| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2012 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Proof |
| Type | Function properties proof |
| Difficulty | Moderate -0.8 This question tests understanding of odd/even function properties through straightforward algebraic manipulation. Part (i) requires direct substitution using the definition f(-x)=-f(x), while part (ii) involves recognizing that the product of two odd functions is even. Both parts are routine applications of definitions with minimal steps, making this easier than average A-level questions which typically require more problem-solving or multi-step reasoning. |
| Spec | 1.02u Functions: definition and vocabulary (domain, range, mapping) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(s(-x)=f(-x)+g(-x)\) | M1 | must have \(s(-x)=\ldots\) |
| \(=-f(x)+-g(x)\) | ||
| \(=-(f(x)+g(x))\) | ||
| \(=-s(x)\) (so \(s\) is odd) | A1 [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(p(-x)=f(-x)g(-x)\) | M1 | must have \(p(-x)=\ldots\) |
| \(=(-f(x))\times(-g(x))\) | ||
| \(=f(x)g(x)=p(x)\) | ||
| so \(p\) is even | A1 [2] | Allow SC1 for showing \(p(-x)=p(x)\) using two specific odd functions, but must still show \(p\) is even; e.g. \(f(x)=x\), \(g(x)=x^3\), \(p(x)=x^4\); \(p(-x)=(-x)^4=x^4=p(x)\), so \(p\) even; condone \(f\) and \(g\) being the same function |
## Question 7(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $s(-x)=f(-x)+g(-x)$ | M1 | must have $s(-x)=\ldots$ |
| $=-f(x)+-g(x)$ | | |
| $=-(f(x)+g(x))$ | | |
| $=-s(x)$ (so $s$ is odd) | A1 **[2]** | |
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## Question 7(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $p(-x)=f(-x)g(-x)$ | M1 | must have $p(-x)=\ldots$ |
| $=(-f(x))\times(-g(x))$ | | |
| $=f(x)g(x)=p(x)$ | | |
| so $p$ is even | A1 **[2]** | Allow SC1 for showing $p(-x)=p(x)$ using two specific odd functions, but must still show $p$ is even; e.g. $f(x)=x$, $g(x)=x^3$, $p(x)=x^4$; $p(-x)=(-x)^4=x^4=p(x)$, so $p$ even; condone $f$ and $g$ being the same function |
---7 You are given that $\mathrm { f } ( x )$ and $\mathrm { g } ( x )$ are odd functions, defined for $x \in \mathbb { R }$.\\
(i) Given that $\mathrm { s } ( x ) = \mathrm { f } ( x ) + \mathrm { g } ( x )$, prove that $\mathrm { s } ( x )$ is an odd function.\\
(ii) Given that $\mathrm { p } ( x ) = \mathrm { f } ( x ) \mathrm { g } ( x )$, determine whether $\mathrm { p } ( x )$ is odd, even or neither.
\hfill \mbox{\textit{OCR MEI C3 2012 Q7 [4]}}