Higher June 2023 Paper 6 Q18
18
(a) Describe fully the graph of \(x^2 + y^2 = 20\). [3]
(b) The graph of \(y = 3x + 10\) intersects the graph of \(x^2 + y^2 = 20\) at two points.
Use an algebraic method to work out the coordinates of the two points.
You must show your working. [6]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Circle | 1 | Accept circular graph | |
| Centre (0, 0) oe | 1 | Accept origin or O for (0, 0) but not turning point (0, 0) | |
| Radius \(\sqrt{20}\) or \(2\sqrt{5}\) or 4.47[2..] or 4.5 | 1 | If their description uniquely defines the circle then award full marks eg After “circle” and “centre (0, 0)”, passes through one correct stated point, scores 3 “circle” and (\(\pm\sqrt{20}\) , 0) and (0, \(\pm\sqrt{20}\) ), scores 3 “circle” and two correct stated points, scores 1 | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (\(-2\), 4) and (\(-4\), \(-2\)) with correct working | 6 | M1 for \(x^2 + (3x + 10)^2 = 20\) M1 for expanding their square term e.g. \(9x^2 + 30x + 30x + 100\) M1 for simplifying their quadratic equation e.g. \(10x^2 + 60x + 100 = 20\) or better M1 for correctly factorising their 3-term quadratic equation or for correct use of quadratic formula for their 3-term quadratic equation or for correct completing the square A1 for one correct point or two correct \(x\)-values If 0 or 1 scored, instead award SC2 for 2 correct points with no or insufficient working If 0 scored SC1 for 1 correct point or 2 correct \(x\)-coordinates or 2 correct \(y\)-coordinates with no or insufficient working | ‘Correct working’ requires evidence of at least M1M1M1 Award equivalent marks if working in terms of \(y\) May be in a grid May be implied by subsequent working Their quadratic must include an \(x\) term Simplified: \(10x^2 + 60x + 80\) [= 0] or \(x^2 + 6x + 8\) [= 0] e.g. \((x + 2)(x + 4)\), \((5x + 10)(2x + 8)\) e.g. reaching \(d(x + e)^2 + f\) |