The number of distinct real roots of the equation $x^{7}-7x-2=0$ is

The sum of the terms of an infinitely decreasing geometric progression is equal to the greatest value of the function $f(x) = x^3 + 3x - 9$ on the interval $[-2, 3]$. If the difference between the first and the second term of the progression is equal to $f'(0)$,then the common ratio of the $G.P.$ is

$A$ cylindrical container is to be made from a certain solid material with the following constraints: It has a fixed inner volume of $V \ mm^3$,has a $2 \ mm$ thick solid wall,and is open at the top. The bottom of the container is a solid circular disc of thickness $2 \ mm$ and has a radius equal to the outer radius of the container. If the volume of the material used to make the container is minimum when the inner radius of the container is $10 \ mm$,then the value of $\frac{V}{250 \pi}$ is

If $x + 2y = 8$,then the maximum value of $xy$ is .......

For the function $f(x)=x^3-6x^2+12x-3$,the point $x=2$ is

Two particles $P$ and $Q$ located at the points $P(t, t^3 - 16t - 3)$ and $Q(t + 1, t^3 - 6t - 6)$ are moving in a plane. The minimum distance between the points during their motion is: