In an oscillation magnetometer, a combination of two bar magnets...

The frequency of oscillation in an oscillation magnetometer is given by $f = \dfrac{1}{2\pi} \sqrt{\dfrac{MB}{I}}$, which implies $f \propto \sqrt{M}$ for a constant moment of inertia $I$ and magnetic field $B$.

When like poles are tied together, the effective magnetic moment is $M_1 + M_2$. The frequency is $f_1 = 10$.

When unlike poles are tied together, the effective magnetic moment is $M_1 - M_2$. The frequency is $f_2 = 2$.

Taking the ratio of the frequencies:

$\dfrac{f_1}{f_2} = \sqrt{\dfrac{M_1 + M_2}{M_1 - M_2}}$

Substituting the given values:

$\dfrac{10}{2} = \sqrt{\dfrac{M_1 + M_2}{M_1 - M_2}}$

$5 = \sqrt{\dfrac{M_1 + M_2}{M_1 - M_2}}$

Squaring both sides:

$25 = \dfrac{M_1 + M_2}{M_1 - M_2}$

$25(M_1 - M_2) = M_1 + M_2$

$25M_1 - 25M_2 = M_1 + M_2$

$24M_1 = 26M_2$

$\dfrac{M_1}{M_2} = \dfrac{26}{24} = \dfrac{13}{12}$

The ratio of the magnetic moments is $13 : 12$.