A long magnet of magnetic moment $\mathrm{M}$ and a pole strength $\mathrm{m}$ is broken into two halves of equal length along the equatorial line. The magnetic moment of each half is

A short bar magnet is placed in the magnetic meridian of the earth with North Pole pointing north. Neutral points are found at a distance of $30 \mathrm{cm}$ from the magnet on the East-West line, drawn through the middle point of the magnet. The magnetic moment of the magnet in ${\mathrm{Am}}^2$ is close to:

(Given $\frac{{\mu }_0}{4\pi }={10}^{-7}$ in SI units and $B_H=$ Horizontal component of earth's magnetic field $=3.6\times {10}^{-5}$ Tesla.)

Two identical short magnetic dipoles of magnetic moments $1.0 \mathrm{A} {\mathrm{m}}^3$ each, placed at a separation of $2 \mathrm{m}$ with their axes perpendicular to each other. The resultant magnetic field at a point midway between the dipole is :

Two short magnets of equal dipole moments $M$, are fastened perpendicularly at their centre (figure). The magnitude of the magnetic field, at a distance $d$ from the centre, on the bisector of the right angle, is 

Represent the union of two sets by Venn diagram for each of the following.

$X=\{x | x$ is a prime number between $80$ and $100\}$

$Y=\{y | y$ is an odd number between $90$ and $100\}$