# Question 5:

## Part (a)(i)
| Working | Mark | Guidance |
|---------|------|----------|
| $\frac{(b-2a)-a}{b-a} = \frac{1}{3}$ or $\frac{b-(b-2a)}{b-a} = \frac{2}{3}$; leading to $b = 4a$ | M1 | For a correct probability equation $= \frac{1}{3}$ (or $= \frac{2}{3}$) in terms of $a$ and $b$ |
| $E(X) = \frac{a+4a}{2} = \frac{5a}{2}$ | M1, A1cso | 2nd M1 for use of $E(X) = \frac{a + '4a'}{2}$ oe; correct solution with no errors seen |

## Part (a)(ii)
| Working | Mark | Guidance |
|---------|------|----------|
| $[P(X > b - 4a) = P(X > 0) =]\ 1$ | B1 | |

## Part (b)(i)
| Working | Mark | Guidance |
|---------|------|----------|
| $\frac{(c-3)^2}{12} = 3c - 9$ | M1 | 1st M1 for $\frac{(c-3)^2}{12} = 3c-9$ |
| $c^2 - 42c + 117 = 0 \rightarrow (c-3)(c-39) = 0$ | M1 | 2nd M1 for rearranging correctly to form $3TQ = 0$ and attempt to solve, or $\frac{c-3}{12} = 3$ oe |
| $\underline{c=39}$ | A1 | If more than one value given and $c=39$ not clearly selected then A0; allow [3,39] |

## Part (b)(ii)
| Working | Mark | Guidance |
|---------|------|----------|
| $P(2Y - 7 < 20 - Y) = P(Y < 9)$ | M1 | For rearranging to (or using) $P(Y < 9)$ |
| $\frac{9-3}{'39'-3} = \frac{1}{6}$ awrt **0.167** | A1ft, A1 | 1st A1ft for a correct probability expression ft their '39'; if (i) incorrect must see correct expression e.g. $\frac{9-3}{"39"-3}$ or $\frac{6}{"36"}$ to ft |

## Part (b)(iii)
| Working | Mark | Guidance |
|---------|------|----------|
| $E(Y^2) = \text{Var}(Y) + [E(Y)]^2 = (3('39')-9) + \left(\frac{'39'+3}{2}\right)^2$ | M1, A1ft | M1 for writing or using $E(Y^2) = \text{Var}(Y) + [E(Y)]^2$; must be $E(Y^2) = \int_3^{'39'} \frac{1}{'39'-3} y^2\,dy$ with $y^2 \to y^3$; condone different letter; if $c$ incorrect must show where $\text{Var}(Y)$ and $E(Y)$ come from; A1ft correct (follow through) expression for $E(Y^2)$, allow $\frac{1}{'39'-3}\left[\frac{y^3}{3}\right]_3^{'39'}$ |
| $E(Y^2) = \mathbf{549}$ | A1 | |

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