Question:
If (i(z^3) + Z^2 - z + i = 0).
Then mod(z) = ?
a)" "Rational

b)" "Integer

c)" "None

Solution -1 :

We can simplify this equation by simply taking i^2 = -1 before z and then factorizing the equation to find a root of this equation.
That is
(i(z^3) + Z^2 + i^2(z) + i) = 0

[z^2(1+iz) + i(1+iz)] = 0

(z^2+i)(1+iz) = 0

This gives
(z^2) = (-i) and z = (-i) and mod(z) = 1 which is an integer. So the option (b) is correct.

Solution-2:
We can put z = (e^(iθ)) in the above equation and then factorize the equation to get the roots.
That is
(i(e^(iθ))^3 + (e^iθ)^2 - (e^iθ) + i) = 0

Upon factoring we get;
(ie^(2iθ)-1)(e^(iθ)-i) = 0

That is
z = i and z^2 = i So Mod(z) = 1 which is an integer.
So option (b) is correct.