If two dice are thrown and if $X$ denotes the sum of the numbers that show up on the faces of the dice, then the mean of the random variable $X$ is

In a university campus, the probability that a person chosen at random is an engineering student is $\frac{1}{5}$. The probability of having atmost two engineering students in a sample of 8 people is

If $A(1,1), B(-1,1)$ and $C(-1,-1)$ are three points and a point $P$ moves such that $(P A)^2=(P B)^2+(P C)^2$, then the equation of the locus of $P$ is

If $x^2=8 a y$ is the transformed equation of $x^2-4 y+6 x+15=0$ when the origin is shifted to the point $(\alpha, \beta)$ by translation of axes, then $2 \alpha+8 \beta^2=$