2 4 4 8i2= 2 8i2(x+ yi)2 = 8ix2 + 2xyi + y2i 2 = 8ix2 y2 = 0...........(1)2xy = 8 y = 4

x..........(2)

x2 4x

2

= 0

x216x2

= 0

x2

x4 16 = 0 (x2 4)(x2 + 4) = 0x2 4 = 0 x2 = 4 x = 2 y = 42 y = 2

x2 + 4 = 0 x2 = 4(2+ 2i)(2 2i)8i

2 = 2 8i2 =2 (2 2i)

2 = i2 (2 2i)

2 = 2+ i

(0 i), (2+ i)

d)Z 2 + 2Z + i(2 i) = 0Z 2 + 2Z + 2i i 2 = 0Z 2 + 2Z + 2i +1= 0

Z = 2 4 (4)(1)(1+ 2i)(2)(1)


x..........(2)

x2 4x

2

= 0

x216x2

= 0

x2

x4 16 = 0 (x2 4)(x2 + 4) = 0x2 4 = 0 x2 = 4 x = 2 y = 42 y = 2

x2 + 4 = 0 x2 = 4(2+ 2i)(2 2i)8i

2 = 2 8i2 =2 (2 2i)

2 = i2 (2 2i)

2 = 2+ i

(0 i), (2+ i)

d)Z 2 + 2Z + i(2 i) = 0Z 2 + 2Z + 2i i 2 = 0Z 2 + 2Z + 2i +1= 0

Z = 2 4 (4)(1)(1+ 2i)(2)(1)


x..........(2)

x2 4x

2

= 0

x216x2

= 0

x2

x4 16 = 0 (x2 4)(x2 + 4) = 0x2 4 = 0 x2 = 4 x = 2 y = 42 y = 2

x2 + 4 = 0 x2 = 4(2+ 2i)(2 2i)8i

2 = 2 8i2 =2 (2 2i)

2 = i2 (2 2i)

2 = 2+ i

(0 i), (2+ i)

d)Z 2 + 2Z + i(2 i) = 0Z 2 + 2Z + 2i i 2 = 0Z 2 + 2Z + 2i +1= 0

Z = 2 4 (4)(1)(1+ 2i)(2)(1)


x..........(2)

x2 4x

2

= 0

x216x2

= 0

x2

x4 16 = 0 (x2 4)(x2 + 4) = 0x2 4 = 0 x2 = 4 x = 2 y = 42 y = 2

x2 + 4 = 0 x2 = 4(2+ 2i)(2 2i)8i

2 = 2 8i2 =2 (2 2i)

2 = i2 (2 2i)

2 = 2+ i

(0 i), (2+ i)

d)Z 2 + 2Z + i(2 i) = 0Z 2 + 2Z + 2i i 2 = 0Z 2 + 2Z + 2i +1= 0

Z = 2 4 (4)(1)(1+ 2i)(2)(1)


x..........(2)

x2 4x

2

= 0

x216x2

= 0

x2

x4 16 = 0 (x2 4)(x2 + 4) = 0x2 4 = 0 x2 = 4 x = 2 y = 42 y = 2

x2 + 4 = 0 x2 = 4(2+ 2i)(2 2i)8i

2 = 2 8i2 =2 (2 2i)

2 = i2 (2 2i)

2 = 2+ i

(0 i), (2+ i)

x2 0

d)Z 2 + 2Z + i(2 i) = 0Z 2 + 2Z + 2i i 2 = 0Z 2 + 2Z + 2i +1= 0

Z = 2 4 (4)(1)(1+ 2i)(2)(1)


x..........(2)

x2 4x

2

= 0

x216x2

= 0

x2

x4 16 = 0 (x2 4)(x2 + 4) = 0x2 4 = 0 x2 = 4 x = 2 y = 42 y = 2

x2 + 4 = 0 x2 = 4(2+ 2i)(2 2i)8i

2 = 2 8i2 =2 (2 2i)

2 = i2 (2 2i)

2 = 2+ i

(0 i), (2+ i)

d)Z 2 + 2Z + i(2 i) = 0Z 2 + 2Z + 2i i 2 = 0Z 2 + 2Z + 2i +1= 0

Z = 2 4 (4)(1)(1+ 2i)(2)(1)


x..........(2)

x2 4x

2

= 0

x216x2

= 0

x2

x4 16 = 0 (x2 4)(x2 + 4) = 0x2 4 = 0 x2 = 4 x = 2 y = 42 y = 2

x2 + 4 = 0 x2 = 4(2+ 2i)(2 2i)8i

2 = 2 8i2 =2 (2 2i)

2 = i2 (2 2i)

2 = 2+ i

(0 i), (2+ i)

d)Z 2 + 2Z + i(2 i) = 0Z 2 + 2Z + 2i i 2 = 0Z 2 + 2Z + 2i +1= 0

Z = 2 4 (4)(1)(1+ 2i)(2)(1)


x..........(2)

x2 4x

2

= 0

x216x2

= 0

x2

x4 16 = 0 (x2 4)(x2 + 4) = 0x2 4 = 0 x2 = 4 x = 2 y = 42 y = 2

x2 + 4 = 0 x2 = 4(2+ 2i)(2 2i)8i

2 = 2 8i2 =2 (2 2i)

2 = i2 (2 2i)

2 = 2+ i

(0 i), (2+ i)

Z2 2Z+i(Z-i)=0Z2 +2Z+2i-i2 =0Z2 -i2 +2Z+2i=0(Z-i)(Z+i)+2(Z+i)=0(Z+i)[(Z-i)+2]=

Z+i=0 Z=-iZ-i=0 Z=-2+i

d)Z 2 + 2Z + i(2 i) = 0Z 2 + 2Z + 2i i 2 = 0Z 2 + 2Z + 2i +1= 0

Z = 2 4 (4)(1)(1+ 2i)(2)(1)


x..........(2)

x2 4x

2

= 0

x216x2

= 0

x2

x4 16 = 0 (x2 4)(x2 + 4) = 0x2 4 = 0 x2 = 4 x = 2 y = 42 y = 2

x2 + 4 = 0 x2 = 4(2+ 2i)(2 2i)8i

2 = 2 8i2 =2 (2 2i)

2 = i2 (2 2i)

2 = 2+ i

(0 i), (2+ i)

...)1(

...)2()1( )2(

)2 (

x= m 2

y= m 42 y= m 2

d)Z 2 + 2Z + i(2 i) = 0Z 2 + 2Z + 2i i 2 = 0Z 2 + 2Z + 2i +1= 0

Z = 2 4 (4)(1)(1+ 2i)(2)(1)


x..........(2)

x2 4x

2

= 0

x216x2

= 0

x2

x4 16 = 0 (x2 4)(x2 + 4) = 0x2 4 = 0 x2 = 4 x = 2 y = 42 y = 2

x2 + 4 = 0 x2 = 4(2+ 2i)(2 2i)8i

2 = 2 8i2 =2 (2 2i)

2 = i2 (2 2i)

2 = 2+ i

(0 i), (2+ i)

Z i +2 = 0


Z+i=0 Z=-iZ-i=0 Z=-2+i


Z+i=0 Z=-iZ-i=0 Z=-2+i

29

d)Z 2 + 2Z + i(2 i) = 0Z 2 + 2Z + 2i i 2 = 0Z 2 + 2Z + 2i +1= 0

Z = 2 4 (4)(1)(1+ 2i)(2)(1)


x..........(2)

x2 4x

2

= 0

x216x2

= 0

x2

x4 16 = 0 (x2 4)(x2 + 4) = 0x2 4 = 0 x2 = 4 x = 2 y = 42 y = 2

x2 + 4 = 0 x2 = 4(2+ 2i)(2 2i)8i

2 = 2 8i2 =2 (2 2i)

2 = i2 (2 2i)

2 = 2+ i

(0 i), (2+ i)

d)Z 2 + 2Z + i(2 i) = 0Z 2 + 2Z + 2i i 2 = 0Z 2 + 2Z + 2i +1= 0

Z = 2 4 (4)(1)(1+ 2i)(2)(1)


x..........(2)

x2 4x

2

= 0

x216x2

= 0

x2

x4 16 = 0 (x2 4)(x2 + 4) = 0x2 4 = 0 x2 = 4 x = 2 y = 42 y = 2

x2 + 4 = 0 x2 = 4(2+ 2i)(2 2i)8i

2 = 2 8i2 =2 (2 2i)

2 = i2 (2 2i)

2 = 2+ i

(0 i), (2+ i)

z

d)Z 2 + 2Z + i(2 i) = 0Z 2 + 2Z + 2i i 2 = 0Z 2 + 2Z + 2i +1= 0

Z = 2 4 (4)(1)(1+ 2i)(2)(1)


x..........(2)

x2 4x

2

= 0

x216x2

= 0

x2

x4 16 = 0 (x2 4)(x2 + 4) = 0x2 4 = 0 x2 = 4 x = 2 y = 42 y = 2

x2 + 4 = 0 x2 = 4(2+ 2i)(2 2i)8i

2 = 2 8i2 =2 (2 2i)

2 = i2 (2 2i)

2 = 2+ i

(0 i), (2+ i)

2+ (2 2i)2 = i

2 (2 2i)2 = 2+ i

{(0 i), (2+ i)}

x R

d)Z 2 + 2Z + i(2 i) = 0Z 2 + 2Z + 2i i 2 = 0Z 2 + 2Z + 2i +1= 0

Z = 2 4 (4)(1)(1+ 2i)(2)(1)


x..........(2)

x2 4x

2

= 0

x216x2

= 0

x2

x4 16 = 0 (x2 4)(x2 + 4) = 0x2 4 = 0 x2 = 4 x = 2 y = 42 y = 2

x2 + 4 = 0 x2 = 4(2+ 2i)(2 2i)8i

2 = 2 8i2 =2 (2 2i)

2 = i2 (2 2i)

2 = 2+ i

(0 i), (2+ i)

d)Z 2 + 2Z + i(2 i) = 0Z 2 + 2Z + 2i i 2 = 0Z 2 + 2Z + 2i +1= 0

Z = 2 4 (4)(1)(1+ 2i)(2)(1)


x..........(2)

x2 4x

2

= 0

x216x2

= 0

x2

x4 16 = 0 (x2 4)(x2 + 4) = 0x2 4 = 0 x2 = 4 x = 2 y = 42 y = 2

x2 + 4 = 0 x2 = 4(2+ 2i)(2 2i)8i

2 = 2 8i2 =2 (2 2i)

2 = i2 (2 2i)

2 = 2+ i

(0 i), (2+ i)

e(e)4Z 2 + 25 = 04Z 2 25i 2 = 0(2Z 5i)(2Z + 5i) = 02Z 5i = 0 Z = 52 i

2Z + 5i = 0 Z = 52 i

f )Z 2 2iz+ 3= 0d =1,b= 2i, c = 3

Z = 2i 4i2 (4)(1)(3)

(2)(1)

= 2i 4 122 =2i 16

2

= 2i 16i2 =2i 4i

2 =22 i = i

2i + 4i2 =

6i2 = 3i

e)4Z 2 + 25 = 04Z 2 25i 2 = 0(2Z 5i)(2Z + 5i) = 02Z 5i = 0 Z = 52 i

2Z + 5i = 0 Z = 52 i

f )Z 2 2iz+ 3= 0d =1,b= 2i, c = 3

Z = 2i 4i2 (4)(1)(3)

(2)(1)

= 2i 4 122 =2i 16

2

= 2i 16i2 =2i 4i

2 =22 i = i

2i + 4i2 =

6i2 = 3i

{m 52 i}

0

f(

e)4Z 2 + 25 = 04Z 2 25i 2 = 0(2Z 5i)(2Z + 5i) = 02Z 5i = 0 Z = 52 i

2Z + 5i = 0 Z = 52 i

f )Z 2 2iz+ 3= 0d =1,b= 2i, c = 3

Z = 2i 4i2 (4)(1)(3)

(2)(1)

= 2i 4 122 =2i 16

2

= 2i 16i2 =2i 4i

2 =22 i = i

2i + 4i2 =

6i2 = 3i

e)4Z 2 + 25 = 04Z 2 25i 2 = 0(2Z 5i)(2Z + 5i) = 02Z 5i = 0 Z = 52 i

2Z + 5i = 0 Z = 52 i

f )Z 2 2iz+ 3= 0d =1,b= 2i, c = 3

Z = 2i 4i2 (4)(1)(3)

(2)(1)

= 2i 4 122 =2i 16

2

= 2i 16i2 =2i 4i

2 =22 i = i

2i + 4i2 =

6i2 = 3i

e)4Z 2 + 25 = 04Z 2 25i 2 = 0(2Z 5i)(2Z + 5i) = 02Z 5i = 0 Z = 52 i

2Z + 5i = 0 Z = 52 i

f )Z 2 2iz+ 3= 0d =1,b= 2i, c = 3

Z = 2i 4i2 (4)(1)(3)

(2)(1)

= 2i 4 122 =2i 16

2

= 2i 16i2 =2i 4i

2 =22 i = i

2i + 4i2 =

6i2 = 3i

a =

e)4Z 2 + 25 = 04Z 2 25i 2 = 0(2Z 5i)(2Z + 5i) = 02Z 5i = 0 Z = 52 i

2Z + 5i = 0 Z = 52 i

f )Z 2 2iz+ 3= 0d =1,b= 2i, c = 3

Z = 2i 4i2 (4)(1)(3)

(2)(1)

= 2i 4 122 =2i 16

2

= 2i 16i2 =2i 4i

2 =22 i = i

2i + 4i2 =

6i2 = 3i

e)4Z 2 + 25 = 04Z 2 25i 2 = 0(2Z 5i)(2Z + 5i) = 02Z 5i = 0 Z = 52 i

2Z + 5i = 0 Z = 52 i

f )Z 2 2iz+ 3= 0d =1,b= 2i, c = 3

Z = 2i 4i2 (4)(1)(3)

(2)(1)

= 2i 4 122 =2i 16

2

= 2i 16i2 =2i 4i

2 =22 i = i

2i + 4i2 =

6i2 = 3i

e)4Z 2 + 25 = 04Z 2 25i 2 = 0(2Z 5i)(2Z + 5i) = 02Z 5i = 0 Z = 52 i

2Z + 5i = 0 Z = 52 i

f )Z 2 2iz+ 3= 0d =1,b= 2i, c = 3

Z = 2i 4i2 (4)(1)(3)

(2)(1)

= 2i 4 122 =2i 16

2

= 2i 16i2 =2i 4i

2 =22 i = i

2i + 4i2 =

6i2 = 3i

e)4Z 2 + 25 = 04Z 2 25i 2 = 0(2Z 5i)(2Z + 5i) = 02Z 5i = 0 Z = 52 i

2Z + 5i = 0 Z = 52 i

f )Z 2 2iz+ 3= 0d =1,b= 2i, c = 3

Z = 2i 4i2 (4)(1)(3)

(2)(1)

= 2i 4 122 =2i 16

2

= 2i 16i2 =2i 4i

2 =22 i = i

2i + 4i2 =

6i2 = 3i

16 i

i,3i{ }

z2 2iz+ 3= 0z2 2iz 3i 2 = 0(z 3i)(z+1) = 0z 3i = 0 z = 3iz+ i = 0 z = i



1

m,L : 2

a (

a)m=1+ 2i, L =1 i (1+ 2i)+ (1 i) = 2+ im+ L m.Lm.L (1+ 2i)(1 i) =1 i + 2i 2i 2 = 3+ ix2 (2+ i)x+ (3+ i) = 0b)m= 3 i1+ i , L = (3 2i)

2

m= 3 i1+ i= 3 i1+ i .

1 i1 i

= 3 3i i + i2

2 =2 4i

2 =1 2i

L = (3 2i)2= 912i + 4i 2 = 512im+ L = (1 2i)+ (512i) = 614i m.L = (1 2i)(512i)= 512i 10i +14i 2= 512i 10i 24= 19 22ix2 (614i)x+ (19 22i) = 0x2 (614i)x (19+ 22i) = 0

m= 1+2i, L = 1 ia)m=1+ 2i, L =1 i (1+ 2i)+ (1 i) = 2+ im+ L m.Lm.L (1+ 2i)(1 i) =1 i + 2i 2i 2 = 3+ ix2 (2+ i)x+ (3+ i) = 0b)m= 3 i1+ i , L = (3 2i)

2

m= 3 i1+ i= 3 i1+ i .

1 i1 i

= 3 3i i + i2

2 =2 4i

2 =1 2i



2

m= 3 i1+ i= 3 i1+ i .

1 i1 i

= 3 3i i + i2

2 =2 4i

2 =1 2i


m= 1+2i, L = 1 i


2

m= 3 i1+ i= 3 i1+ i .

1 i1 i

= 3 3i i + i2

2 =2 4i

2 =1 2i



2

m= 3 i1+ i= 3 i1+ i .

1 i1 i

= 3 3i i + i2

2 =2 4i

2 =1 2i


m= 1+2i, L = 1 i

b(


2

m= 3 i1+ i= 3 i1+ i .

1 i1 i

= 3 3i i + i2

2 =2 4i

2 =1 2i



2

m= 3 i1+ i= 3 i1+ i .

1 i1 i

= 3 3i i + i2

2 =2 4i

2 =1 2i



2

m= 3 i1+ i= 3 i1+ i .

1 i1 i

= 3 3i i + i2

2 =2 4i

2 =1 2i



2

m= 3 i1+ i= 3 i1+ i .

1 i1 i

= 3 3i i + i2

2 =2 4i

2 =1 2i



2

m= 3 i1+ i= 3 i1+ i .

1 i1 i

= 3 3i i + i2

2 =2 4i

2 =1 2i



2

m= 3 i1+ i= 3 i1+ i .

1 i1 i

= 3 3i i + i2

2 =2 4i

2 =1 2i



2

m= 3 i1+ i= 3 i1+ i .

1 i1 i

= 3 3i i + i2

2 =2 4i

2 =1 2i


24i 2


2

m= 3 i1+ i= 3 i1+ i .

1 i1 i

= 3 3i i + i2

2 =2 4i

2 =1 2i



2

m= 3 i1+ i= 3 i1+ i .

1 i1 i

= 3 3i i + i2

2 =2 4i

2 =1 2i


2

: a (a)-6i(x+yi)2 =-6ix2 +2xyi+y2i2 =-6i+0x2 -y2 ....(1)2xy=6 y= 62x y=

3x ...(2)

x2 - 3x

2

=0

x2 - 9x2 =0

x4 -9=0 (x2 -3)(x2 +3)=0x2 =-3x2 =3 x=m 3 y=m 3

( 3+ 3i),(- 3- 3i)

a)-6i(x+yi)2 =-6ix2 +2xyi+y2i2 =-6i+0x2 -y2 ....(1)2xy=6 y= 62x y=

3x ...(2)

x2 - 3x

2

=0

x2 - 9x2 =0

x4 -9=0 (x2 -3)(x2 +3)=0x2 =-3x2 =3 x=m 3 y=m 3

( 3+ 3i),(- 3- 3i)

x2 y2 = 0 ...(1) 2xy = 6 y = 62x y =

3x

...(2)

)1( )2( x2 3x

2

= 0

x2 0 x2 9x= 0

x2 0

a)-6i(x+yi)2 =-6ix2 +2xyi+y2i2 =-6i+0x2 -y2 ....(1)2xy=6 y= 62x y=

3x ...(2)

x2 - 3x

2

=0

x2 - 9x2 =0

x4 -9=0 (x2 -3)(x2 +3)=0x2 =-3x2 =3 x=m 3 y=m 3

( 3+ 3i),(- 3- 3i)

x2 = 3 x = m 3 y = m 3 ( 3 3i), ( 3+ 3i)b(b)7+ 24i(x+ yi)2 = 7+ 24ix2 + 2xyi + y2i 2 = 7+ 24ix2 y2 = 72xy = 24 y = 242x y =

12x

x2 12x

2

= 7

x2 144x2

= 7

x2 144 = 7x2x4 7x2 144 = 0(x2 16)(x2 + 9) = 0(x R )x2 = 9 x2 + 9 = 0x = 4 x2 =16 x2 16 = 0 y = 124 y = 3(4 3i)

b)7+ 24i(x+ yi)2 = 7+ 24ix2 + 2xyi + y2i 2 = 7+ 24ix2 y2 = 72xy = 24 y = 242x y =

12x

x2 12x

2

= 7

x2 144x2

= 7


)1( )2(

...)1(

...)2(


12x

x2 12x

2

= 7

x2 144x2

= 7



12x

x2 12x

2

= 7

x2 144x2

= 7


x4


12x

x2 12x

2

= 7

x2 144x2

= 7



12x

x2 12x

2

= 7

x2 144x2

= 7


x2 = 9


12x

x2 12x

2

= 7

x2 144x2

= 7



12x

x2 12x

2

= 7

x2 144x2

= 7



12x

x2 12x

2

= 7

x2 144x2

= 7


x2 = 9 x2 = 9


12x

x2 12x

2

= 7

x2 144x2

= 7


(x R ) )2(

x2 0

x2 144x2

=7

c(c) 41 3i4

1 3i.1+ 3i1+ 3i

= 4+ 4 3i4 =4(1+ 4 3i)

4 =1+ 3i

(x+ yi)2 =1+ 3ix2 y2 + 2xyi =1+ 3i x2 y2 =1

2xy = 3 y = 32x

x2 32x

2

=1

c) 41 3i4

1 3i.1+ 3i1+ 3i

= 4+ 4 3i4 =4(1+ 4 3i)

4 =1+ 3i

(x+ yi)2 =1+ 3ix2 y2 + 2xyi =1+ 3i x2 y2 =1

2xy = 3 y = 32x

x2 32x

2

=1

c) 41 3i4

1 3i.1+ 3i1+ 3i

= 4+ 4 3i4 =4(1+ 4 3i)

4 =1+ 3i

(x+ yi)2 =1+ 3ix2 y2 + 2xyi =1+ 3i x2 y2 =1

2xy = 3 y = 32x

x2 32x

2

=1

c) 41 3i4

1 3i.1+ 3i1+ 3i

= 4+ 4 3i4 =4(1+ 4 3i)

4 =1+ 3i

(x+ yi)2 =1+ 3ix2 y2 + 2xyi =1+ 3i x2 y2 =1

2xy = 3 y = 32x

x2 32x

2

=1

c) 41 3i4

1 3i.1+ 3i1+ 3i

= 4+ 4 3i4 =4(1+ 4 3i)

4 =1+ 3i

(x+ yi)2 =1+ 3ix2 y2 + 2xyi =1+ 3i x2 y2 =1

2xy = 3 y = 32x

x2 32x

2

=1

:

...)1(

...)2(

)1( )2(

x2 34x2 =1 4x4 3= 4x2

4x4 4x2 3= 0(2x2 3)(2x2 +1) = 02x2 = 1 2x2 +1= 0

x = 32 x2 = 32 2x

2 = 3 2x2 3= 0

y = 32. 32

= 32 3

2

= 3. 22 3

= 22. 2

= 12

( 32 1

2i)

x2 34x2 =1 4x4 3= 4x2

4x4 4x2 3= 0(2x2 3)(2x2 +1) = 02x2 = 1 2x2 +1= 0

x = 32 x2 = 32 2x

2 = 3 2x2 3= 0

y = 32. 32

= 32 3

2

= 3. 22 3

= 22. 2

= 12

( 32 1

2i)

x2 34x2 =1 4x4 3= 4x2

4x4 4x2 3= 0(2x2 3)(2x2 +1) = 02x2 = 1 2x2 +1= 0

x = 32 x2 = 32 2x

2 = 3 2x2 3= 0

y = 32. 32

= 32 3

2

= 3. 22 3

= 22. 2

= 12

( 32 1

2i)

x2 34x2 =1 4x4 3= 4x2

4x4 4x2 3= 0(2x2 3)(2x2 +1) = 02x2 = 1 2x2 +1= 0

x = 32 x2 = 32 2x

2 = 3 2x2 3= 0

y = 32. 32

= 32 3

2

= 3. 22 3

= 22. 2

= 12

( 32 1

2i)

x2 34x2 =1 4x4 3= 4x2

4x4 4x2 3= 0(2x2 3)(2x2 +1) = 02x2 = 1 2x2 +1= 0

x = 32 x2 = 32 2x

2 = 3 2x2 3= 0

y = 32. 32

= 32 3

2

= 3. 22 3

= 22. 2

= 12

( 32 1

2i)

x2 34x2 =1 4x4 3= 4x2

4x4 4x2 3= 0(2x2 3)(2x2 +1) = 02x2 = 1 2x2 +1= 0

x = 32 x2 = 32 2x

2 = 3 2x2 3= 0

y = 32. 32

= 32 3

2

= 3. 22 3

= 22. 2

= 12

( 32 1

2i)

x2 34x2 =1 4x4 3= 4x2

4x4 4x2 3= 0(2x2 3)(2x2 +1) = 02x2 = 1 2x2 +1= 0

x = 32 x2 = 32 2x

2 = 3 2x2 3= 0

y = 32. 32

= 32 3

2

= 3. 22 3

= 22. 2

= 12

( 32 1

2i)

x2 34x2 =1 4x4 3= 4x2

4x4 4x2 3= 0(2x2 3)(2x2 +1) = 02x2 = 1 2x2 +1= 0

x = 32 x2 = 32 2x

2 = 3 2x2 3= 0

y = 32. 32

= 32 3

2

= 3. 22 3

= 22. 2

= 12

( 32 1

2i)

x2 34x2 =1 4x4 3= 4x2

4x4 4x2 3= 0(2x2 3)(2x2 +1) = 02x2 = 1 2x2 +1= 0

x = 32 x2 = 32 2x

2 = 3 2x2 3= 0

y = 32. 32

= 32 3

2

= 3. 22 3

= 22. 2

= 12

( 32 1

2i)

x2 34x2 =1 4x4 3= 4x2

4x4 4x2 3= 0(2x2 3)(2x2 +1) = 02x2 = 1 2x2 +1= 0

x = 32 x2 = 32 2x

2 = 3 2x2 3= 0

y = 32. 32

= 32 3

2

= 3. 22 3

= 22. 2

= 12

( 32 1

2i)

x2 34x2 =1 4x4 3= 4x2

4x4 4x2 3= 0(2x2 3)(2x2 +1) = 02x2 = 1 2x2 +1= 0

x = 32 x2 = 32 2x

2 = 3 2x2 3= 0

y = 32. 32

= 32 3

2

= 3. 22 3

= 22. 2

= 12

( 32 1

2i)

x2 34x2 =1 4x4 3= 4x2

4x4 4x2 3= 0(2x2 3)(2x2 +1) = 02x2 = 1 2x2 +1= 0

x = 32 x2 = 32 2x

2 = 3 2x2 3= 0

y = 32. 32

= 32 3

2

= 3. 22 3

= 22. 2

= 12

( 32 1

2i)

(x R )

x2 34x2 =1 4x4 3= 4x2

4x4 4x2 3= 0(2x2 3)(2x2 +1) = 02x2 = 1 2x2 +1= 0

x = 32 x2 = 32 2x

2 = 3 2x2 3= 0

y = 32. 32

= 32 3

2

= 3. 22 3

= 22. 2

= 12

( 32 1

2i)

4x2x2 04x2

x2 0 x2 34x2 =1 4x

4 3= 4x2

4x4 4x2 3= 0(2x2 3)(2x2 +1) = 02x2 = 1 2x2 +1= 0

x = 32 x2 = 32 2x

2 = 3 2x2 3= 0

y = 32. 32

= 32 3

2

= 3. 22 3

= 22. 2

= 12

( 32 1

2i)

) (

)2(

y = 3

2( 33 )= 3( 2

2 3) = 2

2. 2= 1

2

x2 34x2 =1 4x4 3= 4x2

4x4 4x2 3= 0(2x2 3)(2x2 +1) = 02x2 = 1 2x2 +1= 0

x = 32 x2 = 32 2x

2 = 3 2x2 3= 0

y = 32. 32

= 32 3

2

= 3. 22 3

= 22. 2

= 12

( 32 1

2i)

: :

a( ) ( -i=

a)ii + (i) = i i = 0i.(i) = i 2 =1x2 (0)x+1= 0 x2 +1= 0

a)ii + (i) = i i = 0i.(i) = i 2 =1x2 (0)x+1= 0 x2 +1= 0

a)ii + (i) = i i = 0i.(i) = i 2 =1x2 (0)x+1= 0 x2 +1= 0

a)ii + (i) = i i = 0i.(i) = i 2 =1x2 (0)x+1= 0 x2 +1= 0

a)ii + (i) = i i = 0i.(i) = i 2 =1x2 (0)x+1= 0 x2 +1= 0

a)ii + (i) = i i = 0i.(i) = i 2 =1x2 (0)x+1= 0 x2 +1= 0

5

i 5)b(b01= )i +5()i 5(62 =1+52 = )i +5()i 5(0 = 62 +x01 2x

( )

i+5 i 5)b01= )i +5()i 5(62 =1+52 = )i +5()i 5(0 = 62 +x01 2x

i 5)b01= )i +5()i 5(62 =1+52 = )i +5()i 5(0 = 62 +x01 2x

i 5)b01= )i +5()i 5(62 =1+52 = )i +5()i 5(0 = 62 +x01 2x

i 5)b 01= )i +5()i 5(62 =1+52 = )i +5()i 5(0 = 62 +x01 2x

i 5)b01= )i +5()i 5(62 =1+52 = )i +5()i 5( 0 = 62 +x01 2x

4i3 + 2 )c(ci3 + 2= 4

2+ 4

3i 4

2 4

i34

2 4

i34

+ 42 +

i34

= 42 2 =2

= 22

2 .21 =

22

4i34

+ 42

i34

+ 612 =9

616111 =1 2x

20 = 6111 +x

)c ( )i3 + 24

i3 + 2= 4

2+ 4

3i 4

2 4

i34

2 4

i34

+ 42 +

i34

= 42 2 =2

= 22

2 .21 =

22

4i34

+ 42

i34

+ 612 =9

616111 =1 2x

20 = 6111 +x

2

43 i 4

2 4

3i 4

+ 42 +

3i 4

4i3 + 2 )ci3 + 2= 4

2+ 4

3i 4

2 4

i34

2 4

i34

+ 42 +

i34

= 42 2 =2

= 22

2 .21 =

22

4i34

+ 42

i34

+ 612 =9

616111 =1 2x

20 = 6111 +x

4i3 + 2 )ci3 + 2= 4

2+ 4

3i 4

2 4

i34

2 4

i34

+ 42 +

i34

= 42 2 =2

= 22

2 .21 =

22

4i34

+ 42

i34

+ 612 =9

616111 =1 2x

20 = 6111 +x

2 ) 2 ()2 (

4i3 + 2 )ci3 + 2= 4

2+ 4

3i 4

2 4

i34

2 4

i34

+ 42 +

i34

= 42 2 =2

= 22

2 .21 =

22

4i34

+ 42

i34

+ 612 =9

616111 =1 2x

20 = 6111 +x

6

4i3 + 2 )ci3 + 2= 4

2+ 4

3i 4

2 4

i34

2 4

i34

+ 42 +

i34

= 42 2 =2

= 22

2 .21 =

22

4i34

+ 42

i34

+ 612 =9

616111 =1 2x

20 = 6111 +x

4i3 + 2 )ci3 + 2= 4

2+ 4

3i 4

2 4

i34

2 4

i34

+ 42 +

i34

= 42 2 =2

= 22

2 .21 =

22

4i34

+ 42

i34

+ 612 =9

616111 =1 2x

2 0 = 6111 +x

5: i+ 0 = )i5 +5( +xa 2x

a

i +3 = 1x2x

2x = 2x.1x

= 2x)i +3(i5 +5

1= 2x

i5 +5. i +3

i 3i 3

= 2x2 i5 i51+ i5 51

01= 2x

i01+02i +2 = 01

i +3 = 1x 2x

2x = 2x.1x

= 2x)i +3(i5 +5

1= 2x

i5 +5. i +3

i 3i 3

= 2x2 i5 i51+ i5 51

01= 2x

i01+02i +2 = 01

i +3 = 1x2x

2x = 2x.1x

= 2x)i +3(i5 +5

1= 2x

i5 +5. i +3

i 3i 3

= 2x2 i5 i51+ i5 51

01= 2x

i01+02i +2 = 01

= 2x + 1xx2x

1a = )i +2()i +3(2 i + i2 + i3 +6 = ai5 +5 = a

- = 2x + 1x

x2x

1a = )i +2()i +3(2 i + i2 + i3 +6 = ai5 +5 = a

= 2x + 1xx2x

1a = )i +2()i +3(2 i + i2 + i3 +6 = ai5 +5 = a

a = i2+5

a = i2+5 a : i+ a

7

( - 1)1:

= 1+)12()3(= 46 (a523 (b

1 = 31 = 1= 1)801(3=

=

3

2=

1 (c21) (23+1

1 = 2)01(3+1

= 211

21) (2+11 =

21) (+1

1 =

21) (21 =42

1 =0+)8()3(

1 =0

1= 11 =

1= 42=..... = 6= 3

142

=42

421=

4) 2+1( (d= 4)(1 = 4)( =

14

=6

42=

2= 2.3= 5= 5+)n3(3= 5+n9 (e2:

+1, 2+1 (a

)+1( +) 2+1( 1=12 = + 2+2

)+1() 2+1() 2()( =

523

1 = 3523

1 = 3

8

0 =1+x + 2x :

(b, 22

2

2

+ 222

2

+)2( =) 22( 2

)2() 22(2 =

4 22 + 2= 3+ 22 2 4

2+2= ) 2+(2 5

+ 2= 2 +5

17

. 22

2

2

=3

= )2() 22(17

+x 71 + 2x10 = 7

7

0 =1+x + 2x7

i3 (c 2

3 ,2

i

.i3 =3

23 +

2

ii .i

3 i3 =i2

2 i) 2+(i3 = i23 + i3 =

i3 =

9

i3 =2

3.2

i

9 =i2

i29 =

x

0 = 9xi3+ 211z3+ 01z3+18z3- 7z3-1

z3 +1)311z3 + 01

8z3 7z3 1

z3 +1 =2+9z3 + 1+9

2+6z3 1+6z3 1

z3 +z3 +1 =2

= 2z3 z3 1) 2z +z(3 +1= ) 2z +z(3 1

3 13 +1

= 42 =12

0 =1+z + 2z

b b = zca4 22

4 1 1 = z2

= 23 1

2= 1z

1 2

3= 2z ,i 2

1+ 2

3i 2

w = 1z 2w = 2z2w = 1zw = 2z

)0 =1+z + 2z(

:

b b =zca4- 2

2= 24 1 1 = z

i3 12

= 1z1-- 2

3= 2z,i 2

1-+ 2

3i 2

2= 2z ,= 1z = 2z , 2= 1z =z

z3+1 =z11z3+ 01

= 8z3- 7z3-12+9z3+ 1+9z3+1 2+6z3-1+6z3-1

3 +3 +1 =2

= 23 3 1) 2+(3 +1= ) 2(3 1

3 13 +3 +1 =3 +1

2

= 23 3 1) 2+(3 +1= ) 2(3 1

3 1-= 42- =3 +1

1 2=z 2

21 - () =

0

: 1 (a

+21

22

2

31 =1

+21

) (2+2

2

+2( = SHL)+2( ) 2

) 2+2()+2(

2

+2 =2 2

3+2 + 22 +4

2

( =2) 223

=2+ 32 4

= 92+2

9

+ =2 2

= 92 1

= 93 = 9

1 SHR = 3

(b1 7+ 41

= 2 5+ 0123

= SHL1 1+)2()3(+ 2+) 4(3

= 2 2+)1()3(+ 1+)3(31 + 2= 2 2+

11 2 1

2 == 3

2SHR = 3

1

2 1 )c(c2

2+

5 +1

81=

.2 1 = SHL 3

22+

5 +1

3

+2 1( = ) 25 +1() 2

) 25 2()2 ( =) 26()3( =

SHR = 81= 381=

2 = 3)+1( + 3) 2+1()d (d

3) 2( + 3)( = = SHL 6 3 =

SHR = 2 =11 =

2

( - 1)

1: .

y i3 2 = 1Z i3 +2 = 1Z

x1p

(1z-)p

) (3,2 = 1P ) (3 ,2 = 1P

i3 1= 2Z i3 +1 = 2Zy

x

2p

(2z-)p

) (3,1 = 2P ) (3 ,1 = 2P

i 1= 3Z

i +1 = 3Z

y

x

(z)p

(z-)p

) (1,1 = 3P

) (1,1 = 3P

i = 4Z

i = 4Zy

x

(z)p

(z-)p

) (1,0 = 4P

) (1,0 = 4P

2: .

i3 +5 = 1Z)3 ,5( = ) 1Z(p

i3 5 = 1Z) (3,5 = ) (1Z Pi3 +5 = 1Z)3 ,5( = ) 1Z(p

y

x

(1z)p

) (1z p

i2 +3 = 2Z) (2,3 = ) (2 Z Pi2 3 = 2 Z) (2,3 = ) (2 Z Pi2 +3 = 2Z) (2,3 = ) (3Z Pi 1 = 3 Z) (1,1 = ) (3 Z P

i2 3 = 2 Z ) (2,3 = ) (2 Z P

y

x

(2z)p

) (2z p

i2 +3 = 2Z) (2,3 = ) (2 Z Pi2 3 = 2 Z) (2,3 = ) (2 Z Pi2 +3 = 2Z) (2,3 = ) (3Z Pi 1 = 3 Z ) (1,1 = ) (3 Z P

i +1= 3 Z ) (1,1 = ) (3 Z P

(z)p

) (3z p y

x

i2 = 4Z ) (2,0 = ) (4Z P

i2 = 4 Z ) (2,0 = ) (4 Z P

(z)p

y

x) (4z p

Z, Z , ZZ = 4+ 2i P Z( ) = 4 ,2( )Z = 4 2iP Z( ) = 4 ,2( )

: Z = 4+ 2i :

p)z(

y

x p z( )

p)-z(

P Z( ) = 4 ,2( )Z = 4 2ip(z) = (4,2)

Z, Z , ZZ = 4+ 2i P Z( ) = 4 ,2( )Z = 4 2iP Z( ) = 4 ,2( )

Z4 = Z1 Z2 = 4 2i( ) 1+ 2i( ) = 4 2i( ) + 1 2i( ) = 3 4i( )P Z1 Z2( ) = P Z3( ) = 3,4( )3Z2 = 3 L = 2i( ) = 36iP 3Z2( ) = 3,6( )2Z1 = 2 4 2i( ) = 8 4iP 2z1( ) = 8,4( ): Z2 =1+ 2i , Z1 = 4 2iZ2 =1+ 2i , Z1 = 4 2i :

Z1 + Z2 , Z1 Z2 ,3Z2 , 2Z1

Z3 = Zi + Z2 = 1+ 2i( ) + 4 2i( ) = 5 + 0i = 5P Z1( ) = 1,2( )P Z2( ) = 4 ,2( )P Z1 + Z2( ) = P Z3( ) = 5, 0( )Z4 = Z1 Z2 = 4 2i( ) 1+ 2i( ) = 4 2i( ) + 1 2i( ) = 3 4i( )P Z1 Z2( ) = P Z3( ) = 3,4( )3Z2 = 3 L = 2i( ) = 36iP 3Z2( ) = 3,6( )2Z1 = 2 4 2i( ) = 8 4iP 2z1( ) = 8,4( )

Z4 = Z1 Z2 = 4 2i( ) 1+ 2i( ) = 4 2i( ) + 1 2i( ) = 3 4i( )P Z1 Z2( ) = P Z3( ) = 3,4( )3Z2 = 3 L = 2i( ) = 36iP 3Z2( ) = 3,6( )2Z1 = 2 4 2i( ) = 8 4iP 2z1( ) = 8,4( )



Z3 = Zi + Z2 = 1+ 2i( ) + 4 2i( ) = 5 + 0i = 5Z3 = Zi + Z2 = 1+ 2i( ) + 4 2i( ) = 5 + 0i = 5Z3 = Zi + Z2 = 1+ 2i( ) + 4 2i( ) = 5 + 0i = 5Z3 = Zi + Z2 = 1+ 2i( ) + 4 2i( ) = 5 + 0i = 5P Z1( ) = 1,2( )P Z2( ) = 4 ,2( )P Z1 + Z2( ) = P Z3( ) = 5, 0( )

P Z1( ) = 1,2( )P Z2( ) = 4 ,2( )P Z1 + Z2( ) = P Z3( ) = 5, 0( )

P Z1( ) = 1,2( )P Z2( ) = 4 ,2( )P Z1 + Z2( ) = P Z3( ) = 5, 0( )

3Z2 = 3 1+ 2i( ) = 3 6i

*

*

*

*

y

x

Z1 + Z2 , Z1 Z2 ,3Z2 , 2Z1

y

x Z1 + Z2 , Z1 Z2 ,3Z2 , 2Z1

Z1 + Z2 , Z1 Z2 ,3Z2 , 2Z1

Z4

5

)1 - 5(

- 1a(

cos 524 + i sin5

24

4

cos+ i sin( )n = cos n + i sin n( )

cos 524 + i sin5

24

4

= cos 4. 524 + i sin 4.5

24

= cos 56 + i sin56

= cos 6

+ i sin

6

= cos 6 + i sin6

= 32 +12 i

cos 712 + i sin7

12

3

cos + i sin( )n = cos n i sin n

cos 712 + i sin7

12

3

= cos 3. 712

i sin 3

712

= cos 74

i sin

74

= cos 74 i sin74

= cos 4 + i sin4

= 12 +12 i

1 i( )7

mod Z = 2, cos = 12 . , sin =1

2arg Z = 74Z n = r n cos n + i sin n ( )1 i( )7 = 2( )7 cos 7. 74

+ i sin 7.

74

Q49

4= 12 + 4

cos 524 + i sin5

24

4


cos 524 + i sin5

24

4

= cos 4. 524 + i sin 4.5

24

= cos 56 + i sin56

= cos 6

+ i sin

6

= cos 6 + i sin6

= 32 +12 i

cos 712 + i sin7

12

3

cos + i sin( )n = cos n i sin n

cos 712 + i sin7

12

3

= cos 3. 712

i sin 3

712

= cos 74

i sin

74

= cos 74 i sin74

= cos 4 + i sin4

= 12 +12 i

1 i( )7

mod Z = 2, cos = 12 . , sin =1

2arg Z = 74Z n = r n cos n + i sin n ( )1 i( )7 = 2( )7 cos 7. 74

+ i sin 7.

74

Q49

4= 12 + 4

cos + i sin( )n

b(

cos 524 + i sin5

24

4


cos 524 + i sin5

24

4

= cos 4. 524 + i sin 4.5

24

= cos 56 + i sin56

= cos 6

+ i sin 6

= cos 6 + i sin6

= 32 +12 i

cos 712 + i sin7

12

3

cos + i sin( )n = cos n i sin

cos 712 + i sin7

12

3

= cos 3. 712

i sin 3 712

= cos 74

i sin 74

= cos 74 i sin74

= cos 4 i sin4

= 12 1

2i

(cos + isin )-n = cos n isin n

6

cos 524 + i sin5

24

4


cos 524 + i sin5

24

4

= cos 4. 524 + i sin 4.5

24

= cos 56 + i sin56

= cos 6

+ i sin 6

= cos 6 + i sin6

= 32 +12 i

cos 712 + i sin7

12

3


cos 712 + i sin7

12

3

= cos 3. 712

i sin 3 712

= cos 74

i sin 74

= cos 74 i sin74

= cos 4 i sin4

= 12 1

2i

cos 524 + i sin5

24

4


cos 524 + i sin5

24

4

= cos 4. 524 + i sin 4.5

24

= cos 56 + i sin56

= cos 6

+ i sin 6

= cos 6 + i sin6

= 32 +12 i

cos 712 + i sin7

12

3


cos 712 + i sin7

12

3

= cos 3. 712

i sin 3 712

= cos 74

i sin 74

= cos 74 i sin74

= cos 4 i sin4

= 12 1

2i

cos 524 + i sin5

24

4


cos 524 + i sin5

24

4

= cos 4. 524 + i sin 4.5

24

= cos 56 + i sin56

= cos 6

+ i sin 6

= cos 6 + i sin6

= 32 +12 i

cos 712 + i sin7

12

3


cos 712 + i sin7

12

3

= cos 3. 712

i sin 3 712

= cos 74

i sin 74

= cos 74 i sin74

= cos 4 i sin4

= 12 1

2i

cos 2 4

i sin 2 4

cos 524 + i sin5

24

4


cos 524 + i sin5

24

4

= cos 4. 524 + i sin 4.5

24

= cos 56 + i sin56

= cos 6

+ i sin 6

= cos 6 + i sin6

= 32 +12 i

cos 712 + i sin7

12

3


cos 712 + i sin7

12

3

= cos 3. 712

i sin 3 712

= cos 74

i sin 74

= cos 74 i sin74

= cos 4 i sin4

= 12 1

2i

cos 524 + i sin5

24

4


cos 524 + i sin5

24

4

= cos 4. 524 + i sin 4.5

24

= cos 56 + i sin56

= cos 6

+ i sin 6

= cos 6 + i sin6

= 32 +12 i

cos 712 + i sin7

12

3


cos 712 + i sin7

12

3

= cos 3. 712

i sin 3 712

= cos 74

i sin 74

= cos 74 i sin74

= cos 4 i sin4

= 12 1

2i

cos 524 + i sin5

24

4


cos 524 + i sin5

24

4

= cos 4. 524 + i sin 4.5

24

= cos 56 + i sin56

= cos 6

+ i sin 6

= cos 6 + i sin6

= 32 +12 i

cos 712 + i sin7

12

3


cos 712 + i sin7

12

3

= cos 3. 712

i sin 3 712

= cos 74

i sin 74

= cos 74 i sin74

= cos 4 i sin4

= 12 1

2i

cos 524 + i sin5

24

4


cos 524 + i sin5

24

4

= cos 4. 524 + i sin 4.5

24

= cos 56 + i sin56

= cos 6

+ i sin 6

= cos 6 + i sin6

= 32 +12 i

cos 712 + i sin7

12

3


cos 712 + i sin7

12

3

= cos 3. 712

i sin 3 712

= cos 74

i sin 74

= cos 74 i sin74

= cos 4 i sin4

= 12 1

2i

cos 524 + i sin5

24

4


cos 524 + i sin5

24

4

= cos 4. 524 + i sin 4.5

24

= cos 56 + i sin56

= cos 6

+ i sin 6

= cos 6 + i sin6

= 32 +12 i

cos 712 + i sin7

12

3


cos 712 + i sin7

12

3

= cos 3. 712

i sin 3 712

= cos 74

i sin 74

= cos 74 i sin74

= cos 4 i sin4

= 12 1

2i

cos 524 + i sin5

24

4


cos 524 + i sin5

24

4

= cos 4. 524 + i sin 4.5

24

= cos 56 + i sin56

= cos 6

+ i sin 6

= cos 6 + i sin6

= 32 +12 i

cos 712 + i sin7

12

3


cos 712 + i sin7

12

3

= cos 3. 712

i sin 3 712

= cos 74

i sin 74

= cos 74 i sin74

= cos 4 i sin4

= 12 1

2i

: : 2

a( 1 i( )7

Z = 1 i Z = 2

cos = 12 , sin = 1

2 = 2

4= 7

4

Z7 = 2( )7 cos 74 + i sin74

7

= 2( )7 cos 494 + i sin49

4

= 2( )7 12 i12

8 2 1

2 i 1

2

= 8 8i= 2( )7 12

12

i

8 2 1

2 1

2i

= 8 8i

7

b(

3 + i( )9

Z = 3 + i Z = 2

cos = 32 , sin = 1

2

= 6

Z 9 = 2( )9 cos 6+ i sin

6

9

= 1512

cos 32 i sin 3

2

= 1512

0 i 1( )( ) = 1512

8i

Z = 3 + i Z = 2

cos = 32 , sin = 1

2

= 6

Z 9 = 2( )9 cos 6+ i sin

6

9

= 1512

cos 32 i sin 3

2

= 1512

0 i 1( )( ) = 1512

8i

Z = 3 + i Z = 2

cos = 32 , sin = 1

2

= 6

Z 9 = 2( )9 cos 6+ i sin

6

9

= 1512

cos 32 i sin 3

2

= 1512

0 i 1( )( ) = 1512

8i

: :

a( cos 2 + i sin 2( )5

cos 3 + i sin 3( )3

=cos + i sin( )2( )5cos + i sin( )9 =

cos + i sin( )10cos + i sin( )9

= cos + i sin( )

cos + i sin( )3( )3

cos 2 + i sin 2( )5cos 3 + i sin 3( )3

=cos + i sin( )2( )5cos + i sin( )9 =


= cos + i sin( )

cos 2 + i sin 2( )5cos 3 + i sin 3( )3

=cos + i sin( )2( )5cos + i sin( )9 =


= cos + i sin( )

cos 2 + i sin 2( )5cos 3 + i sin 3( )3

=cos + i sin( )2( )5cos + i sin( )9 =


= cos + i sin( )

8

b(

cos 2 + i sin 2( )5cos 3 + i sin 3( )3

=cos + i sin( )2( )5cos + i sin( )9

=cos + i sin( )( )10cos + i sin( )9

= cos + i sin( )cos + i sin( )8 cos i sin( )4

= cos + i sin( )8 cos + i sin( )1( )4= cos + i sin( )8 cos + i sin( )4

= cos + i sin( )4= cos 4 + i sin 4

cos 2 + i sin 2( )5cos 3 + i sin 3( )3

=cos + i sin( )2( )5cos + i sin( )9

=cos + i sin( )( )10cos + i sin( )9

= cos + i sin( )cos + i sin( )8 cos i sin( )4

= cos + i sin( )8 cos + i sin( )1( )4= cos + i sin( )8 cos + i sin( )4

= cos + i sin( )4= cos 4 + i sin 4

Z = 1+ 3i Z = 2

cos = 12

,sin = 32

arg z = 23

Z12 = 2( )

12 cos

23+ 2k2

+ i sin23+ 2k2

k = 0

Z12 = 2 cos

3+ i sin

3

= 2 12+ i 3

2

Z

12 = 1

2 i 3

2

k = 1

Z12 = 2 cos 4

3+ i sin 4

3

= 2 cos 3 i sin

3

Z12 = 2 1

2 i 3

2

Z

12 = 1

2 i 3

2

: ] 1-[

-:

Z = 1+ 3i Z = 2

cos = 12

,sin = 32

arg z = 23

Z12 = 2( )

12 cos

23+ 2k2

+ i sin23+ 2k2

k = 0

Z12 = 2 cos

3+ i sin

3

= 2 12+ i 3

2

Z

12 = 1

2 i 3

2

k = 1

Z12 = 2 cos 4

3+ i sin 4

3

= 2 cos 3 i sin

3

Z12 = 2 1

2 i 3

2

Z

12 = 1

2 i 3

2

Z = 1+ 3i Z = 2

cos = 12

,sin = 32

arg z = 23

Z12 = 2( )

12 cos

23+ 2k2

+ i sin23+ 2k2

k = 0

Z12 = 2 cos

3+ i sin

3

= 2 12+ i 3

2

Z

12 = 1

2 i 3

2

k = 1

Z12 = 2 cos 4

3+ i sin 4

3

= 2 cos 3 i sin

3

Z12 = 2 1

2 i 3

2

Z

12 = 1

2 i 3

2

Z = 1+ 3i Z = 2

cos = 12

,sin = 32

arg z = 23

Z12 = 2( )

12 cos

23+ 2k2

+ i sin23+ 2k2

k = 0

Z12 = 2 cos

3+ i sin

3

= 2 12+ i 3

2

Z

12 = 1

2 i 3

2

k = 1

Z12 = 2 cos 4

3+ i sin 4

3

= 2 cos 3 i sin

3

Z12 = 2 1

2 i 3

2

Z

12 = 1

2 i 3

2

1

9

Z = 1+ 3i Z = 2

cos = 12

,sin = 32

arg z = 23

Z12 = 2( )

12 cos

23+ 2k2

+ i sin23+ 2k2

k = 0

Z12 = 2 cos

3+ i sin

3

= 2 12+ i 3

2

Z

12 = 1

2 i 3

2

k = 1

Z12 = 2 cos 4

3+ i sin 4

3

= 2 cos 3 i sin

3

Z12 = 2 1

2 i 3

2

Z

12 = 1

2 i 3

2

1 +

Z = 1+ 3i Z = 2

cos = 12

,sin = 32

arg z = 23

Z12 = 2( )

12 cos

23+ 2k2

+ i sin23+ 2k2

k = 0

Z12 = 2 cos

3+ i sin

3

= 2 12+ i 3

2

Z

12 = 1

2 i 3

2

k = 1

Z12 = 2 cos 4

3+ i sin 4

3

= 2 cos 3 i sin

3

Z12 = 2 1

2 i 3

2

Z

12 = 1

2 i 3

22 2

2

Z = 1+ 3i Z = 2

cos = 12

,sin = 32

arg z = 23

Z12 = 2( )

12 cos

23+ 2k2

+ i sin23+ 2k2

k = 0

Z12 = 2 cos

3+ i sin

3

= 2 12+ i 3

2

Z

12 = 1

2 i 3

2

k = 1

Z12 = 2 cos 4

3+ i sin 4

3

= 2 cos 3 i sin

3

Z12 = 2 1

2 i 3

2

Z

12 = 1

2 i 3

2

Z = 1+ 3i Z = 2

cos = 12

,sin = 32

arg z = 23

Z12 = 2( )

12 cos

23+ 2k2

+ i sin23+ 2k2

k = 0

Z12 = 2 cos

3+ i sin

3

= 2 12+ i 3

2

Z

12 = 1

2 i 3

2

k = 1

Z12 = 2 cos 4

3+ i sin 4

3

= 2 cos 3 i sin

3

Z12 = 2 1

2 i 3

2

Z

12 = 1

2 i 3

2

] 1-[ :

x + yi( )2 = 1+ 3ix2 + 2xyi y2 = 1+ 3ix2 y2 = 1...........1

2xy = 3 y = 32x

........2

) 1( ) 2(

X 2 32x

2

= 1

X 2 34x2

= 1

4x4 + 4x2 3 = 02x2 1( ) 2x2 + 3( ) = 0

2x2 1 = 0 x2 = 12 x = m 1

2

2x2 + 3 = 0 x2 = 32

y = 3

2 m12

= m 3

2

m12

m32

i

4x2 0

X 2 32x

2

= 1

X 2 34x2

= 1

4x4 + 4x2 3 = 02x2 1( ) 2x2 + 3( ) = 0

2x2 1 = 0 x2 = 12 x = m 1

2

2x2 + 3 = 0 x2 = 32

y = 3

2 m12

= m 3

2

m12

m32

i

4x2 0

X 2 32x

2

= 1

X 2 34x2

= 1

4x4 + 4x2 3 = 02x2 1( ) 2x2 + 3( ) = 0

2x2 1 = 0 x2 = 12 x = m 1

2

2x2 + 3 = 0 x2 = 32

y = 3

2 m12

= m 3

2

m12

m32

i

4x2 0

50

X 2 32x

2

= 1

X 2 34x2

= 1

4x4 + 4x2 3 = 02x2 1( ) 2x2 + 3( ) = 0

2x2 1 = 0 x2 = 12 x = m 1

2

2x2 + 3 = 0 x2 = 32

y = 3

2 m12

= m 3

2

m12

m32

i

4x2 0

X 2 32x

2

= 1

X 2 34x2

= 1

4x4 + 4x2 3 = 02x2 1( ) 2x2 + 3( ) = 0

2x2 1 = 0 x2 = 12 x = m 1

2

2x2 + 3 = 0 x2 = 32

y = 3

2 m12

= m 3

2

m12

m32

i

4x2 0

) 2(

X 2 32x

2

= 1

X 2 34x2

= 1

4x4 + 4x2 3 = 02x2 1( ) 2x2 + 3( ) = 0

2x2 1 = 0 x2 = 12 x = m 1

2

2x2 + 3 = 0 x2 = 32

y = 3

2 m12

= m 3

2

m12

m32

i

4x2 0

X 2 32x

2

= 1

X 2 34x2

= 1

4x4 + 4x2 3 = 02x2 1( ) 2x2 + 3( ) = 0

2x2 1 = 0 x2 = 12 x = m 1

2

2x2 + 3 = 0 x2 = 32

y = 3

2 m12

= m 3

2

m12

m32

i

4x2 0 27i : 5

0

z3 = 27i z = 27i( )13

27i = 27i( ) = 27(cos 2 + sin2 )

z = [27(cos 2 + i sin2 )]

13

z = 3(cos 2 + i sin2 )

13

z = 3[cos + 2k3 + i sin + 2k

3 ]

z3 = 27i z = 27i( )13

27i = 27i( ) = 27(cos 2 + sin2 )

z = [27(cos 2 + i sin2 )]

13

z = 3(cos 2 + i sin2 )

13

z = 3[cos + 2k3 + i sin + 2k

3 ]

z3 = 27i z = 27i( )13

27i = 27i( ) = 27(cos 2 + sin2 )

z = [27(cos 2 + i sin2 )]

13

z = 3(cos 2 + i sin2 )

13

z = 3[cos + 2k3 + i sin + 2k

3 ]

z3 = 27i z = 27i( )13

27i = 27i( ) = 27(cos 2 + sin2 )

z = [27(cos 2 + i sin2 )]

13

z = 3(cos 2 + i sin2 )

13

z = 3[cos + 2k3 + i sin + 2k

3 ]

27 i( )

z3 = 27i z = 27i( )13

27i = 27i( ) = 27(cos 2 + sin2 )

z = [27(cos 2 + i sin2 )]

13

z = 3(cos 2 + i sin2 )

13

z = 3[cos + 2k3 + i sin + 2k

3 ]

z3 = 27i z = 27i( )13

27i = 27i( ) = 27(cos 2 + sin2 )

z = [27(cos 2 + i sin2 )]

13

z = 3(cos 2 + i sin2 )

13

z = 3[cos + 2k3 + i sin + 2k

3 ]2+ 2k

2+ 2k

51

k=0 z = 3 cos 6+ i sin

6

= 3 3

2+ 1

2i

=

3 32

+ 32

i

z = 3 cos 56+ i sin 5

6

= 3 32

+ 12

i

= 3 3

2+ 3

2i

z = 3 cos2+ 4

3+ i sin

2+ 4

3

= 3 cos 96+ i sin 9

6

= 3 cos 3

2+ i sin 3

2

= 3 0 i[ ] = 3i

3i, 3 32

+ 32

i,3 3

2+ 3

2i

1 z = 3 cos 6+ i sin

6

= 3 3

2+ 1

2i

=

3 32

+ 32

i

z = 3 cos 56+ i sin 5

6

= 3 32

+ 12

i

= 3 3

2+ 3

2i

z = 3 cos2+ 4

3+ i sin

2+ 4

3

= 3 cos 96+ i sin 9

6

= 3 cos 3

2+ i sin 3

2

= 3 0 i[ ] = 3i

3i, 3 32

+ 32

i,3 3

2+ 3

2i

z = 3 cos 6+ i sin

6

= 3 3

2+ 1

2i

=

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