Higher June 2024 Paper 1 Q24
24 A polygon has \(n\) sides, where \(n \gt 5\)
The interior angles of the polygon form an arithmetic sequence.
The smallest angle of the polygon is 84°
The common difference of the sequence is 4°
Work out the sum of the interior angles of the polygon.
Show clear algebraic working.
(6)
| Scheme | Marks |
|---|---|
\(\dfrac{n}{2}\big[2(84) + (n - 1)(4)\big]\) or \(\dfrac{n}{2}\big[168 + 4n - 4\big]\) or \(\dfrac{n}{2}\big[164 + 4n\big]\) oe or \(82n + 2n^2\) oe | M1 |
\(\dfrac{n}{2}\big[2(84) + (n - 1)(4)\big] = (n - 2) \times 180\) or \(\dfrac{n}{2}\big[164 + 4n\big] = (n - 2) \times 180\) oe or \(82n + 2n^2 = (n - 2) \times 180\) oe | M1 |
| eg \(n^2 - 49n + 180 (= 0)\) oe Allow \(n^2 - 49n = -180\) | M1 |
eg \((n - 45)(n - 4)(= 0)\) \(n = \dfrac{--49 \pm \sqrt{(-49)^2 - 4 \times 1 \times 180}}{2}\) e.g. \(\left(n - \dfrac{49}{2}\right)^2 - \left(\dfrac{49}{2}\right)^2 = -180\) | M1ft |
\((\text{``}{45}\text{''} - 2) \times 180\) or \(\dfrac{\text{``}{45}\text{''}}{2}\big[2(84) + (\text{``}{45}\text{''} - 1)(4)\big]\) oe or \((\text{``}{44}\text{''} - 2) \times 180\) or \(\dfrac{\text{``}{44}\text{''}}{2}\big[2(84) + (\text{``}{44}\text{''} - 1)(4)\big]\) oe | M1 |
| Working required Answer: 7740 | A1 |
| (6) | |
| (6 marks) |
Notes
M1: dep on M2 for multiplying out and collecting terms, forming a three term quadratic in any form of \(an^2 + bn + c\) (= 0) where at least 2 coefficients (\(a\) or \(b\) or \(c\)) are correct
M1: dep on previous M1
NB \(n \gt 5\)
A1: dep on M5
Accept 7560 or 7480
24 ALT
| Scheme | Marks |
|---|---|
\(\dfrac{n}{2}\big[2(96) + (n - 1)(-4)\big]\) or \(\dfrac{n}{2}\big[192 - 4n + 4\big]\) or \(\dfrac{n}{2}\big[196 - 4n\big]\) oe or \(98n - 2n^2\) oe | M1 |
\(\dfrac{n}{2}\big[2(96) + (n - 1)(-4)\big] = 360\) or \(\dfrac{n}{2}\big[196 - 4n\big] = 360\) oe or \(98n - 2n^2 = 360\) oe | M1 |
| eg \(2n^2 - 98n + 360 (= 0)\) \(n^2 - 49n + 180 (= 0)\) oe Allow \(n^2 - 49n = -180\) | M1 |
e.g \((n - 45)(n - 4)(= 0)\) \(n = \dfrac{--49 \pm \sqrt{(-49)^2 - 4 \times 1 \times 180}}{2}\) e.g. \(\left(n - \dfrac{49}{2}\right)^2 - \left(\dfrac{49}{2}\right)^2 = -180\) | M1ft |
\((\text{``}{45}\text{''} - 2) \times 180\) or \(\dfrac{\text{``}{45}\text{''}}{2}\big[2(84) + (\text{``}{45}\text{''} - 1)(4)\big]\) oe or \((\text{``}{44}\text{''} - 2) \times 180\) or \(\dfrac{\text{``}{44}\text{''}}{2}\big[2(84) + (\text{``}{44}\text{''} - 1)(4)\big]\) oe | M1 |
| Working required Answer: 7740 | A1 |
| (6 marks) |
Notes
M1: for correctly substituting into \(S_n = \dfrac{n}{2}\big[2a + (n - 1)d\big]\) using exterior angles
M1: dep on M2 for multiplying out and collecting terms, forming a three term quadratic in any form of \(an^2 + bn + c\) (= 0) where at least 2 coefficients (\(a\) or \(b\) or \(c\)) are correct
M1: dep on previous M1
NB \(n \gt 5\)
A1: dep on M5
Accept 7560 or 7480