## Question 3:

| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| Either equation of motion | M1 | All terms required |
| Motion up the road: $F_U = R + 900g\sin\theta$ | A1 | One equation correct unsimplified |
| Motion down the road: $F_D = R - 900g\sin\theta$ | A1 | Both equations correct unsimplified |
| Use of $P = Fv$ to form at least one equation in $R$ and $v$ | M1 | |
| $\dfrac{10800}{v} = R + 900g\sin\theta$ and $\dfrac{10800}{2v} = R - 900g\sin\theta$ | A1 | Correct unsimplified $(90g\sin\theta = 180)$ |
| Solve simultaneous equations $\Rightarrow \dfrac{10800}{2v} = 1800g \times \frac{1}{49}$ | DM1 | Solve simultaneous equations, both containing $R$ and $v$, for $R$ or $v$. Dependent on 2 preceding M marks |
| $v = 15$ only | A1 | One correct value |
| $R = 540$ only | A1 | Both values correct |
| **Total** | **(8)** | **[8]** |

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## Question 4a:

| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| Mass ratio: large $16\pi$, small $4\pi$, $L$: $12\pi$; c of m from $B$: large $4a$, small $3a$, $L$: $x$ | B1 | Correct ratios and distances |
| Moments about $B$: $16\pi \times 4a - 4\pi \times 3a = 12\pi \times x$ | M1 | Or equivalent from another point. Require all terms. Condone sign errors. Dimensionally correct |
| Correct unsimplified | A1 | |
| $x = \dfrac{64a - 12a}{12} = \dfrac{52a}{12} = \dfrac{13}{3}a$ **\*Given Answer\*** | A1 | Obtain **given answer** from correct working |
| **Total** | **(4)** | |

## Question 4b:

| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| Moments about $B$: $\dfrac{13}{3}a\cos\theta \times M = 4\sqrt{2}a\cos(45+\theta) \times M(1+k)$ | M1 | Require all terms. Dimensionally correct. Condone sign errors |
| Unsimplified equation with at most one error | A1 | |
| Correct unsimplified | A1 | |
| Substitute for trig and solve: $\dfrac{13}{3} \times \dfrac{4}{5} = (1+k) \times 4\sqrt{2}\left(\dfrac{1}{\sqrt{2}}\times\dfrac{4}{5} - \dfrac{1}{\sqrt{2}}\times\dfrac{3}{5}\right)$ | DM1 | |
| $\dfrac{13}{3} = 1 + k$, $\ k = \dfrac{10}{3}$ | A1 | (3.3 or better) |
| **Total** | **(5)** | |

**4b alt:**

| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| Moments about $B$: $\dfrac{13}{3}a\cos\theta \times M = a\cos\theta \times M(1+k)$ | M1 | Require all terms. Dimensionally correct. Condone sign errors |
| Unsimplified equation with at most one error | A1 | |
| Correct unsimplified | A1 | |
| Substitute for trig and solve for $k$: $\dfrac{13}{3} = 1 + k$ | DM1 | |
| $k = \dfrac{10}{3}$ | A1 | (3.3 or better) |

**4b alt (Moments about $O$):**

| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| Moments about $O$: $-\dfrac{1}{3}a \times M + kM \times 4a = (kM + M)\bar{x}$ | M1 | Require all terms. Dimensionally correct. Condone sign errors |
| $(k+1)\bar{x} = \left(4k - \dfrac{1}{3}\right)a$ | A1 | Correct unsimplified |
| $\dfrac{\bar{x}}{OD} = \dfrac{3}{4}$ | DM1 | |
| $k = \dfrac{10}{3}$ (3.3 or better) | A1 | |

**4b alt (New c of m at $G$, Moments about $G$):**

| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| New c of m at $G$ where $\dfrac{OG}{OD} = \tan\theta$, $OG = 3a$ | M1 | |
| Moments about $G$: $Mg\left(3a + \dfrac{a}{3}\right) = kMg(4a - 3a)$ | DM1 | Require all terms. Dimensionally correct. Condone sign errors |
| Unsimplified equation with at most one error | A1 | |
| Correct unsimplified | A1 | |
| $k = \dfrac{10}{3}$ (3.3 or better) | A1 | |

**4b alt (Moments about $G$, Moments about $D$):**

| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| New c of m at $G$ where $\dfrac{OG}{OD} = \tan\theta$, $OG = 3a$ | M1 | |
| Moments about $D$: $Mg\left(3a + \dfrac{a}{3}\right)\cos\theta = kMg(4a - 3a)\cos\theta$ | DM1 | Require all terms. Dimensionally correct. Condone sign errors |
| Unsimplified equation with at most one error | A1 | |
| Correct unsimplified | A1 | |
| $k = \dfrac{10}{3}$ (3.3 or better) | A1 | |

**4b alt (Moments about $G$, general):**

| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| Moments about $G$: $M\left(\dfrac{a}{3} + OG\right) = kM(4a - OG)$ | M1 | Require all terms. Dimensionally correct. Condone sign errors |
| Unsimplified equation with at most one error | A1 | |
| $OG(1+k) = a\left(4k - \dfrac{1}{3}\right)$, $\left(OG = \dfrac{a(12k-1)}{3(k+1)}\right)$ | A1 | Correct unsimplified |
| $\dfrac{OG}{OD} = \dfrac{3}{4}$ | DM1 | |
| $k = \dfrac{10}{3}$ | A1 | |
| Other alternatives: Moments equation M1A1A1; Use angle and solve for $k$: M1A1 | | |
| **Total** | **[9]** | |