| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2009 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find intersection points |
| Difficulty | Moderate -0.8 This is a straightforward multi-part question testing basic function composition, inverse functions, and modulus functions. Part (i) requires simple composition and solving a linear equation; part (ii) uses the standard result that y=g(x) meets y=g^{-1}(x) on the line y=x; part (iii) involves routine case analysis of modulus equations. All techniques are standard C3 material with no problem-solving insight required, making it easier than average. |
| Spec | 1.02l Modulus function: notation, relations, equations and inequalities1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence |
| Answer | Marks | Guidance |
|---|---|---|
| Either: Show correct process for comp'n | M1 | |
| Obtain \(y = 3(3x + 7) - 2\) | A1 | |
| Obtain \(x = -\frac{19}{9}\) | A1 | 3 or exact equiv; condone absence of \(y = 0\) |
| Or: Use fg(x) = 0 to obtain \(g(x) = \frac{2}{3}\) | B1 | |
| Attempt solution of \(g(x) = \frac{2}{3}\) | M1 | |
| Obtain \(x = -\frac{19}{9}\) | A1 | (3) or exact equiv; condone absence of \(y = 0\) |
| Answer | Marks | Guidance |
|---|---|---|
| Attempt formation of one of the equations | M1 | or equiv |
| \(3x + 7 = \frac{x-7}{3}\) or \(3x + 7 = x\) or \(\frac{x-7}{3} = x\) | M1 | or equiv |
| Obtain \(x = -\frac{2}{3}\) | A1 | or equiv |
| Obtain \(y = -\frac{2}{3}\) | A1\3 | or equiv; following their value of \(x\) |
| Answer | Marks | Guidance |
|---|---|---|
| Attempt solution of modulus equation | M1 | squaring both sides to obtain 3-term quadratics or forming linear equation with different on each signs of 3x different on each side |
| Obtain \(-12x + 4 = 42x + 49\) or \(3x - 2 = -3x - 7\) | A1 | or equiv |
| Obtain \(x = -\frac{5}{6}\) | A1 | or exact equiv; as final answer |
| Obtain \(y = \frac{2}{3}\) | A1\4 | or equiv; and no other pair of answers |
## Part (i)
Either: Show correct process for comp'n | M1 |
Obtain $y = 3(3x + 7) - 2$ | A1 |
Obtain $x = -\frac{19}{9}$ | A1 | 3 or exact equiv; condone absence of $y = 0$
Or: Use fg(x) = 0 to obtain $g(x) = \frac{2}{3}$ | B1 |
Attempt solution of $g(x) = \frac{2}{3}$ | M1 |
Obtain $x = -\frac{19}{9}$ | A1 | (3) or exact equiv; condone absence of $y = 0$
## Part (ii)
Attempt formation of one of the equations | M1 | or equiv
$3x + 7 = \frac{x-7}{3}$ or $3x + 7 = x$ or $\frac{x-7}{3} = x$ | M1 | or equiv
Obtain $x = -\frac{2}{3}$ | A1 | or equiv
Obtain $y = -\frac{2}{3}$ | A1\3 | or equiv; following their value of $x$
## Part (iii)
Attempt solution of modulus equation | M1 | squaring both sides to obtain 3-term quadratics or forming linear equation with different on each signs of 3x different on each side
Obtain $-12x + 4 = 42x + 49$ or $3x - 2 = -3x - 7$ | A1 | or equiv
Obtain $x = -\frac{5}{6}$ | A1 | or exact equiv; as final answer
Obtain $y = \frac{2}{3}$ | A1\4 | or equiv; and no other pair of answers
---The functions f and g are defined for all real values of $x$ by
$$f(x) = 3x - 2 \quad \text{and} \quad g(x) = 3x + 7.$$
Find the exact coordinates of the point at which
\begin{enumerate}[label=(\roman*)]
\item the graph of $y = fg(x)$ meets the $x$-axis, [3]
\item the graph of $y = g(x)$ meets the graph of $y = g^{-1}(x)$, [3]
\item the graph of $y = |f(x)|$ meets the graph of $y = |g(x)|$. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR C3 2009 Q5 [10]}}