October 2021 Paper 2 Q8
8 The number \(K\) is defined by \(K = n^3 + 1\), where \(n\) is an integer greater than 2.
(a) Given that \(n^3 + 1 \equiv (n + 1)(n^2 + bn + c)\), find the constants \(b\) and \(c\). [1]
(b) Prove that \(K\) has at least two distinct factors other than 1 and \(K\). [5]
| Scheme | Marks |
|---|---|
| \(b = -1\), \(c = 1\) | B1 |
| [1] |
Notes
or \((n + 1)(n^2 - n + 1)\)
| Scheme | Marks |
|---|---|
| \(n + 1\) (or \(n^2 - n + 1\)) is a factor of \(K\) | B1 |
| \(n \gt 2\) so \(n + 1 \gt 1\) or \(n + 1 \gt 3\) or \(n + 1 \neq 1\) (\(n^2 - n + 1\) is a factor of \(K\) and \(n^2 - n + 1 \gt 1\) or \(\neq 1\)) | B1 |
| Assume these factors are equal Let \(n^2 - n + 1 = n + 1\) \(\Rightarrow n^2 - 2n = 0\) | M1 |
| \(n = 0\) or \(2\). | A1 |
| \(n \gt 2\) so both invalid; hence 2 distinct factors Ignore attempted proofs that either factor \(\neq K\) | A1 |
| [5] |
Notes
B1: Stated. Allow \(x\) instead of \(n\)
NOT \(n^3 + 1\) can be expressed as \((n + 1)(n^2 - n + 1)\)
B1: Must see \(n \gt 2\)
Allow omission of this step
A1: Conclusion stated, from correct working seen.
Dep at least B1M1 and correct reasoning given
SC: \((n + 1) \gt 1\) or \(n + 1 \gt 3\) (or \(n^2 - n + 1 \gt 1\)) B1
\((n + 1)\) & \((n^2 - n + 1)\) are factors of \(K\) B1