What makes a question "hard"
Hard questions often require: (1) reading carefully for command words like "prove", "hence" or "show that"; (2) combining two or more topics (e.g. calculus and trigonometry); (3) knowing exactly how much working to show. Top students break the question into steps, state what they’re doing, and do a quick sanity check at the end.
Example: proof question
Prove that 7ⁿ − 3ⁿ is divisible by 4 for all positive integers n.
Approach
Use mathematical induction. Base case n=1: 7−3=4 ✓. Assume 7ᵏ−3ᵏ is divisible by 4. For 7ᵏ⁺¹−3ᵏ⁺¹, write as 7·7ᵏ−3·3ᵏ and subtract a multiple of (7ᵏ−3ᵏ) to leave a term clearly divisible by 4. Conclude with the standard induction wording.
Exam tips for hard questions
- • Read the question twice and underline command words.
- • If stuck, try working backwards from the required result or check the formula booklet.
- • "Hence" means you must use the previous part; "hence or otherwise" means you can use another method.