| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2019 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trig Graphs & Exact Values |
| Type | Find coordinates of turning points |
| Difficulty | Moderate -0.3 This is a multi-part question that tests basic understanding of trig graphs. Part (a) requires identifying a minimum point of cos 2x (straightforward). Part (b) involves sketching sin x (routine). Parts (c)(i)-(ii) require counting intersections over extended intervals, which is slightly more demanding than typical but still follows a standard pattern-recognition approach. Overall, slightly easier than average due to the visual/graphical nature and limited algebraic manipulation required. |
| Spec | 1.05f Trigonometric function graphs: symmetries and periodicities1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(x = \frac{\pi}{2}\) | B1 | Either coordinate correct. Accept \(90°\) or awrt \(1.57\) for \(x\)-coordinate |
| \(y = -1\) | B1 | Coordinates may be stated separately. If only one coordinate stated, must be clear if \(x\) or \(y\). SC: \(\left(-1, \frac{\pi}{2}\right)\) scores B1B0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Sine curve through \((0,0)\) with max/min of \(\pm 1\) | M1 | At least one cycle starting at/through origin, same max/min \(y\)-values as \(\cos 2x\) curve. Condone poor period and symmetry. Condone V-shaped turning points |
| Fully correct sketch of \(\sin x\) between \(-\frac{\pi}{2}\) and \(3\pi\) | A1 | Turning points must appear curved. Not acceptable if graph goes diagonally across square at turning points. Crossings of \(x\)-axis must be within half a square of correct points |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| (i) 30 | B1ft | Follow through on \(10\times\) number of solutions \(0 \to 2\pi\). "Hence or otherwise" so may get B1 for 30 even if graph doesn't suggest that number |
| (ii) 32 | B1 |
## Question 5:
### Part (a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $x = \frac{\pi}{2}$ | B1 | Either coordinate correct. Accept $90°$ or awrt $1.57$ for $x$-coordinate |
| $y = -1$ | B1 | Coordinates may be stated separately. If only one coordinate stated, must be clear if $x$ or $y$. SC: $\left(-1, \frac{\pi}{2}\right)$ scores B1B0 |
### Part (b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Sine curve through $(0,0)$ with max/min of $\pm 1$ | M1 | At least one cycle starting at/through origin, same max/min $y$-values as $\cos 2x$ curve. Condone poor period and symmetry. Condone V-shaped turning points |
| Fully correct sketch of $\sin x$ between $-\frac{\pi}{2}$ and $3\pi$ | A1 | Turning points must appear curved. Not acceptable if graph goes diagonally across square at turning points. Crossings of $x$-axis must be within half a square of correct points |
### Part (c):
| Answer | Mark | Guidance |
|--------|------|----------|
| (i) 30 | B1ft | Follow through on $10\times$ number of solutions $0 \to 2\pi$. "Hence or otherwise" so may get B1 for 30 even if graph doesn't suggest that number |
| (ii) 32 | B1 | |
---5.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{c8f8d35d-c2dd-4a1f-a4bb-a4fa06413d12-10_677_1036_260_456}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
Figure 2 shows a plot of part of the curve with equation $y = \cos 2 x$ with $x$ being measured in radians.
The point $P$, shown on Figure 2, is a minimum point on the curve.
\begin{enumerate}[label=(\alph*)]
\item State the coordinates of $P$.
A copy of Figure 2, called Diagram 1, is shown at the top of the next page.
\item Sketch, on Diagram 1, the curve with equation $y = \sin x$
\item Hence, or otherwise, deduce the number of solutions of the equation
\begin{enumerate}[label=(\roman*)]
\item $\cos 2 x = \sin x$ that lie in the region $0 \leqslant x \leqslant 20 \pi$
\item $\cos 2 x = \sin x$ that lie in the region $0 \leqslant x \leqslant 21 \pi$
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{c8f8d35d-c2dd-4a1f-a4bb-a4fa06413d12-11_693_1050_301_447}
\captionsetup{labelformat=empty}
\caption{\\
Diagram 1}\}\end{center}
\end{figure}
\textbackslash section*\{Diagram 1
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{Edexcel P1 2019 Q5 [6]}}