$A$ right circular cone is inscribed in a sphere of radius $3$ units. If the volume of the cone is maximum, then the semi-vertical angle of the cone is

On the interval $[0,1]$,the function $f(x) = x^{25}(1-x)^{75}$ takes its maximum value at the point

$A$ rectangular sheet of fixed perimeter with sides having their lengths in the ratio $8:15$ is converted into an open rectangular box by folding after removing squares of equal area from all four corners. If the total area of removed squares is $100$,the resulting box has maximum volume. The lengths of the sides of the rectangular sheet are:
$(A)$ $24$
$(B)$ $32$
$(C)$ $45$
$(D)$ $60$

Let $P(x)$ be a polynomial of degree $3$ having an extreme value at $x=1$. If $\lim _{x \rightarrow 0}\left(\frac{P(x)+4}{x^2}+2\right)=6$, then $\left(\frac{d P}{d x}\right)_{x=\frac{1}{2}}=$

Let $AD$ and $BC$ be two vertical poles at $A$ and $B$ respectively on a horizontal ground. If $AD = 8 \ m$,$BC = 11 \ m$ and $AB = 10 \ m$,then the distance (in meters) of point $M$ on $AB$ from the point $A$ such that $MD^2 + MC^2$ is minimum,is

$A$ stone moving vertically upwards has its equation of motion $s = 490t - 4.9t^2$. The maximum height reached by the stone is