Step 1 :Given the function \(f(x) = 3x^2 - 6x + k\), and the remainder when \(f(x)\) is divided by \(x - 1\) is 1.
Step 2 :According to the Remainder Theorem, if a polynomial \(f(x)\) is divided by \(x - c\), then the remainder is \(f(c)\). In this case, we have \(c = 1\) and the remainder is 1.
Step 3 :Set up the equation \(f(1) = 1\) and solve for \(k\):
Step 4 :\(f(1) = 3(1)^2 - 6(1) + k = 1\)
Step 5 :\(k - 3 = 1\)
Step 6 :\(k = 4\)
Step 7 :\(\boxed{k = 4}\)