Asking questions is fundamental, but without a systematic framework, it remains a matter of intuition rather than design. The Generalized Coordinate System (GCS) was initially proposed for analyzing and generating rhetorical modes. In this paper, we apply the GCS to form an inquiry design framework—the GCS-based 10-dimensional inquiry generation framework: treating a question as a coordinate point across ten axes, so that we have potentially a billion ways to ask questions. The five low-dimensional axes (Thing, Feature, Quantitative Attribute, Qualitative Attribute, Formal Attribute) determine what and how the question expresses; the two mediating axes (Basic Element, Rhetorical Mode) transform a raw inquiry into a communicable question package; the three high-dimensional axes (Cognitive Function, Epistemic Purpose, Expression Staircase) determine what mental operation, why, and at what developmental level. This GCS-based 10-dimensional inquiry generation transforms questioning from an intuitive art into a designable, transferable, and evaluable cognitive methodology, and is potentially useful in applications such as education, research, communication, and language modeling.