Most existing diagnostic models are developed to detect whether students have mastered a set of skills of interest, but few have focused on identifying what scientific misconceptions students possess. This article developed a general dual-purpose model for simultaneously estimating students' overall ability and the presence and absence of misconceptions. The expectation-maximization algorithm was developed to estimate the model parameters. A simulation study was conducted to evaluate to what extent the parameters can be accurately recovered under varied conditions. A set of real data in science education was also analyzed to examine the viability of the proposed model in practice.