Solve

Concept (1–2 lines)

• State the relevant definition, theorem, or formula
• Name the proof strategy: direct proof, contradiction, induction, counterexample, case analysis
• Do NOT derive the concept — just name it and write the formula

Step-by-step Working

• Number each step or use clear subheadings
• Show every algebraic manipulation that changes the expression's form
• Skip only trivial arithmetic (e.g. $2 \times 3 = 6$)
• Use $\Longrightarrow$ or "so" to connect logical steps
• Show intermediate forms before final simplification

Result

• Box final answers with $\boxed{...}$
• For "show that" questions: end with explicit confirmation (e.g. "$= 0$ as required $\checkmark$")
• State geometric/physical interpretation if the question asks for one

Formatting Rules

• LaTeX throughout: $...$ inline, $$...$$ display
• Vectors in bold: $\mathbf{r}$, $\mathbf{e}_R$
• Separate each question with ---
• Headings: ## Question N, ### Part (i)
• Reference earlier results explicitly (e.g. "Using $d\mathbf{r}/dt$ from part (i)...")

How Much Detail

Always include: formula being applied, every differentiation/integration/algebraic step, intermediate simplifications, final boxed answer, constraint checking

Keep minimal: concept explanations (name the formula, don't derive it), physical interpretation (one sentence unless asked), motivation

Omit: full derivations of standard results, lengthy prose between steps, restatement of things already shown

Checklist

• Every part of every question is answered
• Each answer starts by identifying the method/formula used
• All working shown step-by-step with no gaps
• Final answers boxed
• "Show that" questions end with explicit confirmation
• Earlier results referenced, not re-derived
• LaTeX renders correctly (matched braces, correct delimiters)
• Constraints and domains respected (e.g. $t \geq 0$)