1. Determine oxidation state and d-electron count

For each complex, first find the metal ion and its $d$-electron configuration.

• $[Cr(H_2O)_6]^{2+}$

  • $H_2O$ is neutral, so oxidation state of Cr is $+2$.
  • Cr: $[Ar]3d^5 4s^1$
  • $Cr^{2+}$: remove two electrons $\Rightarrow [Ar]3d^4$

• $[Fe(H_2O)_6]^{2+}$

  • Fe is $+2$.
  • Fe: $[Ar]3d^6 4s^2$
  • $Fe^{2+}$: $[Ar]3d^6$

• $[Mn(H_2O)_6]^{2+}$

  • Mn is $+2$.
  • Mn: $[Ar]3d^5 4s^2$
  • $Mn^{2+}$: $[Ar]3d^5$

• $[CoCl_4]^{2-}$

  • Let oxidation state of Co be $x$.
  • $x+4(-1)=-2 \Rightarrow x=+2$
  • Co: $[Ar]3d^7 4s^2$
  • $Co^{2+}$: $[Ar]3d^7$

2. Determine spin state / number of unpaired electrons

Use ligand strength and geometry:

• $H_2O$ is a weak-field ligand, so octahedral aqua complexes here are high spin.
• $[CoCl_4]^{2-}$ is a tetrahedral complex with weak-field ligand $Cl^-$, so it is also high spin.

Now count unpaired electrons:

(i) $[Cr(H_2O)_6]^{2+}$

$d^4$ high spin octahedral: $t_{2g}^3 e_g^1$ Unpaired electrons $n=4$.

(ii) $[Fe(H_2O)_6]^{2+}$

$d^6$ high spin octahedral: $t_{2g}^4 e_g^2$ Unpaired electrons $n=4$.

(iii) $[Mn(H_2O)_6]^{2+}$

$d^5$ high spin octahedral: $t_{2g}^3 e_g^2$ Unpaired electrons $n=5$.

(iv) $[CoCl_4]^{2-}$

$d^7$ high spin tetrahedral. For tetrahedral high spin $d^7$, unpaired electrons $n=3$.

3. Magnetic moment comparison

Spin-only magnetic moment is: $\mu=\sqrt{n(n+2)}\ \text{BM}$

• For $n=4$: $\mu=\sqrt{4(4+2)}=\sqrt{24}$
• For $n=5$: $\mu=\sqrt{5(7)}=\sqrt{35}$
• For $n=3$: $\mu=\sqrt{3(5)}=\sqrt{15}$

So the same magnetic moment occurs for the pair having the same number of unpaired electrons, i.e. both with $n=4$:

$[Cr(H_2O)_6]^{2+} \text{ and } [Fe(H_2O)_6]^{2+}$

4. Check options

• A: $[Cr(H_2O)_6]^{2+}$ and $[Fe(H_2O)_6]^{2+}$
  Both have $4$ unpaired electrons. Correct.

• B: $[Mn(H_2O)_6]^{2+}$ and $[Cr(H_2O)_6]^{2+}$
  $5$ and $4$ unpaired electrons. Incorrect.

• C: $[CoCl_4]^{2-}$ and $[Fe(H_2O)_6]^{2+}$
  $3$ and $4$ unpaired electrons. Incorrect.

• D: $[Cr(H_2O)_6]^{2+}$ and $[CoCl_4]^{2-}$
  $4$ and $3$ unpaired electrons. Incorrect.

5. Final answer

The pair having the same magnetic moment is: $\boxed{A}$