June 2022 Paper 2 Q8
8 A curve has equation \(\quad y = x^3 - 5x^2\)
At two points on the curve, the rate of change of \(y\) with respect to \(x\) is 4
(a) Work out an equation, in terms of \(x\), to represent this information.
Give your answer in the form \(\quad ax^2 + bx + c = 0 \quad\) where \(a\), \(b\) and \(c\) are integers. [2 marks]
(b) Hence, work out the two possible values of \(x\).
Give your answers to 3 significant figures. [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(3x^2\) or \(-10x\) | M1 | oe eg \(3 \times x^{3 - 1}\) or \(-2 \times 5x^1\) |
| \(3x^2 - 10x - 4 = 0\) or \(-3x^2 + 10x + 4 = 0\) | A1 | must show = 0 |
Additional guidance
| M1 may be awarded for correct work with no, or incorrect answer, even if this is seen amongst multiple attempts | |
| Ignore extra terms eg \(3x^2 - 10x + c\) | M1 |
| \(3x^2 - 10x = 4\) (even if \(3x^2 - 10x - 4 = 0\) in (b)) | M1A0 |
| \(3x^2 - 10x - 4\) (even if \(3x^2 - 10x - 4 = 0\) in (b)) | M1A0 |
| \(3x^2 - 10x - 4 = 0\) seen in working with \(3x^2 - 10x - 4\) on answer line | M1A1 |
| Condone for M1 \(\;y = 3x^2 \ldots\) etc (may still score A1 if recovered) | |
| Answer \(y = 3x^2 - 10x - 4 = 0\) | M1A0 |
| Answer | Mark | Comments |
|---|---|---|
| \(\dfrac{- -10 \pm \sqrt{(-10)^2 - 4 \times 3 \times -4}}{2 \times 3}\) or \(\dfrac{10 \pm \sqrt{148}}{6}\) or \(\dfrac{5}{3} \pm \sqrt{\dfrac{37}{9}}\) or two correct solutions with at least one not to 3 sf | M1 | oe eg \(\dfrac{5 \pm \sqrt{37}}{3}\) correct attempt to solve their \(ax^2 + bx + c\;(= 0)\) from (a) \(a\), \(b\) and \(c\) all non-zero eg 3.69(4…) and \(-0.36(09\ldots)\) or 3.7 and \(-0.36(09\ldots)\) |
| 3.69 and \(-0.361\) | A1ft | correct or ft any answers that have at least 4 sf must be rounded to 3 sf at least one answer must have at least 4 sf |
Additional guidance
| \(-10^2\) used for \((-10)^2\) is M0 unless recovered | |
| \(10^2\) is equivalent to \((-10)^2\) | |
| Not using \(\pm\) is M0 unless recovered | |
| A short dividing line or a short square root symbol is M0 unless recovered | |
| √((–10)\(^2\) – 4 × 3 × –4) is correct for \(\sqrt{(-10)^2 - 4 \times 3 \times -4}\) | |
| Correct factorisation of their \(ax^2 + bx + c\;(= 0)\) from (a) scores at least M1 | |
| (a) \(3x^2 - 10x + 4 = 0 \qquad\) (b) \(\dfrac{- -10 \pm \sqrt{(-10)^2 - 4 \times 3 \times 4}}{2 \times 3}\) 2.87 and 0.465 | M1A1ft |
| (a) \(3x^2 - 10x = 4 \qquad\) (b) up to 2 marks can be scored if using \(3x^2 - 10x - 4 = 0\) | |
| (a) \(3x^2 - 10x - 8 \qquad\) (b) up to 2 marks can be scored if using \(3x^2 - 10x - 8 = 0\) | |
| One solution correct does not imply M1 | |
| Both solutions seen in working but only one on answer line | M1A0 |
| 3.69 and \(-0.361\) in working with \(-3.69\) and 0.361 on answer line | M1A0 |