| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Sketch reciprocal function graphs |
| Difficulty | Standard +0.3 This is a multi-part question involving reciprocal trig functions (cosec), but each part uses standard techniques: (i) reflecting negative portions of a graph for modulus, (ii) solving simultaneous equations from given points, (iii) solving a basic cosec equation. While it requires understanding of cosec graphs and asymptotes, the steps are routine for C3 level with no novel problem-solving required. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.09a Sign change methods: locate roots |
5.\\
\includegraphics[max width=\textwidth, alt={}, center]{5dd332a5-56d9-407a-8ff6-fa59294b358d-2_520_787_246_479}
The diagram shows the graph of $y = \mathrm { f } ( x )$. The graph has a minimum at $\left( \frac { \pi } { 2 } , - 1 \right)$, a maximum at $\left( \frac { 3 \pi } { 2 } , - 5 \right)$ and an asymptote with equation $x = \pi$.\\
(i) Showing the coordinates of any stationary points, sketch the graph of $y = | \mathrm { f } ( x ) |$.
Given that
$$\mathrm { f } : x \rightarrow a + b \operatorname { cosec } x , \quad x \in \mathbb { R } , \quad 0 < x < 2 \pi , \quad x \neq \pi$$
(ii) find the values of the constants $a$ and $b$,\\
(iii) find, to 2 decimal places, the $x$-coordinates of the points where the graph of $y = \mathrm { f } ( x )$ crosses the $x$-axis.\\
\hfill \mbox{\textit{OCR C3 Q5 [8]}}