Higher January 2020 Paper 2 Q22
22 The line with equation \(\;y = x + 2\;\) intersects the curve with equation \(\;x^2 + y^2 - 2y = 24\;\) at the points \(A\) and \(B\).
Find the coordinates of \(A\) and \(B\).
Show clear algebraic working.
(5)
| Scheme | Marks |
|---|---|
| \(x^2 + (x + 2)^2 - 2(x + 2) = 24\) | M1 |
| \(2x^2 + 2x - 24\;(= 0)\) or \(x^2 + x - 12\;(= 0)\) or \(2x^2 + 2x = 24\) or \(x^2 + x = 12\) | A1 |
\((x + 4)(x - 3)\;(= 0)\) or \(x = \dfrac{-1 \pm \sqrt{1^2 - (4 \times 1 \times -12)}}{2 \times 1}\) or \(\left(x - \dfrac{1}{2}\right)^2 - \left(\dfrac{1}{2}\right)^2 - 12 = 0\) | M1ft |
| \(x = -4\) and \(x = 3\) | A1 |
| \((-4, -2)\) and \((3, 5)\) Working required Answer: \((-4, -2)\) and \((3, 5)\) | A1 |
| (5) | |
| (5 marks) |
Notes
M1: for substituting linear equation into the quadratic equation
A1: for a correct equation in the form \(ax^2 + bx + c = 0\) or \(ax^2 + bx = -c\)
M1ft: dep on M1 for solving their quadratic equation using any correct method (allow one sign error and some simplification – allow as far as \(\dfrac{-1 \pm \sqrt{1 + 48}}{2}\)) or if factorising, allow brackets which expanded give 2 out of 3 terms correct)
A1: for both \(x\) values dep on M1
A1: for both solutions dep on M1
Alternative mark scheme for 22
| Scheme | Marks |
|---|---|
| \((y - 2)^2 + y^2 - 2y = 24\) | M1 |
| \(2y^2 - 6y - 20\;(= 0)\) or \(y^2 - 3y - 10\;(= 0)\) \(2y^2 - 6y = 20\) or \(y^2 - 3y = 10\) | A1 |
\((y - 5)(y + 2) = 0\) or \(y = \dfrac{--3 \pm \sqrt{(-3)^2 - (4 \times 1 \times -10)}}{2 \times 1}\) or \(\left(y - \dfrac{3}{2}\right)^2 - \left(\dfrac{3}{2}\right)^2 - 10 = 0\) | M1ft |
| \(y = 5\) and \(y = -2\) | A1 |
| \((-4, -2)\) and \((3, 5)\) Working required Answer: \((-4, -2)\) and \((3, 5)\) | A1 |
Notes
M1: for substituting linear equation into the quadratic equation
A1: for a correct equation in the form \(ay^2 + by + c = 0\) or \(ay^2 + by = -c\)
M1ft: dep on M1 for solving their quadratic equation using any correct method (allow one sign error and some simplification – allow as far as \(\dfrac{3 \pm \sqrt{9 + 40}}{2}\)) or if factorising, allow brackets which expanded give 2 out of 3 terms correct
A1: for both \(y\) values dep on M1
A1: for both solutions dep on M1