Let $$M$$ and $$N$$ be the number of points on the curve $$y^{5}-9 x y+2 x=0$$, where the tangents to the curve are parallel to $$x$$-axis and $$y$$-axis, respectively. Then the value of $$M+N$$ equals ___________.

Let the function $$f(x)=2 x^{2}-\log _{\mathrm{e}} x, x>0$$, be decreasing in $$(0, \mathrm{a})$$ and increasing in $$(\mathrm{a}, 4)$$. A tangent to the parabola $$y^{2}=4 a x$$ at a point $$\mathrm{P}$$ on it passes through the point $$(8 \mathrm{a}, 8 \mathrm{a}-1)$$ but does not pass through the point $$\left(-\frac{1}{a}, 0\right)$$. If the equation of the normal at $$P$$ is : $$\frac{x}{\alpha}+\frac{y}{\beta}=1$$, then $$\alpha+\beta$$ is equal to ________________.

The sum of the maximum and minimum values of the function $$f(x)=|5 x-7|+\left[x^{2}+2 x\right]$$ in the interval $$\left[\frac{5}{4}, 2\right]$$, where $$[t]$$ is the greatest integer $$\leq t$$, is ______________.

A hostel has 100 students. On a certain day (consider it day zero) it was found that two students are infected with some virus. Assume that the rate at which the virus spreads is directly proportional to the product of the number of infected students and the number of non-infected students. If the number of infected students on 4^th day is 30, then number of infected students on 8^th day will be __________.