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MathidiaMathematics Academy
Further Mathematics synthesis
Lesson 75 of 75
Review

Further Mathematics review

Objective. Demonstrate complete high-attainment mathematical fluency.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

A Further Mathematics review draws together complete fluency across complex numbers, matrices, calculus, series, mechanics, statistics and discrete mathematics, demonstrating the ability to select, combine and justify the correct techniques for any given problem without being told which topic it belongs to. High-attainment work at this stage shows not just correct answers but efficient method selection, clear justification of each step, and accurate handling of exact values, including surds, fractions and complex numbers, without unnecessary rounding. For example, recognising that a problem phrased about rotations might require complex number multiplication or matrix transformations, and choosing whichever is more efficient, is a hallmark of this level of fluency.

Worked example

A review question asks for the exact value of cos(3θ) in terms of cos θ, using whichever Further Mathematics technique is most efficient. Derive this using De Moivre's theorem.

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Illustrated card showing 3
  1. 1By De Moivre's theorem, (cos θ + i sin θ)³ = cos(3θ) + i sin(3θ)
  2. 2Expand the left side using the binomial theorem: cos³θ + 3cos²θ(i sin θ) + 3cos θ(i sin θ)² + (i sin θ)³
  3. 3Collect real parts: cos³θ − 3cos θ sin²θ (since i²=−1 and i³=−i affect only imaginary terms)
  4. 4Substitute sin²θ = 1−cos²θ: cos³θ − 3cos θ(1−cos²θ) = 4cos³θ − 3cos θ

Answer: cos(3θ) = 4cos³θ − 3cos θ

Concept mastery

Based on your past attempts at this lesson — the question types to practise again come first.

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Practice

Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.

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Common mistake. Students revert to the most recently practised topic's method even when a problem is more naturally suited to a different technique learned earlier in the course. Always reconsider the full toolkit of methods before committing to one, especially under exam conditions.

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