C3 January 2010 Q7
7.
Given that \(y = \mathrm{e}^{2x}\sec 3x\),
The curve with equation \(y = \mathrm{e}^{2x}\sec 3x\), \(-\frac{\pi}{6} \lt x \lt \frac{\pi}{6}\), has a minimum turning point at \((a, b)\).
| Scheme | Marks |
|---|---|
| \(y = \sec x = \dfrac{1}{\cos x} = (\cos x)^{-1}\) | |
| \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = -1(\cos x)^{-2}(-\sin x)\) | M1 A1 |
| \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = \left\{\dfrac{\sin x}{\cos^2 x}\right\} = \underline{\underline{\left(\dfrac{1}{\cos x}\right)\left(\dfrac{\sin x}{\cos x}\right)}} = \underline{\underline{\sec x\tan x}}\) | A1 AG |
| (3) |
Notes
M1: \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = \pm\left((\cos x)^{-2}(\sin x)\right)\)
A1: \(-1(\cos x)^{-2}(-\sin x)\) or \((\cos x)^{-2}(\sin x)\)
A1 AG: Convincing proof. Must see both underlined steps.
| Scheme | Marks |
|---|---|
| \(y = \mathrm{e}^{2x}\sec 3x\) | |
| \(\left\{\begin{aligned} u &= \mathrm{e}^{2x} & v &= \sec 3x \\ \frac{\mathrm{d}u}{\mathrm{d}x} &= 2\mathrm{e}^{2x} & \frac{\mathrm{d}v}{\mathrm{d}x} &= 3\sec 3x\tan 3x \end{aligned}\right\}\) | M1 A1 |
| \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = 2\mathrm{e}^{2x}\sec 3x + 3\mathrm{e}^{2x}\sec 3x\tan 3x\) | M1 A1 isw |
| (4) |
Notes
M1: Either \(\mathrm{e}^{2x} \to 2\mathrm{e}^{2x}\) or \(\sec 3x \to 3\sec 3x\tan 3x\)
A1: Both \(\mathrm{e}^{2x} \to 2\mathrm{e}^{2x}\) and \(\sec 3x \to 3\sec 3x\tan 3x\)
(Seen or implied)
M1: Applies \(vu' + uv'\) correctly for their \(u, u', v, v'\)
A1 isw: \(2\mathrm{e}^{2x}\sec 3x + 3\mathrm{e}^{2x}\sec 3x\tan 3x\)
| Scheme | Marks |
|---|---|
| Turning point \(\Rightarrow \dfrac{\mathrm{d}y}{\mathrm{d}x} = 0\) | |
| Hence, \(\mathrm{e}^{2x}\sec 3x(2 + 3\tan 3x) = 0\) | M1 |
| {Note \(\mathrm{e}^{2x} \ne 0\), \(\sec 3x \ne 0\), so \(2 + 3\tan 3x = 0\),} | |
| giving \(\tan 3x = -\tfrac{2}{3}\) | M1 |
| \(\Rightarrow 3x = -0.58800 \Rightarrow x = \{a\} = -0.19600\ldots\) | A1 |
| Hence, \(y = \{b\} = \mathrm{e}^{2(-0.196)}\sec(3 \times -0.196)\) | |
| \(= 0.812093\ldots = 0.812\ (3\text{sf})\) | A1 cao |
| (4) | |
| (11 marks) |
Notes
M1: Sets their \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = 0\) and factorises (or cancels) out at least \(\mathrm{e}^{2x}\) from at least two terms.
M1: \(\tan 3x = \pm k\); \(k \ne 0\)
A1: Either awrt \(-0.196^{c}\) or awrt \(-11.2^\circ\)
A1 cao: 0.812
Part (c): If there are any EXTRA solutions for \(x\) (or \(a\)) inside the range \(-\frac{\pi}{6} \lt x \lt \frac{\pi}{6}\), ie. \(-0.524 \lt x \lt 0.524\) or ANY EXTRA solutions for \(y\) (or \(b\)), (for these values of \(x\)) then withhold the final accuracy mark. Also ignore EXTRA solutions outside the range \(-\frac{\pi}{6} \lt x \lt \frac{\pi}{6}\), ie. \(-0.524 \lt x \lt 0.524\).