The height of a cone with semi-vertical angle $\pi / 3$ is increasing at the rate of $2 \text{ units/min}$. The rate at which the radius of the cone is to be decreased so as to have a fixed volume always is

If the sides of a triangle vary slightly in such a way that its circumradius remains constant,then $\frac{da}{\cos A} + \frac{db}{\cos B} + \frac{dc}{\cos C}$ is equal to

The distance $s$ moved by a particle in time $t$ is given by $s = f(t) = t^3 - 6t^2 + 9t$,where $s$ is in metres and $t$ is in seconds. The velocity of the particle at $t = 2 \ s$ is: (in $m/s$)

$A$ man of height $2 \ m$ walks at a uniform speed of $5 \ km/hr$ away from a lamp post which is $6 \ m$ high. Find the rate at which the length of his shadow increases in $km/hr$.

The radius of a right circular cylinder increases at a constant rate. Its altitude is a linear function of the radius and increases three times as fast as the radius. When the radius is $1 \, cm$,the altitude is $6 \, cm$. When the radius is $6 \, cm$,the volume is increasing at the rate of $1 \, cm^3/sec$. When the radius is $36 \, cm$,the volume is increasing at a rate of $n \, cm^3/sec$. The value of $n$ is equal to:

$A$ particle starts moving from rest from a fixed point in a fixed direction. The distance $s$ from the fixed point at a time $t$ is given by $s = t^{2} + at - b + 17$, where $a$ and $b$ are real numbers. If the particle comes to rest after $5 \ s$ at a distance of $s = 25$ units from the fixed point, then the values of $a$ and $b$ are respectively: