If a line having slope 2 is a tangent to the curve $y=x^4-6 x^3+13 x^2-12 x+5$ at points $P\left(x_1, y_1\right)$ and $Q\left(x_2, y_2\right), x_1, x_2 \in N$, then $x_1 x_2-y_1 y_2=$

Let $m$ be the slope of the normal $L$ drawn at $(1,2)$ to the curve $x=t^2-7 t+7, y=t^2-4 t-10$ and $a x+b y+c=0$ be the equation of the normal $L$. If GCD of $(a, b, c)$ is 1 , then $m(a+b+c)=$

If the function $f(x)=x e^{-x}, x \in R$ attains its maximum value $\beta$ at $x=\alpha$, then $(\alpha, \beta)=$

The diameter of a sphere is measured as 42 cm . If there is an error of $1 / 77 \mathrm{~cm}$ in measuring it, then the error involved in the volume of that sphere (in cubic centimeters) is