We apply I/O analysis to investigate multi-modal instability mechanisms in Mach 6 flow past a cold-wall sharp cone and blunt cones with and without a swept fin. I/O analysis, based on the resolvent operator, considers the linear dynamical response of an entire flow system to small externally imposed fluctuations. The global view taken by I/O analysis allows it to handle complex geometry seamlessly. All of the cases we consider involve multi-mode interaction, which is difficult for traditional stability analysis to capture. We demonstrate that I/O analysis captures both modal growth and non-modal growth in hypersonic boundary layers, and thus provides an ideal platform to study multi-mode instability. As the cases that we consider become progressively more complex we show that such interactions become increasingly important.