4MA0 Higher January 2013 Paper 4 Q26
26 Solve the simultaneous equations
\[y = 3x + 2\]\[x^2 + y^2 = 20\]Show clear algebraic working.
(6)
| Scheme | Marks |
|---|---|
| \(x^2 + (3x + 2)^2 = 20\) | M1 |
| \(x^2 + 9x^2 + 6x + 6x + 4 = 20\) or \(x^2 + 9x^2 + 12x + 4 = 20\) | M1 |
| \(5x^2 + 6x - 8 (= 0)\) or \(10x^2 + 12x - 16 (= 0)\) | A1 |
\((5x - 4)(x + 2) (= 0)\) or \((10x - 8)(x + 2) (= 0)\) or \((5x - 4)(2x + 4) (= 0)\) or \(5x(x + 2) - 4(x + 2) (= 0)\) or \(x(5x - 4) + 2(5x - 4) (= 0)\) or \(\dfrac{-6 \pm \sqrt{6^2 - 4 \times 5 \times -8}}{2 \times 5}\) or better or \(\dfrac{-12 \pm \sqrt{12^2 - 4 \times 10 \times -16}}{2 \times 10}\) or better | M1 |
| \(x = \tfrac{4}{5}\) or \(x = -2\) | A1 |
| \(x = \tfrac{4}{5}\), \(y = 4\tfrac{2}{5}\) \(x = -2\), \(y = -4\) | A1 |
| (6) |
Notes
M1: or \(\left(\dfrac{y - 2}{3}\right)^2 + y^2 = 20\)
M1: (indep) for correct expansion of \((3x + 2)^2\) or \(\left(\dfrac{y - 2}{3}\right)^2\) even if unsimplified, eg \(\left(\dfrac{y^2 - 4y + 4}{9}\right) + y^2 = 20\) or \(\left(\dfrac{y^2 - 2y - 2y + 4}{9}\right) + y^2 = 20\)
A1: \(5y^2 - 2y - 88 (= 0)\) oe
Condone omission of ‘= 0’
M1: \((5y - 22)(y + 4) (= 0)\) oe
or \(\dfrac{2 \pm \sqrt{(-2)^2 - 4 \times 5 \times -88}}{2 \times 5}\) or better
Condone omission of ‘= 0’
A1: \(y = 4\tfrac{2}{5}\) or \(y = -4\)
dep on all preceding marks
A1: No marks for \(x = -2\), \(y = -4\) with no working