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 ________________.

Let $$\mathrm{Q}$$ and $$\mathrm{R}$$ be two points on the line $$\frac{x+1}{2}=\frac{y+2}{3}=\frac{z-1}{2}$$ at a distance $$\sqrt{26}$$ from the point $$P(4,2,7)$$. Then the square of the area of the triangle $$P Q R$$ is ___________.

Three masses $$M=100 \mathrm{~kg}, \mathrm{~m}_{1}=10 \mathrm{~kg}$$ and $$\mathrm{m}_{2}=20 \mathrm{~kg}$$ are arranged in a system as shown in figure. All the surfaces are frictionless and strings are inextensible and weightless. The pulleys are also weightless and frictionless. A force $$\mathrm{F}$$ is applied on the system so that the mass $$\mathrm{m}_{2}$$ moves upward with an acceleration of $$2 \mathrm{~ms}^{-2}$$. The value of $$\mathrm{F}$$ is :

( Take $$\mathrm{g}=10 \mathrm{~ms}^{-2}$$ )

A parallel beam of light of wavelength $$900 \mathrm{~nm}$$ and intensity $$100 \,\mathrm{Wm}^{-2}$$ is incident on a surface perpendicular to the beam. The number of photons crossing $$1 \mathrm{~cm}^{2}$$ area perpendicular to the beam in one second is :