题目
MATH1061/1002/1021 (ND) Early Feedback Task: MATH1061 Canvas Quiz 1
单项选择题
Which of the following options lists all of the solutions to the following equation? 𝑧 3 = − 1 + 3 𝑖 .
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标准答案
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思路分析
Question restatement: The problem asks for all solutions to z^3 = -1 + 3i. The provided answer list appears to include three candidate roots, but there is no separate list of answer choices labeled A, B, C, etc. We will evaluate the given options as potential cube roots.
Step-by-step evaluation:
- First, convert the target complex number to polar form. The complex number -1 + 3i has magnitude r = sqrt((-1)^2 + 3^2) = sqrt(10), and an argument theta = arctan(3/(-1)) relative to the positive real axis places it in the second quadran......Login to view full explanation登录即可查看完整答案
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类似问题
( 6 − 6 𝑖 − 9 3 − 27 𝑖 ) 256 = _ _ _ _ _ _ _ _ _ _ Hints: Convert the complex numbers to polar form. If 𝑧 = 𝑟 ( cos 𝜃 + 𝑖 sin 𝜃 ) then 𝑧 𝑛 = 𝑟 𝑛 ( cos 𝑛 𝜃 + 𝑖 sin 𝑛 𝜃 ) . If 𝑧 1 = 𝑟 1 ( cos 𝜃 1 + 𝑖 sin 𝜃 1 ) and 𝑧 2 = 𝑟 2 ( cos 𝜃 2 + 𝑖 sin 𝜃 2 ) then: 𝑧 1 𝑧 2 = 𝑟 1 𝑟 2 [ cos ( 𝜃 1 + 𝜃 2 ) + 𝑖 sin ( 𝜃 1 + 𝜃 2 ) ] and 𝑧 1 𝑧 2 = 𝑟 1 𝑟 2 [ cos ( 𝜃 1 − 𝜃 2 ) + 𝑖 sin ( 𝜃 1 − 𝜃 2 ) ] .
The polar form of 𝑧 = − 6 7 + 3 2 7 𝑖 is: Hint: 𝑧 = 𝑎 + 𝑏 𝑖 = 𝑟 ( cos 𝜃 + 𝑖 sin 𝜃 ) where 𝑟 = 𝑎 2 + 𝑏 2 and 𝜃 = tan − 1 𝑏 𝑎 . Also, don't forget to plot the complex number on the Argand diagram.
Given four complex numbers 𝑧 1 = 2 + 3 𝑖 , 𝑧 2 = − 7 − 5 𝑖 , 𝑧 3 = − 9 + 7 𝑖 , 𝑧 4 = 2 + 5 𝑖 . Calculate | 𝑧 3 ¯ 𝑧 1 + 𝑧 2 𝑧 4 | . Formulae: If 𝑧 = 𝑎 + 𝑏 𝑖 then 𝑧 ¯ = 𝑎 − 𝑏 𝑖 and | 𝑧 | = 𝑎 2 + 𝑏 2 . 𝑎 + 𝑏 𝑖 𝑐 + 𝑑 𝑖 = ( 𝑎 + 𝑏 𝑖 ) ( 𝑐 − 𝑑 𝑖 ) ( 𝑐 + 𝑑 𝑖 ) ( 𝑐 − 𝑑 𝑖 ) = ⋯
The figure shows the Argand diagram together with the complex number \(z\). If \(d=13\) and the imaginary part of \(z\) is \(-5\), find \(z\).
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