June 2024 Paper 1 Q19
19 A curve has equation
\[y^3\mathrm{e}^{2x} + 2y - 16x = k\]where \(k\) is a constant.
The curve has a stationary point on the \(y\)-axis.
Determine the value of \(k\) [7 marks]
| Scheme | Marks | AO |
|---|---|---|
| Uses implicit differentiation, with \(Ay^2\dfrac{\mathrm{d}y}{\mathrm{d}x}\) or \(2\dfrac{\mathrm{d}y}{\mathrm{d}x}\) seen. | M1 | 3.1a |
| Uses product rule to differentiate \(y^3\mathrm{e}^x\) and obtains \(Ay^2\mathrm{e}^{2x}\dfrac{\mathrm{d}y}{\mathrm{d}x} + By^3\mathrm{e}^{2x}\) | M1 | 3.1a |
| Obtains correctly \(3y^2\mathrm{e}^{2x}\dfrac{\mathrm{d}y}{\mathrm{d}x} + 2y^3\mathrm{e}^{2x} + 2\dfrac{\mathrm{d}y}{\mathrm{d}x} - 16 = 0\) | A1 | 1.1b |
| Substitutes \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = 0, x = 0\) into their differentiated equation or rearranged equation to obtain a value for \(y\). Their equation needs to have contained either \(Ay^2\dfrac{\mathrm{d}y}{\mathrm{d}x}\) or \(2\dfrac{\mathrm{d}y}{\mathrm{d}x}\) and involve \(\mathrm{e}^{2x}\) | M1 | 3.1a |
| Obtains \(y\) = 2 Must have achieved M1M1A1M1 so far. PI substituting \(y = \dfrac{2}{\mathrm{e}^{\frac{2x}{3}}}\) and \(x\)=0 into \(y^3\mathrm{e}^{2x} + 2y - 16x\) | A1 | 1.1b |
| Substitutes \(x = 0\) and their \(y\) = 2 into \(y^3\mathrm{e}^{2x} + 2y - 16x\) to obtain a value of \(k\) | M1 | 3.1a |
| Deduces \(k\) =12 Must have achieved all previous marks | R1 | 2.2a |
| (7 marks) |