| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2015 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Solve equation involving composites |
| Difficulty | Standard +0.3 This is a straightforward multi-part question on composite and inverse functions requiring standard techniques: evaluating compositions, finding inverses, and solving logarithmic equations. All parts follow routine procedures with no novel insight needed, making it slightly easier than the average A-level question. |
| Spec | 1.02v Inverse and composite functions: graphs and conditions for existence1.06d Natural logarithm: ln(x) function and properties1.06f Laws of logarithms: addition, subtraction, power rules |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Obtain 6 or \(2+4\) at any stage for application of f | B1 | |
| Attempt composition of functions the right way round | M1 | |
| Obtain \(a=\frac{1}{4}\) or \(\frac{9}{36}\) or equiv | A1 | |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Obtain expression involving \(e^{y-2}\) or \(e^{x-2}\) | M1 | |
| Obtain \(e^{x-2}-3\) | A1 | |
| State \(x\geq2+\ln3\) or equiv | B1 | Not for \(>\); not for decimal equiv; using \(x\) |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Apply f once to obtain \(2+N\) | B1 | |
| Apply f to their expression involving \(N\) | M1 | |
| Obtain \(2+\ln(N+5)\) or \(2+\ln(2+N+3)\) | A1 | |
| Attempt solution of equation of form \(2+\ln(pN+q)=\ln(53e^2)\) | M1 | Involving manipulation so that value of \(N\) is apparent |
| Obtain 48 from correct work | A1 |
# Question 8(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Obtain 6 or $2+4$ at any stage for application of f | B1 | |
| Attempt composition of functions the right way round | M1 | |
| Obtain $a=\frac{1}{4}$ or $\frac{9}{36}$ or equiv | A1 | |
| **[3]** | | |
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# Question 8(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Obtain expression involving $e^{y-2}$ or $e^{x-2}$ | M1 | |
| Obtain $e^{x-2}-3$ | A1 | |
| State $x\geq2+\ln3$ or equiv | B1 | Not for $>$; not for decimal equiv; using $x$ |
| **[3]** | | |
---
# Question 8(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Apply f once to obtain $2+N$ | B1 | |
| Apply f to their expression involving $N$ | M1 | |
| Obtain $2+\ln(N+5)$ or $2+\ln(2+N+3)$ | A1 | |
| Attempt solution of equation of form $2+\ln(pN+q)=\ln(53e^2)$ | M1 | Involving manipulation so that value of $N$ is apparent |
| Obtain 48 from correct work | A1 | |8 The functions $f$ and $g$ are defined as follows:
$$\begin{gathered}
\mathrm { f } ( x ) = 2 + \ln ( x + 3 ) \text { for } x \geqslant 0 \\
\mathrm {~g} ( x ) = a x ^ { 2 } \text { for all real values of } x , \text { where } a \text { is a positive constant. }
\end{gathered}$$
(i) Given that $\operatorname { gf } \left( \mathrm { e } ^ { 4 } - 3 \right) = 9$, find the value of $a$.\\
(ii) Find an expression for $\mathrm { f } ^ { - 1 } ( x )$ and state the domain of $\mathrm { f } ^ { - 1 }$.\\
(iii) Given that $\mathrm { ff } \left( \mathrm { e } ^ { N } - 3 \right) = \ln \left( 53 \mathrm { e } ^ { 2 } \right)$, find the value of $N$.
\hfill \mbox{\textit{OCR C3 2015 Q8 [11]}}