| Exam Board | OCR |
|---|---|
| Module | H240/02 (Pure Mathematics and Statistics) |
| Year | 2023 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Express and solve equation |
| Difficulty | Standard +0.3 This is a structured multi-part question on differentiation of trigonometric functions and curve sketching. Parts (a)(i) and (a)(ii) involve routine differentiation and solving standard trig equations. Part (b) requires connecting derivatives to graph features (stationary/inflection points). Parts (c)(i) and (c)(ii) test understanding of the relationship between f'(x) and increasing/decreasing behavior. While it requires multiple techniques and careful reasoning, all components are standard A-level material with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03e Complete the square: find centre and radius of circle1.03f Circle properties: angles, chords, tangents1.08e Area between curve and x-axis: using definite integrals |
In this question you must show detailed reasoning.
The function f is defined by $\text{f}(x) = \cos x + \sqrt{3} \sin x$ with domain $0 \leqslant x \leqslant 2\pi$.
\begin{enumerate}[label=(\alph*)]
\item Solve the following equations.
\begin{enumerate}[label=(\roman*)]
\item $\text{f}'(x) = 0$ [4]
\item $\text{f}''(x) = 0$ [3]
\end{enumerate}
The diagram shows the graph of the gradient function $y = \text{f}'(x)$ for the domain $0 \leqslant x \leqslant 2\pi$.
\includegraphics{figure_5}
\item Use your answers to parts (a)(i) and (a)(ii) to find the coordinates of points $A$, $B$, $C$ and $D$. [2]
\item \begin{enumerate}[label=(\roman*)]
\item Explain how to use the graph of the gradient function to find the values of $x$ for which f(x) is increasing. [1]
\item Using set notation, write down the set of values of $x$ for which f(x) is increasing in the domain $0 \leqslant x \leqslant 2\pi$. [2]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{OCR H240/02 2023 Q5 [12]}}