Higher June 2019 Paper 3 Q9
9 Towns \(P\), \(Q\) and \(R\) are connected by roads \(PQ\), \(PR\) and \(QR\).
\(PR\) is 10 km longer than \(PQ\).
\(QR\) is twice as long as \(PR\).
The total length of the three roads is 170 km
Not drawn accurately
Work out the length of \(PQ\). [4 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 – PQ as the unknown | ||
| \(x + 10\) or \(2(x + 10)\) | M1 | any unknown |
| \(x + x + 10 + 2(x + 10) = 170\) | M1dep | oe any consistent unknown \(x\) + their two expressions (with at least one correct) = 170 |
| \(4x + 30 = 170\) | M1dep | oe \(4x = 140\) must be correct |
| 35 | A1 | |
| Alternative method 2 – PR as the unknown | ||
| \(x - 10\) or \(2x\) | M1 | any unknown |
| \(x + x - 10 + 2x = 170\) | M1dep | oe any consistent unknown \(x\) + their two expressions (with at least one correct) = 170 |
| \(4x - 10 = 170\) or \(x = 45\) | M1dep | oe \(4x = 180\) must be correct |
| 35 | A1 | |
| Alternative method 3 – QR as the unknown | ||
| \(\dfrac{x}{2}\) or \(\dfrac{x}{2} - 10\) | M1 | any unknown |
| \(x + \dfrac{x}{2} + \dfrac{x}{2} - 10 = 170\) | M1dep | oe any consistent unknown \(x\) + their two expressions (with at least one correct) = 170 |
| \(2x - 10 = 170\) or \(x = 90\) | M1dep | oe \(2x = 180\) must be correct |
| 35 | A1 | |
| Alternative method 4 – trial and improvement with addition of three lengths | ||
| A correctly evaluated trial witha difference of 10 (km) between the two shorter lengthsandthe longest length twice the length of the middle length | M1 | may be seen as a subtraction of three numbers from 170 |
| A different correctly evaluated trial witha difference of 10 (km) between the two shorter lengthsandthe longest length twice the length of the middle length | M1dep | may be seen as a subtraction of three numbers from 170 |
| 35, 45 and 90 | A1 | |
| 35 | A1 | |
| Alternative method 5 – trial and improvement with subtraction from 170 | ||
| A correctly evaluated trial of two lengths subtracted from 170 witha difference of 10 (km) between the two lengthsorone length twice the length of the other | M1 | |
| A different correctly evaluated trial of two lengths subtracted from 170 witha difference of 10 (km) between the two lengthsorone length twice the length of the other | M1dep | |
| 35, 45 and 90 | A1 | |
| 35 | A1 | |
Additional guidance
| If the student attempts more than one method, mark each method and award the highest mark | |
| Alt 1 \(\ PQ + PQ + 10 + 2(PQ + 10) = 170\) | M1M1 |
| Alt 1 \(\ PQ + PQ + 10 + 2PR = 170\) | M1 |
| Alt 2 \(\ x,\ x + 10\) and \(2x\) seen on diagram, \(4x + 10 = 170\) | M1M1M0A0 |
| Alt 4 35 + 45 + 90 with no choice made | M1M1A1A0 |
| Alt 4 170 – 30 – 40 – 80 = 20 | M1 |
| Alt 4 170 – 30 – 40 – 60 = 40 incorrect number is doubled | M0 |
| Alt 5 170 – 30 – 60 = 80 | M1 |