(i) The sum of all probabilities must equal 1:

\(0.24 + 0.35 + 2k + k + 0.05 = 1\)

\(0.64 + 3k = 1\)

\(3k = 0.36\)

\(k = 0.12\)

(ii) The mode is the value of \(x\) with the highest probability. Here, \(P(X = 1) = 0.35\) is the highest, so the mode is 1.

(iii) Calculate the mean:

\(\text{mean} = 1 \times 0.35 + 2 \times 0.24 + 3 \times 0.12 + 4 \times 0.05\)

\(\text{mean} = 0.35 + 0.48 + 0.36 + 0.20 = 1.39\)

Find \(P(X > 1.39) = P(X = 2) + P(X = 3) + P(X = 4)\)

\(P(X = 2) = 2k = 0.24\)

\(P(X = 3) = k = 0.12\)

\(P(X = 4) = 0.05\)

\(P(X > 1.39) = 0.24 + 0.12 + 0.05 = 0.41\)