| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2014 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trig Graphs & Exact Values |
| Type | Solve trigonometric equation with exact values |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question on transformed trigonometric functions requiring standard techniques: solving a basic cosine equation for an exact value, identifying range from amplitude/vertical shift, sketching a transformed cosine graph, and finding an inverse function. All parts are routine applications with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02m Graphs of functions: difference between plotting and sketching1.02v Inverse and composite functions: graphs and conditions for existence1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{1}{2}x = \frac{1}{3}\pi, x = \frac{2}{3}\pi\) | M1, M1, A1 [3] | Makes \(\cos\left(\frac{1}{2}x\right)\) the subject.; Looks up "\(\frac{1}{2}x\)" before \(\times 2\); co (120° gets A0 – decimals A0) |
| (ii) Range is \(2 \lesssim f(x) \lesssim 10\) | B1, B1 [2] | condone \(<\) |
| (iii) | B1, B1 [2] | Point of inflexion at \(\pi\) Fully correct |
| Answer | Marks | Guidance |
|---|---|---|
| \(f^{-1}(x) = 2\cos^{-1}\left(\frac{6-x}{4}\right)\) | M1, M1, A1 [3] | Makes \(\cos\left(\frac{1}{2}x\right)\) the subject; Order of operations correct (M marks allowed if + for −); oe – needs to be a function of \(x\) not \(y\) |
$f: x \mapsto 6 - 4\cos\left(\frac{1}{2}x\right)$
**(i)** $6 - 4\cos\left(\frac{1}{2}x\right) = 4 \rightarrow 4\cos\left(\frac{1}{2}x\right) = 2$
$\frac{1}{2}x = \frac{1}{3}\pi, x = \frac{2}{3}\pi$ | M1, M1, A1 [3] | Makes $\cos\left(\frac{1}{2}x\right)$ the subject.; Looks up "$\frac{1}{2}x$" before $\times 2$; co (120° gets A0 – decimals A0)
**(ii)** Range is $2 \lesssim f(x) \lesssim 10$ | B1, B1 [2] | condone $<$
**(iii)** | B1, B1 [2] | Point of inflexion at $\pi$ Fully correct
**(iv)** $\cos\left(\frac{1}{2}x\right) = \frac{1}{4}(6-y)$
$\frac{1}{2}x = \cos^{-1}\left(\frac{1}{4}(6-y)\right)$
$f^{-1}(x) = 2\cos^{-1}\left(\frac{6-x}{4}\right)$ | M1, M1, A1 [3] | Makes $\cos\left(\frac{1}{2}x\right)$ the subject; Order of operations correct (M marks allowed if + for −); oe – needs to be a function of $x$ not $y$The function $f : x \mapsto 6 - 4\cos(\frac{1}{2}x)$ is defined for $0 \leqslant x \leqslant 2\pi$.
\begin{enumerate}[label=(\roman*)]
\item Find the exact value of $x$ for which $f(x) = 4$. [3]
\item State the range of $f$. [2]
\item Sketch the graph of $y = f(x)$. [2]
\item Find an expression for $f^{-1}(x)$. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2014 Q11 [10]}}