Which of the following points is the likely position of the center of mass of the system shown in the figure?

1. $A$
2. $B$
3. $C$
4. $D$

In the $\mathrm{HCl}$ molecule, the separation between the nuclei of the two atoms is about $1.27~\mathring{\text A}~(1~\mathring{\text A}=10^{10}~\text m).$ Then the approximate location of the CM of the molecule is:
(Given that a chlorine atom is about $35.5$ times as massive as a hydrogen atom and nearly all the mass of an atom is concentrated in its nucleus).

If $\theta$ is the angle between two vectors $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$, and $|\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}|=\overrightarrow{\mathrm{a}}.\overrightarrow{\mathrm{b}}$, then $\theta$ is equal to:
1. $0^\circ$
2. $180^\circ$
3. $135^\circ$
4. $45^\circ$

A vector $\overrightarrow A$ points vertically upward and $\overrightarrow B$ points towards north. The vector product $\overrightarrow A\times\overrightarrow B$ is:

For a body, with angular velocity $\vec{\omega }=\hat{i}-2\hat{j}+3\hat{k}$  and radius vector $\vec{r }=\hat{i}+\hat{j}++\hat{k},$  its velocity will be:
1. $-5\hat{i}+2\hat{j}+3\hat{k}$
2. $-5\hat{i}+2\hat{j}-3\hat{k}$
3. $-5\hat{i}-2\hat{j}+3\hat{k}$
4. $-5\hat{i}-2\hat{j}-3\hat{k}$

A body is in pure rotation. The linear speed $v$ of a particle, the distance $r$ of the particle from the axis and the angular velocity $\omega$ of the body are related as $w=\dfrac{v}{r}$. Thus:
1. $w\propto\dfrac{1}{r}$
2. $w\propto\ r$
3. $w=0$
4. $w$ is independent of $r$

If $\vec F$ is the force acting on a particle having position vector $\vec r$ and $\vec \tau$ be the torque of this force about the origin, then: