AQA A-Level Mathematics Paper 3, June 2025: Question 5

2 marks · Easy difficulty · Short Answer

Shade the region R defined by the inequalities (x - 3)(x + 4) ≤ y ≤ 2x and x ≤ 0 on the given diagram.

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Question

A Cartesian coordinate plane showing the straight line y = 2x passing through the origin and a quadratic curve y = (x - 3)(x + 4) with x-intercepts at -4 and 3. The line and curve intersect at two points, one in the third quadrant and one in the first quadrant. The question instructs to shade region R defined by (x - 3)(x + 4) ≤ y ≤ 2x and x ≤ 0.

Mark scheme

Show the mark scheme Mark scheme for question 5 showing a method mark M1 for shading one, two, or all three of regions I, II, or III where region I is the enclosed region for x > 0, region II is below the x-axis to the right of the y-axis, and region III is the enclosed area below the line y = 2x, above the curve, and to the left of the y-axis (x ≤ 0). An accuracy mark A1 is awarded for shading the correct region III only.

How to answer it

Shading Graphical Inequalities in 2D

📋 What This Question Tests
  • Simultaneous non-linear inequalities: Understanding compound inequalities involving quadratic curves and straight lines.
  • Boundary identification: Recognising that y ≥ f(x) defines regions on or above a curve, while y ≤ g(x) defines regions on or below a line.
  • Vertical boundaries: Interpreting domain restrictions such as x ≤ 0 (the y-axis and everything to its left).
  • Intersection of regions: Shading only the precise overlapping region satisfying all conditions simultaneously.

Question 5 (2 Marks Total)

Identifying and shading the region R defined by compound inequalities

📐 Step-by-Step Inequality Analysis

  1. Decompose the compound inequality:
    The statement (x - 3)(x + 4) ≤ y ≤ 2x splits into two distinct conditions:
    • Condition 1: y ≥ (x - 3)(x + 4)
    • Condition 2: y ≤ 2x
  2. Apply Condition 1 [ y ≥ (x - 3)(x + 4) ]:
    Points must lie on or above the parabola.
  3. Apply Condition 2 [ y ≤ 2x ]:
    Points must lie on or below the straight line through the origin.
  4. Combine conditions 1 and 2:
    The points satisfying both lie entirely inside the enclosed region between the parabola and the line (from the lower intersection at x = -3 to the upper intersection at x = 4 ).
  5. Apply Condition 3 [ x ≤ 0 ]:
    Restricts the region strictly to the left of, or on, the y-axis.

✅ Correct Answer & Description

The shaded region R must be:

  • Upper boundary: The line segment y = 2x from x = -3 up to the origin (0, 0) .
  • Lower/curved boundary: The arc of the parabola y = (x - 3)(x + 4) from x = -3 down through the minimum to the y-intercept at (0, -12) .
  • Right boundary: The vertical segment of the y-axis from (0, -12) up to (0, 0) .

In the mark scheme schematic, this corresponds exclusively to Region III (the third-quadrant loop to the left of the y-axis).

💡 Key Knowledge

  • Testing a point: If unsure which side of a boundary to shade, pick a test point not on the boundary. For example, test (-1, -4) :
    • Parabola: (-1 - 3)(-1 + 4) = (-4)(3) = -12 . Since -4 ≥ -12 , this is TRUE.
    • Line: 2(-1) = -2 . Since -4 ≤ -2 , this is TRUE.
    • Vertical: -1 ≤ 0 , which is TRUE.
    Thus, (-1, -4) is inside R.
  • Solid vs. dashed lines: Non-strict inequalities ( ≤ , ≥ ) include the boundary curves (drawn solid).

🧠 Exam Technique & Mark Scheme Breakdown

  • M1 (AO 1.1a) : Awarded for identifying the enclosed region between the curve and the line, shading at least one of the major bounded sections (Region I, II, or III) or combinations of them.
  • A1 (AO 1.1b) : Fully correct region shaded — Region III only. No other regions shaded or ambiguous markings left uncrossed.
  • Be neat and unambiguous: Use clear hatching or cross-hatching. If you make a mistake, cross out the incorrect shading cleanly so the examiner knows which region is your final response.

❌ Common Traps & Misconceptions

  • Shading the entire enclosed region: Forgetting the condition x ≤ 0 and shading both the positive and negative sides of the enclosed region between the line and parabola.
  • Confusing upper and lower boundaries: Mistakenly shading above the line y = 2x rather than below it.
  • Stopping at the x-axis: Incorrectly treating y ≤ 0 as a boundary when the question actually states x ≤ 0 (the vertical y-axis, not the horizontal x-axis).
  • Failing the boundary test: Shading outside the parabola in the third quadrant rather than inside the bowl.
Total: 2 Marks • Method mark (1.1a) for correctly identifying the enclosed space between the curve and the line; Accuracy mark (1.1b) for restricting precisely to x ≤ 0 .

Topics

Pure Mathematics · B: Algebra and functions

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.