| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2013 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trig Graphs & Exact Values |
| Type | Sketch two trig curves and count intersections/solutions |
| Difficulty | Moderate -0.3 This question requires sketching two standard trigonometric curves and identifying intersections, which are routine A-level skills. Part (ii)(a) is straightforward horizontal line intersection, while (ii)(b) requires recognizing that intersections of the two sketched curves solve the equation. The question tests graph sketching and interpretation rather than algebraic manipulation or novel problem-solving, making it slightly easier than average. |
| Spec | 1.02m Graphs of functions: difference between plotting and sketching1.02q Use intersection points: of graphs to solve equations1.05f Trigonometric function graphs: symmetries and periodicities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Graph: \(y = \sin 2x\) correct shape | B1 | \(y = \sin 2x\) has 2 cycles, starts and finishes on \(x\)-axis, max comes first |
| From \(+1\) to \(-1\), smooth curves | DB1 B1 | \(y = \cos x - 1\) has one cycle, starts and finishes on \(x\)-axis, with a minimum point |
| From \(0\) to \(-2\), smooth curve, flattens | DB1 [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| (a) \(\sin 2x = -\frac{1}{2} \rightarrow 4\) solutions | B1\(\sqrt{}\) [1] | \(\sqrt{}\) for their curve |
| (b) \(\sin 2x + \cos x + 1 = 0 \rightarrow 3\) solutions | B1\(\sqrt{}\) [1] | \(\sqrt{}\) for intersections of their curves |
## Question 5:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Graph: $y = \sin 2x$ correct shape | B1 | $y = \sin 2x$ has 2 cycles, starts and finishes on $x$-axis, max comes first |
| From $+1$ to $-1$, smooth curves | DB1 B1 | $y = \cos x - 1$ has one cycle, starts and finishes on $x$-axis, with a minimum point |
| From $0$ to $-2$, smooth curve, flattens | DB1 [4] | |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| **(a)** $\sin 2x = -\frac{1}{2} \rightarrow 4$ solutions | B1$\sqrt{}$ [1] | $\sqrt{}$ for their curve |
| **(b)** $\sin 2x + \cos x + 1 = 0 \rightarrow 3$ solutions | B1$\sqrt{}$ [1] | $\sqrt{}$ for intersections of their curves |
---5 (i) Sketch, on the same diagram, the curves $y = \sin 2 x$ and $y = \cos x - 1$ for $0 \leqslant x \leqslant 2 \pi$.\\
(ii) Hence state the number of solutions, in the interval $0 \leqslant x \leqslant 2 \pi$, of the equations
\begin{enumerate}[label=(\alph*)]
\item $2 \sin 2 x + 1 = 0$,
\item $\sin 2 x - \cos x + 1 = 0$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2013 Q5 [6]}}