Let \( f \) be an injective map with domain x, y, z and range 1, 2, 3 such that exactly one of the following statements is correct and the remaining are false.} [I.] \( f(x) = 1 \Rightarrow f(y) = 1, f(z) = 2 \) [II.] \( f(x) = 2 \Rightarrow f(y) = 1, f(z) = 1 \) [III.] \( f(x) = 1, f(y) = 1, f(z) = 2 \) Then the value of \( f(1) \) is:
For what value of \( k \) does the following pair of equations yield a unique solution for \( x \), such that the solution is positive?
\(x^2 - 3y^2 = 0 \)\(x^2 - 6y^2 + k = 0 \)