| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2016 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Solve p(exponential) = 0 |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question testing standard Factor Theorem application, polynomial factorization, and basic logarithms. Part (i) is routine substitution, part (ii) requires algebraic division and analyzing roots (one real, two complex), and part (iii) is a simple logarithm application. While it requires multiple techniques across 8 marks, each step follows standard procedures without requiring problem-solving insight or novel approaches, making it slightly easier than average. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.06f Laws of logarithms: addition, subtraction, power rules1.06g Equations with exponentials: solve a^x = b |
The polynomial $\mathrm{p}(x)$ is defined by
$$\mathrm{p}(x) = ax^3 + 3x^2 + 4ax - 5,$$
where $a$ is a constant. It is given that $(2x - 1)$ is a factor of $\mathrm{p}(x)$.
\begin{enumerate}[label=(\roman*)]
\item Use the factor theorem to find the value of $a$. [2]
\item Factorise $\mathrm{p}(x)$ and hence show that the equation $\mathrm{p}(x) = 0$ has only one real root. [4]
\item Use logarithms to solve the equation $\mathrm{p}(6^x) = 0$ correct to 3 significant figures. [2]
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2016 Q4 [8]}}