Do I calculate weights of assets correctly?

I solved attached question but I am not sure whether I did part a and c correctly. Is there a way to calculate weights of A and B by just knowing their standard deviation and correlation's value?

To find the weights in the question (a) you should write your portfolio expected excess return and variance as: $$E[R_p^e] = w_A R_A + w_b R_B - R_f \\ \sigma^2[R_p^e] = \sigma^2[w_A R_A + w_b R_B - R_f] = w_A^2\sigma_A^2 + w_B^2 \sigma_B^2 + 2 \rho_{AB}\sigma_A\sigma_B$$ The sharpe ratio is given by: $$S(w_A,w_B) = \frac{E[R_p^e]}{\sigma[R_p^e]}$$ So, to find the weights which maximize Sharpe ratio, you should solve the equation: $$\nabla S |_{w_A+w_B=1} = 0$$