Since M can be any symmetric matrix with eigenvalues as 1,2,3. And N can be any matrix with real eigen values. So N can also be a symmetric matrix as real symmetric matrices have real eigenvalues.
Let $p,q,r$ be the eigenvalues of N. So we have,
$M=\begin{bmatrix} 1 & 0 &0 \\ 0&2 & 0\\ 0& 0 &3 \end{bmatrix}, N=\begin{bmatrix} p & 0 &0 \\ 0&q & 0\\ 0& 0 &r \end{bmatrix}$
$MN+N^TM=3I$
$\begin{bmatrix} 1 & 0 &0 \\ 0&2 & 0\\ 0& 0 &3 \end{bmatrix}\begin{bmatrix} p & 0 &0 \\ 0&q & 0\\ 0& 0 &r \end{bmatrix}+\begin{bmatrix} p & 0 &0 \\ 0&q & 0\\ 0& 0 &r \end{bmatrix}\begin{bmatrix} 1 & 0 &0 \\ 0&2 & 0\\ 0& 0 &3 \end{bmatrix} = 3\begin{bmatrix} 1 & 0 &0 \\ 0&1 & 0\\ 0& 0 &1 \end{bmatrix}$
$\begin{bmatrix} 2p & 0 &0 \\ 0&4q & 0\\ 0& 0 &6r \end{bmatrix} = \begin{bmatrix} 3 & 0 &0 \\ 0&3 & 0\\ 0& 0 &3 \end{bmatrix}$
$p = 3/2, q=3/4, r=1/2$
So, options A and D are correct choices