Numeracy Higher June 2025 Unit 2 Q9
9. NoomAir is a cargo plane company based in Nice.
Two of their planes have been hired to deliver cargo to the cities of Rome, Valencia and Tunis.
One of the planes will fly from Nice to Rome, and then on to Tunis.
The pilot knows the following information.
- From Nice, Rome is 470 km on a bearing of 115º.
- From Nice, Tunis is 805 km on a bearing of 162º.
Diagram not drawn to scale
Calculate the distance from Rome to Tunis. [4]
The second plane will fly from Nice to Valencia, and then on to Tunis.
The pilot of this plane knows the following information.
- Valencia is on a bearing of 234º from Nice.
- The distance between Valencia and Tunis is 970 km.
Calculate the bearing of Tunis from Valencia. [6]
NoomAir sells toy replicas of one of its cargo planes.
The two replicas shown below are mathematically similar.
Diagrams not drawn to scale
The volume of a large toy plane is 6 times the volume of a small toy plane.
The length of a large toy plane is 30 cm.
Calculate the length of a small toy plane. [3]
NoomAir makes 200 of the large toy planes. It takes a random sample of 5 of these 200 toy planes to check the quality of their production.
The following numbers are taken from a table of random digits.
| 649 | 133 | 016 | 249 | 585 | 520 | 869 | 239 |
| 489 | 849 | 727 | 133 | 580 | 337 | 182 | 125 |
| 957 | 755 | 510 | 563 | 017 | 205 | 810 | 119 |
Use these numbers to choose 5 toy planes from the 200 toy planes.
You must start with the first number in the list.
Describe clearly how you use the numbers to select the sample. [3]
The numbers of the 5 toy planes in the sample are:
…………… …………… …………… …………… ……………
From the formula list: In any triangle \(ABC\): Cosine rule \(a^2 = b^2 + c^2 - 2bc\cos A\)
| Scheme | Mark | Comments |
|---|---|---|
| (i) Angle RNT = (162 – 115 =) 47 (º) | B1 | May be shown on the diagram or implied in further working |
| (Distance between Rome and Tunis =) \(\sqrt{470^2 + 805^2 - 2 \times 470 \times 805 \times \cos(162 - 115)}\) or \(\sqrt{470^2 + 805^2 - 2 \times 470 \times 805 \times \cos 47}\) (= √352856.8…) | M2 | FT ‘their Angle RNT = 47’ provided it is < 90 M1 for \(470^2 + 805^2 - 2 \times 470 \times 805 \times \cos(162 - 115)\) or \(470^2 + 805^2 - 2 \times 470 \times 805 \times \cos 47\) |
| = 594(.017…) (km) | A1 | CAO If no marks awarded, SC1 for an answer of 1260(.392…) or 1260.4 (km) from \(\sqrt{470^2 + 805^2 - 2 \times 470 \times 805 \times \cos 162}\) |
| (ii) Angle VNT = (234 – 162 =) 72 (º) | B1 | May be shown on the diagram or implied in further working |
| (Angle NVT =) \(\sin^{-1}\left(\dfrac{\sin 72}{970} \times 805\right)\) | M2 | FT ‘their Angle VNT = 72’ provided it is < 120 M1 for \(\dfrac{\sin(\text{Angle NVT})}{805} = \dfrac{\sin 72}{970}\) or equivalent |
| = 52.1(181…) (º) | A1 | CAO Mark final answer Accept 52 (º) provided no incorrect work seen |
| (Bearing of Tunis from Valencia =) 180 – (360 – 234) + 52.1(181…) or 54 + 52.1(181…) | m1 | FT ‘their 52.1(181…)’ provided M2 or M1 previously awarded |
| = 106(.118…)(º) | A1 | On FT, if their bearing < 100, then their answer needs to be written using 3 figures, allowing decimals |
| Scheme | Mark | Comments |
|---|---|---|
| Appropriate sight of \(\sqrt[3]{6}\) or \(\sqrt[3]{\dfrac{1}{6}}\) (= 1.817…) (= 0.550…) | B1 | May be embedded in further working |
| (Length of small toy plane =) 30 ÷ \(\sqrt[3]{6}\) or 30 ÷ 1.8(171…) OR 30 × \(\sqrt[3]{\dfrac{1}{6}}\) or 30 × 0.55(0321…) | M1 | Or equivalent e.g. \(\sqrt[3]{4500}\) The award of M1 implies B1 |
| = 16.5(096…) or 16.51 (cm) | A1 | Penalise an answer of 16.5 if evidence of premature approximation e.g. 30 ÷ 1.82 = 16.48 with a final answer of 16.5 would be awarded B1M1A0 |
| Scheme | Mark | Comments |
|---|---|---|
Statements required:
| E2 | All 4 needed for E2 E1 for any 2 or 3 correct statements The numbering (00)1 to 200 can be implied by the range of numbers they state they will choose from Any alternative numbering system e.g. (00)0 to 199 needs to be clearly stated Allow the 2nd statement to be implied by their numbering of the toy planes (from 001) AND their use of 3-digit numbers in their answer OR 3-digit numbers used in their answer and e.g. 016 seen Do not allow ‘Use numbers less than 201’ or ‘Ignore numbers greater than 200’ if they have numbered the toy planes from 001 to 200 without referencing ignoring 000 |
| (Working in rows would give toy planes) 133, (0)16, OR (Working in columns would give toy planes) 133, (0)16, | B1 | ISW Allow these numbers in any order |
Alternative
Alternative method:
| Scheme | Mark | Comments |
|---|---|---|
Statements required:
| E2 | All 4 needed for E2 E1 for any 2 or 3 correct statements The numbering system can be implied by the division by 200 and how they say they will deal with the remainder of (00)0 Allow the 2nd statement to be implied by their numbering of the toy planes (from 001) AND their use of 3-digit numbers in their answer OR 3-digit numbers used in their answer and e.g. 049 seen If (00)0 to 199 used, when the remainder is 0, toy plane (00)0 is selected |
| (Working in rows would give toy planes) (0)49, 133, (0)16, OR (Working in columns would give toy planes) (0)49, (0)89, 157, 133, | B1 | ISW Allow these numbers in any order |