Higher June 2023 Paper 1R Q21
21 The curve T has equation \(\;y = x^3 - 2x^2 - 9x + 15\)
(a) Find \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\) (2)
(b) Find the range of values of \(x\) for which T has a positive gradient.
Give your values correct to 3 significant figures.
Show your working clearly. (4)
Give your values correct to 3 significant figures.
Show your working clearly. (4)
| Scheme | Marks |
|---|---|
| \(3x^2\) or \(-2 \times 2x\) or \(-4x\) or \(-9\) oe | M1 |
| \(3x^2 - 4x - 9\) | A1 |
| (2) |
Notes
M1: for differentiating one term correctly
A1: for a correct expression
Allow \(3x^2 - 2 \times 2x - 9\)
| Scheme | Marks |
|---|---|
\((x =)\;\dfrac{4 \pm \sqrt{(-4)^2 - (4 \times 3 \times -9)}}{2 \times 3}\) or \(3\left[\left(x - \dfrac{2}{3}\right)^2 - \left(\dfrac{2}{3}\right)^2\right] - 9\;(= 0)\) | M1 |
| \(-1.19\) and 2.52 | A1 |
| \(x \lt -1.19\) | A1 |
| \(x \gt 2.52\) | A1 |
| (4) | |
| (6 marks) |
Notes
M1: for finding the critical values for a 3-term quadratic using any correct method - if using formula or completing the square allow one sign error and some simplification
– allow as far as eg \(\dfrac{4 \pm \sqrt{16 + 108}}{6}\) oe
or eg \(3\left(x - \dfrac{2}{3}\right)^2 - 10\dfrac{1}{3}\) oe)
A1: for critical values of \(-1.19\) and 2.52 or better
(for this A1 mark allow \(-1.2\) or \(-1.18\) and 2.5
or \(\dfrac{2 \pm \sqrt{31}}{3}\) oe)
A1: awrt \(-1.19\)
A1: awrt 2.52