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 ) = ⋯单项选择题

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Question text Let \( z_1 = 4+2i \) and \( z_2=-1+3i \). Write \( z_1 z_2 \) in exponential form \( re^{i \theta} \) using the principal argument for \( \theta \). Then \( r^2 = \) Answer 1 Question 3[input]. Writing \( \theta = \frac{p \pi}{q} \), where \(p,q\) are integers and \( \frac{p}{q} \) is a fraction in simplest form, then \( p = \) Answer 2 Question 3[input] and \( q = \) Answer 3 Question 3[input].

Question text When \[ 2 \, \text{cis} \left( - \frac{\pi}{3} \right) \, \text{cis} \left( \frac{\pi}{6} \right) \] is expressed in Cartesian form, it can be written as \( \sqrt{a} - b i \), where \(a\) and \(b\) are integers. Then \( a = \) Answer 1 Question 2[input] and \( b = \) Answer 2 Question 2[input].

Question text Write \[ \frac{i^{17}}{1+i} \] in Cartesian form as \[ \frac{a}{b} + \frac{c}{d} i, \] where \( a,b,c,d \) are integers and \( \frac{a}{b}, \frac{c}{d} \) are fractions in simplest form. Then \( a = \) Answer 1 Question 1[input], \( b = \) Answer 2 Question 1[input], \( c = \) Answer 3 Question 1[input], and \( d = \) Answer 4 Question 1[input].

Let 𝑢 = 1 − 3 𝑖  and  𝑣 = 1 + 𝑖 . Which of the following is the Cartesian form of the complex number 𝑢 + 𝑢 𝑣 ?

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