Higher November 2020 Paper 1 Q23
23 Curve C has equation \(y = px^3 - mx\) where \(p\) and \(m\) are positive integers.
Find the range of values of \(x\), in terms of \(p\) and \(m\), for which the gradient of C is negative.
(4)
| Scheme | Marks |
|---|---|
| \(\left(\dfrac{\mathrm{d}y}{\mathrm{d}x} =\right) 3px^2 - m\) | M1 |
| \(3px^2 - m \lt 0\) oe | M1 |
| \(\pm\sqrt{\dfrac{m}{3p}}\) | B1 |
Correct answer scores full marks (unless from obvious incorrect working) Answer: \(-\sqrt{\dfrac{m}{3p}} \lt x \lt \sqrt{\dfrac{m}{3p}}\) | A1 |
| (4) | |
| (4 marks) |
Notes
M1: for \(3px^2\) or \(-m\)
M1: ft dep on M1 for setting up an inequality with their ‘\(3px^2\)’ – ‘\(m\)’ must be a two-term expression in the form \(apx^2 \pm m\)
B1: for both critical values
A1: may be seen as two separate inequalities