AQA AS Level Mathematics Paper 1, June 2025: Question 7

Mark scheme

Show the mark scheme

7(a) States 7 1.1b B1 7

Subtotal 1

Q Marking instructions AO Marks Typical solution

7(b) Obtains b < their radius from 2.2a M1

part (a) or b > their negative –7 < b < 7

radius from part (a), condone

non-strict inequality

Obtains –7 < b < 7 2.2a A1F

FT their radius from part (a)

Subtotal 2

Question 7 Total 3

How to answer it

Circle Equations and Axis Intersections

📌 What this question tests

Standard form of a circle: Identifying the centre (a, b) and radius r from the Cartesian equation (x − a)² + (y − b)² = r².
Geometric interpretation of intersections: Understanding that the perpendicular distance from the centre to the x-axis is |b|.
Inequalities: Determining the condition for a circle to intersect a coordinate axis at two distinct points (|b| < r).

Question 7 (a)

State the radius of the circle [1 mark]

✅ Correct Answer

Radius = 7

1 Mark: [B1] for correctly stating 7 without ambiguity.

💡 Key Knowledge

The standard equation of a circle is:

(x − a)² + (y − b)² = r²

Here, r² = 49, so r = √49 = 7.

❌ Common Errors

Writing 49 (forgetting to take the square root of r²).
Writing ±7 (radius is a geometric length and must be strictly positive: r > 0).

Question 7 (b)

Find the range of possible values for b given two distinct intersections with the x-axis [2 marks]

✅ Correct Answer

−7 < b < 7 (or |b| < 7 )

Mark Breakdown:
• [M1] (AO 2.2a): Identifying that b < 7 or b > −7 (condones non-strict inequalities like ≤ or ≥).
• [A1F] (AO 2.2a): Fully correct strict double inequality −7 < b < 7 (Follow-through from part a).

🧠 Exam Technique & Visualising the Problem

Geometric Method (Fastest):

The centre of the circle is at point (a, b) .
The distance from the centre to the x-axis (line y = 0) is |b| .
For the circle to cross the x-axis in two distinct points, the centre must be closer to the axis than the radius: |b| < r .
If |b| = 7, the circle touches the axis once (tangent). If |b| > 7, it does not touch or intersect at all.

📐 Alternative Method: Step-by-Step Algebraic Solution

Set y = 0 for intersection with the x-axis:
(x − a)² + (0 − b)² = 49 ⟹ (x − a)² + b² = 49
Rearrange for (x − a)²:
(x − a)² = 49 − b²
Apply the condition for two distinct real solutions:
For there to be two distinct values of x, the right-hand side must be strictly greater than zero:
49 − b² > 0
Solve the quadratic inequality:
b² < 49 ⟹ −7 < b < 7

❌ Common Misconceptions & Traps

Non-strict inequalities: Writing −7 ≤ b ≤ 7 loses the final A1 mark because at b = ±7, the circle is tangent to the x-axis (only 1 point of intersection, not two distinct points).
Ignoring negative values: Stating only b < 7 or 0 < b < 7 . The centre can lie below the x-axis (negative y-coordinate) as long as it is within 7 units of the axis.
Confusing a and b: Setting up inequalities in terms of a instead of b. The horizontal position a has no effect on whether the circle crosses the horizontal axis.

💡 Examiner Insight: Summary of Conditions

Relationship	Geometric Outcome
\|b\| < 7	2 distinct intersection points
\|b\| = 7	1 point of contact (tangent)
\|b\| > 7	0 intersection points

Topics

Pure Mathematics · B: Algebra and functions · C: Coordinate geometry in the (x, y) plane