## Question 11:

| Answer/Working | Mark | Guidance |
|---|---|---|
| *EITHER:* Find 2 indep. eqns for $R_B$, $F_B$ only: | M1 | |
| Moments for $BA$ about $A$: $F_B\cdot 2a\sin\beta - R_B\cdot 2a\cos\beta = Wa\sin\beta$ | M1 A1 | |
| Moments for system about $C$: $F_B\cdot 6a\sin\beta + R_B\cdot 2a\cos\beta = 9Wa\sin\beta$ | M1 A1 | |
| Add equations to find $F_B$: $F_B = 5W/4$ **A.G.** | M1 A1 | |
| *OR:* If $R_C$, $F_C$ introduced, resolve vertically: $F_B + F_C = 3W$ | (B1) | |
| Any 2 moment eqns indep. of above resln. e.g.: | | |
| For $BA$ about $A$: $F_B\cdot 2a\sin\beta - R_B\cdot 2a\cos\beta = Wa\sin\beta$ | | |
| For $CA$ about $A$: $F_C\cdot 4a\sin\beta - R_C\cdot 4a\cos\beta = 4Wa\sin\beta$ | | |
| For system about $B$: $F_C\cdot 6a\sin\beta - R_C\cdot 2a\cos\beta = 9Wa\sin\beta$ | | |
| (2 system eqns are equiv. to vert. resolution) | $(2\times$ M1 A1) | |
| Solve eqns using $R_B = R_C$ to find $F_B$: $F_B = 5W/4$ **A.G.** | (M1 A1) | |
| Find $F_C$ by eg vertical resolution for rods: $F_C = 7W/4$ | B1 | |
| Find $R_B$ [or $R_C$] from a moment eqn: $R_B\,[= R_C] = \frac{3}{4}W\tan\beta$ | B1 | |
| Find one of $F_B/R_B$, $F_C/R_C$ eg: $F_B/R_B = 5/(3\tan\beta)$ | M1 A1 | |
| Find other eg: $F_C/R_C = 7/(3\tan\beta)$ | A1 | |
| Find set of possible values of $\mu$: $\mu > 7/(3\tan\beta)$ (allow $\geq$) | M1 A1 | **Total: 14** |

# Question 11:

## Part 1 (4 marks) - Finding P(T₅₀ < 70)

| Answer/Working | Mark | Guidance |
|---|---|---|
| $E(T_{50}) = 50 \times 1.5 = 75$ and $\text{Var}(T_{50}) = 50 \times 0.4^2 = 8$ | B1 | Both required |
| By central limit theorem *or* since $n$ [or 50] is large | B1 | State valid justification (A.E.F.) |
| $\Phi\left(\frac{70-75}{\sqrt{8}}\right)$ | M1 | Find Normal approximation to $P(T_{50} < 70)$ |
| $= 1 - \Phi(1.768) = 0.0385 \pm 0.0001$ | A1 | |

## Part 2 (6 marks) - Finding maximum n such that P(Tₙ < 70) > 0.9

| Answer/Working | Mark | Guidance |
|---|---|---|
| $E(T_n) = 1.5n$ and $\text{Var}(T_n) = 0.4^2 n$ | B1 | Both required |
| $\Phi\left(\frac{70 - 1.5n}{0.4\sqrt{n}}\right) \geq 0.9$ | M1 | Use Normal approximation for $P(T_n < 70) > 0.9$ |
| $\frac{70 - 1.5n}{0.4\sqrt{n}} \geq 1.282$ | A1 | Invert function |
| $(\sqrt{n})^2 + 0.3419\sqrt{n} - \frac{140}{3} \leq 0$ | M1 | Rearrange as quadratic expression in $\sqrt{n}$ |
| $44.4$ | A1 | Find positive root when expression is zero |
| $n_{max} = 44$ | A1 | Find greatest integer value of $n$ |

## Part 3 (4 marks) - Finding P(A - H > 0)

| Answer/Working | Mark | Guidance |
|---|---|---|
| $E(A-H) = 75 - 65 = 10$ and $\text{Var}(A-H) = 8 + 12.5 = 20.5$ | M1 A1 | For difference in times, find $E(A-H)$ and $\text{Var}(A-H)$ |
| $\Phi\left(\frac{10}{\sqrt{20.5}}\right) = \Phi(2.209) = 0.986$ | M1 A1 | Find Normal approximation to $P(A-H>0)$ |

**Total: [14]**