List of top Statistics Questions on Multivariate Normal Distribution

Given that \(X \sim N_3(\mu_1, \Sigma)\), where \(\mu_1 = (0\ 1\ 0)'\) and \(Y \sim N_3(\mu_2, \Sigma)\), where \(\mu_2 = (0\ -1\ 0)'\), then which of the following statements are true?
Statement 1: \((X+Y)^2\) has a central Chi-square distribution.
Statement 2: \((X-1)^2 + (Y+1)^2\) has a central Chi-square distribution.

Given that \(X=(X_1\ X_2\ X_3)' \sim N_3(\mu, \Sigma)\), where \(\mu=(0\ 0\ 0)'\) and \(\Sigma = \begin{pmatrix}1 & 0 & 0\\0 & 4 & -1\\0 & -1 & 1\end{pmatrix}\).
Which of the following is true?

Which of the following expressions of the multivariate normal density function describes the shape of the density curve?