Question 178761
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#18.


*[tex \Large z * conj(z) = (a + bi)(a - bi) = a^2 - (bi)^2 = a^2 - (b^2)(i^2) = a^2 - (b^2)(-1)= a^2 + b^2 = \[modulus(z)\]^2], hence B true.


*[tex \Large z + conj(z) = (a + bi) + (a - bi) = 2a + 0bi = 2a], hence C true.


*[tex \Large z - conj(z) = (a + bi) - (a - bi) = 0a + 2bi = 2bi], hence D true.


I have an objection to selecting answer A, however.  Answer A says "All of the following.  Since Answer E follows Answer A, the word "All" in Answer A includes Answer E.  But...  You get the idea.  It is sort of like trying to determine the truth value of the sentence: "This statement is false."


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#19.


For the first four givens, either <i>a</i> or <i>b</i> in *[tex \Large a + bi] is zero, so the 'size' of the complex number reduces to the absolute value of the remaining part.  The first 4 answers have a size of 3, 7, 4, and 2 respectively.  The only thing you now have to determine is whether the size of Answer E is smaller than 7 or not.


*[tex \Large |z| = |4 + 4i| = sqrt{4^2 + 4^2} = sqrt{32} = 4 sqrt{2}]


Since *[tex \Large sqrt{2} < 1.5], *[tex \Large 4 sqrt{2} < 6], hence 7 is the largest magnitude of the given numbers.  Answer B.




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