Step 1: Understanding the structure of the complex.
In the given molecular formula \([Cu(H_2O)_6]^{2+}\) SO\(_4^{2-}\) \(\cdot\) H\(_2\)O, \) the copper ion (Cu\(^{2+}\)) is surrounded by six water molecules coordinated to it. Additionally, there is one water molecule hydrogen-bonded to the complex, not directly coordinated to the copper ion.
Step 2: Analyzing the options.
(A) 4: This is incorrect. Only one water molecule is hydrogen-bonded, not four.
(B) 3: This is incorrect. Only one water molecule is hydrogen-bonded.
(C) 1: Correct — There is one water molecule hydrogen-bonded to the complex.
(D) 5: This is incorrect. Only one water molecule is hydrogen-bonded.
Step 3: Conclusion.
The correct answer is (C) 1.