| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2007 |
| Session | January |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Equations & Modelling |
| Type | Simple exponential equation solving |
| Difficulty | Easy -1.3 This is a straightforward C2 question testing basic index laws with a common base (8). Part (a) requires simple recall of index rules with no problem-solving, while part (b) involves one additional step of algebraic manipulation. The entire question is routine bookwork with no conceptual challenges beyond recognizing powers of 8. |
| Spec | 1.02a Indices: laws of indices for rational exponents |
| Answer | Marks | Guidance |
|---|---|---|
| (a)(i) | \(\{p = \} 2\) | B1 |
| (ii) | \(ar^2 = 3\) | B1 |
| \(r^2 = \frac{1}{16} \Rightarrow r = -\frac{1}{4}\) or \(r = \frac{1}{4}\) | A1 | CSO AG Full valid completion. SC Clear explicit verification (max B2 out of 3.) |
| 4 | ||
| (b)(i) | \(a = -192\) | B1 |
| (ii) | \(\frac{a}{1-r} = \frac{a}{1-(-\frac{1}{4})}\) | M1 |
| \(S_\infty = \frac{-768}{5} (= -153.6)\) | A1ft | Ft on candidate's value for \(a\). i.e. \(\frac{4}{5}a\) SC candidate uses \(r = 0.25\), gives \(a = 192\) and sum to infinity \(= 256\). (max. B0 M1A1) |
| Total: 7 |
(a)(i) | $\{p = \} 2$ | B1 | For either. OE |
| | | |
(ii) | $ar^2 = 3$ | B1 | Elimination of $a$ OE |
| | | |
| $r^2 = \frac{1}{16} \Rightarrow r = -\frac{1}{4}$ or $r = \frac{1}{4}$ | A1 | CSO AG Full valid completion. SC Clear explicit verification (max B2 out of 3.) |
| | **4** | |
(b)(i) | $a = -192$ | B1 | 1 |
| | | |
(ii) | $\frac{a}{1-r} = \frac{a}{1-(-\frac{1}{4})}$ | M1 | $\frac{a}{1-r}$ used |
| | | |
| $S_\infty = \frac{-768}{5} (= -153.6)$ | A1ft | Ft on candidate's value for $a$. i.e. $\frac{4}{5}a$ SC candidate uses $r = 0.25$, gives $a = 192$ and sum to infinity $= 256$. (max. B0 M1A1) |
| | **Total: 7** | |
---3
\begin{enumerate}[label=(\alph*)]
\item Write down the values of $p , q$ and $r$ given that:
\begin{enumerate}[label=(\roman*)]
\item $64 = 8 ^ { p }$;
\item $\frac { 1 } { 64 } = 8 ^ { q }$;
\item $\sqrt { 8 } = 8 ^ { r }$.
\end{enumerate}\item Find the value of $x$ for which
$$\frac { 8 ^ { x } } { \sqrt { 8 } } = \frac { 1 } { 64 }$$
\end{enumerate}
\hfill \mbox{\textit{AQA C2 2007 Q3 [5]}}