Problem Solving and Mathematical Reasoning
Core ideas
This domain tests the ability to select and combine techniques from across the syllabus. Candidates should be able to translate a word problem into algebra, identify the quantity to be optimised or solved, set up equations or inequalities, and interpret the answer in context. Proof and reasoning appear: showing an identity, disproving by counterexample, and justifying each step. Candidates must also handle problems involving rates, proportional reasoning, and multi-step geometry, presenting clear working with correct units.
Traps
- Answering the algebraic question but not the contextual one (e.g. giving x when the question asks for a cost).
- Ignoring units or mixing them.
- Failing to reject physically impossible solutions such as negative lengths.
- Assuming a statement is true from a few examples rather than proving it.
- Losing marks for missing intermediate working even when the final answer is right.
Worked example
A rectangle has perimeter 40 cm. Show that the area is maximised when it is a square. Let sides be x and 20 − x. Area A = x(20 − x) = 20x − x². Then dA/dx = 20 − 2x = 0 gives x = 10, and d²A/dx² = −2 < 0, a maximum. Both sides are 10 cm, so the rectangle is a square with area 100 cm².
Rule of thumb
After solving, reread the question: check what was asked for, whether units match, and whether the value is physically sensible.