AQA A-Level Mathematics Paper 1, June 2025: Question 4

2 marks · Easy difficulty · Short Answer

Determine the validity range and the coefficient of the linear term for the binomial expansion of (1 - 8x)^(1/2).

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Question

Question 4 states: The first three terms, in ascending powers of x, of the binomial expansion of (1 - 8x)^(1/2) are 1 + nx - 8x^2, where n is a constant. Part (a) asks to state the range of values of x for which the expansion is valid by circling an answer from: |x| > -8, |x| > -1/8, |x| < 1/8, |x| < 8. Part (b) asks to state the value of the constant n by circling an answer from: -16, -4, 1/2, 4. Each part is worth 1 mark.
Question text

4 The first three terms, in ascending powers of x, of the binomial expansion

of (1 – 8x)2 are

1 + nx – 8x2

where n is a constant.

4 (a) State the range of values of x for which the expansion is valid.

Circle your answer.

[1 mark]

│x│> –8 │ x│> – │ x│< │x│< 8

4 (b) State the value of the constant n

Circle your answer.

[1 mark]

–16 –4 4

Mark scheme

Show the mark scheme Mark scheme for Question 4. Part 4(a): Circles third option, AO 1.1b, mark B1, typical solution |x| < 1/8. Part 4(b): Circles second option, AO 1.1b, mark B1, typical solution -4. Total 2 marks.

Q Marking instructions AO Marks Typical solution

4(a) Circles third option 1.1b B1 1

x <

Subtotal 1

4(b) Circles second option 1.1b B1 –4

Subtotal 1

Question 4 Total 2

How to answer it

Binomial Expansion: Validity & Linear Coefficient

📌 What this question tests

This question assesses your understanding of the infinite binomial series expansion for rational powers, specifically:

  • Finding the interval of validity for an expansion of the form (1 + kx)p where p is not a positive integer.
  • Applying the formula (1 + X)n = 1 + nX + ... to find the coefficient of the linear x term.
  • Handling negative signs accurately inside bracketed terms.

Part (a): Validity of the Expansion

State the range of values of x for which the expansion is valid

✅ Correct Answer

Circle the third option:

|x| < 1/8

Mark: 1 mark (B1, AO 1.1b)

💡 Key Knowledge

  • An infinite binomial expansion of (1 + X)p (where p ∉ ℕ) converges if and only if:
    |X| < 1
  • Here, the substitution is X = −8x.
  • Since |−8x| = 8|x|, the inequality becomes:
    8|x| < 1 ⇒ |x| < 1/8

📐 Step-by-Step Breakdown

  1. Identify the variable term: In (1 − 8x)1/2, the term added to 1 is (−8x).
  2. Apply the condition: |−8x| < 1.
  3. Simplify the modulus: |−8| × |x| < 1 ⇒ 8|x| < 1.
  4. Divide by 8: |x| < 1/8 (or −1/8 < x < 1/8).

❌ Common Errors

  • Confusing < and >: Choosing |x| > −1/8. A modulus cannot be less than 0, let alone compared as > negative for convergence.
  • Multiplying instead of dividing: Selecting |x| < 8 by writing |x| < 1 × 8.
  • Keeping the negative sign: Incorrectly stating |x| < −1/8. Modulus quantities are strictly non-negative.

Part (b): Finding the Constant n

State the value of the constant n

✅ Correct Answer

Circle the second option:

−4

Mark: 1 mark (B1, AO 1.1b)

💡 Key Knowledge

For any real index p:

(1 + X)p = 1 + pX + [p(p − 1)/2!]X² + ...

Comparing terms in ascending powers:

  • Constant term: 1
  • Linear term: pX

📐 Step-by-Step Calculation

  1. Identify parameters: Power p = 1/2, and inner term X = −8x.
  2. Write the first two terms:
    (1 − 8x)1/2 = 1 + (1/2)(−8x) + ...
  3. Multiply:
    (1/2) × (−8x) = −4x
  4. Match with given expansion:
    Given form is 1 + nx − 8x²
    ⇒ nx = −4x, so n = −4.

🧠 Exam Technique & Sanity Check

  • Check the sign: The expression is (1 minus 8x), so the linear term must carry a negative sign. This immediately eliminates 1/2 and 4.
  • Check the x² term to confirm method:
    Term 3 = [(1/2)(−1/2) / 2] × (−8x)²
    = (−1/8) × 64x² = −8x².
    This perfectly matches the given −8x² in the question stem!
  • Sign slip trap: Selecting +4 is the most common student error due to dropping the minus sign inside (−8x).

Topics

Pure Mathematics · D: Sequences and series

Question and mark scheme from the AQA A-Level Mathematics examination, Paper 1, June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.