AQA A-Level Mathematics Paper 3, June 2025: Question 6

4 marks · Easy difficulty · Multi-step Problem

Find the coefficients in a binomial expansion of (2 - 3x)^5 and use it to calculate an approximation for 1.94^5.

Practise this question

Question

Question 6 states that the first four terms in ascending powers of x of the binomial expansion of (2 - 3x)^5 are given by 32 + px + qx^2 - 1080x^3, where p and q are constants. Part (a) asks to find the value of p and the value of q for 2 marks. Part (b) asks to hence find an approximation for 1.94^5, giving the answer to five decimal places, for 2 marks.

Mark scheme

Show the mark scheme Mark scheme for Question 6. For 6(a), M1 is awarded for using the binomial coefficient formula to find the x or x^2 term, and A1 is awarded for obtaining p = -240 and q = 720. For 6(b), M1 is awarded for substituting x = 0.02 into the expansion with their p and q, and A1 is awarded for obtaining the correct approximation 27.47936.

How to answer it

Binomial Expansion & Decimal Approximation

WHAT THIS QUESTION TESTS

This question assesses your mastery of pure algebra skills in Year 1 / AS Mathematics:

  • Expanding a binomial expression of the form (a + bx)ⁿ where n is a positive integer.
  • Correctly dealing with negative signs and bracketed powers such as (-3x)².
  • Using the command word "Hence" to evaluate a numerical power approximation by equating base terms.
  • Maintaining accuracy to a specified number of decimal places without premature rounding.
PART (a)

Finding Coefficients p and q

Expansion of (2 - 3x)⁵ in ascending powers of x (2 Marks)

💡 Key Knowledge

The general binomial formula for a positive integer n is:

(a + b)ⁿ = aⁿ + ⁿC₁ aⁿ⁻¹b + ⁿC₂ aⁿ⁻²b² + ...

Here, identify the components carefully:

  • a = 2
  • b = -3x (ensure the negative sign is included!)
  • n = 5

📐 Step-by-Step Calculation

1 Term in x (find p):

⁵C₁ × (2)⁴ × (-3x)¹

= 5 × 16 × (-3x) = -240x  &implies;  p = -240

2 Term in x² (find q):

⁵C₂ × (2)³ × (-3x)²

= 10 × 8 × (9x²) = 720x²  &implies;  q = 720

3 Spot Check with Given Term:
⁵C₃ × (2)² × (-3x)³ = 10 × 4 × (-27x³) = -1080x³. This matches the question, confirming the signs and powers are set up correctly.

✅ Correct Answers & Mark Allocation

p = -240

q = 720

[M1] Method mark for attempting either ⁵C₁ × 2⁴ × (-3x) or ⁵C₂ × 2³ × (-3x)² (seen with or without x).
[A1] Accuracy mark for obtaining both correct values: p = -240 and q = 720.

❌ Common Errors to Avoid

  • Bracket Neglect: Writing -3x² instead of (-3x)² = +9x², resulting in q = -240 or -720.
  • Lost Negative Sign: Forgetting the minus sign on the x term, giving p = +240.
  • Leaving the Variable: Writing p = -240x and q = 720x². While examiners condoned this here, coefficients are constants, so strictly omit x.
PART (b)

Approximating 1.94⁵

Substitution and Decimal Precision (2 Marks)

🧠 Exam Technique: "Hence"

  • The word "Hence" means you must use your expansion from part (a).
  • Simply typing 1.94⁵ into your calculator gives 27.4793574... but earns 0 marks if the substitution is not shown!
  • Equate bases to find x:
    2 - 3x = 1.94
    3x = 2 - 1.94 = 0.06
    x = 0.02

📐 Step-by-Step Calculation

1 Substitute x = 0.02 into the expansion:

32 + px + qx² - 1080x³

= 32 - 240(0.02) + 720(0.02)² - 1080(0.02)³

2 Evaluate each term separately:

  • Constant term: 32
  • Linear term: -240(0.02) = -4.8
  • Quadratic term: 720(0.0004) = +0.288
  • Cubic term: -1080(0.000008) = -0.00864

3 Sum terms:

32 - 4.8 + 0.288 - 0.00864 = 27.47936

✅ Correct Answer

1.94⁵ ≈ 27.47936

[M1] Substituting x = 0.02 into the expansion using their values of p and q.
[A1] Correct Answer Only (CAO): 27.47936 (must be stated to 5 d.p.).

💡 Examiner Insights & Tips

  • Exact vs Rounded: The evaluated sum terminates naturally at 5 decimal places. Do not add trailing zeroes unless required, and do not round prematurely!
  • ISW (Ignore Subsequent Working): Once the correct value of 27.47936 is seen, unnecessary further rounding won't penalise you, but always write the exact 5 d.p. value first.

Topics

Pure Mathematics · D: Sequences and series

Question and mark scheme from the AQA A-Level Mathematics examination, Paper 3, June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.