June 2023 Paper 2 Q17
17 In this question you must show detailed reasoning.
Solve the equation \(2\sin x + \sec x = 4\cos x\), where \(-\pi < x < \pi\). [6]
| Scheme | Marks | AO |
|---|---|---|
| divide through by \(\cos x\) to obtain \(2\tan x + \sec^2 x = 4\) | B1 | 2.1 |
| \(2\tan x + \tan^2 x + 1 = 4\) | M1* | 3.1a |
| \(\tan^2 x + 2\tan x - 3\ [= 0]\) | A1 | 1.1 |
| \(\tan x = 1\) or \(-3\) | M1*dep | 1.1 |
| \([x =]\ -1.24905\) to \(-1.249\) or \(-1.25\) or \(-1.2\) \([x =]\ 1.8925\) to \(1.893\) or \(1.89\) or \(1.9\) | A1 | 3.2a |
| \([x =]\ \frac{\pi}{4}\) or 0.785 to 0.7854 or 0.79 \([x =]\ -\frac{3\pi}{4}\) or \(-2.3562\) to \(-2.356\) or \(-2.36\) or \(-2.4\) | A1 | 2.2a |
| [6] |
Notes
M1*: use of Pythagoras to obtain equation in \(\tan x\) only; allow 1 sign error
M1*dep: 2 values obtained for \(\tan x\) from their quadratic
A1: any two correct
A1: all four correct and no extra values in range; ignore correct extra values outside range but A0 if incorrect values outside range
alternatively
| Scheme | Marks |
|---|---|
| multiply through by \(\cos x\) to obtain \(2\sin x\cos x + 1 = 4\cos^2 x\) | B1 |
| \(\sin 2x + 1 = 2\cos 2x + 2\) | M1* |
| \(5\cos^2 2x + 4\cos 2x\ [= 0]\) NB square both sides: \(\sin^2 2x = 4\cos^2 2x + 4\cos 2x + 1\) oe | A1 |
| \(\cos 2x = 0\) or \(-0.8\) 2 values obtained for \(\cos 2x\) from their quadratic | M1dep* |
| \([x =]\ -1.24905\) to \(-1.249\) or \(-1.25\) or \(-1.2\) \([x =]\ 1.8925\) to \(1.893\) or \(1.89\) or \(1.9\) | A1 |
| \([x =]\ \frac{\pi}{4}\) or 0.785 to 0.7854 or 0.79 \([x =]\ -\frac{3\pi}{4}\) or \(-2.3562\) to \(-2.356\) or \(-2.36\) or \(-2.4\) | A1 |
M1*: use of double angle formulae, allow 1 sign error
A1: or \(\sqrt{5}\cos(2x + 0.4636\ldots) = -1\)
or \(\sqrt{5}\sin(2x - 1.1071\ldots) = 1\)
M1dep*: \(\cos(2x + 0.4636\ldots) = -\frac{1}{\sqrt{5}}\) or \(\sin(2x - 1.1071\ldots) = \frac{1}{\sqrt{5}}\)
A1: any two correct
A1: all four correct and no extra values in range; ignore correct extra values outside range but A0 if incorrect values outside range