| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2011 |
| Session | June |
| Marks | 8 |
| 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 testing standard A-level content: finding range of a trigonometric function, solving a basic equation using exact values (cos x = ±1/√2), sketching a cosine transformation, and identifying whether a function is one-to-one. All parts are routine applications of well-practiced techniques 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 existence1.05f Trigonometric function graphs: symmetries and periodicities1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Other limit is \(3\) | B1 B1 [2] | co irrespective of inequalities. co irrespective of inequalities. |
| Answer | Marks | Guidance |
|---|---|---|
| \(\to x = \frac{1}{4}\pi\) or \(\frac{3}{4}\pi\) | M1 A1 A1∨ [3] | Makes \(\cos x\) the subject. co (radians). \(\sqrt{for "\)\pi\( – (1st answer)"\) ("exact" means that decimal answers only earn A0 A1∨)} |
| (ii)(a) | B1 B1 [2] | Joins (0, −1) to \((\pi, 7)\), providing increasing function. Not a line, flattens at extremities-needs inflexion. |
| (b) \(f\) has an inverse since it is 1:1 or increasing or no turning points. | B1 [1] | co independent of part (i) |
**(i)(a)** $f(x) = 3 - 4\cos^2x$.
One limit is $-1$
Other limit is $3$ | B1 B1 [2] | co irrespective of inequalities. co irrespective of inequalities.
**(b)** $3 - 4\cos^2x = 1 \to \cos^2x = \frac{1}{2}$
$\to \cos x = \pm\frac{1}{\sqrt{2}}$
$\to x = \frac{1}{4}\pi$ or $\frac{3}{4}\pi$ | M1 A1 A1∨ [3] | Makes $\cos x$ the subject. co (radians). $\sqrt{for "$\pi$ – (1st answer)"$ ("exact" means that decimal answers only earn A0 A1∨)}
**(ii)(a)** | B1 B1 [2] | Joins (0, −1) to $(\pi, 7)$, providing increasing function. Not a line, flattens at extremities-needs inflexion.
**(b)** $f$ has an inverse since it is 1:1 or increasing or no turning points. | B1 [1] | co independent of part (i)The function $f$ is such that $f(x) = 3 - 4\cos^k x$, for $0 \leq x \leq \pi$, where $k$ is a constant.
\begin{enumerate}[label=(\roman*)]
\item In the case where $k = 2$,
\begin{enumerate}[label=(\alph*)]
\item find the range of $f$, [2]
\item find the exact solutions of the equation $f(x) = 1$. [3]
\end{enumerate}
\item In the case where $k = 1$,
\begin{enumerate}[label=(\alph*)]
\item sketch the graph of $y = f(x)$, [2]
\item state, with a reason, whether $f$ has an inverse. [1]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2011 Q9 [8]}}