The equation of motion of a particle is given by $s = at^2 + bt + c$. If the displacement after $1 \ s$ is $20 \ m$,the velocity after $2 \ s$ is $30 \ m/s$,and the acceleration is $10 \ m/s^2$,then which of the following is true?

The volume of a spherical balloon is increasing at the rate of $30 \ cm^3/min$. Find the rate of change of the surface area of the balloon when its radius is $6 \ cm$.

$A$ is a point on the circle with radius $8$ and centre at $O$. $A$ particle $P$ is moving on the circumference of the circle starting from $A$. $M$ is the foot of the perpendicular from $P$ on $OA$ and $\angle POM = \theta$. When $OM = 4$ and $\frac{d\theta}{dt} = 6 \text{ radians/sec}$,then the rate of change of $PM$ is (in units/sec)

$A$ stone is dropped in a quiet lake and it is observed that waves move in circles. If the radius of a circular wave increases at the rate of $2 \text{ cm/sec}$, then the rate of increase in its area at the instant when its radius is $10 \text{ cm}$, is in $\text{cm}^2\text{/sec}$: (in $\pi$)

$A$ ladder $10 \, m$ long rests against a vertical wall with the lower end on the horizontal ground. The lower end of the ladder is pulled along the ground away from the wall at the rate of $3 \, cm/sec$. The height of the upper end while it is descending at the rate of $4 \, cm/sec$ is (in $, m$)

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 ....... $sq \, cm$.