(1)
i 2666 = ( i 4 ) 666 i 2 = − 1 i^{2666}=(i^4)^{666}i^2=-1 i 2666 = ( i 4 ) 666 i 2 = − 1 i 145 = ( i 4 ) 36 i = i i^{145}=(i^4)^{36}i=i i 145 = ( i 4 ) 36 i = i (2)
z 1 = − i − 1 + i = e i 3 π 2 ( 2 2 e − i 3 π 4 ) = 2 2 e i 3 π 4 = 2 2 ( cos 3 π 4 + i sin 3 π 4 ) = − 1 2 + i ( 1 2 ) z 2 = 1 + i 1 − i = 2 e i π 4 ( 2 2 e i π 4 ) = e i π 2 = cos π 2 + i sin π 2 = i z_1=\dfrac{-i}{-1+i}=e^{i{3\pi \over 2}}(\dfrac{\sqrt{2}}{2}e^{-i{3\pi \over 4}})\\
=\dfrac{\sqrt{2}}{2}e^{i{3\pi \over 4}}=\dfrac{\sqrt{2}}{2}(\cos\dfrac{3\pi}{4}+i\sin\dfrac{3\pi}{4})=-\dfrac{1}{2}+i(\dfrac{1}{2})\\
z_2=\dfrac{1+i}{1-i}=\sqrt{2}e^{i{\pi \over 4}}(\dfrac{\sqrt{2}}{2}e^{i{\pi \over 4}})\\
=e^{i{\pi \over 2}}=\cos\dfrac{\pi}{2}+i\sin\dfrac{\pi}{2}=i z 1 = − 1 + i − i = e i 2 3 π ( 2 2 e − i 4 3 π ) = 2 2 e i 4 3 π = 2 2 ( cos 4 3 π + i sin 4 3 π ) = − 2 1 + i ( 2 1 ) z 2 = 1 − i 1 + i = 2 e i 4 π ( 2 2 e i 4 π ) = e i 2 π = cos 2 π + i sin 2 π = i
Since Z3 is not given, we will assume that it is equal to 1
Part i
z 1 ∗ z 3 z 2 = 2 2 e i 3 π 4 e i π 2 = 2 2 e i π 4 = 2 2 ( cos π 4 + i sin π 4 ) = 1 2 + i ( 1 2 ) \dfrac{z_1*z_3}{z_2}=\dfrac{\dfrac{\sqrt{2}}{2}e^{i{3\pi \over 4}}}{e^{i{\pi \over 2}}}=\dfrac{\sqrt{2}}{2}e^{i{\pi \over 4}}\\
=\dfrac{\sqrt{2}}{2}(\cos\dfrac{\pi}{4}+i\sin\dfrac{\pi}{4})=\dfrac{1}{2}+i(\dfrac{1}{2})\\ z 2 z 1 ∗ z 3 = e i 2 π 2 2 e i 4 3 π = 2 2 e i 4 π = 2 2 ( cos 4 π + i sin 4 π ) = 2 1 + i ( 2 1 )
Part ii
z 1 ∗ z 2 z 3 = 2 2 e i 3 π 4 ∗ e i π 2 = 2 2 e i 5 π 4 = 2 2 ( cos 5 π 4 + i sin 5 π 4 ) = − 1 2 − i ( 1 2 ) \dfrac{z_1*z_2}{z_3}=\dfrac{\sqrt{2}}{2}e^{i{3\pi \over 4}}*{e^{i{\pi \over 2}}}=\dfrac{\sqrt{2}}{2}e^{i{5\pi \over 4}}\\
=\dfrac{\sqrt{2}}{2}(\cos\dfrac{5\pi}{4}+i\sin\dfrac{5\pi}{4})=-\dfrac{1}{2}-i(\dfrac{1}{2})\\ z 3 z 1 ∗ z 2 = 2 2 e i 4 3 π ∗ e i 2 π = 2 2 e i 4 5 π = 2 2 ( cos 4 5 π + i sin 4 5 π ) = − 2 1 − i ( 2 1 )
Part iii
z 2 z 3 ∗ z 1 = e i π 2 2 2 e i 3 π 4 = 2 e − i π 4 = 2 ( cos ( − π 4 ) + i sin ( − π 4 ) ) = 1 − i \dfrac{z_2}{z_3*z_1}=\dfrac{e^{i{\pi \over 2}}}{\dfrac{\sqrt{2}}{2}e^{i{3\pi \over 4}}}=\sqrt{2}e^{-i{\pi \over 4}}\\
=\sqrt{2}(\cos(-\dfrac{\pi}{4})+i\sin(-\dfrac{\pi}{4}))
=1-i z 3 ∗ z 1 z 2 = 2 2 e i 4 3 π e i 2 π = 2 e − i 4 π = 2 ( cos ( − 4 π ) + i sin ( − 4 π )) = 1 − i
(3)
z 1 = − i = cos ( − π 2 ) + i sin ( − π 2 ) z_1=-i=\cos(-\dfrac{\pi}{2})+i\sin(-\dfrac{\pi}{2}) z 1 = − i = cos ( − 2 π ) + i sin ( − 2 π ) z 2 = − 1 − i 3 = 2 ( cos ( − 2 π 3 ) + i sin ( − 2 π 3 ) ) z_2=-1-i\sqrt{3}=2(\cos(-\dfrac{2\pi}{3})+i\sin(-\dfrac{2\pi}{3})) z 2 = − 1 − i 3 = 2 ( cos ( − 3 2 π ) + i sin ( − 3 2 π )) z 3 = − 3 + i = 2 ( cos ( 5 π 6 ) + i sin ( 5 π 6 ) ) z_3=-\sqrt{3}+i=2(\cos(\dfrac{5\pi}{6})+i\sin(\dfrac{5\pi}{6})) z 3 = − 3 + i = 2 ( cos ( 6 5 π ) + i sin ( 6 5 π )) ( z 3 ) 4 = 16 ( cos ( 10 π 3 ) + i sin ( 10 π 3 ) ) (z_3)^4=16(\cos(\dfrac{10\pi}{3})+i\sin(\dfrac{10\pi}{3})) ( z 3 ) 4 = 16 ( cos ( 3 10 π ) + i sin ( 3 10 π )) ( z 1 ) 2 = cos ( − π ) + i sin ( − π ) ) (z_1)^2=\cos(-\pi)+i\sin(-\pi)) ( z 1 ) 2 = cos ( − π ) + i sin ( − π )) ( z 2 ) − 1 = 1 2 ( cos ( 2 π 3 ) + i sin ( 2 π 3 ) ) (z_2)^{-1}=\dfrac{1}{2}(\cos(\dfrac{2\pi}{3})+i\sin(\dfrac{2\pi}{3})) ( z 2 ) − 1 = 2 1 ( cos ( 3 2 π ) + i sin ( 3 2 π )) ( z 3 ) 4 ( z 1 ) 2 ⋅ ( z 2 ) − 1 = 8 ( cos ( 5 π ) + i sin ( 5 π ) ) = − 8 \dfrac{(z_3)^4}{(z_1)^2}\cdot(z_2)^{-1}=8(\cos(5\pi)+i\sin(5\pi))=-8 ( z 1 ) 2 ( z 3 ) 4 ⋅ ( z 2 ) − 1 = 8 ( cos ( 5 π ) + i sin ( 5 π )) = − 8 (a) P ( z ) = z 2 + a , a > 0 P(z)=z^2+a, a>0 P ( z ) = z 2 + a , a > 0
z 2 + a = 0 = > z 1 = − i a , z 2 = i a z^2+a=0=>z_1=-i\sqrt{a}, z_2=i\sqrt{a} z 2 + a = 0 => z 1 = − i a , z 2 = i a (b) P ( z ) = z 3 − z 2 + z − 1 P(z)=z^3-z^2+z-1 P ( z ) = z 3 − z 2 + z − 1
z 3 − z 2 + z − 1 = 0 z^3-z^2+z-1=0 z 3 − z 2 + z − 1 = 0 z 2 ( z − 1 ) + ( z − 1 ) = 0 z^2(z-1)+(z-1)=0 z 2 ( z − 1 ) + ( z − 1 ) = 0 z 1 = 1 , z 2 = − i , z 3 = i z_1=1, z_2=-i, z_3=i z 1 = 1 , z 2 = − i , z 3 = i (c) z 3 − 1 = 0 z^3-1=0 z 3 − 1 = 0
( z − 1 ) ( z 2 + z + 1 ) = 0 (z-1)(z^2+z+1)=0 ( z − 1 ) ( z 2 + z + 1 ) = 0 z 1 = 1 , z 2 , 3 = − 1 ± i 3 2 z_1=1, z_{2,3}=\dfrac{-1\pm i\sqrt{3}}{2} z 1 = 1 , z 2 , 3 = 2 − 1 ± i 3 z 1 = 1 , z 2 = − 1 2 − i 3 2 , z 3 = − 1 2 + i 3 2 z_1=1, z_2=-\dfrac{1}{2}-i\dfrac{\sqrt{3}}{2}, z_3=-\dfrac{1}{2}+i\dfrac{\sqrt{3}}{2} z 1 = 1 , z 2 = − 2 1 − i 2 3 , z 3 = − 2 1 + i 2 3 (d)
8 − 8 i = 8 2 ( cos ( − π 4 ) + i sin ( − π 4 ) ) 8-8i=8\sqrt{2}(\cos(-\dfrac{\pi}{4})+i\sin(-\dfrac{\pi}{4})) 8 − 8 i = 8 2 ( cos ( − 4 π ) + i sin ( − 4 π )) k = 0 : 2 7 / 6 ( cos ( − π 12 ) + i sin ( − π 12 ) ) k=0: 2^{7/6}(\cos(-\dfrac{\pi}{12})+i\sin(-\dfrac{\pi}{12})) k = 0 : 2 7/6 ( cos ( − 12 π ) + i sin ( − 12 π )) k = 1 : 2 7 / 6 ( cos ( 7 π 12 ) + i sin ( 7 π 12 ) ) k=1: 2^{7/6}(\cos(\dfrac{7\pi}{12})+i\sin(\dfrac{7\pi}{12})) k = 1 : 2 7/6 ( cos ( 12 7 π ) + i sin ( 12 7 π )) k = 2 : 2 7 / 6 ( cos ( 5 π 4 ) + i sin ( 5 π 4 ) ) = − 2 2 / 3 − i ( 2 2 / 3 ) k=2: 2^{7/6}(\cos(\dfrac{5\pi}{4})+i\sin(\dfrac{5\pi}{4}))=-2^{2/3}-i(2^{2/3}) k = 2 : 2 7/6 ( cos ( 4 5 π ) + i sin ( 4 5 π )) = − 2 2/3 − i ( 2 2/3 ) (e)
w = 1 + i = 2 ( cos ( π 4 ) + i sin ( π 4 ) ) w=1+i=\sqrt{2}(\cos(\dfrac{\pi}{4})+i\sin(\dfrac{\pi}{4})) w = 1 + i = 2 ( cos ( 4 π ) + i sin ( 4 π )) w 3 = 2 3 / 2 ( cos ( 3 π 4 ) + i sin ( 3 π 4 ) ) w^3=2^{3/2}(\cos(\dfrac{3\pi}{4})+i\sin(\dfrac{3\pi}{4})) w 3 = 2 3/2 ( cos ( 4 3 π ) + i sin ( 4 3 π )) z 4 = w 3 z^4=w^3 z 4 = w 3
k = 0 : 2 3 / 8 ( cos ( 3 π 16 ) + i sin ( 3 π 16 ) ) k=0: 2^{3/8}(\cos(\dfrac{3\pi}{16})+i\sin(\dfrac{3\pi}{16})) k = 0 : 2 3/8 ( cos ( 16 3 π ) + i sin ( 16 3 π )) k = 1 : 2 3 / 8 ( cos ( 11 π 16 ) + i sin ( 11 π 16 ) ) k=1: 2^{3/8}(\cos(\dfrac{11\pi}{16})+i\sin(\dfrac{11\pi}{16})) k = 1 : 2 3/8 ( cos ( 16 11 π ) + i sin ( 16 11 π )) k = 2 : 2 3 / 8 ( cos ( 19 π 16 ) + i sin ( 19 π 16 ) ) k=2: 2^{3/8}(\cos(\dfrac{19\pi}{16})+i\sin(\dfrac{19\pi}{16})) k = 2 : 2 3/8 ( cos ( 16 19 π ) + i sin ( 16 19 π )) k = 3 : 2 3 / 8 ( cos ( 27 π 16 ) + i sin ( 27 π 16 ) ) k=3: 2^{3/8}(\cos(\dfrac{27\pi}{16})+i\sin(\dfrac{27\pi}{16})) k = 3 : 2 3/8 ( cos ( 16 27 π ) + i sin ( 16 27 π )) (4)
P ( z ) = z 2 + λ z − 6 P(z)=z^2+\lambda z-6 P ( z ) = z 2 + λ z − 6 z = i z=i z = i
( i ) 2 + λ i − 6 = 0 (i)^2+\lambda i-6=0 ( i ) 2 + λi − 6 = 0 λ = − 7 i \lambda=-7i λ = − 7 i
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