# Question 3:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Use of $\mathbf{I} = m\mathbf{v} - m\mathbf{u}$ | M1 | Accept equivalent e.g. $\mathbf{I}+m\mathbf{u}=m\mathbf{v}$. Dimensionally correct, must be using subtraction. Use of 7 in place of velocity in impulse-momentum equation is M0 unless they recover |
| $0.2(\mathbf{v} - 4\mathbf{i} + 3\mathbf{j}) = \lambda(\mathbf{i}+\mathbf{j})$ leading to $((x-4)\mathbf{i}+(y+3)\mathbf{j} = 5\lambda\mathbf{i}+5\lambda\mathbf{j})$ | A1 | Correct unsimplified vector equation or pair of separate equations for $\mathbf{i}$ and $\mathbf{j}$ components. Condone column vectors |
| Use of Pythagoras for the speed | M1 | Correct use of Pythagoras and 49 for their speed |
| $x^2 + y^2 = 49$ | A1 | Correct unsimplified equation for their $x$, $y$ |
| Form quadratic in $x$, $y$ or $\lambda$ and solve for $\lambda$ | DM1 | Dependent on both previous M marks. $x^2 + -(x-7)^2 = 49$ or $(y+7)^2 + y^2 = 49$ or $(5\lambda+4)^2 + (5\lambda-3)^2 = 49$ |
| $\lambda = \dfrac{3}{5}$ or $\lambda = -\dfrac{4}{5}$ | A1 | Or equivalent |
| **Total: (6)** | | |

### Alternative Method (Question 3):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Form vector triangle | M1 | Dimensionally correct. Allow incorrect configuration |
| Correct triangle and correct lengths | A1 | In speeds or momentum but not a mixture |
| Use scalar product to find cosine of angle | M1 | Or equivalent method |
| $\cos\theta = -\dfrac{1}{5\sqrt{2}}$ | A1 | Allow $\pm$ |
| Form equation in $\lambda$: $(2\lambda^2 + .4\lambda - 0.96 = 0)$ | DM1 | e.g. by use of cosine rule. Dependent on the first 2 M marks |
| $\lambda = \dfrac{3}{5}$ or $\lambda = -\dfrac{4}{5}$ | A1 | Or equivalent |
| **Total: (6)** | | |

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