Question

is
3. If 8 iz^3 +12z^2 - 18z +27i = 0, then the value of |z|
(a) 3/2
(b) 2/3
(c) 1
(d) 3/4​

Answer 1

Answer:

The product of the complex roots of az3+bz2+cz+daz3+bz2+cz+d is given by −d/a−d/a. In your case, the product of the roots is z1z2z3=(27i)/(8i)=27/8z1z2z3=(27i)/(8i)=27/8. Now, assuming that each of the roots has the same modulus |z1|=|z2|=|z3|=|z||z1|=|z2|=|z3|=|z|, we get |z|3=27/8|z|3=27/8, hence |z|=3/2|z|=3/2 and so 4|z|2=94|z|2=9.

The reason I assumed that |z1|=|z2|=|z3||z1|=|z2|=|z3| was that the question asked for the value of 4|z|24|z|2, which implies that this value doesn't depend on which root you choose. It is not true in general that the modulus of all the roots of a cubic equation are equal.

Answer 2

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