$A$ spherical iron ball of radius $10 \ cm$ is coated with a layer of ice of uniform thickness. The ice melts at a rate of $50 \ cm^3/min$. When the thickness of the ice is $5 \ cm$,the rate at which the thickness of the ice decreases is:

The radius of a sphere is measured to be $20 \, cm$ with a possible error of $0.02 \, cm$. The consequent error in the surface area of the sphere is ....... $sq \, cm$.

$A$ point moves along the arc of the parabola $y = 2x^2$. Its abscissa increases uniformly at the rate of $2 \text{ units/sec}$. At the instant the point is passing through $(1, 2)$,its distance from the origin is increasing at the rate of

The rate of change of the volume of a sphere with respect to its radius when $r = 7 \ cm$ is . . . . . . .

The radius of a circular plate is increasing at the rate of $0.01 \text{ cm/sec}$. When the radius is $12 \text{ cm}$, the rate at which the area increases is (in $\text{cm}^2/\text{sec}$): (in $\pi$)

$A$ 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: