AQA A-Level Mathematics Paper 2, June 2025: Question 6

6 marks · Medium difficulty · Multi-step Problem

Use the trapezium rule to approximate the area under the curve y = e^(x/2)/(x - 3) between x = 4 and x = 8, and evaluate a claim about whether the approximation is an under- or overestimate.

Practise this question

Question

Question 6 displays the graph of y = e^(x/2)/(x - 3) with a shaded area between x = 4 and x = 8. A table lists values of x from 4 to 8 in steps of 0.8 with corresponding y-values, where the value for x = 6.4 is missing. Part (a)(i) asks for the missing y-value. Part (a)(ii) asks to use the trapezium rule with 5 strips to find an approximate value for the area to five significant figures. Part (b) presents a student's improved approximation of 29.759 using 11 ordinates and asks to explain without calculation whether their claim that the exact area is greater than 29.759 is correct.
Question text

6 The curve with equation y = is shown in the diagram.

x – 3

y x

e2

y =

x – 3

O 4 8 x

The region shaded is bounded by the curve, the x‑axis, and the lines x = 4 and x = 8

The trapezium rule with six ordinates (5 strips) is to be used to find an approximate

value for the area of the shaded region.

Some of the values required to obtain this approximation are shown in the table below.

x 4 4.8 5.6 6.4 7.2 8

y 7.3891 6.1240 6.3249 8.7139 10.9196

6 (a) (i) Find the y‑value that is missing from the table.

[1 mark]

6 (a) (ii) Use the trapezium rule with six ordinates (5 strips) to find an approximate value for the

area of the shaded region.

Give your answer to five significant figures.

[3 marks]

(08)

6 (b) A student finds an improved approximation for the area of the shaded region by using

the trapezium rule with 11 ordinates.

The student correctly obtains 29.759 as their improved approximation.

The student claims that the exact area must be greater than 29.759

Without further calculation, explain whether or not the student is correct.

[2 marks]

Mark scheme

Show the mark scheme Mark scheme for Question 6: 6(a)(i) awards B1 for 7.2155 (AWFW [7.2154, 7.2155]). 6(a)(ii) awards B1 for h = 0.8, M1 for substituting all ordinates into the trapezium rule formula, and A1 for 30.026. 6(b) awards E1 for stating that the curve is convex / concave upwards (or d²y/dx² > 0) and E1 for explaining that the trapezium rule gives an overestimate, deducing that the student is incorrect.

Q Marking instructions AO Marks Typical solution

6(a)(i) Obtains AWFW [7.2154, 7.2155] 1.1b B1 7.2155

May be seen in the table

Subtotal 1

6(a)(ii) States or uses h = 0.8 OE 1.1b B1

8 − 4

h =

Substitutes all the given y 1.1a M1 5

values and their answer from 0 8. 7 3891.+10 9196. +

A =

6(a)(i) to achieve 2 2 6 1240(. + 6 3249. + 7 2155. + 8 7139.)

7 3891.+10 9196. +

= 30 026.

2(6 124. + 6 3249. + 7 2155. + 8 7139.)

FT their 7.2155

PI 75.0653

Accept correct exact values or

values to more than 4 decimal

places.

Condone omission of the

bracket after 8.7139

Obtains AWRT 30.026 1.1b A1

Subtotal 3

6(b) States that the curve is convex 2.4 E1 The curve is convex, so the trapezium

or concave upwards rule gives an over-estimate.

Accept concave up Therefore, the student is incorrect.

Or

d2 y

States that 2 > 0

dx

Explains that the trapezium rule 2.4 E1

gives an over-estimate

and

deduces that the student is

incorrect

Subtotal 2

Question 6 Total 6

How to answer it

The Trapezium Rule & Curve Nature

AQA A-Level Mathematics • Numerical Methods • Integration

What this question tests

This question assesses your ability to apply numerical integration techniques and critically evaluate approximations:

  • Evaluating exponential/rational functions accurately at specified points.
  • Applying the Trapezium Rule formula with correct strip width ( h ) and ordinates.
  • Rounding to a specified level of accuracy (5 significant figures).
  • Understanding curve concavity/convexity ( d²y/dx² > 0 ) and explaining whether an approximation gives an underestimate or overestimate.

Part (a)(i) — Missing Ordinate Value

Calculate the missing y-value in the table when x = 6.4 [1 Mark]

📐 Calculations

The curve has equation y = e^(x/2) / (x − 3) .

  1. Substitute x = 6.4 :
    y = e^(6.4 / 2) / (6.4 − 3)
  2. Simplify numerator and denominator:
    y = e^(3.2) / 3.4
  3. Evaluate:
    y = 24.53253... / 3.4 = 7.21545...
  4. Round to match the table (4 decimal places):
    y = 7.2155

✅ Correct Answer & Mark Scheme

y = 7.2155 (accept any value in range [7.2154, 7.2155] )

[B1] Awarded for correct value within tolerance. May be seen written in the table or working.

❌ Common Errors

  • Premature rounding: Rounding e^(3.2) to 24.5 before dividing, giving 7.2059 (inaccurate).
  • Bracket errors on calculator: Typing e^6.4/2/(6.4 - 3) without appropriate grouping.

Part (a)(ii) — Trapezium Rule Approximation

Approximate the shaded area to 5 significant figures using 5 strips [3 Marks]

💡 Key Knowledge

The Trapezium Rule is given by:

Area ≈ (h / 2) × [ y₀ + yₙ + 2(y₁ + y₂ + ... + yₙ₋₁) ]

  • Number of strips n = 5
  • Number of ordinates n + 1 = 6
  • Strip width: h = (b − a) / n = (8 − 4) / 5 = 0.8

📐 Step-by-Step Calculation

  1. Strip width:
    h = (8 − 4) / 5 = 0.8
  2. Sum the end ordinates (y₀ + y₅):
    7.3891 + 10.9196 = 18.3087
  3. Sum the interior ordinates (y₁ to y₄):
    6.1240 + 6.3249 + 7.2155 + 8.7139 = 28.3783
  4. Combine using formula:
    Area ≈ (0.8 / 2) × [ 18.3087 + 2(28.3783) ]
    Area ≈ 0.4 × [ 18.3087 + 56.7566 ] = 0.4 × 75.0653
    Area ≈ 30.02612
  5. Round to 5 significant figures:
    Area ≈ 30.026

✅ Mark Scheme Breakdown

  • B1: Stating or using h = 0.8 (or equivalent).
  • M1: Correct substitution of all y-values into the formula:
    0.8/2 × [ 7.3891 + 10.9196 + 2(6.1240 + 6.3249 + 7.2155 + 8.7139) ] (follow through their 7.2155).
  • A1: Final answer of 30.026 (AWRT). Must be 5 significant figures.

❌ Common Errors & Pitfalls

  • Incorrect h: Using h = (8 − 4) / 6 = 0.667 by dividing by the number of ordinates instead of the number of strips.
  • Ignoring accuracy requirements: Giving 30.03 (4 s.f.) or 30.0 (3 s.f.) loses the final A1 mark.
  • Mispairing y-values: Doubling all ordinates or forgetting to multiply interior ordinates by 2.

Part (b) — Evaluation of Approximation (Over/Underestimate)

Explain whether the student's claim that exact area > 29.759 is correct [2 Marks]

💡 Key Mathematical Principle

Whether the trapezium rule overestimates or underestimates depends on the curvature:

  • Convex curve (concave upwards / d²y/dx² > 0): The straight line chords connecting points lie above the curve. The trapezium rule produces an over-estimate.
  • Concave curve (concave downwards / d²y/dx² < 0): Chords lie below the curve. The rule produces an under-estimate.

🧠 Exam Technique & Structure

To gain both explanation marks, your reasoning must follow a strict two-step deductive chain:

  1. Describe the shape: State that the curve is convex (or concave upwards, or d²y/dx² > 0 ).
  2. Deduce and conclude: State that the trapezium rule gives an over-estimate, which means the exact area is less than 29.759. Therefore, the student is incorrect.

✅ Model Answer

The curve is convex (concave upwards). Therefore, the straight chords lie above the curve, so the trapezium rule gives an over-estimate of the true area.

This means the exact area must be less than 29.759. Consequently, the student's claim is incorrect.

[E1] Stating the curve is convex / concave upwards / concave up / d²y/dx² > 0 .
[E1] Explaining that the approximation is an over-estimate and concluding the student is incorrect.

❌ Examiner Commentary & Lost Marks

  • Saying "increasing": Mentioning that the curve is "increasing" or has a minimum does NOT explain why it over-estimates. The crucial property is convexity/curvature.
  • Confusing concave vs convex: Calling it "concave" without specifying "upwards" loses the first mark under AQA conventions.
  • Missing the conclusion: Explaining that it is an overestimate but omitting the explicit deduction that "the student is incorrect".

Topics

Pure Mathematics · I: Numerical methods

Question and mark scheme from the AQA A-Level Mathematics examination, Paper 2, June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.