Higher June 2024 Paper 2R Q22
22 Solve the simultaneous equations
\[\begin{aligned} x^2 + y^2 &= y + 11 \\ y &= 3x - 1 \end{aligned}\]Show clear algebraic working.
(5)
| Scheme | Marks |
|---|---|
eg \(x^2 + (3x - 1)^2 = 3x - 1 + 11\) or eg \(\left(\dfrac{y + 1}{3}\right)^2 + y^2 = y + 11\) | M1 |
| \(10x^2 - 9x - 9 (= 0)\) oe or \(10x^2 - 9x = 9\) oe or \(10x^2 = 9 + 9x\) oe or \(10y^2 - 7y - 98 (= 0)\) oe or \(10y^2 - 7y = 98\) oe or \(10y^2 = 98 + 7y\) oe | M1ft |
\((5x + 3)(2x - 3) (= 0)\) or \((x =)\, \dfrac{9 \pm \sqrt{(-9)^2 - 4 \times 10 \times (-9)}}{2 \times 10}\) or \(10\left[\left(x - \dfrac{9}{20}\right)^2 - \left(\dfrac{9}{20}\right)^2\right] - 9 (= 0)\) or \(x = -0.6\) and \(x = 1.5 = \dfrac{3}{2}\) or \((5y + 14)(2y - 7) (= 0)\) or \((y =)\, \dfrac{7 \pm \sqrt{(-7)^2 - 4 \times 10 \times (-98)}}{2 \times 10}\) or \(10\left[\left(y - \dfrac{7}{20}\right)^2 - \left(\dfrac{7}{20}\right)^2\right] - 98 (= 0)\) or \(y = -2.8 = -\dfrac{14}{5}\) and \(y = 3.5 = \dfrac{7}{2}\) | M1ft |
(\(y\) =) 3 × “−0.6” – 1 (= −2.8) and (\(y\) =) 3 × “1.5” – 1 (= 3.5) or \((x =)\, \dfrac{\text{``}{-2.8}\text{''} + 1}{3} (= -0.6)\) and \((x =)\, \dfrac{\text{``}{3.5}\text{''} + 1}{3} (= 1.5)\) | M1ft |
| Working required Answer: \(x = -0.6\) \(y = -2.8\) \(x = 1.5\) \(y = 3.5\) | A1 |
| (5) | |
| (5 marks) |
Notes
M1: for substitution of \(y = 3x - 1\) (or \(x = \dfrac{\pm y \pm 1}{3}\)) into \(x^2 + y^2 = y + 11\) to obtain an equation in \(x\) only (or \(y\) only)
M1ft: 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 first M1 method to solve their 3 term quadratic using any correct method (allow one sign error and some simplification – allow as far as eg \(\dfrac{9 \pm \sqrt{81 + 360}}{20}\) or \(\dfrac{7 \pm \sqrt{49 + 3920}}{20}\)
or if factorising allow brackets which expanded give 2 out of 3 terms correct)
or correct values for \(x\)
or correct values for \(y\)
(Allow incorrect labels for \(x\) or \(y\) for this mark only)
M1ft: dep on previous M1 for substituting their 2 found values of \(x\) or \(y\) into one of the two given equations
or fully correct values for the other variable
A1: oe dep on M2 for 4 correct values correctly labelled or correctly shown as coordinates