June 2023 Paper 1 Q8

8 A circle has centre \((0, 0)\) and radius 5

A straight line has equation \(\quad 2y = x + 5\)

Work out the coordinates of the two points where the circle and straight line intersect.

Do not use trial and improvement.

You must show your working. [6 marks]

Mark scheme

Answer	Mark	Comments
Alternative method 1
\(x^2 + y^2 = 25\)	B1	oe equation of circle eg \(x^2 + y^2 = 5^2\) eg \((x - 0)^2 + (y - 0)^2 = 25\)
\((2y - 5)^2 + y^2\)	M1	must be \(2y - 5\) substituted into \(x^2 + y^2\)
\((4y^2 - 20y + 25) + y^2\)	M1	oe ft their \(2y - 5\) correctly expanded
\(5y^2 - 20y = 0\) or \(y^2 - 4y = 0\) or \(5y(y - 4) = 0\) or \(y(y - 4) = 0\)	M1dep	oe (dependent on B1 M2) must be fully correct (no ft)
\(y = 0\) and \(y = 4\) or \((-5, 0)\) or \((3, 4)\)	A1
\((-5, 0)\) and \((3, 4)\)	A1
Alternative method 2
\(x^2 + y^2 = 25\)	B1	oe equation of circle eg \(x^2 + y^2 = 5^2\) eg \((x - 0)^2 + (y - 0)^2 = 25\)
\(x^2 + \left(\dfrac{x + 5}{2}\right)^2\)	M1	must be \(\left(\dfrac{x + 5}{2}\right)\) substituted into \(x^2 + y^2\)
\(x^2 + \left(\dfrac{x^2 + 10x + 25}{4}\right)\) or \(4x^2 + x^2 + 10x + 25 \;(= 100)\)	M1	oe ft their \(\left(\dfrac{x + 5}{2}\right)\) correctly expanded
\(5x^2 + 10x - 75 = 0\) or \(x^2 + 2x - 15 = 0\) or \((x - 3)(x + 5) = 0\)	M1dep	oe (dependent on B1 M2) must be fully correct (no ft)
\(x = 3\) and \(x = -5\) or \((-5, 0)\) or \((3, 4)\)	A1
\((-5, 0)\) and \((3, 4)\)	A1

Answer could be written as \(x = -5\), \(y = 0\) and \(x = 3\), \(y = 4\)	B1 M3 A2
Answer with no working (for M3) or from trying to draw an accurate graph or T&I	B0 M0 A0
Incorrect circle equation can still score 2 marks eg. \(x^2 + y^2 = 10\) \(x^2 + \left(\dfrac{x + 5}{2}\right)^2\) \(4x^2 + x^2 + 10x + 25 = 40\) \(5x^2 + 10x - 15 = 0\) \((x + 3)(x - 1) = 0\) \(x = -3\) and \(x = 1\) \(y = 1\) and \(y = 3\)	B0 M2 M0 A0 A0 B0 M1 M1 M0 A0 A0
Correct answer can be gained by using the incorrect substitution \(x = 5 - 2y\)	B1 M0 M1 M0 A0
Candidates that rearrange \(x^2 + y^2 = 25\) correctly and substitute into \(2y = x + 5\) will gain first M mark. They should then follow main scheme for other marks	B1 M1 …..