| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2013 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Solve equation with inverses |
| Difficulty | Standard +0.3 This is a slightly easier than average C3 question. Part (i) requires completing the square to find the range of a quadratic (standard technique), worth 4 marks. Part (ii) involves function composition, solving a quadratic equation, and finding where an inverse function equals x (solving g(x)=x), all routine procedures. The multi-step nature adds some complexity, but no novel insight is required—just systematic application of well-practiced techniques. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02v Inverse and composite functions: graphs and conditions for existence |
| Answer | Marks | Guidance |
|---|---|---|
| Attempt completion of square at least as far as \((x + 2a)^2\) or differentiation to find stationary point at least as far as linear equation involving two terms | *M1 | or equiv but \(a\) must be present |
| Obtain \((x + 2a)^2 - 3a^2\) or \((-2a, -3a^2)\) | A1 | |
| Attempt inequality involving appropriate \(y\)-value | M1 | dep *M; allow \(<\), \(>\) or \(\le\) here; allow use of \(x\); or unsimplified equiv |
| State \(y \ge -3a^2\) or \(f(x) \ge -3a^2\) | A1 | now with \(\ge\); here \(x \ge -3a^2\) is A0 |
| Answer | Marks | Guidance |
|---|---|---|
| Attempt composition of \(f\) and \(g\) the right way round | *M1 | algebraic or (part) numerical; need to see \(4x - 2a\) replacing \(x\) at least once |
| Obtain or imply \(16x^2 - 3a^2\) or \(144 - 3a^2\) | A1 | or less simplified equiv but with at least the brackets expanded correctly |
| Attempt to find \(a\) from fg(3) = 69 | M1 | dep *M |
| Obtain at least \(a = 5\) | A1 | |
| Attempt to solve \(4x - 10 = x\) or \(\frac{1}{4}(x+10) = x\) or \(4x - 10 = \frac{1}{4}(x+10)\) | M1 | for their \(a\); must be linear equation in one variable; condone sign slip in finding inverse of \(g\) |
| Obtain \(\frac{10}{3}\) | A1 | and no other answer |
## (i)
Attempt completion of square at least as far as $(x + 2a)^2$ or differentiation to find stationary point at least as far as linear equation involving two terms | *M1 | or equiv but $a$ must be present
Obtain $(x + 2a)^2 - 3a^2$ or $(-2a, -3a^2)$ | A1 |
Attempt inequality involving appropriate $y$-value | M1 | dep *M; allow $<$, $>$ or $\le$ here; allow use of $x$; or unsimplified equiv
State $y \ge -3a^2$ or $f(x) \ge -3a^2$ | A1 | now with $\ge$; here $x \ge -3a^2$ is A0
**Total: [4]**
## (ii)
Attempt composition of $f$ and $g$ the right way round | *M1 | algebraic or (part) numerical; need to see $4x - 2a$ replacing $x$ at least once
Obtain or imply $16x^2 - 3a^2$ or $144 - 3a^2$ | A1 | or less simplified equiv but with at least the brackets expanded correctly
Attempt to find $a$ from fg(3) = 69 | M1 | dep *M
Obtain at least $a = 5$ | A1 |
Attempt to solve $4x - 10 = x$ or $\frac{1}{4}(x+10) = x$ or $4x - 10 = \frac{1}{4}(x+10)$ | M1 | for their $a$; must be linear equation in one variable; condone sign slip in finding inverse of $g$
Obtain $\frac{10}{3}$ | A1 | and no other answer
**Total: [6]**
---The functions f and g are defined for all real values of $x$ by
$$\text{f}(x) = x^2 + 4ax + a^2 \text{ and } \text{g}(x) = 4x - 2a,$$
where $a$ is a positive constant.
\begin{enumerate}[label=(\roman*)]
\item Find the range of f in terms of $a$. [4]
\item Given that fg(3) = 69, find the value of $a$ and hence find the value of $x$ such that $\text{g}^{-1}(x) = x$. [6]
\end{enumerate}
\hfill \mbox{\textit{OCR C3 2013 Q8 [10]}}