| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 9 |
| 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 that requires identifying function parameters from a graph and solving a cosecant equation. Part (a) is straightforward reflection of negative portions. Part (b) requires solving simultaneous equations using known points (standard technique). Part (c) involves routine equation solving. While it covers several skills, each step follows standard procedures with no novel insight required, making it slightly easier than average. |
| Spec | 1.02l Modulus function: notation, relations, equations and inequalities1.02n Sketch curves: simple equations including polynomials1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs |
| Answer | Marks |
|---|---|
| (a) Graph showing: stationary point at \((\frac{8}{3}, 1)\), stationary point at \((\frac{3\pi}{5}, 5)\) | B3 |
| Answer | Marks | Guidance |
|---|---|---|
| \((\frac{3\pi}{5}, -5) \Rightarrow -5 = a-b\) | B1 | |
| Adding, \(-6 = 2a \therefore a = -3, b = 2\) | M1 A1 | |
| (c) \(-3+2\cos x = 0\), \(\cos x = \frac{3}{2}\), \(\sin x = \frac{\sqrt{5}}{2}\) | M1 | |
| \(x = 0.73, \pi-0.7297\), \(x = 0.73, 2.41\) (2dp) | A2 | (9) |
**(a)** Graph showing: stationary point at $(\frac{8}{3}, 1)$, stationary point at $(\frac{3\pi}{5}, 5)$ | B3
**(b)** $(\frac{8}{3}, -1) \Rightarrow -1 = a+b$
$(\frac{3\pi}{5}, -5) \Rightarrow -5 = a-b$ | B1
Adding, $-6 = 2a \therefore a = -3, b = 2$ | M1 A1
**(c)** $-3+2\cos x = 0$, $\cos x = \frac{3}{2}$, $\sin x = \frac{\sqrt{5}}{2}$ | M1
$x = 0.73, \pi-0.7297$, $x = 0.73, 2.41$ (2dp) | A2 | **(9)**4.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{3bd9d8a3-a324-4649-9357-392a48a4a1de-3_508_771_255_488}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 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 $x = \pi$.
\begin{enumerate}[label=(\alph*)]
\item Showing the coordinates of any stationary points, sketch the graph of $y = | \mathrm { f } ( x ) |$.
Given that
$$f : x \rightarrow a + b \operatorname { cosec } x , \quad x \in \mathbb { R } , \quad 0 < x < 2 \pi , \quad x \neq \pi ,$$
\item find the values of the constants $a$ and $b$,
\item find, to 2 decimal places, the $x$-coordinates of the points where the graph of $y = \mathrm { f } ( x )$ crosses the $x$-axis.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 Q4 [9]}}