If $f(x)$ is a polynomial of degree $n$ with rational coefficients and $1+2i, 2-\sqrt{3}$ and $5$ are three roots of $f(x)=0$,then the least value of $n$ is

Let $z \in \mathbb{C}$ be such that $\frac{z^2+3i}{z-2+i}=2+3i$. Then the sum of all possible values of $z^2$ is

If $x = 1 + 2i$,then the value of $x^3 + 7x^2 - x + 16$ is

If $e^{i \theta} = \operatorname{cis} \theta$, then find the value of $\sum_{n=0}^{\infty} \frac{\cos (n \theta)}{2^n}$.

Let $A = \{\theta \in [0, 2\pi] : 1 + 10 \operatorname{Re}\left(\frac{2 \cos \theta + i \sin \theta}{\cos \theta - 3i \sin \theta}\right) = 0\}$. Then $\sum_{\theta \in A} \theta^2$ is equal to

Let $z$ be a complex number with a non-zero imaginary part. If $\frac{2+3z+4z^2}{2-3z+4z^2}$ is a real number,then the value of $|z|^2$ is: