1. The preliminary concept of a mathematical investigation involves defining the main idea or problem you want to explore. 2. This step includes understanding the context, familiarizing yourself with necessary definitions, and outlining what you aim to discover or prove. 3. A conjecture is an educated guess or hypothesis based on observations, which you believe might be true but requires proof. 4. To formulate conjectures, analyze patterns from examples, test initial cases, and then state your hypothesis clearly. 5. The goal of preliminary concepts and conjectures is to guide your investigation and focus your proofs or counterexamples. In summary, start by clearly defining your problem and related concepts, then use observations to form thoughtful conjectures before proceeding with rigorous mathematical proof or analysis.