| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2012 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Evaluate composite at point |
| Difficulty | Moderate -0.8 This question tests basic understanding of composite functions, inverse functions, and Simpson's rule application. Part (i) requires simple table lookup: ff(6) = f(14) = 26 and f^(-1)(8) = 4. Part (ii) is a standard reflection in y=x. Part (iii) is routine Simpson's rule with values provided. All parts are direct application of learned techniques with no problem-solving or insight required, making this easier than average. |
| Spec | 1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence1.02w Graph transformations: simple transformations of f(x)1.09f Trapezium rule: numerical integration |
| \(x\) | 2 | 4 | 6 | 8 | 10 | 12 | 14 |
| \(y\) | 1 | 8 | 14 | 19 | 23 | 25 | 26 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State \(26\) | B1 | |
| State \(4\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Sketch (more or less) correct curve | B1 | With approx correct curvatures and curve going through second quadrant but not fourth; allow if sketch does not meet given curve on line \(y=x\) |
| Refer to reflection in \(y=x\) or symmetrical about \(y=x\) or mirrored in \(y=x\) | B1 | Explicit reference needed, not just line \(y=x\) shown on sketch |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Attempt calculation \(k(y + 4y + 2y + \ldots)\) | M1 | Any constant \(k\); with \(y\)-values from table and coefficients 1, 2 and 4 occurring at least once each |
| Obtain \(k(1+32+28+76+46+100+26)\) | A1 | Or (unsimplified) equiv |
| Use \(k = \frac{1}{3} \times 2\) | A1 | |
| Obtain \(206\) | A1 |
# Question 5(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State $26$ | B1 | |
| State $4$ | B1 | |
---
# Question 5(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Sketch (more or less) correct curve | B1 | With approx correct curvatures and curve going through second quadrant but not fourth; allow if sketch does not meet given curve on line $y=x$ |
| Refer to reflection in $y=x$ or symmetrical about $y=x$ or mirrored in $y=x$ | B1 | Explicit reference needed, not just line $y=x$ shown on sketch |
---
# Question 5(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Attempt calculation $k(y + 4y + 2y + \ldots)$ | M1 | Any constant $k$; with $y$-values from table and coefficients 1, 2 and 4 occurring at least once each |
| Obtain $k(1+32+28+76+46+100+26)$ | A1 | Or (unsimplified) equiv |
| Use $k = \frac{1}{3} \times 2$ | A1 | |
| Obtain $206$ | A1 | |
---5\\
\includegraphics[max width=\textwidth, alt={}, center]{89e54367-bb83-483a-add5-0527b71a5cac-3_844_837_242_621}
It is given that f is a one-one function defined for all real values. The diagram shows the curve with equation $y = \mathrm { f } ( x )$. The coordinates of certain points on the curve are shown in the following table.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | }
\hline
$x$ & 2 & 4 & 6 & 8 & 10 & 12 & 14 \\
\hline
$y$ & 1 & 8 & 14 & 19 & 23 & 25 & 26 \\
\hline
\end{tabular}
\end{center}
(i) State the value of $\mathrm { ff } ( 6 )$ and the value of $\mathrm { f } ^ { - 1 } ( 8 )$.\\
(ii) On the copy of the diagram, sketch the curve $y = \mathrm { f } ^ { - 1 } ( x )$, indicating how the curves $y = \mathrm { f } ( x )$ and $y = \mathrm { f } ^ { - 1 } ( x )$ are related.\\
(iii) Use Simpson's rule with 6 strips to find an approximation to $\int _ { 2 } ^ { 14 } \mathrm { f } ( x ) \mathrm { d } x$.
\hfill \mbox{\textit{OCR C3 2012 Q5 [8]}}