Answer: (a) \(-729729\). (b) \(a=3\).

(a) The general term in the expansion is

\({}^{15}C_r(x^2)^{15-r}\left(-\frac{3}{x^4}\right)^r.\)

The power of \(x\) is

\(2(15-r)-4r=30-6r.\)

For the term independent of \(x\), set \(30-6r=0\), giving \(r=5\).

The required coefficient is therefore

\({}^{15}C_5(-3)^5=3003(-243)=-729729.\)

(b) In \((1+ax)^9\), the coefficient of \(x^3\) is

\({}^9C_3a^3=84a^3.\)

The coefficient of \(x^2\) is

\({}^9C_2a^2=36a^2.\)

Given that the coefficient of \(x^3\) is \(7\) times the coefficient of \(x^2\),

\(84a^3=7(36a^2).\)

Since \(a\) is positive, divide by \(84a^2\):

\(a=3.\)