A-level exam technique Key command words Exact: Indicates that answers should not be rounded (e.g. 𝜋 rather than 3.14159). Analytic working is required rather than direct calculation, though note Casio scientific calculators often give exact answers. Hence: Indicates that the previous part of the question should be used. (Hence or otherwise is also a big hint that the previous part of the question is useful) Find, Solve, Calculate: Working may be necessary to answer the question but no justification required. A solution could be obtained by efficient use of a calculator. Give, State, Write down: Neither working nor justification is required.

Prove, Show that, Verify, Determine, Detailed Reasoning Each of the above indicate that working and justification are required. Prove: requires a high level of mathematical detail, clearly defining variables and always giving a concise conclusion. Show that: Explanation has to be sufficiently detailed to cover every step of the working, including the conclusion. Verify: A little less than above, uses a clear substitution of a given value to justify the statement. Determine: Justification should be given for any results, including working where appropriate. Steps may involve using a calculator to find numerical solutions. “In this question you must show detailed reasoning”: A full solution that leads to a conclusion, showing a detailed and complete analytical method. Note, this command word means you cannot use GDC as a method step (though it might still be useful for checking your answer).

Answer the following questions according to the command words Verify that 𝑒 4 is a root of ln 𝑥 4 =12 ln 𝑥 2 +64 (3) Find the roots of ln 𝑥 4 =12 ln 𝑥 2 +64 (3) Find the exact roots of ln 𝑥 4 =12 ln 𝑥 2 +64 (5) In this question you must show detailed reasoning Find all roots of ln 𝑥 4 =12 ln 𝑥 2 +64 (6)

You can do this! Clear mathematical working. Careful use of notation. Some “free” marks using common sense. Otherwise lots of practice!

Integration Solve each of the following integrals: 𝑥+1 𝑥+2 𝑑𝑥 𝑥 𝑒 𝑥 2 𝑑𝑥 𝑥 3 ln 𝑥 𝑑𝑥 𝑒 4𝑥 1−𝑒 2𝑥 𝑑𝑥

Bonus problems