The probability distribution of a discrete random variable $X$ is given below

$$ \begin{array}{lllll} \hline X=x & -1 & 0 & 1 & 2 \\ \hline P(X=x) & \frac{1}{3} & \frac{1}{6} & \frac{1}{6} & \frac{1}{3} \\ \hline \end{array} $$

Then, the value of $6 \sum\left(x^2\right) P(X=x)-\operatorname{var}(X)=$

If the average number of accidents occurring at a particular junction on a highway in a week is 5 , then the probability that atmost one accident occurs in a particular week is

Let $A(5,4)$ and $B(5,-4)$ be two points.

If $P$ is a point in the coordinate plane such that $\sqrt{A P B}=\frac{\pi}{4}$, then the point $P$ lies on the curve

When the axes are rotated through an angle $\theta$ about origin in anti-clockwise direction and then translated to the new origin $(2,-2)$, if the transformed equation the equation of $x^2+y^2=4$ is $X^2+Y^2+a X+b Y+c=0$ then $a+b+c=$