# Question 5:

## Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $HB = 2r\cos\theta$ and $HP = a - 2r\cos\theta$ | B1 B1 | |
| $V = -\lambda mg(HP) - mg(HB)\cos\theta$ | M1 | Attempt at $V$ with their $HP$ and $HB$ |
| $V = -\lambda mg(a - 2r\cos\theta) - 2mgr\cos^2\theta$ | A1 | One term correct |
| $V = mg(2\lambda r\cos\theta - 2r\cos^2\theta - \lambda a)$ | A1 **[5]** | **AG www** correctly obtained. If M0 then SC B1 for one term correct |

## Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\dfrac{dV}{d\theta} = -2\lambda mgr\sin\theta + 4mgr\cos\theta\sin\theta = 0$ | M1 | Attempt at differentiation |
| $2mgr\sin\theta(2\cos\theta - \lambda) = 0$ | A1 | Correct derivative and equal to zero |
| $\frac{1}{2} \leq \cos\theta < 1$ | M1 | Comparing their $\cos\theta < 1$ or $\cos\theta \geq \frac{1}{2}$ - condone $\cos\theta \leq 1$ |
| $\Rightarrow 1 \leq \lambda < 2$ | A1 **[4]** | |

## Part (iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\dfrac{d^2V}{d\theta^2} = -2\left(\frac{3}{2}\right)mgr\cos\theta + 4mgr\cos2\theta$ | *M1 A1 | M1 Attempt to differentiate $V'$ (or first derivative test) – allow in terms of $\lambda$ |
| | M1 dep* | M1 sub. their first angle into $V''$ (maybe implied by later working) |
| $\sin\theta = 0 \Rightarrow V'' = mgr > 0\ \therefore$ stable | A1 | A1 correct (unsimplified) value of $V''$ and $> 0$ |
| | M1 dep* | M1 sub. their second angle into $V''$ (maybe implied by later working) |
| $\cos\theta = \frac{3}{4} \Rightarrow V'' = -\frac{7}{4}mgr < 0\ \therefore$ unstable | A1 **[6]** | A1 correct (unsimplified) value of $V''$ and $< 0$. If both values of $V''$ correct and correct conclusion then award B1 if no consideration of sign seen |

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