A random variable $X$ has the following probability distribution

$$ \begin{array}{llllllllll} \hline X=\mathbf{x}_i & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline P\left(X=\mathbf{x}_i\right) & 10 k & 9 k & 8 k & 8 k & 6 k & 5 k & 4 k & 3 k & k \\ \hline \end{array} $$

where $k$ is a real number.

If $A=\left\{x_i \mid x_i\right.$ is a prime number $\}$ and $B=\left\{x_i \mid x_i>5\right\}$ are two events, then $P(A \cup B)=$

If $X$ is a Poisson variate such that $\frac{5}{3} k=P(X=2) =P(X=3)$, then $P(X=5)=$

$A(-4,0)$ and $B(4,0)$ are two fixed points. $C$ and $D$ are two points on $Y$ - axis such that $C D=4$ and $C$ is a point below $D$. Then, the locus of the point of intersection of the lines $A C$ and $B D$ is

By rotating the axes through an angle of $30^{\circ}$ in the anti-clockwise direction about the origin, the equation $4 x^2+12 x y+9 y^2+6 x+9 y+2=0$ becomes $a x^2+2 h x y+b y^2+2 g x+2 f y+c=0$ becomes, then