| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2013 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Function Transformations |
| Type | Identify transformation from equations |
| Difficulty | Moderate -0.8 This is a straightforward C2 question on function transformations and logarithms. Parts (i)-(iii) require only basic recall: identifying a horizontal translation, substituting into log₂x, and solving a simple logarithmic equation. Part (iv) involves setting up log₂c - log₂(c-3) = 4 and solving, which is routine manipulation. All parts are standard textbook exercises with no problem-solving insight required. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.06c Logarithm definition: log_a(x) as inverse of a^x1.06d Natural logarithm: ln(x) function and properties |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Translation of 3 units in positive \(x\)-direction | B1 | Must be 'translation' and not 'move', 'slide', 'shift' etc |
| State or imply 3 units in positive \(x\)-direction | B1 | Independent of first B1. Allow vector notation, but not a coordinate ie \((3,0)\). Worded descriptions must give clear intention of direction, so B0 for just '\(x\)-direction' or 'parallel to \(x\)-axis' unless \(+3\) also stated (as '\(+\)' implies the direction). For the direction, allow 'in the positive \(x\)-direction', 'parallel to the positive \(x\)-axis' or 'to the right'. Do not allow 'in the positive \(x\)-axis' or 'along the positive \(x\)-axis' even if combined with correct statement eg 'right'. Allow '3' or '3 units' but not '3 places', '3 squares', 'sf 3'. Ignore irrelevant statements, but penalise contradictions. B0 B0 if second transformation also given |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(a = 8\) | B1 | Allow \(x\) not \(a\). Allow implied value eg \((8,3)\) or \(\log_2 8 = 3\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(b - 3 = 2^{1.8}\), \(b = 6.48\) | B1 | State or imply \(b - 3 = 2^{1.8}\). Allow \(x\) not \(b\) |
| Obtain \(6.48\), or better | B1 | More accurate answer is \(6.482202253...\). Answer only can gain B2 as long as accurate |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\log_2 c - \log_2(c-3) = 4\) | M1 | Equate difference in \(y\)-coordinates to \(\pm 4\). Allow in terms of \(x\) not \(c\). Allow any equiv eg \(\log_2 c = \log_2(c-3)+4\). Brackets must be seen or implied by later working. Allow if subtraction is other way around, but M0 if two log terms are summed. Allow as part of Pythagoras attempt eg \(\sqrt{\{(c-c)^2+(\log_2 c - \log_2(c-3))^2\}}=4\) |
| Use \(\log a - \log b = \log \frac{a}{b}\) | M1 | Could be implied if \(\log_2\) dealt with at same time. Must be used on difference not sum. Starting with \(\log_2 c = \log_2(c-3)\), rearranging to equal 0 and then using a log law could get M1. Allow if 4 is attempted as \(\log_2 k\) \((k \neq 4)\) and then combined with at least one of the other two terms |
| Obtain \(\frac{c}{c-3} = 2^4\) | A1 | Any correct equation in a form not involving logs |
| Obtain \(\frac{16}{5}\) oe | A1 | Allow \(3.2\), or unsimplified fraction. SR B2 for answer only or T&I |
## Question 8(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Translation of 3 units in positive $x$-direction | B1 | Must be 'translation' and not 'move', 'slide', 'shift' etc |
| State or imply 3 units in positive $x$-direction | B1 | Independent of first B1. Allow vector notation, but not a coordinate ie $(3,0)$. Worded descriptions must give clear intention of direction, so B0 for just '$x$-direction' or 'parallel to $x$-axis' unless $+3$ also stated (as '$+$' implies the direction). For the direction, allow 'in the positive $x$-direction', 'parallel to the positive $x$-axis' or 'to the right'. Do not allow 'in the positive $x$-axis' or 'along the positive $x$-axis' even if combined with correct statement eg 'right'. Allow '3' or '3 units' but not '3 places', '3 squares', 'sf 3'. Ignore irrelevant statements, but penalise contradictions. **B0 B0** if second transformation also given |
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## Question 8(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $a = 8$ | B1 | Allow $x$ not $a$. Allow implied value eg $(8,3)$ or $\log_2 8 = 3$ |
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## Question 8(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $b - 3 = 2^{1.8}$, $b = 6.48$ | B1 | State or imply $b - 3 = 2^{1.8}$. Allow $x$ not $b$ |
| Obtain $6.48$, or better | B1 | More accurate answer is $6.482202253...$. Answer only can gain B2 as long as accurate |
---
## Question 8(iv):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\log_2 c - \log_2(c-3) = 4$ | M1 | Equate difference in $y$-coordinates to $\pm 4$. Allow in terms of $x$ not $c$. Allow any equiv eg $\log_2 c = \log_2(c-3)+4$. Brackets must be seen or implied by later working. Allow if subtraction is other way around, but M0 if two log terms are summed. Allow as part of Pythagoras attempt eg $\sqrt{\{(c-c)^2+(\log_2 c - \log_2(c-3))^2\}}=4$ |
| Use $\log a - \log b = \log \frac{a}{b}$ | M1 | Could be implied if $\log_2$ dealt with at same time. Must be used on difference not sum. Starting with $\log_2 c = \log_2(c-3)$, rearranging to equal 0 and then using a log law could get M1. Allow if 4 is attempted as $\log_2 k$ $(k \neq 4)$ and then combined with at least one of the other two terms |
| Obtain $\frac{c}{c-3} = 2^4$ | A1 | Any correct equation in a form not involving logs |
| Obtain $\frac{16}{5}$ oe | A1 | Allow $3.2$, or unsimplified fraction. **SR B2** for answer only or T&I |
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\includegraphics[max width=\textwidth, alt={}, center]{87012792-fa63-4003-875d-b8e7739037f1-4_489_697_274_667}
The diagram shows the curves $y = \log _ { 2 } x$ and $y = \log _ { 2 } ( x - 3 )$.\\
(i) Describe the geometrical transformation that transforms the curve $y = \log _ { 2 } x$ to the curve $y = \log _ { 2 } ( x - 3 )$.\\
(ii) The curve $y = \log _ { 2 } x$ passes through the point ( $a , 3$ ). State the value of $a$.\\
(iii) The curve $y = \log _ { 2 } ( x - 3 )$ passes through the point ( $b , 1.8$ ). Find the value of $b$, giving your answer correct to 3 significant figures.\\
(iv) The point $P$ lies on $y = \log _ { 2 } x$ and has an $x$-coordinate of $c$. The point $Q$ lies on $y = \log _ { 2 } ( x - 3 )$ and also has an $x$-coordinate of $c$. Given that the distance $P Q$ is 4 units find the exact value of $c$.
\hfill \mbox{\textit{OCR C2 2013 Q8 [9]}}