# Question 1:

## Part (a)

| Answer/Working | Mark | Guidance |
|---|---|---|
| $P(X > 4) = 1 - F(4)$ | M1 | $1 - F(4)$ seen or used |
| $= 1 - \frac{3}{5} = \frac{2}{5}$ | A1 | $\frac{2}{5}$ or $0.4$ |

**[2 marks total]**

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## Part (b)

| Answer/Working | Mark | Guidance |
|---|---|---|
| $P(3 < X < a) = 0.642$ | | |
| $F(a) - F(3) = 0.642$ | M1 o.e. | $F(a) - F(3) = 0.642$ |
| $F(a) - \frac{1}{20}(3^2 - 4) = 0.642 \Rightarrow F(a) = 0.892$ | A1 o.e. | Correct equation |
| $\frac{1}{5}(2a-5) - \frac{1}{20}(3^2-4) = 0.642 \Rightarrow a = \ldots$ | dM1 | Solving this equation o.e., leading to $a = \ldots$ Follow through their $F(3)$ |
| $\frac{1}{5}(2a-5) = 0.892 \Rightarrow a = 4.73$ | A1 cao | $a = 4.73$ (or $x = 4.73$) |

**Alternative Method for Part (b):**

| Answer/Working | Mark | Guidance |
|---|---|---|
| $\int_3^4 \left(\frac{1}{10}x\right)\{dx\}$ | M1 | Correct expression for finding probability between $x=3$ and $x=4$ |
| $= \left[\frac{x^2}{20}\right]_3^4 = \frac{4^2}{20} - \frac{3^2}{20} = \frac{7}{20}$ | A1 | Correct $\frac{4^2}{20} - \frac{3^2}{20}$, simplified or un-simplified |
| $\int_3^4 \left(\frac{1}{10}x\right)\{dx\} + \int_4^a \left(\frac{2}{5}\right)\{dx\} = 0.642 \Rightarrow a = \ldots$ | dM1 | Writes correct equation and attempts to solve leading to $a = \ldots$ |
| $\frac{7}{20} + \frac{2}{5}a - \frac{8}{5} = 0.642 \Rightarrow a = 4.73$ | A1 cao | $a = 4.73$ (or $x = 4.73$) |

**[4 marks total]**

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## Part (c)

| Answer/Working | Mark | Guidance |
|---|---|---|
| $f(x) = \frac{d}{dx}\left(\frac{1}{20}(x^2-4)\right) = \frac{1}{10}x$ | M1 | Attempt at differentiation |
| At least one of $\frac{1}{10}x$ or $\frac{2}{5}$ | A1 | |
| Both $\frac{1}{10}x$ and $\frac{2}{5}$ | A1 | |
| $f(x) = \begin{cases} \frac{1}{10}x, & 2 \leq x \leq 4 \\ \frac{2}{5}, & 4 < x \leq 5 \\ 0, & \text{otherwise} \end{cases}$ | dB1ft | Dependent on M1; all three lines with limits correctly followed through from their $F'(x)$ |

**[4 marks total]**

**[10 marks total for Question 1]**

# Question 1:

## Part (a)

| Answer/Working | Mark | Guidance |
|---|---|---|
| $1 - F(4)$ seen or used | M1 | Can be implied by $1-\frac{3}{5}$ or $1-\frac{1}{5}(2(4)-5)$ or $1-\frac{1}{20}(4^2-4)$; probability statements $1-P(X \leq 4)$ or $1-P(X<4)$ not sufficient |
| $\frac{2}{5}$ or $0.4$ | A1 | Give M1A1 for correct answer from no working |

## Part (b)

| Answer/Working | Mark | Guidance |
|---|---|---|
| $F(a) - F(3) = 0.642$ or equivalent | M1 | $F(a)$ can be written as $P(X<a)$ or $P(X \leq a)$; $F(3)$ as $P(X<3)$ or $P(X \leq 3)$ |
| $F(a) - \frac{1}{20}(3^2-4) = 0.642$ or equivalent (un-simplified) | A1 | Give $1^{\text{st}}$ M1 $1^{\text{st}}$ A1 for $F(a)=0.892$ or $P(X \geq a)=0.108$ |
| Allow SC: $\frac{1}{20}(a^2-4) - \frac{1}{20}(3^2-4) = 0.642$ | SC | — |
| Give $1^{\text{st}}$ M0 for $F(a-1)-F(3)=0.642$ without correct acceptable statement | Note | — |
| Attempts to solve $\frac{1}{5}(2a-5) - \text{their } F(3)'' = 0.642$ leading to $a=\ldots$ | dM1 | Dependent on FIRST method mark; dM1 can also be given for $\frac{1}{5}(2a-5)=0.892$ or $1-\frac{1}{5}(2a-5)=0.108$ |
| $a = 4.73$ (or $x=4.73$) | A1 cao | Give M0A0M0A0 for $F(a)-(1-F(3))=0.642 \Rightarrow F(a)=1.392$; Give M0A0M0A0 for $\int_3^a \left(\frac{1}{10}x\right)dx=0.642$ (solves to give awrt 4.67) |

## Part (c)

| Answer/Working | Mark | Guidance |
|---|---|---|
| At least one of: $\frac{1}{20}(x^2-4) \to \pm\alpha x \pm \beta,\ \alpha\neq 0,\ \beta$ can be 0; or $\frac{1}{5}(2x-5) \to \pm\delta,\ \delta\neq 0$ | M1 | — |
| At least one of $\frac{1}{10}x$ or $\frac{2}{5}$ | $1^{\text{st}}$ A1 | Can be simplified or un-simplified |
| Both $\frac{1}{10}x$ and $\frac{2}{5}$ | $2^{\text{nd}}$ A1 | Can be simplified or un-simplified |
| All three lines with limits correctly followed through from their $F'(x)$ | dB1ft | Dependent on FIRST method mark; condone use of $<$ rather than $\leq$; $0$ otherwise equivalent to $0, x<2$ **and** $0, x>5$; accept $f$ expressed consistently in another variable e.g. $u$ |

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