Let $z_1$ and $z_2$ be two complex numbers such that $z_1 + z_2 = 5$ and $z_1^3 + z_2^3 = 20 + 15i$. Then $|z_1^4 + z_2^4|$ equals-

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

If $z \in \mathbb{C}$,then the minimum value of $|z| + |2z - 3| + |z - 1|$ is

Let $a, b$ be two real numbers such that $ab < 0$. If the complex number $\frac{1+ai}{b+i}$ is of unit modulus and $a+ib$ lies on the circle $|z-1|=|2z|$,then a possible value of $\frac{1+[a]}{4b}$,where $[t]$ is the greatest integer function,is:

If the set $R = \{(a, b) : a + 5b = 42, a, b \in N\}$ has $m$ elements and $\sum_{n=1}^m (1 - i^{n!}) = x + iy$,where $i = \sqrt{-1}$,then the value of $m + x + y$ is:

If $z$ is a complex number satisfying $|z|^2 - |z| - 2 < 0$,then the value of $|z^2 + z \sin \theta|$,for all values of $\theta$,is