## Question 4(a):

| Working/Answer | Mark | Guidance |
|---|---|---|
| $M\bar{y} = (\rho)\pi\int_0^a y(a^2 - y^2)\,dy$ | M1 A1 | — |
| $M\bar{y} = (\rho)\pi\left[\frac{a^2y^2}{2} - \frac{y^4}{4}\right]_0^a$ | A1 | — |
| $\left(M\bar{y} = (\rho)\pi\frac{a^4}{4}\right)$ | — | — |
| $\bar{y} = \frac{(\pi\rho)\frac{a^4}{4}}{(\pi\rho)\frac{2a^3}{3}} = \frac{3a}{8}$ | M1A1* | S.C. Clear use of $r=1$ with $a$ substituted at end can score max M1A0M1A0 |

**(5)**

## Question 4(b):

| Working/Answer | Mark | Guidance |
|---|---|---|
| $(k+1)m\bar{x} = kma \left(\Rightarrow \bar{x} = \frac{ka}{k+1}\right)$ | M1A1 | — |
| $(k+1)m\bar{y} = m\times\frac{3}{8}a \left(\Rightarrow \bar{y} = \frac{3a}{8(k+1)}\right)$ | A1 | — |
| $\tan\alpha = \frac{\bar{x}}{\bar{y}}$ | M1 | — |
| $k = \frac{3\sqrt{3}}{8}$ | A1 | — |

**(5) [10]**

**Alt b:**

| Working/Answer | Mark | Guidance |
|---|---|---|
| Moments about $O$: $m\times\frac{3a}{8}\sin\alpha = kma\times\cos\alpha$ | M1A2 | — |
| $k = \frac{3\sqrt{3}}{8}$ | M1A1 | — |

# Question (Centre of Mass - Part a):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $(\pi)\int_0^r x^2 y \, dy$ | M1 | With or without $\pi$. Must be dimensionally correct with integrand of form $y(a^2 - y^2)$. Limits not needed. |
| $\int_0^a y(a^2 - y^2) \, dx$ | A1 | Correct integral. Limits not needed. |
| Correct integration | A1 | Limits not needed. |
| $\bar{y} = \dfrac{\int_0^a \pi x^2 y \, dy}{\frac{2}{3}\pi a^3}$ | M1 | $\pi$ in numerator and denominator or in neither. |
| Correct given result with no errors seen | A1 | **Cso** |

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# Question (Centre of Mass - Part b):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Dimensionally correct moments equation for $\bar{x}$ or $\bar{y}$ | M1 | First 3 marks can be awarded for equations only seen as part of a vector equation. |
| Correct equation in $\bar{x}$ or $\bar{y}$ | A1 | Give BOD on use of $x$ and/or $y$. |
| Correct equations for $\bar{x}$ and $\bar{y}$ | A1 | |
| Dividing to form equation in $\tan\alpha$ and solve for $k$ | M1 | Trig must be right. |
| $k = \dfrac{3\sqrt{3}}{8}$ | A1 | Any correct exact form. |

**ALT b) - Taking moments about O:**

| Answer/Working | Mark | Guidance |
|---|---|---|
| Dimensionally correct moments equation about $O$ | M1 | |
| Moments equation with at most one error | A1 | |
| Fully correct equation | A1 | |
| Solve for $k$ | M1 | |
| c.a.o. | A1 | |

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