What is the physical quantity of the time rate of the angular momentum?

$A$ pulley of radius $2 \ m$ is rotated about its axis by a force $F = (20t - 5t^2) \ N$ (where $t$ is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is $10 \ kg \cdot m^2$,the number of rotations made by the pulley before its direction of motion is reversed is:

At time $t = 0$,a $2\, kg$ particle has position vector $\vec{r} = (4\hat{i} - 2\hat{j})\, m$ relative to the origin. Its velocity is given by $\vec{v} = 2t^2\hat{i}\, m/s$. The torque acting on the particle about the origin at $t = 2\, s$ is ........ $\hat{k}\, N\cdot m$.

$A$ disc of radius $0.4$ m and mass $1$ kg rotates about an axis passing through its centre and perpendicular to its plane. The angular acceleration is $10$ rad/s$^2$. The tangential force applied to the rim of the disc is (in $N$)

$A$ ceiling fan rotates about its own axis with some angular velocity. When the fan is switched off,the angular velocity becomes $\left(\frac{1}{4}\right)^{th}$ of the original in time $t$ and $n$ revolutions are made in that time. The number of revolutions made by the fan during the time interval between switch off and rest is (Angular retardation is uniform):

Which of the following are correct expressions for torque acting on a body?
$A. \ \vec{\tau}=\vec{ r } \times \vec{ L }$
$B. \ \vec{\tau}=\frac{ d }{ dt }(\vec{ r } \times \vec{ p })$
$C. \ \vec{\tau}=\vec{ r } \times \frac{ d \vec{ p }}{ dt }$
$D. \ \vec{\tau}= I \vec{\alpha}$
$E. \ \vec{\tau}=\vec{ r } \times \vec{ F }$
($\vec{ r }=$ position vector; $\vec{ p }=$ linear momentum;
$\vec{ L }=$ angular momentum; $\vec{\alpha}=$ angular acceleration;
$I=$ moment of inertia; $\vec{ F }=$ force; $t =$ time)
Choose the correct answer from the options given below: