4MA0 Higher January 2018 Paper 4 Q22
22 Solve the simultaneous equations
\[\begin{aligned} y + 2x &= 3 \\ x^2 + y^2 &= 18 \end{aligned}\]Show clear algebraic working.
(6)
| Scheme | Marks |
|---|---|
| \(x^2 + (3 - 2x)^2 = 18\) | M1 |
| \(9 - 6x - 6x + 4x^2\) | M1 |
| \(5x^2 - 12x - 9\;(= 0)\) | A1 |
\((5x + 3)(x - 3)\;(= 0)\) or \(\dfrac{--12 \pm \sqrt{(-12)^2 - 4 \times 5 \times -9}}{2 \times 5}\) | M1 |
| \(x = 3\) and \(x = -0.6\) | A1 |
| \(x = 3\), \(y = -3\) and \(x = -0.6\), \(y = 4.2\) | A1 |
| (6) |
Notes
M1: for elimination of one variable
e.g. \(y^2 + \left(\dfrac{3 - y}{2}\right)^2 = 18\)
M1: indep for correct expansion
e.g. \(\dfrac{y^2 - 3y - 3y + 9}{4}\)
A1: for correct simplified quadratic equation
(terms may not all be ‘on the same side’)
e.g. \(5y^2 - 6y - 63\;(= 0)\)
NB dep on first M1
M1: ft if first M1 awarded and equation is a 3 term quadratic for correct factorisation or correct substitution into formula e.g. \((5y - 21)(y + 3) = 0\) or \(\dfrac{--6 \pm \sqrt{(-6)^2 - 4 \times 5 \times -63}}{2 \times 5}\)
A1: or \(y = -3\) and \(y = 4.2\)
correct \(x\) or \(y\) values implies previous M1
dep on first A1 awarded
A1: Values for \(x\) and \(y\) must be correctly paired
dep on first A1 awarded