Higher June 2017 Paper 2 Q21
21 Solve \(\quad 5x^2 = 10x + 4\)
Give your answers to 2 decimal places. [4 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| \(5x^2 - 10x - 4\ (= 0)\) or \(-5x^2 + 10x + 4\ (= 0)\) | B1 | If no rearrangement seen implied by \(a = 5,\ b = -10,\ c = -4\) or \(a = -5,\ b = 10,\ c = 4\) seen or used correctly |
| \(\dfrac{--10 \pm \sqrt{(-10)^2 - 4 \times 5 \times -4}}{2 \times 5}\) | M1 | ft their 3-term quadratic (equation) seen Allow one sign error Allow \(10^2\) for \((-10)^2\) (do not count as a sign error) Allow recovery of invisible brackets Conceptual error (omission of square root, incomplete square root symbol, \(\pm\) not included, short fraction line) is M0 unless recovered |
| \(\dfrac{--10 \pm \sqrt{(-10)^2 - 4 \times 5 \times -4}}{2 \times 5}\) or \(\dfrac{10 \pm \sqrt{100 + 80}}{10}\) or \(\dfrac{10 \pm \sqrt{180}}{10}\) or \(\dfrac{10 \pm 6\sqrt{5}}{10}\) or 2.341(…) or 2.342 and \(-0.341(\ldots)\) or \(-0.342\) | A1ft | Fully correct substitution ft their 3-term quadratic (equation) seen oe eg \(\dfrac{5 \pm 3\sqrt{5}}{5}\) Allow \(10^2\) for \((-10)^2\) Allow recovery of invisible brackets Two correct solutions > 2 dp for their 3-term quadratic equation |
| 2.34 and \(-0.34\) | A1ft | ft B0M1A1ft ft answers must be rounded to 2 dp |
| Alternative method 2 | ||
| \(5\left(x^2 - 2x - \dfrac{4}{5}\right)\ (= 0)\) or \(x^2 - 2x - \dfrac{4}{5}\ (= 0)\) or \(5(x^2 - 2x) = 4\) or \(x^2 - 2x = \dfrac{4}{5}\) | B1 | May be implied |
| \(5\left[(x - 1)^2 - 1^2 - \dfrac{4}{5}\right]\ (= 0)\) or \((x - 1)^2 - 1^2 - \dfrac{4}{5}\ (= 0)\) or \(5[(x - 1)^2 - 1^2] = 4\) or \((x - 1)^2 - 1^2 = \dfrac{4}{5}\) | M1 | ft their 3-term quadratic (equation) seen Allow one sign error but \((x - 1)^2\) must be correct |
| \(1 \pm \sqrt{1^2 + \dfrac{4}{5}}\) or 2.341(…) or 2.342 and \(-0.341(\ldots)\) or \(-0.342\) | A1ft | Fully correct ft their 3-term quadratic (equation) seen oe eg \(\dfrac{5 \pm 3\sqrt{5}}{5}\) Two correct solutions > 2 dp for their 3-term quadratic equation seen |
| 2.34 and \(-0.34\) | A1ft | ft B0M1A1ft ft answers must be rounded to 2 dp |
Additional guidance
| Do not count a sign error in \(a\) (or \(b\)) as two sign errors eg If \(a\) should be \(-5\) but \(a = 5\) is used in both \(4ac\) and \(2a\), only count as one sign error | |
| Final A1 mark can be awarded if both answers seen in working but only one is written on answer line | |
| \(5x^2 + 10x - 4\ (= 0)\) seen with solutions \(-2.34\) and 0.34 (no incorrect method seen) | B0M1A1ftA1ft |
| \(5x^2 - 10x + 4\ (= 0)\) seen with solutions 0.55 and 1.45 (no incorrect method seen) | B0M1A1ftA1ft |
| \(5x^2 + 10x + 4\ (= 0)\) seen with solutions \(-0.55\) and \(-1.45\) (no incorrect method seen) | B0M1A1ftA1ft |
| Note that the pairs of solutions seen in the three rows above can come from incorrect method so will not always score 3 marks | |
| 2.34 and \(-0.34\) with no working or from T & I | 4 marks |
| 2.34 or \(-0.34\) with no working or from T & I | Zero |
| 2.3 and/or \(-0.3\) with no working or from T & I | Zero |