1. Identify the geometry of the complex

The complex is $[\mathrm{Co(CN)_6}]^{3-}$ in $K_3[\mathrm{Co(CN)_6}]$.

Since there are $6$ ligands around cobalt, the geometry is octahedral.

2. Determine which d-orbitals point directly toward ligands in an octahedral field

In an octahedral complex, the six ligands approach along the Cartesian axes: $+x, -x, +y, -y, +z, -z$.

The d-orbitals that have lobes lying directly along these axes are:

$d_{x^2-y^2} \quad \text{and} \quad d_{z^2}$

These form the $e_g$ set in an octahedral crystal field.

3. Why not the other orbitals?

The orbitals

$d_{xy},\ d_{xz},\ d_{yz}$

have their lobes oriented between the axes, not directly along the ligand directions. These form the $t_{2g}$ set.

So they are not directly facing the ligands.

4. Match with the options

• A: $d_{x^2-y^2}$ and $d_{z^2}$ ✅
• B: $d_{xy}, d_{xz}, d_{yz}$ ❌
• C: $d_{xz}, d_{yz}, d_{z^2}$ ❌
• D: $d_{xy}$ and $d_{x^2-y^2}$ ❌

Therefore, the correct answer is:

$\boxed{\text{A}}$

5. Comparison with stored correct answer

Stored correct answer = A

My derived answer = A

They agree.