A Balanced Yet Demanding Paper

AQA's June 2022 Paper 1 for AS Further Mathematics provides a thorough test of core pure competencies. Heavy emphasis on rational functions, algebraic manipulation, and coordinate systems marks this as a standard-to-challenging paper. While early questions offer quick marks, the later multi-step questions demand a high level of rigor and structured problem-solving.

Where the Marks are Won and Lost

The absolute core of this paper lies in Further Algebra and Functions, which makes up more than 35% of the total marks. In particular, rational graph sketching, finding asymptotes, and analyzing quadratic denominators for specific asymptote behaviors (as in Question 14) are high-tariff areas. Students who master discriminants and quadratic theory in the context of functions can easily secure these marks. On the other hand, marks are frequently lost on the Proof by Induction question due to poor communication of the inductive hypothesis or failure to write a complete concluding statement. Loci problems in Complex Numbers also saw significant mark drops where geometric reasoning was required to find maximum bounds.

Examiner Pitfalls to Avoid

  • Induction Rigor: Always state the base case explicitly, assert the inductive hypothesis clearly for \( n=k \), show the step for \( n=k+1 \), and finish with a complete mathematical induction conclusion.
  • Tricky Hyperbolic Relationships: In hyperbolic quadratic equations, remember that the sum of the roots of the quadratic equation in \( \sinh \theta \) is \( \sinh \theta_1 + \sinh \theta_2 \), not \( \theta_1 + \theta_2 \). You must solve for \( \theta \) first using logarithmic forms before summing.
  • Geometric Loci: For maximum value of \( |w| \) questions, use exact geometric trigonometry on the Argand diagram rather than guessing algebraic inequalities.

Strategy & Prediction for Next Series

Given the heavy algebraic focus of this series, expect future papers to rebalance with a stronger emphasis on Further Calculus, particularly integration involving hyperbolic substitutions or areas under polar curves. Ensure you are highly fluent with matrix transformations and vector intersection techniques, as these represent stable, recurring marks.