| Exam Board | OCR MEI |
|---|---|
| Module | Paper 3 (Paper 3) |
| Year | 2022 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Determine domain for composite |
| Difficulty | Moderate -0.3 This is a straightforward composite function question requiring students to find fg(x) = √(25-x²), determine where 25-x² ≥ 0 (giving -5 ≤ x ≤ 5), and find the range [0,5]. While it involves multiple parts, each step follows standard procedures with no novel insight required, making it slightly easier than average. |
| Spec | 1.02u Functions: definition and vocabulary (domain, range, mapping) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\sqrt{25 - x^2}\) | B1 | Isw |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(25 - x^2 \geq 0\) | M1 | Implied by B2 |
| \(-5 \leq x \leq 5\) o.e. | B1 | At least one part of the domain correct, condone missing \(=\) for this mark, e.g. \(x < 5\). Can score M0 B1 |
| B1 | Domain completely correct and correctly expressed. Eg allow \(x \geq -5\) and \(x \leq -5\) but not \(x \geq -5\) or \(x \leq -5\) or other ambiguous notation. Condone \(x \leq \pm 5\) seen in working if final answer is correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(0 \leq \text{fg}(x) \leq 5\) or \(0 \leq y \leq 5\) | B1 | At least one part of the range correct, condone missing \(=\) for this mark, e.g. \(y < 5\), allow use of \(f(x)\) or \(g(x)\) notation here but not \(x\) |
| B1 | Range completely correct and correctly expressed. See notes in 2(b)(i) |
# Question 2(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\sqrt{25 - x^2}$ | B1 | Isw |
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# Question 2(b)(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| $25 - x^2 \geq 0$ | M1 | Implied by B2 |
| $-5 \leq x \leq 5$ o.e. | B1 | At least one part of the domain correct, condone missing $=$ for this mark, e.g. $x < 5$. Can score M0 B1 |
| | B1 | Domain completely correct and correctly expressed. Eg allow $x \geq -5$ and $x \leq -5$ but not $x \geq -5$ or $x \leq -5$ or other ambiguous notation. Condone $x \leq \pm 5$ seen in working if final answer is correct |
---
# Question 2(b)(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $0 \leq \text{fg}(x) \leq 5$ or $0 \leq y \leq 5$ | B1 | At least one part of the range correct, condone missing $=$ for this mark, e.g. $y < 5$, allow use of $f(x)$ or $g(x)$ notation here but not $x$ |
| | B1 | Range completely correct and correctly expressed. See notes in 2(b)(i) |
---2 The function $\mathrm { f } ( x ) = \sqrt { x }$ is defined on the domain $x \geqslant 0$.\\
The function $\mathrm { g } ( x ) = 25 - x ^ { 2 }$ is defined on the domain $\mathbb { R }$.
\begin{enumerate}[label=(\alph*)]
\item Write down an expression for $\mathrm { fg } ( x )$.
\item \begin{enumerate}[label=(\roman*)]
\item Find the domain of $\mathrm { fg } ( x )$.
\item Find the range of $\mathrm { fg } ( x )$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR MEI Paper 3 2022 Q2 [6]}}