Higher June 2024 Paper 4 Q6
6 A regular polygon has \(n\) sides.
The interior angle of the polygon is 15 times its exterior angle.
Find the value of \(n\). [4]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 32 nfww | 4 | B2 for 11.25 M1 for 360 ÷ their 11.25 seen OR B1 for \(15a\) and \(a\) or \(16a\) M1 for \(15a + a = 180\) oe M1 for 360 ÷ their 11.25 seen | \(a\) = exterior angle and allow any consistent single letter B1 M1 implied by \(16a = 180\) or \(\frac{180}{15 + 1}\) oe Alternative for number of sides: e.g. M1 for \(\frac{180(n - 2)}{n} = 180 -\) their11.25 |
| Alternative method 1: M2 for \(15 \times 360 = 180(n - 2)\) or better or M1 for \(\frac{180(n - 2)}{n} = 15 \times \frac{360}{n}\) and M1 for \(180n = 5400 + 360\) oe | |||
| Alternative method 2: Use of trials by choosing a value for \(n\). M1 for each correct trial up to a maximum of M3 for using two of these [exterior angle] = \(\frac{360}{n}\) (formula A) [interior angle] = \(\frac{180(n - 2)}{n}\) (formula B) interior + exterior = 180 and for checking that interior = 15 × exterior if 0 scored SC1 for one of formula A or B seen or used | If they get 32 from any number of trials they score 4 marks. Trials can be seen from a calculation or a list. see appendix for likely results | ||
Appendix: Question 6
| sides | interior | exterior |
|---|---|---|
| 5.00 | 108.00 | 72.00 |
| 6.00 | 120.00 | 60.00 |
| 7.00 | 128.57 | 51.43 |
| 8.00 | 135.00 | 45.00 |
| 9.00 | 140.00 | 40.00 |
| 10.00 | 144.00 | 36.00 |
| 11.00 | 147.27 | 32.73 |
| 12.00 | 150.00 | 30.00 |
| 13.00 | 152.31 | 27.69 |
| 14.00 | 154.29 | 25.71 |
| 15.00 | 156.00 | 24.00 |
| 16.00 | 157.50 | 22.50 |
| 17.00 | 158.82 | 21.18 |
| 18.00 | 160.00 | 20.00 |
| 19.00 | 161.05 | 18.95 |
| 20.00 | 162.00 | 18.00 |
| 21.00 | 162.86 | 17.14 |
| 22.00 | 163.64 | 16.36 |
| 23.00 | 164.35 | 15.65 |
| 24.00 | 165.00 | 15.00 |
| 25.00 | 165.60 | 14.40 |
| 26.00 | 166.15 | 13.85 |
| 27.00 | 166.67 | 13.33 |
| 28.00 | 167.14 | 12.86 |
| 29.00 | 167.59 | 12.41 |
| 30.00 | 168.00 | 12.00 |
| 31.00 | 168.39 | 11.61 |
| 32.00 | 168.75 | 11.25 |
| 33.00 | 169.09 | 10.91 |
| 34.00 | 169.41 | 10.59 |
| 35.00 | 169.71 | 10.29 |