June 2023 Paper 3 Q6
6 The first, third and fourth terms of an arithmetic progression are \(u_1\), \(u_3\) and \(u_4\) respectively, where
\(u_1 = 2\sin\theta, \qquad u_3 = -\sqrt{3}\cos\theta, \qquad u_4 = \frac{7}{2}\sin\theta,\)
and \(\frac{1}{2}\pi \lt \theta \lt \pi\).
| Scheme | Marks | AO |
|---|---|---|
| \(u_1 = a = 2\sin\theta\), \(u_3 = a + 2d = -\sqrt{3}\cos\theta\) and \(u_4 = a + 3d = \frac{7}{2}\sin\theta\) (for reference) | ||
| \(d = \frac{7}{2}\sin\theta + \sqrt{3}\cos\theta\) | B1* | 2.1 |
| \(-\sqrt{3}\cos\theta = 2\sin\theta + 2\left(\frac{7}{2}\sin\theta + \sqrt{3}\cos\theta\right) \Rightarrow\) \(\tan\theta = -\dfrac{\sqrt{3}}{3}\) | M1dep* | 3.1a |
| \(\theta = \frac{5}{6}\pi\) | A1 | 2.2a |
| [3] |
Notes
B1*: Forming a correct expression for \(d\) (or a correct equation containing \(d\)) e.g. \((d =)\,\frac{1}{2}(-\sqrt{3}\cos\theta - 2\sin\theta)\), \((d =)\,\frac{1}{3}\left(\frac{7}{2}\sin\theta - 2\sin\theta\right)(= 0.5\sin\theta)\)
Can be implied e.g. \(\frac{7}{2}\sin\theta = 2\sin\theta + 3(\ldots)\) seen
Can be implied by a correct equation for \(\theta\)
M1dep*: Obtaining an equation of the form \(\tan\theta = k\) from a trigonometric equation which initially had 3 sine and 1 cosine terms or 2 sine and 2 cosine terms e.g. if correct \(\frac{7}{2}\sin\theta = 2\sin\theta + 3\left(\frac{7}{2}\sin\theta + \sqrt{3}\cos\theta\right)\)
A1: Condone \(-\dfrac{\pi}{6}\) stated too but A0 if any other value given in the interval \(\frac{1}{2}\pi \lt \theta \lt \pi\) (but ignore any values that are given outside this range)
Exact answer must be seen at some stage
| Scheme | Marks | AO |
|---|---|---|
| \(S_{100} = \frac{100}{2}[2(2\sin\theta) + (100 - 1)d]\) | B1ft | 1.2 |
| \(d = \frac{7}{2}\sin\left(\frac{5}{6}\pi\right) + \sqrt{3}\cos\left(\frac{5}{6}\pi\right)\ \left(= \frac{1}{4}\right)\) | B1ft | 1.1 |
| \(S_{100} = 1337.5\) | B1 | 2.2a |
| [3] |
Notes
B1ft: Correct formula for the sum of an AP with \(a = 2\sin\theta\) (with either \(\theta\) or their value of \(\theta\) substituted) and either \(d\) or their value of \(d\) substituted or their expression for \(d\)
Follow through their values of \(\theta\) and \(d\) if used provided \(\frac{100}{2}[2(2\sin\theta) + (100 - 1)d]\) implied
B1ft: Correct expression for \(d\) using their \(\theta\) (e.g. \(d = \frac{1}{2}(-\sqrt{3}\cos\theta - 2\sin\theta)\), \(d = \frac{1}{3}\left(\frac{7}{2}\sin\theta - 2\sin\theta\right)\))
Follow through their value of \(\theta\) only
B1: www – must have come from \(\theta = \frac{5}{6}\pi\) correctly derived in (a) oe (not for 1338 or 1340 unless 1337.5 seen so isw once 1337.5 (oe e.g. \(\frac{2675}{2}\)) seen)
Correct answer with no working scores all 3 marks