Higher June 2018 Paper 1R Q19
19 The curve shown in the diagram has equation
\(y = x^3 - 27x + k\) where \(k\) is a positive constant with \(k \lt 54\)
The curve has a maximum point at \(A\,(a, b)\)
The curve has a minimum point at \(B\,(c, d)\)
Diagram NOT accurately drawn
Using differentiation, find the value of \(b - d\)
Show your working clearly.
(6)
| Scheme | Marks |
|---|---|
| \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = 3x^2 - 27\) | M1 |
| \(3x^2 - 27 = 0\) | M1 |
| \(x = \pm 3\) | A1 |
| When \(x = -3\), \(b = (-3)^3 - 27(-3) + k\) (\(= 54 + k\)) When \(x = 3\), \(d = 3^3 - 27(3) + k\) (\(= -54 + k\)) | M1 |
| \(b - d = 54 + k - (-54 + k)\) | M1 |
| 108 | A1 |
| (6) | |
| (6 marks) |
Notes
M1: for at least one of \(3x^2\) or 27
M1: (dep) for a 2 or 3 term quadratic = 0
M1: for either substituting \(x = 3\) or \(x = -3\) into the \(y\) expression.
Only award this mark if \(k\) or a number representing \(k\) is in the expression for \(b\) or \(d\)
M1: dep on all previous M marks
Expressions for \(b\) and \(d\) must have \(k\) or the same number representing \(k\)