FP2 June 2017 Q7
7.
(a) Find, in the form \(y = \mathrm{f}(x)\), the general solution of the equation \[\cos x\frac{\mathrm{d}y}{\mathrm{d}x} + y\sin x = 2\cos^3 x\sin x + 1, \quad 0 \lt x \lt \frac{\pi}{2}\] (8)
Given that \(y = 5\sqrt{2}\) when \(x = \dfrac{\pi}{4}\)
(b) find the value of \(y\) when \(x = \dfrac{\pi}{6}\), giving your answer in the form \(a + b\sqrt{3}\), where \(a\) and \(b\) are rational numbers to be found. (3)
| Scheme | Marks |
|---|---|
| \(\cos x\dfrac{\mathrm{d}y}{\mathrm{d}x} + y\sin x = 2\cos^3 x\sin x + 1\) | |
| \(\dfrac{\mathrm{d}y}{\mathrm{d}x} + y\tan x = 2\cos^2 x\sin x + \dfrac{1}{\cos x}\) Divides by \(\cos x\) LHS both terms divided RHS min 1 term divided | M1 |
| \(I = \mathrm{e}^{\int\tan x\,\mathrm{d}x} = \mathrm{e}^{\ln\sec x} = \sec x\) M1: Attempt integrating factor \(\mathrm{e}^{\int\tan x\,\mathrm{d}x}\) needed A1: Correct integrating factor, \(\sec x\) or \(\dfrac{1}{\cos x}\) | dM1A1 |
| \(y\sec x = \displaystyle\int\left(2\sin x\cos x + \sec^2 x\right)\mathrm{d}x\) Multiply through by their IF and integrate LHS (integration may be done later) \(yI = \displaystyle\int(\text{their RHS})I\,\mathrm{d}x\) | M1 |
| \(y\sec x = -\dfrac{1}{2}\cos 2x + \tan x\ (+c)\) M1: Attempt integration of at least one term on RHS (provided both sides have been multiplied by their IF.) OR \(\sec^2 x \to K\tan x\) A1: \(-\dfrac{1}{2}\cos 2x\) or equivalent integration of \(2\sin x\cos x\) (\(\sin^2 x\) or \(-\cos^2 x\)) A1: \(\tan x\) constant not needed. | M1A1A1 |
| \(y = \left(-\dfrac{1}{2}\cos 2x + \tan x + c\right)\cos x\) \(y = \left(-\cos^2 x + \tan x + c\right)\cos x\) \(y = \left(\sin^2 x + \tan x + c\right)\cos x\) Include the constant and deal with it correctly. Must start \(y = \ldots\) Or equivalent eg \(y = -\dfrac{1}{2}\cos 2x\cos x + \sin x + c\cos x\) Follow through from the line above | A1ft |
| (8) |
| Scheme | Marks |
|---|---|
| \(x = \dfrac{\pi}{4} \Rightarrow 5\sqrt{2} = \ldots\ldots \Rightarrow c = \ldots\ldots\) Substitutes for \(x\) and \(y\) and solves for \(c\) (If substitution not shown award for at least one term evaluated correctly.) | M1 |
| \(x = \dfrac{\pi}{6} \Rightarrow y = \ldots\ldots\) Substitutes \(x = \dfrac{\pi}{6}\) to find a value for \(y\) | M1 |
| \(y = \dfrac{1}{2} + \dfrac{35}{8}\sqrt{3}\) or \(y = 0.5 + 4.375\sqrt{3}\) Must be in given form. Equivalent fractions allowed. | A1cao |
| NB (b) There may be no working shown due to use of calculator. In such cases: Final answer correct (and in required form with no decimals instead of \(\sqrt{3}\) seen), score 3/3. Final answer incorrect (or decimals instead of \(\sqrt{3}\) seen), score 0/3. This applies whether (a) is correct or not. | |
| (3) | |
| (11 marks) |