3 - Complex Numbers Questions Answers

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what is the angular difference between the +j and -j ?


+4j and -4j  ?


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Joshi sir comment

180 degree

let z be a complex number satisfying |z-3|=|z-4i|, then find the least possible value of 10|z|.

Asked By: AMIT DAS
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Joshi sir comment


In |z-3|=|z-4i|, z represents all points lying on the perpendicular bisector of the line joining 3 and 4i in x-y plane. In this perpendicular bisector, if we draw a perpendicular then it will be the minimum magnitude of z. Construct this as a diagram we will get

sinθ = |z| / (7/8)

or |z| = (7/8)(4/5) = 7/10   

The number of roots of equations z15 = 1 and Iarg zI< π/2 is

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Joshi sir comment

let z = r(cosθ+i sinθ) and it is given that z15 = 1  =>  r15(cosθ+i sinθ)15 = 1

or r15(cos15θ+i sin15θ) = 1+0i                   (Here we use Demoivre's theorem)

on comparison we get sin 15θ = 0 => 15θ = nπ          here n is an integer

for n = -7 to +7 we will get 15 answers for which θ will lie between -π/2 to +π/2


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