$x_1, x_2 \in \mathbf{N}$. If a line having slope 2 is a tangent to the curve $y=x^4-6 x^3+13 x^2-10 x+5$ at points $P\left(x_1, y_1\right)$ and $Q\left(x_2, y_2\right)$, then $x_1 x_2+y_1 y_2=$

Consider the following statements

Statement I If $a_0+\frac{a_1}{2}+\frac{a_2}{3}+\ldots .+\frac{a_n}{n+1}=0$, where $a_0, a_1, \ldots, a_n$ are real numbers, then the polynomial $a_0+a_1 x+a_2 x^2+\ldots .+a_n x^n$ has a zero in the interval $(0,1)$.

Statement II If $f:[a, b] \rightarrow \mathbf{R}$ is continuous on $[a, b]$ and $f$ is differentiable in $(a, b)$, where $a>0$ and if $\frac{f(a)}{a}=\frac{f(b)}{b}$, then there exists $c \in(a, b)$, such that $c f^{\prime}(c)=f(c)$.

Which one of the following options is true?

If $\frac{k}{\alpha^3}$ is the length of the sub normal at any point $P(\alpha, y)$ on the curve $x^2-a^2=\frac{x^2 y^2}{a^2}$, then $k=$

A tank in the shape of a rectangular parallelopiped has volume 27 cubic meters. This tank is filled with water such that the rate of change of level of the water is thrice the rate of change water quantity falling in the tank, then the height of the tank (in meters) is