$A$ particle moves in a straight line in such a way that its velocity at any point is given by $v^2 = 2 - 3x$,where $x$ is measured from a fixed point. The acceleration is

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

$A$ particle moves along the curve $y = x^{3/2}$ in the first quadrant in such a way that its distance from the origin increases at the rate of $11$ units per second. The value of $\frac{dx}{dt}$ when $x = 3$ is

An object is moving in the clockwise direction around the unit circle $x^2+y^2=1$. As it passes through the point $\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$,its $y$-coordinate is decreasing at the rate of $3 \text{ units/sec}$. The rate at which the $x$-coordinate changes at this point is

$A$ man of height $1.8 \ m$ is moving away from a lamp post at the rate of $1.2 \ m/sec$. If the height of the lamp post is $4.5 \ m$,then the rate at which the shadow of the man is lengthening is .......... $m/sec$.

The rate of change of the volume of a sphere of radius $r$ with respect to its surface area is: