Standard +0.8 This question requires finding when dy/dx = 6x² + 6mx - 9m has no real roots, leading to a discriminant inequality. While the differentiation is routine, setting up and solving the quadratic discriminant condition b² - 4ac < 0 correctly requires careful algebraic manipulation and understanding of when stationary points exist, making it moderately challenging but still within standard A-level scope.
8 The equation of a curve is \(y = 2 x ^ { 3 } + 3 m x ^ { 2 } - 9 m x + 4\).
Determine the range of values of \(m\) for which the curve has no stationary values.
Differentiation of all 4 terms with 3 of the 4 terms differentiated correctly
(all correct)
A1
\((6m)^2 - 4 \times 6 \times (-9m)\) oe seen
M1dep*
Discriminant for their \(6m\) and \(-9m\); may see \((2m)^2 - 4 \times 2 \times (-3m)\)
Two values of \(m\) obtained from their discriminant: \(36m^2 + 216m < 0\) or \(36m^2 + 216m = 0\) oe
M1
Dependent on obtaining discriminant from their derivative; M0 for use of their discriminant \(> 0\) or \(\geq 0\). NB \(4m^2 + 24m < 0\) or \(4m^2 + 24m = 0\)
\(0\) and \(-6\) identified
A1
\(-6 < m < 0\) oe
A1
Inequality or interval must be strict
## Question 8:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{dy}{dx} = 6x^2 + 6mx - 9m$ | M1* | Differentiation of all 4 terms with 3 of the 4 terms differentiated correctly |
| (all correct) | A1 | |
| $(6m)^2 - 4 \times 6 \times (-9m)$ oe seen | M1dep* | Discriminant for their $6m$ and $-9m$; may see $(2m)^2 - 4 \times 2 \times (-3m)$ |
| Two values of $m$ obtained from their discriminant: $36m^2 + 216m < 0$ or $36m^2 + 216m = 0$ oe | M1 | Dependent on obtaining discriminant from their derivative; **M0** for use of their discriminant $> 0$ or $\geq 0$. **NB** $4m^2 + 24m < 0$ or $4m^2 + 24m = 0$ |
| $0$ and $-6$ identified | A1 | |
| $-6 < m < 0$ oe | A1 | Inequality or interval must be strict |
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8 The equation of a curve is $y = 2 x ^ { 3 } + 3 m x ^ { 2 } - 9 m x + 4$.
Determine the range of values of $m$ for which the curve has no stationary values.
\hfill \mbox{\textit{OCR MEI Paper 2 2024 Q8 [6]}}