AQA A-Level Mathematics Paper 1, June 2025: Question 8

5 marks · Medium difficulty · Multi-step Problem

Show that for small values of x, the trigonometric rational function can be approximated by a linear equation of the form y = ax + 1.

Practise this question

Question

Question 8 asks to show that, for small values of x, the graph with equation y = (4 + sin 5x - 3 cos 2x) / (1 + 2 tan x) can be approximated by the straight line with equation of the form y = ax + 1, where a is a constant to be found. Total marks: 5.
Question text

8 Show that, for small values of x, the graph with equation

4 + sin 5x – 3cos 2x

y =

1 + 2tan x

can be approximated by the straight line with equation of the form

y = ax + 1

where a is a constant to be found.

[5 marks]

Mark scheme

Show the mark scheme Mark scheme for Question 8 detailing 5 marks: M1 for replacing sin 5x with 5x or tan x with x; M1 for replacing cos 2x with 1 - (2x)^2 / 2; A1 for obtaining the complete unsimplified algebraic fraction (4 + 5x - 3(1 - 2x^2)) / (1 + 2x); M1 for using a valid method such as factorising (6x^2 + 5x + 1) to (2x + 1)(3x + 1) to simplify the fraction; R1 for a fully reasoned conclusion obtaining y = 3x + 1.

Q Marking instructions AO Marks Typical solution

8 Replaces sin 5x with 5x

Or 2

(2x)

Replaces tan x with x 1.1a M1 4 + 5x − 3 1 −

4 + sin5x − 3cos2x 2

In

1+ 2 tanx 1+ 2x

(2x) 4 + 5x − 3 1(− 2x )

Obtains 3 1− =

2 1+ 2x

1.1a M1

6x2 + 5x +1

2x2

Condone 3 1− =

1+ 2x

(2x +1)(3x +1)

2 =

(2x) 1+ 2x

4 +5x −3 1 −

2 = 3x +1

1.1b A1

Obtains

1+ 2x

OE y = 3x +1

Uses a valid method to simplify

their fraction to a linear form.

May see evidence of factorising, 3.1a M1

algebraic division or comparing

coefficients.

Completes a reasoned

argument to obtain y = 3x +1 2.1 R1

How to answer it

Small Angle Approximations with Algebraic Simplification

📋 What this question tests

This question assesses your ability to apply trigonometric small angle approximations (when θ is small and measured in radians: sin θ ≈ θ, cos θ ≈ 1 − θ²/2, tan θ ≈ θ) to compound arguments (such as 5x and 2x), expand and collect like terms to form a quadratic expression, and simplify a rational algebraic function into a linear form using polynomial factorisation or algebraic long division.

Question 8 (5 Marks)

Linear Approximation of a Trigonometric Fraction

AQA A-Level Mathematics • Pure Mathematics • Trigonometry & Algebra

💡 Key Knowledge

  • For small angles in radians:
    • sin θ ≈ θ ⇒ sin(5x) ≈ 5x
    • cos θ ≈ 1 − θ²/2 ⇒ cos(2x) ≈ 1 − (2x)²/2 = 1 − 2x²
    • tan θ ≈ θ ⇒ tan(x) ≈ x
  • Always replace θ completely with brackets around multiple terms (e.g. (2x)² = 4x², not 2x²).
  • Simplifying rational functions: factorise the quadratic numerator to find a factor that cancels with the linear denominator.

📐 Step-by-Step Calculation

  1. Substitute small angle formulas:
    y ≈ [4 + 5x − 3(1 − (2x)²/2)] / [1 + 2(x)]
  2. Simplify the cosine term:
    1 − (2x)²/2 = 1 − 4x²/2 = 1 − 2x²
    −3(1 − 2x²) = −3 + 6x²
  3. Collect terms in numerator:
    Numerator = 4 + 5x − 3 + 6x² = 6x² + 5x + 1
    y ≈ (6x² + 5x + 1) / (2x + 1)
  4. Factorise the quadratic:
    6x² + 5x + 1 = (2x + 1)(3x + 1)
  5. Cancel common factor:
    y ≈ [(2x + 1)(3x + 1)] / (2x + 1) = 3x + 1

✅ Correct Answer & Mark Breakdown

The straight line equation is:
y = 3x + 1
(where constant a = 3).

M1 (1.1a): Replaces sin 5x with 5x OR replaces tan x with x.
M1 (1.1a): Obtains 3(1 − (2x)²/2) or condones 3(1 − 2x²/2).
A1 (1.1b): Obtains fully correct unsimplified fraction: [4 + 5x − 3(1 − (2x)²/2)] / [1 + 2x].
M1 (3.1a): Uses a valid method to simplify fraction to linear form (factorisation, polynomial division, or equating coefficients).
R1 (2.1): Completes a reasoned argument showing full working to conclude y = 3x + 1.

❌ Common Traps & Misconceptions

  • Bracket Squaring Error: Writing cos(2x) ≈ 1 − 2x²/2 instead of 1 − (2x)²/2. Squaring only x instead of (2x) leads to 1 − x², corrupting the entire numerator quadratic.
  • Sign Errors: Mishandling the negative expansion −3(1 − 2x²). It expands to −3 + 6x², NOT −3 − 6x².
  • Illegal Cancelling: Trying to cancel terms individually before factorising (e.g. dividing 2x in the denominator into terms in the numerator).
  • Missing the Final Statement: Reaching 3x + 1 without writing the complete equation y = 3x + 1 or stating a = 3.

🧠 Exam Technique & Examiner Commentary

  • "Show that" questions require complete rigor: Because the target form y = ax + 1 gives away the constant term (+1), examiners do not award the final reasoning mark (R1) if intermediate steps are missing. Show the expansion of brackets, the unsimplified quadratic, and the explicit factorised form (2x + 1)(3x + 1) .
  • Self-Checking Mechanism: The target is ax + 1. If your denominator is (2x + 1), you know in advance that (2x + 1) must be a factor of the numerator quadratic, and the other bracket must end in +1 so that 1 × 1 = 1. This provides an immediate sanity check on your algebra!

Topics

Pure Mathematics · E: Trigonometry · B: Algebra and functions

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.