If the probability distribution of a random variable $X$ is as follows, then the mean of $X$ is

$$ \begin{array}{ccccc} \hline \boldsymbol{X}=\boldsymbol{x}_{\boldsymbol{i}} & -1 & 0 & 1 & 2 \\ \hline \boldsymbol{P}\left(\boldsymbol{X}=\boldsymbol{x}_{\boldsymbol{i}}\right) & \boldsymbol{k}^3 & 2 \boldsymbol{k}^3+\boldsymbol{k} & 4 \boldsymbol{k}-10 \boldsymbol{k}^2 & 4 \boldsymbol{k}-1 \\ \hline \end{array} $$

If $X$ is a binomial variate with mean $\frac{16}{5}$ and variance $\frac{48}{25}$, then $P(X \leq 2)=$

$A(a, 0)$ is a fixed point and $\theta$ is a parameter such that $0<\theta<2 \pi$. If $P(a \cos \theta, a \sin \theta)$ is a point on the circle $x^2+y^2=a^2$ and $Q(b \sin \theta,-b \cos \theta)$ is a point on the circle $x^2+y^2=b^2$, then the locus of the centroid of the $\triangle A P Q$ is

The point $P(4,1)$ undergoes the following transformations in succession :

(i) origin is shifted to the point $(1,6)$ by translation of axes.

(ii) translation through a distance of 2 units along the positive direction of $X$-axis.

(iii) rotation of axes through an angle of $90^{\circ}$ in the positive direction.

Then, the coordinates of the point $P$ in its final position are