Higher November 2021 Paper 5 Q22
22 \(n\) is a positive integer.
Prove that \((2n + 1)(n - 3)(n + 2) + 3n(n + 7)\) is always even. [6]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(2n^3 + 2n^2 + 8n - 6\) or \(2(n^3 + n^2 + 4n - 3)\) | M5 | M3 for \(2n^3 + n^2 - 6n^2 + 4n^2 + 2n - 12n - 3n - 6\) oe or better or condone one error in coefficients in simplified expression \(2n^3 + 2n^2 + 8n - 6\) | \(2n^3 - n^2 - 13n - 6\) when simplified |
or M2 for one correct expanded pair or M1 for 3 correct terms out of 4 in expanded pair (term in \(n\) counts as 2 terms) | ie \(2n^2 + n - 6n - 3\) or better or \(2n^2 + 4n + n + 2\) or better or \(n^2 + 2n - 3n - 6\) or better | ||
| M1 for \(3n^2 + 21n\) | |||
| \(2(n^3 + n^2 + 4n - 3)\) | A1 | Accept e.g. each term is divisible by 2 or \((2n^3 + 2n^2 + 8n - 6) \div 2 = n^3 + n^2 + 4n - 3\) | |
| Alt method Alt method for 6 marks Fully correct reasoning with even and odds for \(n\) and for each term when \(n\) is even and when \(n\) is odd \((2n + 1)(n - 3)(n + 2)\) and \(3n(n + 7)\) e.g. if \(n\) is even, \((2n + 1)\) is odd etc | Must include e.g. if \(n\) is even, \(2n + 1\) is odd, \(n - 3\) is odd, \(n + 2\), is even then odd × odd × even = even and \(3n\) is even and \(n + 7\) is odd, even × odd = even, then even + even = even and repeat when \(n\) = odd | ||
| or M3 for fully correct reasoning with even and odds but only considering \(n\) is even or \(n\) is odd not both | |||