# Question 9:

## Part (a):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $a=2$ **or** $b=7$ | B1 | Uses model to obtain correct value for $a$ or $b$ |
| $a=2$ **and** $b=7$ | B1 | Uses model to obtain correct values for both $a$ and $b$ |

## Part (b):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $V=(\pi)\int x^2\frac{dy}{dt}dt=\int(2+3\sin 2t)^2(-7\sin t)\,dt$ | M1 | |
| $=-7(\pi)\int(4\sin t+12\sin 2t\sin t+9\sin^2 2t\sin t)\,dt$ $=-7(\pi)\int(4\sin t+24\sin^2 t\cos t+36\sin^3 t\cos^2 t)\,dt$ | M1 | |
| $=-7(\pi)\left[-4\cos t+8\sin^3 t-12\cos^3 t+\frac{36}{5}\cos^5 t\right]$ | A1ft, A1 | |
| Cylinder volume $=\pi\times 2^2\times 4.5=(18\pi)$ | B1 | |
| Total Volume $= V +$ cylinder volume $=-7\pi\left[-4\cos t+8\sin^3 t-12\cos^3 t+\frac{36}{5}\cos^5 t\right]_{\frac{\pi}{2}}^{0}+\pi\times 2^2\times 4.5$ | ddM1 | NB this is $\frac{588}{5}\pi+18\pi$ |
| $426\ (\text{cm}^3)$ | A1 | |

## Part (c):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Any one of: the vase may not be completely smooth; the vase may not be symmetrical; the measurements may not be accurate; the equation of the curve may not be a suitable model; the thickness of the sides has not been considered; the base may have a dimple (may not be completely flat) | B1 | |

## Question (b):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Uses parametric curve and applies $(\pi)\int x^2 \frac{dy}{dt}\,dt$ | **M1** | $\pi$ symbol may be missing; missing $dt$ at end condoned. Must see squaring of $x$ and finding $\frac{dy}{dt}$ where $x=(a+3\sin 2t)$ and $\frac{dy}{dt}=k\sin t$, with their $a$ and $b$ or letters $a$ and $b$ in integral |
| Expands and applies $\sin 2t = 2\sin t\cos t$ at least once, making progress to integrable form: $\pm(\pi)\int(a^2b\sin t + 6ab\sin 2t\sin t + 9b\sin^2 2t\sin t)\,dt$ $= \pm(\pi)\int a^2b\sin t + 12ab\sin^2 t\cos t + 36b\sin^3 t\cos^2 t\,dt$ | **M1** | |
| With correct values of $a$ and $b$: $\pm(\pi)\int(28\sin t + 84\sin 2t\sin t + 63\sin^2 2t\sin t)\,dt$ $= \pm(\pi)\int(28\sin t + 168\sin^2 t\cos t + 252\sin^3 t\cos^2 t)\,dt$ | **A1ft** | Dependent on **both** M marks. At least 2 terms integrated correctly; follow through on $a$ and $b$, but must now be numerical |
| $\pm(\pi)\left[-a^2b\cos t + 4ab\sin^3 t - \frac{36}{3}b\cos^3 t + \frac{36b}{5}\cos^5 t\right]$ $= \pm(\pi)\left[28\cos t - 56\sin^3 t + 84\cos^3 t - \frac{252}{5}\cos^5 t\right]$ | **A1** | All correct from correct values of $a$ and $b$ |
| Uses model to deduce correct volume of cylinder; need not be simplified. Cylinder volume must have been obtained from $\pi\times2^2\times4.5$, condone $\pi\times4^2\times4.5$. Both volumes must be positive when combined. | **B1, ddM1** | **ddM1** dependent on **both** previous M marks. Applies correct limits (either way round) and adds to volume of cylinder |
| Correct volume, awrt $426\,(\text{cm}^3)$ | **A1** | Units not required but if given must be correct |

---

## Question (c):

| Answer/Working | Mark | Guidance |
|---|---|---|
| See scheme. Award for a correct statement about the shape. | **B1** | If more than one statement, ignore incorrect statements as long as none contradict the correct statement. Comments **not** worthy of marks: cross section not a circle; vase is not solid; cylinder might not be vertical; depends on material; vase may not have same density throughout |