Let $\psi_j$ be the eigenfunctions of the operator $Q$ . Let $q_j$ be the eigenvalue associated with $\psi_j$ . Expand $\psi$ in terms of the eigenstates $\psi_j$ of $Q$
$\begin{align*}\psi = \sum_{j = 0}^{\infty} c_j \psi_j\end{align*}$
Then
$\begin{align*}\psi^* = \sum_{j = 0}^{\infty} c_j^{*} \psi_j^*\end{align*}$
So,
$\begin{align*}\int \psi^* Q \psi = \sum_{j = 0}^{\infty} \left|c_j \right|^2 q_j\end{align*}$
which is the expectation value of the physical observable that $Q$ represents. Therefore, answer (C) is correct.

Solution to 1986 Problem 56