June 2022 Paper 2 Q21
21 Work out the values of \(x\) between \(0^\circ\) and \(360^\circ\) for which
\[2\tan^2 x = 3\]Give your answers to 1 decimal place.
You must show your working. [4 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| \(\tan x = \sqrt{\dfrac{3}{2}}\) or \(\tan x = \dfrac{\sqrt{6}}{2}\) or [50.7, 50.8] or 51 or [230.7, 230.8] or 231 | M1 | oe eg \(\tan^{-1}\sqrt{\dfrac{3}{2}}\) allow [1.22, 1.225] for \(\sqrt{\dfrac{3}{2}}\) |
| \(\tan x = -\sqrt{\dfrac{3}{2}}\) or \(\tan x = -\dfrac{\sqrt{6}}{2}\) or [\(-50.8\), \(-50.7\)] or \(-51\) or [129.2, 129.3] or 129 or [309.2, 309.3] or 309 | M1 | oe eg \(\tan^{-1} -\sqrt{\dfrac{3}{2}}\) allow [\(-1.225\), \(-1.22\)] for \(-\sqrt{\dfrac{3}{2}}\) |
| 50.8 and 129.2 and 230.8 and 309.2 with no other angles in range [0, 360] | A2 | A1 [50.7, 50.8] or 51 and [230.7, 230.8] or 231 or [129.2, 129.3] or 129 and [309.2, 309.3] or 309 |
| Alternative method 2 | ||
| \(\sin x = \sqrt{\dfrac{3}{5}}\) or \(\sin x = \dfrac{\sqrt{15}}{5}\) or [50.7, 50.8] or 51 [129.2, 129.3] or 129 | M1 | oe eg \(\sin^{-1}\sqrt{\dfrac{3}{5}}\) allow [0.77, 0.775] for \(\sqrt{\dfrac{3}{5}}\) |
| \(\sin x = -\sqrt{\dfrac{3}{5}}\) or \(\sin x = -\dfrac{\sqrt{15}}{5}\) or [\(-50.8\), \(-50.7\)] or \(-51\) [230.7, 230.8] or 231 or [309.2, 309.3] or 309 | M1 | oe eg \(\sin^{-1} -\sqrt{\dfrac{3}{5}}\) allow [\(-0.775\), \(-0.77\)] for \(-\sqrt{\dfrac{3}{5}}\) |
| 50.8 and 129.2 and 230.8 and 309.2 with no other angles in range [0, 360] | A2 | A1 [50.7, 50.8] or 51 and [129.2, 129.3] or 129 or [230.7, 230.8] or 231 and [309.2, 309.3] or 309 |
| Alternative method 3 | ||
| \(\cos x = \sqrt{\dfrac{2}{5}}\) or \(\cos x = \dfrac{\sqrt{10}}{5}\) or [50.7, 50.8] or 51 or [309.2, 309.3] or 309 | M1 | oe eg \(\cos^{-1}\sqrt{\dfrac{2}{5}}\) allow [0.63, 0.6325] for \(\sqrt{\dfrac{2}{5}}\) |
| \(\cos x = -\sqrt{\dfrac{2}{5}}\) or \(\cos x = -\dfrac{\sqrt{10}}{5}\) or [129.2, 129.3] or 129 or [230.7, 230.8] or 231 | M1 | oe eg \(\cos^{-1} -\sqrt{\dfrac{2}{5}}\) allow [\(-0.6325\), \(-0.63\)] for \(-\sqrt{\dfrac{2}{5}}\) |
| 50.8 and 129.2 and 230.8 and 309.2 with no other angles in range [0, 360] | A2 | A1 [50.7, 50.8] or 51 and [309.2, 309.3] or 309 or [129.2, 129.3] or 129 and [230.7, 230.8] or 231 |
Additional guidance
| Allow t for \(\tan x\) etc | |
| \(\tan x = \pm\sqrt{\dfrac{3}{2}}\) | M1M1 |
| Ignore any solutions outside the range [0, 360] | |
| All four solutions with extra solutions in range [0, 360] scores M1M1A1 eg 50.8 and 230.8 and 129.2 and 309.2 and 180 and 60 | M1M1A1 |
| For A1 there may be extra solutions in range eg1 50.77 and 230.8 and 180 eg2 50.8 and 230.8 and 129.2 and 90 | M1M0A1 M1M1A1 |
| If answer line is blank, award any marks gained in the working lines | |
| If correct angles are found in the working lines but only some are listed on the answer line award any M marks gained from the working lines award any A marks gained from the answer line eg1 working lines \(\tan x = \pm\sqrt{\dfrac{3}{2}} \quad\) 50.8 230.8 129.2 309.2 answer line 50.76 230.76 129.2 eg2 working lines \(\tan x = \sqrt{\dfrac{3}{2}} \quad\) 50.8 230.8 answer line 50.8 eg3 working lines \(\tan x = \sqrt{\dfrac{3}{2}} \quad\) 50.8 230.8 \(\tan x = -\sqrt{\dfrac{3}{2}} \quad\) 129.2 answer line 129.2 | M1M1 A1 M1M0A0 M1M1 A0 |
| Answers only (with no extra solutions in range) can score up to 4 marks 4 correct → 4 marks 3 correct → 3 marks 2 correct → 2 marks 1 correct → 1 mark | |
| M1M0A1 or M0M1A1 are possible eg1 \(\tan x = \sqrt{\dfrac{3}{2}} \quad\) 50.76 230.8 eg2 \(\tan x = -\sqrt{\dfrac{3}{2}} \quad\) 129.2 and 309.2 | M1M0A1 M0M1A1 |
| Embedded answers can score up to M1M1A1 |