| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Topic | Laws of Logarithms |
| Type | Two unrelated log parts: one non-log algebraic part |
| Difficulty | Standard +0.3 This is a slightly easier than average A-level question. Part (a) requires setting up and solving a quadratic equation from the recurrence relation (routine algebra), part (b) is trivial recall of logarithm definition, and part (c) applies standard log laws mechanically. The recurrence relation adds mild complexity but the overall demand is below typical multi-step C2 questions. |
| Spec | 1.04e Sequences: nth term and recurrence relations1.06f Laws of logarithms: addition, subtraction, power rules |
The sequence $u_1, u_2, u_3, \ldots, u_n$ is defined by the recurrence relation
$$u_{n+1} = pu_n + 5, \quad u_1 = 2, \text{ where } p \text{ is a constant.}$$
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 Q35 [9]}}