few p6 questions

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#1
1.Solve Z^3 = 8i givin each of roots in the form re^(itheta) wher r>0

2. |z-1-i| = 4 find the min and max values of |z| for points on the locus
0
15 years ago
#2
(Original post by chetan)
1.Solve Z^3 = 8i givin each of roots in the form re^(itheta) wher r>0

2. |z-1-i| = 4 find the min and max values of |z| for points on the locus
1. express 8i as 2^3 (cos(pi/2)+isin(pi/2)) and then apply De Movre's theorem

2. it is equivalent to |z-(1+i)| = 4 and so its a circle with centre 1+i and radius of 4
draw it on the Argand Diagram
the complex numbers corresponding to the min and max values of |z| should lies on the line joining the centre and the origin
the min is in the South-West direction and the max North-East
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#3
how do u knw its pi/2 tho?
0
15 years ago
#4
(Original post by chetan)
how do u knw its pi/2 tho?
all complex number a+bi can be expressed in form of R(cosx+isinx)
the modulus, R = root(a^2 + b^2)
so Rcosx = a and Rsinx = a
x = arcos(a/R) and x = arsin(a/R)
x can have two values for each of the above equations
the same value of x coming from the two equations is the argument

follow it and u can find pi/2
0
15 years ago
#5
(1)
z^3 = 8e^(i pi/2)
=> z = 2e^(i pi/6) or 2e^(5i pi/6) or 2e^(9i pi/6).

(2)
The min and max both occur on the line through 0 and 1 + i.

min = 4 - sqrt(2),
max = 4 + sqrt(2).
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