| Exam Board | OCR MEI |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2014 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discriminant and conditions for roots |
| Type | Find discriminant, state roots |
| Difficulty | Moderate -0.8 This is a straightforward algebraic manipulation question requiring expansion of brackets and collecting like terms to get 3n²+6n+5, followed by basic reasoning about odd/even properties. The algebra is routine C1 level with no problem-solving insight needed, making it easier than average but not trivial due to the two-part structure and the need to reason about parity. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(3n^2 + 6n + 5\) isw | B2 | M1 for a correct expansion of at least one of \((n+1)^2\) and \((n+2)^2\) |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Odd numbers with valid explanation | B2 | Marks depend on 9(i) correct or starting again; for B2 must see at least odd \(\times\) odd \(=\) odd [for \(3n^2\)] (or when \(n\) is odd, \([3]n^2\) is odd) and odd \([+\) even\(]\ +\) odd \(=\) even soi; condone lack of odd \(\times\) even \(=\) even for \(6n\); condone no consideration of \(n\) being even |
| Or B2 for deductive argument such as: \(6n\) is always even [and 5 is odd] so \(3n^2\) must be odd so \(n\) is odd | ||
| B1 for odd numbers with a correct partial explanation or a partially correct explanation | ||
| Or B1 for an otherwise fully correct argument for odd numbers but with conclusion positive odd numbers or conclusion negative odd numbers | ||
| B0 for just a few trials and conclusion | ||
| [2] |
## Question 9(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $3n^2 + 6n + 5$ isw | B2 | M1 for a correct expansion of at least one of $(n+1)^2$ and $(n+2)^2$ |
| | **[2]** | |
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## Question 9(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Odd numbers with valid explanation | B2 | Marks depend on 9(i) correct or starting again; for B2 must see at least odd $\times$ odd $=$ odd [for $3n^2$] (or when $n$ is odd, $[3]n^2$ is odd) and odd $[+$ even$]\ +$ odd $=$ even soi; condone lack of odd $\times$ even $=$ even for $6n$; condone no consideration of $n$ being even | Accept a full valid argument using odd and even from starting again; ignore numerical trials or examples in this part — only a generalised argument can gain credit |
| | Or B2 for deductive argument such as: $6n$ is always even [and 5 is odd] so $3n^2$ must be odd so $n$ is odd | |
| | B1 for odd numbers with a correct partial explanation or a partially correct explanation | |
| | Or B1 for an otherwise fully correct argument for odd numbers but with conclusion positive odd numbers or conclusion negative odd numbers | |
| | B0 for just a few trials and conclusion | |
| | **[2]** | |9 You are given that $n , n + 1$ and $n + 2$ are three consecutive integers.\\
(i) Expand and simplify $n ^ { 2 } + ( n + 1 ) ^ { 2 } + ( n + 2 ) ^ { 2 }$.\\
(ii) For what values of $n$ will the sum of the squares of these three consecutive integers be an even number? Give a reason for your answer.
Section B (36 marks)
\hfill \mbox{\textit{OCR MEI C1 2014 Q9 [4]}}