A paramagnetic sample has a relative permeability of 1.0006 at an...

The relative permeability ($\mu_r$) of a magnetic material is related to its magnetic susceptibility ($\chi$) by the relation:
$\mu_r = 1 + \chi$

At $T_1 = 300\text{ K}$, the initial susceptibility is:
$\chi_1 = \mu_{r1} - 1 = 1.0006 - 1 = 0.0006$

According to Curie's law for paramagnetic materials, susceptibility is inversely proportional to absolute temperature ($\chi \propto \dfrac{1}{T}$).

Therefore, the ratio of susceptibilities is:
$\dfrac{\chi_2}{\chi_1} = \dfrac{T_1}{T_2}$

$\chi_2 = \chi_1 \times \dfrac{T_1}{T_2} = 0.0006 \times \dfrac{300}{200}$
$\chi_2 = 0.0006 \times 1.5 = 0.0009$

The new relative permeability at $T_2 = 200\text{ K}$ is:
$\mu_{r2} = 1 + \chi_2 = 1 + 0.0009 = 1.0009$

Answer: $1.0009$