| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Laws of Logarithms |
| Type | Two unrelated log parts: one non-log algebraic part |
| Difficulty | Moderate -0.3 This is a multi-part C2 question combining recurrence relations and logarithms. Part (a) requires setting up and solving a quadratic equation from the recurrence relation (routine algebra), part (b) is trivial recall of log laws, and part (c) applies standard logarithm rules. The question is slightly easier than average as it's mostly procedural with clear signposting and no conceptual challenges beyond applying learned techniques. |
| Spec | 1.04e Sequences: nth term and recurrence relations1.06f Laws of logarithms: addition, subtraction, power rules |
| Answer | Marks | Guidance |
|---|---|---|
| \(u_2 = 2p + 5\) | B1 | |
| \(u_3 = p(2p + 5) + 5\) | M1 | |
| \(8 = 2p^2 + 5p + 5\) or \(2p^2 + 5p - 3 = 0\) | M1 | |
| \((2p - 1)(p + 3) = 0\) | M1 | |
| \(p = -3\) or \(\frac{1}{2}\) | A1, B1 cso | (5 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\log_2 \left(\frac{1}{2}\right) = \log_2 2^{-1} = -1\) | B1 | (1 mark) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\log_2 \left(\frac{p^3}{\sqrt{q}}\right) = \log_2 p^3 - \log_2 \sqrt{q}\) | Use of \(\log a - \log b\) | M1 |
| \(b\) | ||
| \(= 3\log_2 p - \frac{1}{2}\log_2 q\) | Use of \(\log a^n\) | M1 |
| \(= -3 - \frac{1}{2}t\) | accept \(3\log_2 p - \frac{1}{2}t\) | A1 ft |
**Part (a)**
$u_2 = 2p + 5$ | B1 |
$u_3 = p(2p + 5) + 5$ | M1 |
$8 = 2p^2 + 5p + 5$ or $2p^2 + 5p - 3 = 0$ | M1 |
$(2p - 1)(p + 3) = 0$ | M1 |
$p = -3$ or $\frac{1}{2}$ | A1, B1 cso | (5 marks)
**Part (b)**
$\log_2 \left(\frac{1}{2}\right) = \log_2 2^{-1} = -1$ | B1 | (1 mark)
**Part (c)**
$\log_2 \left(\frac{p^3}{\sqrt{q}}\right) = \log_2 p^3 - \log_2 \sqrt{q}$ | Use of $\log a - \log b$ | M1 |
$b$ | | |
$= 3\log_2 p - \frac{1}{2}\log_2 q$ | Use of $\log a^n$ | M1 |
$= -3 - \frac{1}{2}t$ | accept $3\log_2 p - \frac{1}{2}t$ | A1 ft | (3 marks)
**Total for Question 6: 9 marks**
---The sequence $u_1, u_2, u_3, \ldots, u_n$ is defined by the recurrence relation
$$u_{n+1} = pu_n + 5, u_1 = 2, \text{ where } p \text{ is a constant.}$$
Given that $u_3 = 8$,
\begin{enumerate}[label=(\alph*)]
\item show that one possible value of $p$ is $\frac{1}{2}$ and find the other value of $p$. [5]
\end{enumerate}
Using $p = \frac{1}{2}$,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item write down the value of $\log_2 p$. [1]
\end{enumerate}
Given also that $\log_2 q = t$,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item express $\log_2 \left(\frac{p^3}{\sqrt{q}}\right)$ in terms of $t$. [3]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q6 [9]}}