= Solution
The <covariance matrix> is
$$
\boxed{V=\begin{pmatrix}
\sigma_1^2&\rho\sigma_1\sigma_2\\
\rho\sigma_1\sigma_2&\sigma_2^2
\end{pmatrix}}.
$$
If $\operatorname{Cov}(X_1,X_2)=0$, the moment-generating function in part (ii) factors into the product of the two marginal moment-generating functions. Thus $X_1$ and $X_2$ are <independent random variables>; this is the <independence of uncorrelated jointly normal variables>.
By bilinearity of <covariance>,
$$
\operatorname{Cov}(X_1,X_2-aX_1)
=\rho\sigma_1\sigma_2-a\sigma_1^2.
$$
Choose
$$
\boxed{a=\rho\frac{\sigma_2}{\sigma_1}},
\qquad Y=X_2-aX_1.
$$
Then $(X_1,Y)$ is bivariate normal by part (i) and has zero covariance, so $Y$ is a normal random variable independent of $X_1$. Hence $X_2=aX_1+Y$.
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