(a) Given \(P(X = 4) = 3P(X = 2)\), we have:

\(4k(4 + a) = 3 \times 2k(2 + a)\)

\(16k + 4ak = 12k + 6ak\)

\(4k = 2ak\)

\(a = 2\)

Using the sum of probabilities:

\(k(1 + a) + 2k(2 + a) + 3k(3 + a) + 4k(4 + a) = 1\)

\(3k + 8k + 15k + 24k = 1\)

\(50k = 1\)

\(k = \frac{1}{50}\)

(b) The probability distribution table is:

(c) To find \(\text{Var}(X)\), use:

\(\text{Var}(X) = \sum (x^2 \cdot P(X = x)) - (E(X))^2\)

\(\text{Var}(X) = \frac{3}{50} \times 1^2 + \frac{8}{50} \times 2^2 + \frac{15}{50} \times 3^2 + \frac{24}{50} \times 4^2 - 3.2^2\)

\(\text{Var}(X) = \frac{3 + 32 + 135 + 384}{50} - 10.24\)

\(\text{Var}(X) = \frac{554}{50} - 10.24\)

\(\text{Var}(X) = 11.08 - 10.24\)

\(\text{Var}(X) = 0.84\)

\(\text{Var}(X) = \frac{21}{25}\)