| Exam Board | OCR |
|---|---|
| Module | H240/01 (Pure Mathematics) |
| Year | 2022 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Topic | Exponential Equations & Modelling |
| Type | Logarithmic equation solving |
| Difficulty | Moderate -0.3 Part (a) tests basic transformation knowledge (translation left 4 units, vertical stretch factor 16), which is routine recall. Part (b) requires applying logarithm laws and solving a resulting quadratic, but follows a standard procedure taught explicitly in the syllabus. The question is slightly easier than average as it's methodical rather than requiring problem-solving insight, though the algebraic manipulation keeps it from being trivial. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.06f Laws of logarithms: addition, subtraction, power rules1.06g Equations with exponentials: solve a^x = b |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| 4 units in the negative \(x\)-direction | M1 | Indicate horizontal translation (in either direction) in some way with magnitude of 4 ('units' not required); B1 for \(\begin{pmatrix}4\\0\end{pmatrix}\); condone informal language as long as intent is clear; M0 if ambiguous e.g. 'in' or 'on' the \(x\)-axis |
| or 4 in negative \(x\)-direction; correct language needed | A1 [2] | B2 for \(\begin{pmatrix}-4\\0\end{pmatrix}\); must now be correct language so A0 for e.g. 'along' the \(x\)-axis or 'left'; allow 'parallel to the \(x\)-axis' or 'horizontal' |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| in the \(y\)-direction with scale factor 16 | B1 | Identify direction - correct language needed; allow '\(x\)-axis invariant', 'parallel to the \(y\)-axis' or 'vertical'; condone 'positive' \(y\)-direction (as given function \(> 0\)) |
| or \(2^4\) | B1 [2] | 'scale factor' or 'factor' needed (condone 'stretch' factor); not dep on previous B1, but must have indicated vertical stretch in some way, including informal language such as 'upwards'; cannot be ambiguous language such as 'in', 'on', 'across' the \(y\)-axis |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\log_2(8x(1-x)) = 1\) | M1 | Correctly combine two correct log terms; or \(\log_2(8x) = \log_2\frac{2}{1-x}\); or \(3 + \log_2(x(1-x)) = 1\); or \(\log_2(4x(1-x)) = 0\); OR use indices base 2 on both sides (i.e. \(8x = 2^{1-\log_2(1-x)}\)) and use rules of indices to split e.g. \(8x = 2 \times 2^{-\log_2(1-x)}\) |
| \(8x(1-x) = 2\) | M1 | Correct method to remove logs; correctly used on equation of form \(\log_2 f(x) = \log_2 g(x)\) or \(\log_2 f(x) = k\); OR correct method to deal with log term — expect \(8x = \frac{2}{1-x}\) |
| e.g. \(8x^2 - 8x + 2 = 0\) or \(8x(1-x) = 2\) or \(8x = \frac{2}{1-x}\) | A1 | Any correct equation not involving logarithms; could still contain brackets and/or fractions |
| \(x = 0.5\) | A1 [4] | Obtain \(x = 0.5\); A0 if additional solutions; DR so no credit for answer only |
# Question 5(a)(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| 4 units in the negative $x$-direction | M1 | Indicate horizontal translation (in either direction) in some way with magnitude of 4 ('units' not required); **B1** for $\begin{pmatrix}4\\0\end{pmatrix}$; condone informal language as long as intent is clear; **M0** if ambiguous e.g. 'in' or 'on' the $x$-axis |
| or 4 in negative $x$-direction; correct language needed | A1 [2] | **B2** for $\begin{pmatrix}-4\\0\end{pmatrix}$; must now be correct language so **A0** for e.g. 'along' the $x$-axis or 'left'; allow 'parallel to the $x$-axis' or 'horizontal' |
---
# Question 5(a)(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| in the $y$-direction with scale factor 16 | B1 | Identify direction - correct language needed; allow '$x$-axis invariant', 'parallel to the $y$-axis' or 'vertical'; condone 'positive' $y$-direction (as given function $> 0$) |
| or $2^4$ | B1 [2] | 'scale factor' or 'factor' needed (condone 'stretch' factor); not dep on previous B1, but must have indicated vertical stretch in some way, including informal language such as 'upwards'; cannot be ambiguous language such as 'in', 'on', 'across' the $y$-axis |
---
# Question 5(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\log_2(8x(1-x)) = 1$ | M1 | Correctly combine two correct log terms; or $\log_2(8x) = \log_2\frac{2}{1-x}$; or $3 + \log_2(x(1-x)) = 1$; or $\log_2(4x(1-x)) = 0$; **OR** use indices base 2 on both sides (i.e. $8x = 2^{1-\log_2(1-x)}$) and use rules of indices to split e.g. $8x = 2 \times 2^{-\log_2(1-x)}$ |
| $8x(1-x) = 2$ | M1 | Correct method to remove logs; correctly used on equation of form $\log_2 f(x) = \log_2 g(x)$ or $\log_2 f(x) = k$; **OR** correct method to deal with log term — expect $8x = \frac{2}{1-x}$ |
| e.g. $8x^2 - 8x + 2 = 0$ or $8x(1-x) = 2$ or $8x = \frac{2}{1-x}$ | A1 | Any correct equation not involving logarithms; could still contain brackets and/or fractions |
| $x = 0.5$ | A1 [4] | Obtain $x = 0.5$; **A0** if additional solutions; **DR** so no credit for answer only |
---5
\begin{enumerate}[label=(\alph*)]
\item The graph of $y = 2 ^ { x }$ can be transformed to the graph of $y = 2 ^ { x + 4 }$ either by a translation or by a stretch.
\begin{enumerate}[label=(\roman*)]
\item Give full details of the translation.
\item Give full details of the stretch.
\end{enumerate}\item In this question you must show detailed reasoning.
Solve the equation $\log _ { 2 } ( 8 x ) = 1 - \log _ { 2 } ( 1 - x )$.
\end{enumerate}
\hfill \mbox{\textit{OCR H240/01 2022 Q5 [8]}}