| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2016 |
| Session | June |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indefinite & Definite Integrals |
| Type | Exponential and logarithmic integration |
| Difficulty | Standard +0.3 This is a structured multi-part question testing standard C3 techniques: finding y-intercepts by substitution, locating turning points via differentiation, integration for area, and function transformations. While part (iv) requires understanding of even functions and symmetry, each step follows predictable methods with clear signposting. The algebraic manipulation is routine for C3 level, making this slightly easier than average. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.06a Exponential function: a^x and e^x graphs and properties1.07j Differentiate exponentials: e^(kx) and a^(kx)1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
Fig. 9 shows the curve $y = f(x)$, where $f(x) = e^{2x} + k e^{-2x}$ and $k$ is a constant greater than 1.
The curve crosses the $y$-axis at P and has a turning point Q.
\includegraphics{figure_9}
\begin{enumerate}[label=(\roman*)]
\item Find the $y$-coordinate of P in terms of $k$. [1]
\item Show that the $x$-coordinate of Q is $\frac{1}{4}\ln k$, and find the $y$-coordinate in its simplest form. [5]
\item Find, in terms of $k$, the area of the region enclosed by the curve, the $x$-axis, the $y$-axis and the line $x = \frac{1}{4}\ln k$. Give your answer in the form $ak + b$. [4]
\end{enumerate}
The function $g(x)$ is defined by $g(x) = f(x + \frac{1}{4}\ln k)$.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{3}
\item \begin{enumerate}[label=(\Alph*)]
\item Show that $g(x) = \sqrt{k}(e^{2x} + e^{-2x})$. [3]
\item Hence show that $g(x)$ is an even function. [2]
\item Deduce, with reasons, a geometrical property of the curve $y = f(x)$. [3]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C3 2016 Q9 [18]}}