## Question 10:
| Crossing points at 0 and near $+2$ and $-2$ | B1 | |
| Rotational symmetry (or close to) | B1 | |
| General shape of positive cubic; do not condone incorrect curvature | B1 | $y=x$ scores B0B1B0; use judgement on candidate's intention |

## Question 11a:
| $x\cos x$ or unsimplified $\sin x + x\cos x - \sin x$ | B1 | Differentiating |
| Setting $x\cos x = 1$ | M1 | |
| Finishing convincingly | A1 | If no differentiation, M1 not available |

## Question 11b:
| Attempting to differentiate $\frac{1}{x} - \cos x$ or $x\cos x - 1$ | M1 | |
| Setting up iteration function **with subscripts** | A1 | Must include subscripts |
| Choosing suitable starting value (accept values between 3.6 and 6.1) | M1 | |
| Root $4.9172$ (4dp) | A1 | Value outside range giving 4.9172 gets M1A1; wrong answer gets M0A0 |

## Question 11c:
| Comment more than just 'it is close to another root' | B1 | |

## Question 12a:
| Differentiating $y$ wrt $\theta$; may appear as $dy/d\theta$ or numerator of $dy/dx$ | M1 | Ignore $dx/d\theta$ unless used incorrectly in subsequent work |
| Putting expression $= 0$ AND using $\sin 2\theta = 2\sin\theta\cos\theta$ or $\cos 2\theta$ in correct form; if $dx/d\theta$ used incorrectly give M0 | M1 | |
| Getting terms on one side and factorising | M1 | Dividing through by $\sin\theta$ costs all A marks |
| $\sin\theta = 0$ and $\cos\theta = -\frac{1}{2}$; 3rd and 4th M1s can be implied (BOD) by 2 correct solutions | M1 | 4th M1 dependent on previous M1 |
| Exact values of $\theta$ and coordinates (www marks) | A marks | e.g. $\frac{4-\sqrt{3}}{2}$ for $2-\frac{\sqrt{3}}{2}$, $\frac{4+\sqrt{3}}{2}$ for $2+\frac{\sqrt{3}}{2}$ |

*Note: CHECK PAGE 11 for working*

## Question 12b:
| $x = 2$ | B1 | Only acceptable answer |