A2 June 2022 Paper 1 Q3
3.
Given that \(y = 3\) when \(x = 0\)
| Scheme | Marks | AO |
|---|---|---|
| \(\dfrac{\mathrm{d}y}{\mathrm{d}x} + y\tan x = \mathrm{e}^{2x}\cos x\) \(\text{IF} = \mathrm{e}^{\int \tan x\,\mathrm{d}x} = \mathrm{e}^{\ln\sec x} = \sec x \Rightarrow \sec x\dfrac{\mathrm{d}y}{\mathrm{d}x} + y\sec x\tan x = \mathrm{e}^{2x}\) \(\Rightarrow y\sec x = \displaystyle\int \mathrm{e}^{2x}\,\mathrm{d}x\) | M1 | 3.1a |
| \(y\sec x = \dfrac{1}{2}\mathrm{e}^{2x}(+c)\) | A1 | 1.1b |
| \(y = \left(\dfrac{1}{2}\mathrm{e}^{2x} + c\right)\cos x\) | A1 | 1.1b |
| (3) |
Notes
M1: Finds the integrating factor and attempts the solution of the differential equation.
Look for \(\text{I.F.} = \mathrm{e}^{\int \tan x\,\mathrm{d}x} \Rightarrow y \times \text{'their I.F.'} = \int \mathrm{e}^{2x}\cos x \times \text{'their I.F.'}\,\mathrm{d}x\)
A1: Correct solution condone missing \(+\,c\)
A1: Correct general solution, Accept equivalents of the form \(y = \mathrm{f}(x)\), such as \(y = \dfrac{\mathrm{e}^{2x}}{2\sec x} + \dfrac{c}{\sec x}\)
| Scheme | Marks | AO |
|---|---|---|
| \(x = 0,\ y = 3 \Rightarrow c = \ldots\{2.5\}\) | M1 | 3.1a |
| \(y = \left(\dfrac{1}{2}\mathrm{e}^{2x} + \dfrac{5}{2}\right)\cos x = 0 \Rightarrow \cos x = 0 \Rightarrow x = \ldots\) | M1 | 1.1b |
| \(x = \dfrac{\pi}{2}\) | A1 | 1.1b |
| (3) | ||
| (6 marks) |
Notes
M1: Uses \(x = 0\) \(y = 3\) to find the constant of integration. Allow if done as part of part (a) and allow for their answer to (a) as long as it has a constant of integration to find.
M1: Sets \(y = 0\) in an equation of the form \(y = \left(A\mathrm{e}^{2x} + c\right)\cos x\) (oe) where \(A\) is 1, 2 or \(\dfrac{1}{2}\), with their \(c\) or constant \(c\) and makes a valid attempt to solve the equation to find a value for \(x\). (Allow even if the constant of integration has not been found).
A1: Depends on both M’s. Awrt 1.57 or \(\dfrac{\pi}{2}\) only. There must have been an attempt to find the constant of integration, but allow from a correct answer to (a) as long as a positive value for \(c\) has been found (can be scored from implicit form).