Using simple special cases in problem-solving

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You’ve seen how mathematical problems can be solved with less, or even no, detailed calculation by looking at what the problem tells you about the shape of the solution.

Another useful method, at least for checking answers, is to look at simple special cases of the problem, or simplified versions of the problem. We saw some of that when we talked about how to check your answers in FP1 and M2.

For example, in M2 problems involving a slope with angle θ you can check whether you have cos and sin in the right places in your equations by thinking about the case θ=0.

With some of the multiple-choice questions in MAT, that sort of argument is enough to give the complete answer.

Try these. Please write in your books, not just your multiple-choice answer, but the argument you used to reach the answer. For hints click here, but only if you are stuck after ten minutes. For at least one of the problems, a symmetry argument like those in the last worksheet is just as good.

1. MAT 2011, B. A rectangle has perimeter P and area A. The values P and A must satisfy:
(a) P3 > A, (b) A2 > 2P + 1, (c) P2 ≥ 16A, (d) PA≥A+P.

2. MAT 2012, J
2012J

3. MAT 2010, B
2010B

4. MAT 2010, D
2010D

5. MAT, 2009, D
2009D

6. (not a MAT question, but same sort of thing)
Without using a calculator, find which is bigger of 99100 and 10099.

7. (a STEP problem: STEP 1 1999, Q.1)

step