A magnetic compass needle oscillates 30 times per minute at a place where the dip is 45 ° and 40 times per…

Let $f_1$ and $f_2$ be the respective oscillation frequencies respectively for $45°$ and $30°$ dip. Then,

$f_1=\frac{1}{2\mathrm{\pi }}\sqrt{\frac{M{B}_1\mathrm{c}\mathrm{o}\mathrm{s}45°}{I}}$

$f_2=\frac{1}{2\mathrm{\pi }}\sqrt{\frac{M{B}_2\cos 30°}{I}}$

$\frac{f_1}{f_2}=\sqrt{\frac{B_1\mathrm{c}\mathrm{o}\mathrm{s}45°}{B_2\mathrm{c}\mathrm{o}\mathrm{s}30°}}$

$\Rightarrow \frac{{30}^2}{{40}^2}=\frac{B_1\mathrm{c}\mathrm{o}\mathrm{s}45°}{B_2\mathrm{c}\mathrm{o}\mathrm{s}30°}\mathrm{ }$

$\Rightarrow \frac{B_1}{B_2}=\frac{900\times \mathrm{c}\mathrm{o}\mathrm{s}30°}{1600\times \mathrm{c}\mathrm{o}\mathrm{s}45°}$

$\Rightarrow \frac{B_1}{B_2}=0.688\approx 0.7$