Let $$F\left( x \right) = \int\limits_x^{{x^2} + {\pi \over 6}} {2{{\cos }^2}t\left( {dt} \right)} $$ for all $$x \in R$$ and $$f:\left[ {0,{1 \over 2}} \right] \to \left[ {0,\infty } \right]$$ be a continuous function. For $$a \in \left[ {0,{1 \over 2}} \right],\,$$ $$F'(a)+2$$ is the area of the region bounded by $$x=0, y=0, y=f(x)$$ and $$x=a,$$ then $$f(0)$$ is

The minimum number of times a fair coin needs to be tossed, so that the probability of getting at least two heads is at least $$0.96,$$ is

In $${R^3},$$ consider the planes $$\,{P_1}:y = 0$$ and $${P_2}:x + z = 1.$$ Let $${P_3}$$ be the plane, different from $${P_1}$$ and $${P_2}$$, which passes through the intersection of $${P_1}$$ and $${P_2}.$$ If the distance of the point $$(0,1, 0)$$ from $${P_3}$$ is $$1$$ and the distance of a point $$\left( {\alpha ,\beta ,\gamma } \right)$$ from $${P_3}$$ is $$2,$$ then which of the following relations is (are) true?

In $${R^3},$$ let $$L$$ be a straight lines passing through the origin. Suppose that all the points on $$L$$ are at a constant distance from the two planes $${P_1}:x + 2y - z + 1 = 0$$ and $${P_2}:2x - y + z - 1 = 0.$$ Let $$M$$ be the locus of the feet of the perpendiculars drawn from the points on $$L$$ to the plane $${P_1}.$$ Which of the following points lie (s) on $$M$$?