AQA A-Level Mathematics: Binomial Expansion and Approximation — mark scheme explained

The question

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.

Model answer

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

⏱Timing: For a typical question involving binomial expansion, allocate about 5-7 minutes to ensure you have enough time to write out the full expansion, simplify, and check your work.

1. (3 - 2x) 4 = Σ r=0 4 (4Cr) 3 4-r (-2x) r1 mark
2. = (4C0) 3 4 (-2x) 0 + (4C1) 3 3 (-2x) 1 + (4C2) 3 2 (-2x) 2 + (4C3) 3 1 (-2x) 3 + (4C4) 3 0 (-2x) 41 mark
3. = 1 × 81 × 1 + 4 × 27 × -2x + 6 × 9 × 4x 2 + 4 × 3 × -8x 3 + 1 × 1 × 16x 41 mark
4. = 81 - 216x + 216x 2 - 96x 3 + 16x 42 marks

Final answer: 81 - 216x + 216x 2 - 96x 3 + 16x 4

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

Common mistakes

• Forgetting to include the factorial in the binomial coefficient formula.

  Why it happens: Students often memorize the formula for nCr but forget to include the factorials, leading to incorrect calculations.

  Fix: Always write out the full formula: nCr = n! / [r!(n-r)!].

• Using the wrong sign when expanding (a - bx) n .

  Why it happens: Students sometimes forget that each term in the expansion of (a - bx) n will have alternating signs.

  Fix: Write out the first few terms to see the pattern: a n , -n × a n-1 × bx, + (n(n-1)/2!) × a n-2 × (bx) 2 , etc.

• Not checking the validity condition for rational n expansions.

  Why it happens: Students may apply the binomial expansion without considering the validity of the approximation, leading to incorrect results. They also wrongly assume |x| n requires |bx/a| < 1.

  Fix: Always verify the validity condition before using the binomial expansion for rational n: for (1 + x) n it is |x| n it is |bx/a| < 1.

• Using the wrong value for 0! in calculations.

  Why it happens: Students sometimes forget that 0! is defined as 1, leading to incorrect factorial calculations.

  Fix: Remember that 0! = 1 and use this in your calculations.

• Forgetting to include the constant term when expanding (a + bx) n .

  Why it happens: Students may focus on the variable terms and forget to include the constant term, which is crucial for the expansion.

  Fix: Always start with the constant term a n and then proceed to the other terms.

• Not simplifying fractions in the binomial coefficients.

  Why it happens: Students may leave fractions unsimplified, leading to more complex calculations and potential errors.

  Fix: Simplify fractions in the binomial coefficients before substituting them into the expansion.

• Using the wrong number of terms for approximation.

  Why it happens: Students may use too few or too many terms when approximating, leading to inaccurate results.

  Fix: Use enough terms to achieve the desired level of accuracy, but not so many that it becomes overly complex.

• Forgetting to apply the binomial theorem for probability calculations.

  Why it happens: Students may use other methods or formulas instead of the binomial theorem when calculating probabilities, leading to incorrect results.

  Fix: Always use the formula P(X = r) = nCr × p r × (1-p) n-r for binomial probability calculations.

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

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