Higher June 2019 Paper 2 Q20
20 The equation of the line L is \(y = 9 - x\)
The equation of the curve C is \(x^2 - 3xy + 2y^2 = 0\)
L and C intersect at two points.
Find the coordinates of these two points.
Show clear algebraic working.
(5)
| Scheme | Marks |
|---|---|
| \(x^2 - 3x(9 - x) + 2(9 - x)^2\) (= 0) or \((9 - y)^2 - 3y(9 - y) + 2y^2\) (= 0) | M1 |
| e.g. \(6x^2 - 63x + 162\) (= 0) or \(2x^2 - 21x + 54\) (= 0) allow \(2x^2 - 21x = -54\) oe or e.g. \(6y^2 - 45y + 81\) (= 0) or \(2y^2 - 15y + 27\) (= 0) allow \(2y^2 - 15y = -27\) oe | A1 |
e.g. \((2x - 9)(x - 6)\) (= 0) \(x = \dfrac{-(-21) \pm \sqrt{(-21)^2 - 4 \times 2 \times 54}}{2 \times 2}\) e.g. \(2\left(\left(x - \dfrac{21}{4}\right)^2 - \left(\dfrac{21}{4}\right)^2\right) = -54\) or e.g. \((2y - 9)(y - 3)\) (= 0) \(y = \dfrac{-(-15) \pm \sqrt{(-15)^2 - 4 \times 2 \times 27}}{2 \times 2}\) e.g. \(2\left(\left(y - \dfrac{15}{4}\right)^2 - \left(\dfrac{15}{4}\right)^2\right) = -27\) (corrected from the printed mark scheme, which has \(x\) in place of \(y\) here) | M1 |
| \(x = 4.5\) and \(x = 6\) or \(y = 4.5\) and \(y = 3\) | A1 |
| Working required Answer: (4.5, 4.5) and (6, 3) | A1 |
| (5) | |
| (5 marks) |
Notes
M1: substitution of linear equation into quadratic
A1: (dep on M1) writing the correct quadratic expression in form \(ax^2 + bx + c\) (= 0)
allow \(ax^2 + bx = c\)
M1: (dep on M1) for a complete method to solve their 3-term quadratic equation (allow one sign error and some simplification – allow as far as \(\dfrac{21 \pm \sqrt{441 - 432}}{4}\))
A1: (dep on M1) both \(x\)-values or both \(y\)-values
A1: (dep on M1) oe
Must be paired correctly
| Scheme | Marks |
|---|---|
| \((x - y)(x - 2y)\) (= 0) | M1 |
| \((x - (9 - x))(x - 2(9 - x))\) (= 0) or \((9 - y - y)(9 - y - 2y)\) (= 0) | A1 |
| \((2x - 9)(3x - 18)\) (= 0) oe or \((9 - 2y)(9 - 3y)\) (= 0) oe | M1 |
| \(x = 4.5\) and \(x = 6\) or \(y = 4.5\) and \(y = 3\) | A1 |
| Working required Answer: (4.5, 4.5) and (6, 3) | A1 |
Notes
M1: for a method to factorise C
A1: (dep M1) substitution of L into their factorised C
M1: (dep on M1)
A1: (dep on M1) both \(x\)-values or both \(y\)-values
A1: (dep on M1) oe
Must be paired correctly