Higher June 2018 Paper 2 Q17
17 \(y = x^3 - 2x^2 - 15x + 5\)
(a) Find \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\) (2)
\(C\) is the curve with equation \(y = x^3 - 2x^2 - 15x + 5\)
(b) Work out the range of values of \(x\) for which \(C\) has a negative gradient. (4)
| Scheme | Marks |
|---|---|
| \(3x^2 - 4x - 15\) | B2 |
| (2) |
Notes
B2: Award B1 for any 2 or 3 of the 4 terms differentiated correctly.
| Scheme | Marks |
|---|---|
| \(3x^2 - 4x - 15 \lt 0\) (or = 0) | M1 |
\((3x + 5)(x - 3)\) (< 0) or \(\dfrac{-(-4) \pm \sqrt{(-4)^2 - 4 \times 3 \times (-15)}}{2 \times 3}\) | M1 |
| \(-\dfrac{5}{3}\), 3 | M1 |
| \(-\dfrac{5}{3} \lt x \lt 3\) | A1oe |
| (4) | |
| (6 marks) |
Notes
M1: ft from (a) ie “their (a)” = 0 (or < 0)
M1: ft from “their (a)” (=0) for 3 term quadratic, for correct factorisation or correct use of quadratic formula to find the two critical values, allow 1 sign error. [−(−4) could be 4 and (−4)² could be 4²] (condone missing brackets)
M1: Both critical values correct
Accept -1.66… rounded or truncated to 3SF.
A1oe: Inequality signs needed
Allow \(x \gt -\dfrac{5}{3}\), \(x \lt 3\)