4400 Higher November 2009 Paper 4 Q22
22 Solve the simultaneous equations \[\begin{aligned} y - 3x &= 4 \\ x^2 + y^2 &= 34 \end{aligned}\]
(7)
| Scheme | Marks |
|---|---|
| \(y = 3x + 4\) | B1 |
| \(x^2 + (3x + 4)^2 = 34\) | M1 |
| \(x^2 + 9x^2 + 12x + 12x + 16 = 34\) or \(x^2 + 9x^2 + 24x + 16 = 34\) | B1 |
| \(10x^2 + 24x - 18\ (= 0)\) | B1 |
\((5x - 3)(2x + 6)\ (= 0)\) or \((5x - 3)(x + 3)\ (= 0)\) or \((10x - 6)(x + 3)\ (= 0)\) or \(\dfrac{-24 \pm \sqrt{1296}}{20}\) or \(\dfrac{-12 \pm \sqrt{324}}{10}\) or \(-\dfrac{12}{10} \pm \dfrac{\sqrt{324}}{10}\) or \(-\dfrac{6}{5} \pm \dfrac{\sqrt{81}}{5}\) | B1 |
| \(x = \frac{3}{5}\) or \(x = -3\) | A1 |
| \(x = \frac{3}{5}\), \(y = 5\frac{4}{5}\) \(x = -3\), \(y = -5\) | A1 |
| (7) | |
| (7 marks) |
Notes
B1: for correct rearrangement
M1: for correct substitution
B1: (indep) for correct expansion of \((3x + 4)^2\) even if unsimplified
B1: for correct simplification
Condone omission of ‘= 0’
B1: for correct factorisation
Condone omission of ‘= 0’
or for correct substitution into the quadratic formula and correct evaluation of ‘\(b^2 - 4ac\)’
or for using square completion correctly as far as indicated
A1: for both values of \(x\)
A1: for complete, correct solutions
Need not be explicitly paired
Alternative (eliminating \(x\)): \(x = \frac{y - 4}{3}\) B1; \(\left(\frac{y - 4}{3}\right)^2 + y^2 = 34\) M1; \(\frac{y^2 - 4y - 4y + 16}{9} + y^2 = 34\) or \(y^2 - 8y + 16 + 9y^2 = 306\) B1 (indep) for correct expansion of \((y - 4)^2\) even if unsimplified; \(10y^2 - 8y - 290\ (= 0)\) B1; \((5y - 29)(y + 5)\ (= 0)\) or \(\frac{8 \pm \sqrt{11664}}{20}\) or \(\frac{4 \pm \sqrt{2916}}{10}\) or \(\frac{2}{5} \pm \frac{\sqrt{729}}{5}\) B1; \(y = 5\frac{4}{5}\) or \(y = -5\) A1 for both values of \(y\); complete, correct solutions A1