# Question 9:

## Part (i)(a):

| Answer | Mark | Guidance |
|--------|------|----------|
| $6\pi - \alpha$ | B1 | Allow unsimplified equiv. Allow in degrees ie $1080 - \alpha$, or equiv. |
| | **[1]** | |

## Part (i)(b):

| Answer | Mark | Guidance |
|--------|------|----------|
| Use period of $6\pi$ to make valid attempt at solution | M1 | Allow any unsimplified equiv. Allow in degrees ie $540 - \alpha$, or equiv. |
| $3\pi - \alpha$ | A1 | Must be simplified, and in radians. Allow $a$ or alpha for $\alpha$. |
| | **[2]** | |

## Part (ii):

| Answer | Mark | Guidance |
|--------|------|----------|
| Correct graph shape for $y = k\sin\frac{1}{3}x$ | M1 | Must be one complete (positive) sin cycle, starting at $(0,0)$ and clearly intended to have a final root at the same $x$-value as the end point of the given curve. Allow the curve to extend beyond this final root. Allow any amplitude. Condone a slightly inaccurate $x$-intercept for the middle root. Condone poor curvature, including overly straight sections and stationary values that are pointed rather than curved. |
| Fully correct graph | A1 | Curve should clearly be intended to have an amplitude that is half of the given curve, but explicit labels of 1 and -1 are not required. A0 if an incorrect scale is given. A smooth, symmetrical curve is now required, with correct $x$-intercepts clearly intended. Ignore any scale, correct or incorrect, on the $x$-axis. |
| | **[2]** | |

## Part (iii):

| Answer | Mark | Guidance |
|--------|------|----------|
| $\tan\frac{1}{3}x = 2$ | B1 | Allow B1 for correct equation even if no, or an incorrect, attempt to solve. Give BOD on notation e.g. $\frac{\sin}{\cos}(\frac{1}{3}x) = 2$. If $\tan\frac{1}{3}x = 2$ is obtained fortuitously from incorrect algebra then mark as B0M1A0A0. |
| $\frac{1}{3}x = 1.107,\ 4.249$ | M1 | Attempt to solve $\tan\frac{1}{3}x = k$. Attempt $3\tan^{-1}(k)$, any non-zero numerical $k$. M0 for $\tan^{-1}(3k)$. Allow if attempted in degrees not radians. M1 could be implied rather than explicit. |
| | A1 | Obtain $3.32$. Must be radians and not degrees. Allow answers in range $[3.32, 3.33]$. A0 for answer given as a multiple of $\pi$. |
| $x = 3.32,\ 12.7$ | A1 | Obtain $12.7$. Must be radians and not degrees. Allow answers in range $[12.7, 12.8]$. A0 for answer given as a multiple of $\pi$. Max of 3/4 if additional solutions given in range $[0, 6\pi]$ but ignore any solutions outside of this range. Answer only with no method shown is 0/4. |
| | | **Alt method:** **B1** Obtain $5\sin^2\frac{1}{3}x = 4$ or $5\cos^2\frac{1}{3}x = 1$. **M1** Attempt to solve $\sin^2\frac{1}{3}x = k$ or $\cos^2\frac{1}{3}x = k$ (allow M1 if just the positive square root used). **A1** Obtain $3.32$. **A1** Obtain $12.7$ (max 3/4 if additional solutions in range). |
| | **[4]** | |