## Question 9:

### Part (i):

| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{2\pi}{a}$ | B1 | State $\frac{2\pi}{a}$ — any exact equiv; allow in degrees i.e. $\frac{360}{a}$; B0 if given as a range |

### Part (ii):

| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{1}{5}\pi a = \pi - \frac{2}{5}\pi a$; hence $a = \frac{5}{6}$ | M1 | Attempt to use symmetry of sine curve, or equiv — allow any correct relationship between two solutions in radians or degrees |
| $k = \frac{1}{2}\sqrt{3}$ | A1 | Obtain $a = \frac{5}{6}$ — any exact equiv; CWO |
| | A1 | Obtain $k = \frac{1}{2}\sqrt{3}$ — any exact equiv, not involving sin; A0 if from incorrect $a$ |
| **Alternative:** $\sin(\frac{4}{5}\pi a) = \sin(\frac{2}{5}\pi a)$; using $\sin 2A$ identity: $2\cos(\frac{4}{5}\pi a)=1$, $a=\frac{5}{6}$, $k=\frac{1}{2}\sqrt{3}$ | M1, A1, A1 | As far as $2\cos(\frac{4}{5}\pi a) = 1$ for M1 |

## Question (iii):

**Main Solution:**

| Answer/Working | Marks | Guidance |
|---|---|---|
| $\tan(ax) = \sqrt{3}$ | B1 | Allow B1 for correct equation even if no, or incorrect, attempt to solve. Give BOD on notation e.g. $\frac{\sin}{\cos}(ax)$ as long as correct equation is seen or implied. Allow $\tan(ax) - \sqrt{3} = 0$ or equiv. Allow B1 for identifying $ax = \frac{\pi}{3}$ or 60° even if equation in $\tan(ax)$ not seen – M1 would then be awarded for an attempt at $x$ |
| Attempt to solve $\tan(ax) = c$ | M1 | Attempt $\frac{1}{a}\tan^{-1}(c)$, any non-zero numerical $c$. M0 for $\tan^{-1}\left(\frac{c}{a}\right)$. Allow if attempted in degrees not radians. M1 could be implied rather than explicit. M1 can be awarded if using a numerical value for $a$ |
| $x = \frac{\pi}{3a}$ | A1 | Must be in radians not degrees. Allow any exact equiv e.g. $\frac{\pi}{a} \cdot$ as long as intention clear – but A0 if this is then given as $\frac{a\pi}{3}$ |
| $x = \frac{4\pi}{3a}$ | A1 | Must be in radians not degrees. Allow any exact equiv e.g. $\frac{\frac{4\pi}{3}}{a}$ as long as intention clear – but A0 if given as $\frac{4a\pi}{3}$. Allow $\frac{\pi}{3a} + \frac{\pi}{a}$ unless then incorrectly simplified. If more than two solutions given, mark the two smallest and ISW the rest. e.g. $\frac{\pi}{3a}, \frac{4\pi}{3a}, \frac{7\pi}{3a}$ would be A1A1 but $\frac{\pi}{3a}, \frac{2\pi}{3a}, \frac{4\pi}{3a}$ would be A1A0 |
| **[4]** | | |

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**Alternative Solution:**

| Answer/Working | Marks | Guidance |
|---|---|---|
| $\sin^2(ax) = 3\cos^2(ax)$, $4\sin^2(ax) = 3$ or $4\cos^2(ax) = 1$, $\sin(ax) = \pm\frac{\sqrt{3}}{2}$ or $\cos(ax) = \pm\frac{1}{2}$ | B1 | Any correct, simplified equation in a single trig ratio |
| Attempt to solve $\sin^2(ax) = c$ or $\cos^2(ax) = c$ | M1 | Allow M1 if just the positive square root used. Attempt $\frac{1}{a}\sin^{-1}(\sqrt{c})$ or $\frac{1}{a}\cos^{-1}(\sqrt{c})$, any non-zero numerical $c$. M0 for $\sin^{-1}\left(\frac{\sqrt{c}}{a}\right)$, M0 for $\cos^{-1}\left(\frac{\sqrt{c}}{a}\right)$. Allow if attempted in degrees not radians. M1 could be implied. M1 can be awarded if using numerical value for $a$ |
| $x = \frac{\pi}{3a}$ | A1 | Must be in radians not degrees. Allow any exact equiv as long as intention clear – but A0 if given as $\frac{a\pi}{3}$ |
| $x = \frac{4\pi}{3a}$ | A1 | Allow any exact equiv as long as intention clear – but A0 if given as $\frac{4a\pi}{3}$. Allow a correct answer still in two terms, unless then incorrectly simplified. If more than two solutions given, mark two smallest and ISW the rest. e.g. $\frac{\pi}{3a}, \frac{4\pi}{3a}, \frac{7\pi}{3a}$ would be A1A1 but $\frac{\pi}{3a}, \frac{2\pi}{3a}, \frac{4\pi}{3a}$ would be A1A0 |