| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2010 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trig Graphs & Exact Values |
| Type | Determine range or set of values |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question on sine transformations requiring standard techniques: solving a basic trig equation for exact values, sketching a transformed sine curve, finding range restrictions, and working with inverse functions. All parts follow routine procedures with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02v Inverse and composite functions: graphs and conditions for existence1.05f Trigonometric function graphs: symmetries and periodicities1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(4 - 3\sin x = 2 \rightarrow \sin x = ⅔ \rightarrow x = 0.730\) or \(2.41\) | M1, A1 A1√ [3] | Makes \(\sin x\) the subject + solution. co. √ for \(\pi\) – first answer. |
| (ii) | B1, B1 [2] | Must be 1 complete oscillation. Shape and position correct, in 1st quadrant, curve not lines. |
| (iii) \(k < 1\), \(k > 7\). | B1 B1 [2] | B1 for \(k = 1, 7\); B1 for answer. Or B1 for \(k < 1\); B1 for \(k > 7\). |
| (iv) \(A = \frac{3\pi}{2}\). | B1 [1] | co. |
| (v) \(\sin x = ⅓\) – or using inverse \(g^{-1}(3) = 2.80\) | M1A1 [2] | M1 for soln of \(3 = 4 - 3\sin x\) or inverse. M1 for soln of \(3 = 4 - 3\sin x\) or inverse. |
$f: x \mapsto 4 - 3\sin x$
(i) $4 - 3\sin x = 2 \rightarrow \sin x = ⅔ \rightarrow x = 0.730$ or $2.41$ | M1, A1 A1√ [3] | Makes $\sin x$ the subject + solution. co. √ for $\pi$ – first answer.
(ii) | B1, B1 [2] | Must be 1 complete oscillation. Shape and position correct, in 1st quadrant, curve not lines.
(iii) $k < 1$, $k > 7$. | B1 B1 [2] | B1 for $k = 1, 7$; B1 for answer. Or B1 for $k < 1$; B1 for $k > 7$.
(iv) $A = \frac{3\pi}{2}$. | B1 [1] | co.
(v) $\sin x = ⅓$ – or using inverse $g^{-1}(3) = 2.80$ | M1A1 [2] | M1 for soln of $3 = 4 - 3\sin x$ or inverse. M1 for soln of $3 = 4 - 3\sin x$ or inverse.The function $f : x \mapsto 4 - 3\sin x$ is defined for the domain $0 \leq x < 2\pi$.
\begin{enumerate}[label=(\roman*)]
\item Solve the equation $f(x) = 2$. [3]
\item Sketch the graph of $y = f(x)$. [2]
\item Find the set of values of $k$ for which the equation $f(x) = k$ has no solution. [2]
\end{enumerate}
The function $g : x \mapsto 4 - 3\sin x$ is defined for the domain $\frac{1}{2}\pi \leq x \leq A$.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{3}
\item State the largest value of $A$ for which $g$ has an inverse. [1]
\item For this value of $A$, find the value of $g^{-1}(3)$. [2]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2010 Q11 [10]}}