October 2020 Paper 3 Q9

OCR MEICurrent spec3 marksProof

9

The questions in this section refer to the article on the Insert. You should read the article before attempting the questions.

The relevant parts of the article “Which is bigger?” are reproduced below; the line numbers are those printed on the Insert.

Lines 4–9
It is often helpful in mathematics to consider simpler examples. It is easy to work out that \(3^4 > 4^3\). In the expression \(3^4\), 3 is the base and 4 is the exponent. Working with integers greater than 1, it is easy to find many examples where \(a^b > b^a\) if \(a < b\). That is, using the smaller base and the larger exponent gives the larger result. This might lead us to conjecture that \(a^b > b^a\) if \(a < b\) and both \(a\) and \(b\) are integers greater than 1. However, it is also possible to find counter examples to this conjecture.

(a) Show that if \(a = 1\) and \(b > 1\) then \(a^b < b^a\). [2]
(b) Find integer values of \(a\) and \(b\) with \(b > a > 1\) and \(a^b\) not greater than \(b^a\) (a counter example to the conjecture given in lines 7–8). [1]
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