Let z be a complex number such that the imaginary part of z is non-zero and $$a\, = \,{z^2} + \,z\, + 1$$ is real. Then a cannot take the value

If $$\overrightarrow a ,\overrightarrow b $$ and $$\overrightarrow c $$ are unit vectors satisfying
$${\left| {\overrightarrow a - \overrightarrow b } \right|^2} + {\left| {\overrightarrow b - \overrightarrow c } \right|^2} + {\left| {\overrightarrow c - \overrightarrow a } \right|^2} = 9,$$ then $$\left| {2\overrightarrow a + 5\overrightarrow b + 5\overrightarrow c } \right|$$ is

A ship is fitted with three engines $${E_1},{E_2}$$ and $${E_3}$$. The engines function independently of each other with respective probabilities $${1 \over 2},{1 \over 4}$$ and $${1 \over 4}$$. For the ship to be operational at least two of its engines must function. Let $$X$$ denote the event that the ship is operational and Let $${X_1},{X_2}$$ and $${X_3}$$ denote respectively the events that the engines $${E_1},{E_2}$$ and $${E_3}$$ are functioning. Which of the following is (are) true?

The total number of ways in which 5 balls of different colours can be distributed among 3 persons so that each person gets at least one ball is