Higher June 2024 Paper 2 Q22
22 The straight line \(L\) has equation \(\;x + y = 5\)
The curve \(C\) has equation \(\;2x^2 + 3y^2 = 210\)
Find the coordinates of the points where \(L\) and \(C\) intersect.
Show clear algebraic working.
(5)
| Scheme | Marks |
|---|---|
eg \(2(5 - y)^2 + 3y^2 = 210\) \(\sqrt{\dfrac{210 - 3y^2}{2}} = 5 - y\) oe or Eg \(2x^2 + 3(5 - x)^2 = 210\) \(\sqrt{\dfrac{210 - 2x^2}{3}} = 5 - x\) oe | M1 |
| eg \(5y^2 - 20y - 160 (= 0)\) or \(y^2 - 4y - 32 (= 0)\) or eg \(5x^2 - 30x - 135 (= 0)\) or \(x^2 - 6x - 27 (= 0)\) | M1 |
eg \((y - 8)(y + 4) (= 0)\) \(y = \dfrac{--4 \pm \sqrt{(-4)^2 - 4 \times 1 \times -32}}{2 \times 1}\) eg \((y - 2)^2 - 2^2 = -32\) (allow incorrect labels for \(x\)/\(y\)) or eg \((x - 9)(x + 3) (= 0)\) \(x = \dfrac{--6 \pm \sqrt{(-6)^2 - 4 \times 1 \times -27}}{2 \times 1}\) eg \((x - 3)^2 - 3^2 - 27 = 0\) (allow incorrect labels for \(x\)/\(y\)) | M1 |
| eg \(x + 8 = 5\) and \(x + -4 = 5\) (correct labels for \(x\)/\(y\)) or eg \(y = 5 - 9\) and \(y = 5 - -3\) (correct labels for \(x\)/\(y\)) | M1ft |
| working required Answer: (9, –4) ( –3, 8) | A1 |
| (5) | |
| (5 marks) |
Notes
M1: substitution of \(x = \pm 5 \pm y\) or \(y = \pm 5 \pm x\) into \(2x^2 + 3y^2 = 210\) or a correct equation formed by using \(x = \pm 5 \pm y\) or \(y = \pm 5 \pm x\) to obtain an equation in \(x\) only or \(y\) only
M1: dep on previous M1 for multiplying out and collecting terms, forming a three term quadratic in any form of \(ax^2 + bx + c\) (= 0) where at least 2 coefficients (\(a\) or \(b\) or \(c\)) are correct
M1ft: dep on previous M1 for substituting their 2 found values of \(x\) or \(y\) in a suitable equation (allow use of quadratic equation)
or fully correct values for the other variable must see substitution for incorrect \(x\)/\(y\) values
A1: (dep on M2)