| Exam Board | OCR |
|---|---|
| Module | H240/02 (Pure Mathematics and Statistics) |
| Year | 2022 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Topic | Exponential Equations & Modelling |
| Type | Simple exponential equation solving |
| Difficulty | Moderate -0.8 This is a straightforward multi-part question testing routine algebraic techniques: (a) requires basic fraction manipulation, (b) is a disguised quadratic via substitution u=x³, and (c) uses simple index laws to equate powers. All three parts are standard textbook exercises with no problem-solving insight required, making this easier than average but not trivial due to the algebraic manipulation involved. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.06g Equations with exponentials: solve a^x = b |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\frac{x(x+2)-(x-1)(x+1)}{(x+1)(x+2)}\) or \(\frac{x^2+2x-x^2+1}{x^2+3x+2}\) oe \((=0)\) | M1, M1 | M1 for \(x(x+2)-(x+1)(x-1)\) oe. Multiply out brackets. Allow one error. Ignore denominator even if "\(=0\)" |
| \(x = -\frac{1}{2}\) | A1 | NB correct with no working: SC B1 |
| Alternative: \(x(x+2)=(x+1)(x-1)\), \(x^2+2x=x^2-1\), or \(2x=-1\), \(x=-\frac{1}{2}\) | M1, M1, A1 | M1 for attempt "cross-multiply". Multiply out brackets. Allow one error |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Solve quadratic in \(\frac{1}{x^3}\) or \(x^3\) or \(u\) \((=x^3\) or \(\frac{1}{x^3})\) using any correct method | M1 | or cubic in \(x\). Condone quadratic in \(x\) with \(x=\frac{1}{x^3}\) or \(x=x^3\). Must see attempt at correct method. Allow arithmetical errors |
| \(\frac{1}{x^3}\) (or \(u\)) \(= 1\) & \(-\frac{1}{8}\) or \(x^3\) (or \(u\)) \(= 1\) & \(-8\) or correct factorisation of quadratic | B1 | Can be scored without M1. Condone \(x=1\), \(-\frac{1}{8}\) or \(x=1,-8\). Ignore \(x^3=0\) if seen |
| \(x=1\) & \(x=-2\) with no extras | B1f | ft their \(x^3\) or \(\frac{1}{x^3}\). If also \(x=0\), B0. NB correct with no working: M0B0B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| eg \((x^2-7)\ln 3 = \ln\frac{1}{243}\) or \(x^2-7=\log_3\!\left(\frac{1}{243}\right)\) or \(3^{x^2-7}=3^{-5}\) or \(x^2-7=-5\) or \(3^{x^2}=3^2\) | M1 | Any correct step after log(both sides) or ANY correct step using indices. Condone incorrect or omitted brackets |
| \(x=\pm\sqrt{2}\) or \(\pm1.41\) (3 sf) | A1 | NB correct with no working or T & I: SC B1 |
# Question 1:
## Part (a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{x(x+2)-(x-1)(x+1)}{(x+1)(x+2)}$ or $\frac{x^2+2x-x^2+1}{x^2+3x+2}$ oe $(=0)$ | M1, M1 | M1 for $x(x+2)-(x+1)(x-1)$ oe. Multiply out brackets. Allow one error. Ignore denominator even if "$=0$" |
| $x = -\frac{1}{2}$ | A1 | NB correct with no working: SC B1 |
| **Alternative:** $x(x+2)=(x+1)(x-1)$, $x^2+2x=x^2-1$, or $2x=-1$, $x=-\frac{1}{2}$ | M1, M1, A1 | M1 for attempt "cross-multiply". Multiply out brackets. Allow one error |
## Part (b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Solve quadratic in $\frac{1}{x^3}$ or $x^3$ or $u$ $(=x^3$ or $\frac{1}{x^3})$ using any correct method | M1 | or cubic in $x$. Condone quadratic in $x$ with $x=\frac{1}{x^3}$ or $x=x^3$. Must see attempt at correct method. Allow arithmetical errors |
| $\frac{1}{x^3}$ (or $u$) $= 1$ & $-\frac{1}{8}$ or $x^3$ (or $u$) $= 1$ & $-8$ or correct factorisation of quadratic | B1 | Can be scored without M1. Condone $x=1$, $-\frac{1}{8}$ or $x=1,-8$. Ignore $x^3=0$ if seen |
| $x=1$ & $x=-2$ with no extras | B1f | ft their $x^3$ or $\frac{1}{x^3}$. If also $x=0$, B0. NB correct with no working: M0B0B1 |
## Part (c):
| Answer | Mark | Guidance |
|--------|------|----------|
| eg $(x^2-7)\ln 3 = \ln\frac{1}{243}$ or $x^2-7=\log_3\!\left(\frac{1}{243}\right)$ or $3^{x^2-7}=3^{-5}$ or $x^2-7=-5$ or $3^{x^2}=3^2$ | M1 | Any correct step after log(both sides) or ANY correct step using indices. Condone incorrect or omitted brackets |
| $x=\pm\sqrt{2}$ or $\pm1.41$ (3 sf) | A1 | NB correct with no working or T & I: SC B1 |
---1 In this question you must show detailed reasoning.
Solve the following equations.
\begin{enumerate}[label=(\alph*)]
\item $\frac { x } { x + 1 } - \frac { x - 1 } { x + 2 } = 0$
\item $\frac { 8 } { x ^ { 6 } } - \frac { 7 } { x ^ { 3 } } - 1 = 0$
\item $3 ^ { x ^ { 2 } - 7 } = \frac { 1 } { 243 }$
\end{enumerate}
\hfill \mbox{\textit{OCR H240/02 2022 Q1 [8]}}