A paramagnetic material has a relative permeability of 1.006 at a...

The initial relative permeability is $\mu_{r1} = 1.006$.
The initial magnetic susceptibility $\chi_1$ is given by:
$\chi_1 = \mu_{r1} - 1 = 1.006 - 1 = 0.006$

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

$\dfrac{\chi_2}{\chi_1} = \dfrac{T_1}{T_2}$

$\chi_2 = 0.006 \times \dfrac{300}{400} = 0.006 \times \dfrac{3}{4} = 0.0045$

The new relative permeability $\mu_{r2}$ at $400$ K is:
$\mu_{r2} = 1 + \chi_2$
$\mu_{r2} = 1 + 0.0045 = 1.0045$

Answer: $1.0045$