The circular scale of a micrometer has $200$ divisions and pitch of $2\,mm$ . Find the measured value of thickness of a thin sheet .......... $mm$

Using screw gauge of pitch $0.1$ $cm$ and $50$ divisions on its circular scale, the thickness of an object is measured. It should correctly be recorded as 

Two full turns of the circular scale of a screw gauge cover a distance of $1\ mm$ on its main scale. The total number of divisions on the circular scale is $50$. Further, it is found that the screw gauge has a zero error of $- 0.03\ mm$. While measuring the diameter of a thin wire, a student notes the main scale reading of $3\ mm$ and the number of circular scale divisions in line with the main scale as $35$. The diameter of the wire is  ....... $mm$

A screw gauge gives the following reading when used to measure the diameter of a wire.

Main scale reading : $0\ mm$

Circular scale reading : $52\ divisions$

Given that $1\ mm$ on main scale corresponds to $100$ divisions of the circular scale. The diameter of wire from the above data is:

$N$ divisions on the main scale of a vernier calliper coincide with $(N + 1 )$ divisions of the vernier scale. If each division of main scale is $a$ units , then the least count of the instrument is