A-Level · Mathematics · AQA · Mark scheme decoded

AQA A-Level Mathematics: Iterative Methods and Newton-Raphson — mark scheme explained

Machine-verifiedchecked against the AQA A-Level Mathematics specificationlast verified 3 July 2026

The short answer

In AQA A-Level Mathematics, the topic of numerical methods includes solving equations approximately using simple iterative methods. This involves understanding how to use recurrence relations and graphical techniques like cobweb and staircase diagrams. Additionally, you will learn about the Newton-Raphson method, a powerful technique for finding roots of equations, and understand its limitations.

The question

Use the simple iterative method to solve the equation x 3 - 2x - 5 = 0. Start with an initial guess of x 0 = 2 and iterate until the difference between successive terms is less than 0.001.

[Paraphrased for study — not reproduced from any exam paper.]

5 marks

Mark scheme, decoded

How the examiner actually awards the marks on this topic.

Gradora's own decode of the marking approach — not the exam board's published mark scheme.

How marks are awarded

For questions involving iterative methods, marks are typically awarded for correct application of the recurrence relation, accurate plotting of diagrams, and verification of convergence. Ensure each step is clearly shown and labeled.

What the command words demand

solve: use an iterative method to find the solution to an equation
draw: create a cobweb or staircase diagram for an iterative process
iterate: perform multiple steps of an iterative method until convergence is achieved
converge: determine if a sequence of approximations converges to a solution

Model answer

A full-mark response to the question above, worked through step by step.

Timing: Allocate about 10-15 minutes for a question on this topic, depending on its complexity. Practice solving problems within this time frame to improve speed and accuracy.

Rewrite the equation as x = g(x): x = (2x + 5) 1/30 marks
Use the recurrence relation: x n+1 = (2x n + 5) 1/31 mark
Start with x 0 = 20 marks
Calculate x 1 : x 1 = (2 × 2 + 5) 1/3 ≈ 2.0801 mark
Calculate x 2 : x 2 = (2 × 2.080 + 5) 1/3 ≈ 2.0941 mark
Calculate x 3 : x 3 = (2 × 2.094 + 5) 1/3 ≈ 2.0960 marks
Check the difference: |x 3 - x 2 | = |2.096 - 2.094| = 0.002 (not less than 0.001)1 mark
Calculate x 4 : x 4 = (2 × 2.096 + 5) 1/3 ≈ 2.0970 marks
Check the difference: |x 4 - x 3 | = |2.097 - 2.096| = 0.001 (less than 0.001)1 mark
The sequence converges to approximately x ≈ 2.0970 marks

Final answer: x ≈ 2.097

Work through every step correctly and you earn all 5 marks.

Another worked example

Use the Newton-Raphson method to solve the equation x 3 - 2x - 5 = 0. Start with an initial guess of x 0 = 2 and iterate until the difference between successive terms is less than 0.001.

6 marks

Use the Newton-Raphson formula: x n+1 = x n - (x n 3 - 2x n - 5) / (3x n 2 - 2)1 mark
Start with x 0 = 20 marks
Calculate x 1 : x 1 = 2 - (2 3 - 2 × 2 - 5) / (3 × 2 2 - 2) ≈ 2.09462 marks
Calculate x 2 : x 2 = 2.0946 - (2.0946 3 - 2 × 2.0946 - 5) / (3 × 2.0946 2 - 2) ≈ 2.0945512 marks
Check the difference: |x 2 - x 1 | = |2.094551 - 2.0946| = 0.000049 (less than 0.001)1 mark
The sequence converges to approximately x ≈ 2.0945510 marks

Final answer: x ≈ 2.094551

Work through every step correctly and you earn all 6 marks.

Common mistakes

Using the wrong recurrence relation for a given equation.
Why it happens: Students may incorrectly rearrange the equation or use an incorrect form of g(x).
Fix: Double-check the rearrangement and ensure it is in the correct form x = g(x).
Not checking for convergence criteria.
Why it happens: Students may stop iterating too early or continue unnecessarily without verifying if the sequence has converged.
Fix: Always check the difference between successive terms to ensure it is below a specified tolerance level.
Using an inappropriate initial guess.
Why it happens: An initial guess that is too far from the root can lead to divergence or slow convergence.
Fix: Choose an initial guess close to the expected root, possibly by sketching the function or using a calculator.
Failing to recognize when the Newton-Raphson method fails.
Why it happens: Students may not be aware of the conditions under which the Newton-Raphson method can fail, such as stationary points or multiple roots.
Fix: Understand the limitations of the method and check for these conditions before applying it.
Incorrectly plotting cobweb or staircase diagrams.
Why it happens: Students may draw lines incorrectly, leading to a misinterpretation of the iterative process.
Fix: Practice drawing these diagrams step by step and ensure each line is drawn accurately according to the rules.
Not using the derivative correctly in the Newton-Raphson method.
Why it happens: Students may forget to calculate or use the derivative of the function, leading to incorrect results.
Fix: Always compute and use the correct derivative when applying the Newton-Raphson formula.
Failing to check for multiple roots in the Newton-Raphson method.
Why it happens: Students may not realize that the method can converge to a different root than intended if there are multiple roots.
Fix: Plot the function and identify all possible roots before applying the method. Choose an initial guess close to the desired root.
Not understanding the difference between cobweb and staircase diagrams.
Why it happens: Students may confuse the two types of diagrams, leading to incorrect visualizations.
Fix: Understand that a cobweb diagram uses vertical and horizontal lines alternately, while a staircase diagram uses steps. Practice drawing both types of diagrams.

Where the marks go

The question types you’ll meet on this topic and the marks each one carries — so you know what to expect and where to focus.

Question type	What you’re asked to do	Marks
Iterative Root Finding	Apply an iterative recurrence relation repeatedly until successive terms converge within a given tolerance.	5
Newton-Raphson Iteration	Apply the Newton-Raphson recurrence from a starting value and iterate until successive terms converge.	6
Cobweb Diagram	Draw a cobweb diagram showing at least three iterations of the given recurrence from x0 = 2.	4
Staircase Diagram	Draw a cobweb or staircase diagram to illustrate iterations converging to a root.	4
Total across these question types		19

Question types and mark tariffs are Gradora’s guidance based on how this topic is typically examined — not the board’s official paper structure.