rag/knowledge_base/solution_templates.md

Solution Templates

General Problem-Solving Strategy

1. Read the problem carefully — identify what is given and what is asked
2. Classify the problem type (algebra, probability, calculus, linear algebra)
3. Identify relevant formulas and theorems
4. Plan the solution approach before computing
5. Execute the computation step by step
6. Verify the answer (substitute back, check units/dimensions, check edge cases)

Algebra Problem Template

1. Identify the type: equation, inequality, expression simplification
2. Standard form: rearrange to standard form if needed
3. Apply appropriate technique: factoring, formula, substitution
4. Solve and simplify
5. Check: substitute solution back into original equation

Probability Problem Template

1. Define the sample space and events clearly
2. Identify the type: counting, conditional, Bayes, distribution
3. Determine if order matters (permutation vs combination)
4. Calculate using appropriate formula
5. Verify: $0 \leq P \leq 1$, probabilities sum to 1

Calculus Problem Template

1. Identify: limit, derivative, integral, or optimization
2. Check for standard forms that allow direct formulas
3. Apply appropriate technique (L'Hôpital, chain rule, substitution, by parts)
4. Simplify the result
5. Verify: dimensional consistency, check special values

Linear Algebra Problem Template

1. Write in matrix form if not already
2. Identify dimensions and check compatibility
3. Apply appropriate operations (row reduction, determinant, inverse)
4. Solve and interpret the result
5. Verify: multiply back to check, verify rank conditions

Optimization Template (JEE Favorite)

1. Define the objective function $f(x)$
2. Find the domain/constraints
3. Compute $f'(x)$ and find critical points ($f'(x) = 0$)
4. Apply second derivative test or first derivative test
5. Compare with boundary values
6. State the maximum/minimum value and where it occurs