An unknown transition metal salt X is subjected to the borax bead...

The characteristic colour changes in the borax bead test identify the metal present in salt $X$.

A blue bead in an oxidizing flame (when cold) that turns opaque red in a reducing flame (when cold) is characteristic of copper.

In the oxidizing flame, copper forms copper(II) metaborate, $\text{Cu}(\text{BO}_2)_2$. The metal cation responsible for the blue colour is $\text{Cu}^{2+}$.

The atomic number of copper is $29$. Its electronic configuration is $[\text{Ar}] 3d^{10} 4s^1$.
For the $\text{Cu}^{2+}$ ion, the electronic configuration is $[\text{Ar}] 3d^9$.

In the $3d^9$ configuration, there is $1$ unpaired electron ($n = 1$).

The spin-only magnetic moment $\mu$ is given by the formula:
$\mu = \sqrt{n(n+2)} \text{ BM}$
$\mu = \sqrt{1(1+2)} = \sqrt{3} \approx 1.73 \text{ BM}$

Answer: $1.73 \text{ BM}$