The sum, of the coefficients of the first 50 terms in the binomial expansion of $$(1-x)^{100}$$, is equal to

If the local maximum value of the function $$f(x)=\left(\frac{\sqrt{3 e}}{2 \sin x}\right)^{\sin ^{2} x}, x \in\left(0, \frac{\pi}{2}\right)$$ , is $$\frac{k}{e}$$, then $$\left(\frac{k}{e}\right)^{8}+\frac{k^{8}}{e^{5}}+k^{8}$$ is equal to

Let the plane P: $$4 x-y+z=10$$ be rotated by an angle $$\frac{\pi}{2}$$ about its line of intersection with the plane $$x+y-z=4$$. If $$\alpha$$ is the distance of the point $$(2,3,-4)$$ from the new position of the plane $$\mathrm{P}$$, then $$35 \alpha$$ is equal to :

The area of the region enclosed by the curve $$y=x^{3}$$ and its tangent at the point $$(-1,-1)$$ is :