| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2007 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Function Transformations |
| Type | Sequence of transformations order |
| Difficulty | Standard +0.8 This question requires understanding the order of transformations (a common source of confusion), factoring to identify the correct sequence, sketching transformed logarithmic curves including modulus, and interpreting the modulus condition algebraically. While systematic, it demands careful reasoning about transformation order and multiple connected parts beyond routine application. |
| Spec | 1.02l Modulus function: notation, relations, equations and inequalities1.02s Modulus graphs: sketch graph of |ax+b|1.02w Graph transformations: simple transformations of f(x)1.02x Combinations of transformations: multiple transformations1.06d Natural logarithm: ln(x) function and properties |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State \(a\) in \(x\)-direction | B1 | or clear equiv |
| State factor 2 in \(x\)-direction | B1 | 2 or clear equiv |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Show (largely) increasing function crossing \(x\)-axis | M1 | with correct curvature |
| Show curve in first and fourth quadrants only | A1 | 2 not touching \(y\)-axis and with no maximum point; ignore intercept |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Show attempt at reflecting negative part in \(x\)-axis | M1 | |
| Show (more or less) correct graph | A1\(\sqrt{}\) | 2 following their graph in (ii) and showing correct curvatures |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Identify \(2a\) as asymptote or \(2a+2\) as intercept | B1 | allow anywhere in question |
| State \(2a < x \leq 2a+2\) | B1 | 2 allow \(<\) or \(\leq\) for each inequality |
# Question 7:
## Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State $a$ in $x$-direction | B1 | or clear equiv |
| State factor 2 in $x$-direction | B1 | **2** or clear equiv |
## Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Show (largely) increasing function crossing $x$-axis | M1 | with correct curvature |
| Show curve in first and fourth quadrants only | A1 | **2** not touching $y$-axis and with no maximum point; ignore intercept |
## Part (iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Show attempt at reflecting negative part in $x$-axis | M1 | |
| Show (more or less) correct graph | A1$\sqrt{}$ | **2** following their graph in (ii) and showing correct curvatures |
## Part (iv):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Identify $2a$ as asymptote or $2a+2$ as intercept | B1 | allow anywhere in question |
| State $2a < x \leq 2a+2$ | B1 | **2** allow $<$ or $\leq$ for each inequality |
---7 The curve $y = \ln x$ is transformed to the curve $y = \ln \left( \frac { 1 } { 2 } x - a \right)$ by means of a translation followed by a stretch. It is given that $a$ is a positive constant.\\
(i) Give full details of the translation and stretch involved.\\
(ii) Sketch the graph of $y = \ln \left( \frac { 1 } { 2 } x - a \right)$.\\
(iii) Sketch, on another diagram, the graph of $y = \left| \ln \left( \frac { 1 } { 2 } x - a \right) \right|$.\\
(iv) State, in terms of $a$, the set of values of $x$ for which $\left| \ln \left( \frac { 1 } { 2 } x - a \right) \right| = - \ln \left( \frac { 1 } { 2 } x - a \right)$.
\hfill \mbox{\textit{OCR C3 2007 Q7 [8]}}