## Question 11(a):

| Answer/Working | Marks | Guidance |
|---|---|---|
| $R_C \cdot 3a = kW \cdot 4a\cos\theta + W \cdot 2a\cos\theta$ | M1 A1 | Take moments about $A$ for rod to find $R_C$ |
| $R_C = (2k+1)(2W/5)$ | A1 | |
| $F_A = R_C\sin\theta = (2k+1)(8W/25)$ | M1 A1 | Resolve horizontal forces on rod to find $F_A$ |
| $R_A = W + kW - R_C\cos\theta = (13k+19)(W/25)$ | M1 A1 | Resolve vertical forces on rod to find $R_A$ |
| *OR:* Take moments about $C$ for rod: | | |
| $F_A \cdot 3a\sin\theta - R_A \cdot 3a\cos\theta = kWa\cos\theta - Wa\cos\theta$ | (M1 A1) | |
| $4F_A - 3R_A = (k-1)W$ | (A1) | |
| Resolve forces parallel to rod: $F_A\cos\theta + R_A\sin\theta = kW\sin\theta + W\sin\theta$ | (M1) | |
| $3F_A + 4R_A = 4(k+1)W$ | (A1) | |
| Combine to find $F_A$, $R_A$: $F_A = (2k+1)(8W/25)$ and $R_A = (13k+19)(W/25)$ | (M1 A1) | |
| Use $F_A \le \mu R_A$ to find $\mu_{\min}$: $\mu_{\min} = 8(2k+1)/(13k+19)$ **A.G.** | M1 A1 | (No use of inequality loses A1) |
| $160k + 80 \le (\text{or} =)\ 117k + 171$ | M1 | Find bound on $k$ (allow use of equality) |
| Bound $= 91/43$ or $2.12$ | A1 | |
| $k \le 91/43$ or $2.12$ | A1 | Correct use of inequality |

**Total: 12 marks**

## Question 11(b):

**Find correlation coefficient $r$:**

$r^2 = (-0.5)(-1.2) = 0.6 \text{ or } 0.775^2$ | M1 A1 |

$r = -0.775$ | *A1 | 3 marks

---

**State both hypotheses (B0 for $r$...):**

$H_0: \rho = 0,\ H_1: \rho \neq 0$ | B1 |

---

**State or use correct tabular one-tail $r$ value:**

$r_{5,\ 2.5\%} = 0.878$ | *B1 |

---

**Valid method for reaching conclusion:**

Accept $H_0$ if $|r| <$ tabular value | M1 |

---

**Correct conclusion (AEF, dep *A1, *B1):**

Coeff. does not differ from zero | A1 | 4 marks

---

**EITHER: Find two indep. eqns for $p, q$:**

$14 + q = (-0.5)(13 + p) + 5 \times 5$ | M1 A1 |

$or\ (14 + q)/5 = (-0.5)(13 + p)/5 + 5$

$13 + p = (-1.2)(14 + q) + 5 \times 7.6$ | |

$or\ (13 + p)/5 = (-1.2)(14 + q)/5 + 7.6$ | A1 |

---

**Combine to find $p, q$:**

$p = 7,\ q = 1$ | A1, A1 |

---

**OR: Combine regression lines to find $\bar{x},\ \bar{y}$:**

$\bar{x} = 4;\ \bar{y} = 3$ | M1 A1; A1 |

**Use sample data to find $p, q$:**

$p = 7,\ q = 1$ | A1, A1 | 5 marks

---

**Total: [12]**