| Exam Board | Edexcel |
|---|---|
| Module | AEA (Advanced Extension Award) |
| Year | 2004 |
| Session | June |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Topic | Modulus function |
| Type | Find range of k for number of roots |
| Difficulty | Challenging +1.8 This AEA question involves the fractional part function and requires multiple techniques: sketching a sawtooth graph, integration with the floor function, solving equations involving both functions, proving an algebraic relationship, and finding when a sequence converges sufficiently. While parts (a)-(d) are relatively standard, part (e) requires careful algebraic manipulation to derive the quadratic equation, and part (f) involves iterative calculation or solving inequalities with the quadratic formula. The multi-step nature and need to connect different parts elevates this above typical A-level questions, though it's more accessible than the most challenging AEA problems. |
| Spec | 1.01a Proof: structure of mathematical proof and logical steps1.02f Solve quadratic equations: including in a function of unknown1.02m Graphs of functions: difference between plotting and sketching1.02n Sketch curves: simple equations including polynomials1.02q Use intersection points: of graphs to solve equations1.08d Evaluate definite integrals: between limits1.08e Area between curve and x-axis: using definite integrals |
$$f(x) = x - [x], \quad x \geq 0$$
where $[x]$ is the largest integer $\leq x$.
For example, $f(3.7) = 3.7 - 3 = 0.7$; $f(3) = 3 - 3 = 0$.
\begin{enumerate}[label=(\alph*)]
\item Sketch the graph of $y = f(x)$ for $0 \leq x < 4$. [3]
\item Find the value of $p$ for which $\int_2^p f(x) dx = 0.18$. [3]
\end{enumerate}
Given that
$$g(x) = \frac{1}{1+kx}, \quad x \geq 0, \quad k > 0,$$
and that $x_0 = \frac{1}{2}$ is a root of the equation $f(x) = g(x)$,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item find the value of $k$. [2]
\item Add a sketch of the graph of $y = g(x)$ to your answer to part $(a)$. [1]
\end{enumerate}
The root of $f(x) = g(x)$ in the interval $n < x < n + 1$ is $x_n$, where $n$ is an integer.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{4}
\item Prove that
$$2 x_n^2 - (2n - 1)x_n - (n + 1) = 0.$$ [4]
\item Find the smallest value of $n$ for which $x_n - n < 0.05$. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel AEA 2004 Q6 [17]}}