If the sum of the diagonal elements of a $$2 \times 2$$ matrix is $$-6$$, then the maximum possible value of determinant of the matrix is ____________.

The maximum value of $$'a'$$ such that the matrix $$\left[ {\matrix{ { - 3} & 0 & { - 2} \cr 1 & { - 1} & 0 \cr 0 & a & { - 2} \cr } } \right]$$ has three linearly independent real eigenvectors is

Two players, $$A$$ and $$B,$$ alternately keep rolling a fair dice. The person to get a six first wins the game. Given that player $$A$$ starts the game, the probability that $$A$$ wins the game is

A random variable $$X$$ has probability density function $$f(x)$$ as given below: $$$\,\,f\left( x \right) = \left\{ {\matrix{ {a + bx} & {for\,\,0 < x < 1} \cr 0 & {otherwise} \cr } } \right.\,\,$$$
If the expected value $$\,\,E\left[ X \right] = 2/3,\,\,$$ then $$\,\,\Pr \left[ {X < 0.5} \right]\,\,$$ is __________.