## Question 15:

### Part (i):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $k^2 - 4k + 5 = (k-2)^2 + \ldots$ | M1 | Starts process of showing expression is positive; completing the square requires $(k-2)^2 \pm \ldots$ |
| $k^2 - 4k + 5 = (k-2)^2 + 1$ so $k^2 - 4k + 5 \geq 1$, so $k^2 - 4k + 5$ is always positive | A1* | Completes proof with no errors; $(k-2)^2 \geq 0$ must be correctly stated |

### Part (ii):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempts to solve any 1 pair of relevant simultaneous equations, e.g. $3x+2y=28$, $2x-5y=1 \Rightarrow x=\ldots, y=\ldots$ | M1 | Any one of the 11 relevant pairs attempted |
| Attempts to solve any 2 pairs with at least one correct and correctly rejected, e.g. $x=\frac{142}{19}$, $y=\frac{53}{19}$ not integers; and $3x+2y=7$, $2x-5y=4 \Rightarrow x=\ldots, y=\ldots$ | dM1 (A1 on EPEN) | At least one correct solution and correctly rejected |
| Attempts all 5 pairs with positive RHS; shows all non-integer or impossible cases rejected | ddM1 | All cases considered systematically |
| All cases considered with correct values and rejected; correct reasons in each case e.g. "not integers"; concluding statement e.g. "hence proven" | A1 | Requires all cases, correct reasons, and conclusion |

# Mark Scheme

## Question (Proof by Contradiction - Simultaneous Equations)

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**Case A:** | $3x+2y=28, \quad 2x-5y=1 \Rightarrow x=\frac{142}{19}, \quad y=\frac{53}{19}$ — Not integers | M1/dM1 | Attempt to solve at least one case

**Case B:** | $3x+2y=7, \quad 2x-5y=4 \Rightarrow x=\frac{43}{19}, \quad y=\frac{2}{19}$ — Not integers | M1/dM1 |

**Case C:** | $3x+2y=4, \quad 2x-5y=7 \Rightarrow x=\frac{34}{19}, \quad y=-\frac{13}{19}$ — Not integers/not positive | M1/dM1 |

**Case D:** | $3x+2y=2, \quad 2x-5y=14 \Rightarrow x=2, \quad y=-2$ — Not positive | M1/dM1 |

**Case E:** | $3x+2y=1, \quad 2x-5y=28 \Rightarrow x=\frac{61}{19}, \quad y=-\frac{82}{19}$ — Not integers/not positive | M1/dM1 |

**Case F:** | $3x+2y=-1, \quad 2x-5y=-28 \Rightarrow x=-\frac{61}{19}, \quad y=\frac{82}{19}$ — Not integers/not positive | (dM1) |

**Case G:** | $3x+2y=-2, \quad 2x-5y=-14 \Rightarrow x=-2, \quad y=2$ — Not positive | (dM1) |

**Case H:** | $3x+2y=-4, \quad 2x-5y=-7 \Rightarrow x=-\frac{34}{19}, \quad y=\frac{13}{19}$ — Not integers/not positive | (dM1) |

**Case I:** | $3x+2y=-7, \quad 2x-5y=-4 \Rightarrow x=-\frac{43}{19}, \quad y=-\frac{2}{19}$ — Not integers/not positive | (dM1) |

**Case J:** | $3x+2y=-14, \quad 2x-5y=-2 \Rightarrow x=-\frac{74}{19}, \quad y=-\frac{22}{19}$ — Not integers/not positive | (dM1) |

**Case K:** | $3x+2y=-28, \quad 2x-5y=-1 \Rightarrow x=-\frac{142}{19}, \quad y=-\frac{53}{19}$ — Not integers/not positive | (dM1) |

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**Overall structure marks:**

| Attempt to solve any one of the 11 possible cases to find a value for $x$ or $y$ | **M1** | Value need not be correct

| Attempt to solve any two cases with at least one correct **and** correctly rejected | **dM1** | Dependent on M1

| Attempts to solve all 5 pairs (A–E) with positive RHS **or** cases A and B justified as only cases needed, using $3x+2y\geq5$ or $3x+2y>2x-5y$ | **ddM1** | Dependent on dM1

| All necessary cases shown impossible with correct values, correct reasons, and minimal conclusion e.g. "hence proved" | **A1** | cso; all 5 pairs may be rejected simultaneously if rejection is sufficiently clear