June 2018 Paper 2 Q22
22
(a) \(8 \cos x + 5 \sin x = 0 \quad\) where \(\ 90^\circ \lt x \lt 180^\circ\)
Work out the size of angle \(x\). [3 marks]
(b) \(6 \sin^2 x + 4 \cos^2 x \equiv \mathrm{A} + \mathrm{B} \cos^2 x \qquad\) where A and B are integers.
Work out the values of A and B.
You must show your working. [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| Divides trigonometric expression by \(\cos x\) or rearranges equation | M1 | eg \(8\,\dfrac{\cos x}{\cos x} + 5\,\dfrac{\sin x}{\cos x}\) or \(8 + 5\,\dfrac{\sin x}{\cos x}\) or \(8 \cos x = -5 \sin x\) or \(\dfrac{\cos x}{\sin x} = -\dfrac{5}{8}\) or \(8\,\dfrac{\cos x}{\cos x} = -5\,\dfrac{\sin x}{\cos x}\) or \(5 \tan x = -8\) |
| \(\tan x = -\dfrac{8}{5}\) or \(\tan x = -1.6\) or \(\cos x = -\dfrac{5}{\sqrt{89}}\) or \(-57.9\ldots\) or \(-58\) | A1 | oe eg \(\tan^{-1} -1.6\) may be implied by final answer |
| 122.(0…) with no other angle | A1 | |
| Alternative method 2 | ||
| Divides trigonometric expression by \(\cos x\) or rearranges equation | M1 | eg \(8\,\dfrac{\cos x}{\cos x} + 5\,\dfrac{\sin x}{\cos x}\) or \(8 + 5\,\dfrac{\sin x}{\cos x}\) or \(8 \cos x = -5 \sin x\) or \(\dfrac{\cos x}{\sin x} = -\dfrac{5}{8}\) or \(8\,\dfrac{\cos x}{\cos x} = -5\,\dfrac{\sin x}{\cos x}\) or \(5 \tan x = -8\) |
| \(\sin x = \dfrac{8}{\sqrt{89}}\) or 57.9… or 58 | A1 | oe eg \(\sin^{-1} \dfrac{8}{\sqrt{89}}\) may be implied by final answer |
| 122.(0…) with no other angle | A1 | |
Additional guidance
| Allow division of expression by \(\mathrm{k}\cos x\) eg (k = 8) \(\ 1 + \dfrac{5 \sin x}{8 \cos x}\) | M1 |
| Answer only 122.(0…) | M1A2 |
| Embedded answer 122.(0…) | M1A1A0 |
| Answer only 121.9 | Zero |
| If working seen, use the alt method for the working seen | |
| Answer only \(-58\) (BOD alt 1) | M1A1A0 |
| Answer only 58 (BOD alt 2) | M1A1A0 |
| Allow cos for \(\cos x\) etc | |
| Allow c for \(\cos x\) etc |
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| \(6(1 - \cos^2 x) + 4 \cos^2 x\) or \(6 - 6 \cos^2 x + 4 \cos^2 x\) or \(2(1 - \cos^2 x) + 4\) or \(6 - 2 \cos^2 x\) | M1 | oe expression in terms of \(\cos^2 x\) |
| \(\mathrm{A} = 6\) and \(\mathrm{B} = -2\) with no incorrect working | A1 | |
| Alternative method 2 | ||
| \(\mathrm{A} \sin^2 x + \mathrm{A} \cos^2 x + \mathrm{B} \cos^2 x\) and \(\mathrm{A} = 6\) and \(\mathrm{A} + \mathrm{B} = 4\) | M1 | |
| \(\mathrm{A} = 6\) and \(\mathrm{B} = -2\) with no incorrect working | A1 | |
Additional guidance
| \(\mathrm{A} = 6\) and \(\mathrm{B} = -2\) with no working | M1A1 |