## Question 5:

### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Cone is uniform | B1 | Accept other suitable descriptions such as consistent density |

### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $y = 4 - \frac{1}{2}x$ | B1 | Correct equation of line |
| $\text{Mass} = \frac{1}{3}\pi \times 4^2 \times 8\rho = \frac{128\pi\rho}{3}$ | B1 | Correct mass. PI by correct completed integral. Condone missing $\rho$, condone cancelled $\pi$ if fraction used from outset |
| $\frac{128\pi\rho}{3} \times \bar{x} = \pi\rho\int_0^8 x\left(4-\frac{1}{2}x\right)^2 dx$ | M1 | Forms equation for centre of mass using their line and mass |
| $\frac{128}{3} \times \bar{x} = \frac{256}{3}$ | A1 | Completes integration, obtains correct equation for $\bar{x}$ |
| $\bar{x} = 2$ | R1 | Rigorous argument with correct notation throughout. Density must be included |

### Part (c)(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\tan\alpha = \frac{4}{2}$ | M1 | Forms equation to find angle. Accept $\tan\alpha = \frac{2}{4}$. PI by angle of 26.6° or 27° |
| $\alpha = 63°$ | A1 | Correct angle |

### Part (c)(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $F = \mu R$ | M1 | Uses expressions for $R$ and $F$ in friction equation or inequality |
| $mg\sin\alpha = \mu mg\cos\alpha$ | | |
| $\mu = \tan\alpha = 2$ | A1F | Correct coefficient. Only follow through $\tan\alpha = \frac{1}{2}$. PI by inequality or equation involving $\mu$ and 2 |
| $\therefore \mu > 2$ | A1 | Correct inequality. NMS 3/3 |

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