June 2025 Paper 2 Q21
21
(a)
\(\theta\) is an angle between \(270^\circ\) and \(360^\circ\)
\(\qquad \cos\theta = \dfrac{1}{5k} \quad\) and \(\quad \tan\theta = -3k \qquad\) where \(k\) is a positive constant.
Work out the value of \(\theta\). [2 marks]
(b)
Work out the values of \(x\) between \(0^\circ\) and \(360^\circ\) for which
\[8\cos^2 x - \cos x = 0\][3 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1: works out \(\sin\theta\) | ||
| \(\sin\theta = -\dfrac{3}{5}\) or \(\sin\theta = -\dfrac{3k}{5k}\) or \([-36.9, -36.8]\) or \(-37\) or [216.8, 216.9] or 217 | M1 | oe eg \(\sin\theta = -3k \div 5k\) or \(\sin\theta = -3k \times \dfrac{1}{5k}\) or \(\sin^{-1} -0.6\) |
| [323.1, 323.1301024] or 323 with no other value | A1 | |
| Alternative method 2: works out \(\tan\theta\) or uses \(\cos^{-1}\) | ||
| \(\tan\theta = -\dfrac{3}{4}\) or \(360 - \cos^{-1} 0.8\) | M1 | oe eg \(\tan\theta = -3 \times \dfrac{1}{4}\) or \(\tan^{-1} -0.75\) or \(360 - [36.8, 36.9]\) |
| [323.1, 323.1301024] or 323 with no other value | A1 | |
Additional guidance
| M1 may be awarded for correct work with no answer or incorrect answer, even if this is seen amongst multiple attempts | |
| Accept any indication of \(\theta\) eg \(x\) | |
| Alt 1 \(\ 5\sin\theta = -3\) with no further correct working | M0 |
| Alt 1 \(\ \cos\theta \times \tan\theta = -\dfrac{3}{5}\) with no further correct working | M0 |
| Answer | Mark | Comments |
|---|---|---|
| 90 and 270 and [82.8, 83] and [277, 277.2] with no other values in the range [0, 360] | B3 | B2 four correct values with one other value in the range [0, 360] or three correct values with no other value in the range [0, 360] B1 four correct values with two other values in the range [0, 360] or three correct values with one other value in the range [0, 360] or two correct values with no other value in the range [0, 360] |
Additional guidance
| Ignore any values outside the range [0, 360] | |
| If values are in the working lines and the answer line is blank, award marks from the working lines | |
| If values are in the working lines but only some are listed on the answer line, award marks from the answer line | |
| Embedded values B3 response becomes B2 B2 response becomes B1 B1 response becomes B0 |