| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2020 |
| Session | October |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Sketch transformed curve from description |
| Difficulty | Moderate -0.3 This P1 question tests standard transformations (horizontal translation and reflection) and basic trigonometry (finding amplitude and solving cos equations). Part (i) requires applying well-rehearsed transformation rules to find new coordinates. Part (ii) involves substituting a point to find k, then solving a standard cosine equation. All techniques are routine for P1 level with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.05f Trigonometric function graphs: symmetries and periodicities1.05g Exact trigonometric values: for standard angles |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Horizontal translation \(\leftarrow\) | B1 | Negative cubic in quadrants 2, 3, 4 only; local max on \(x\)-axis where \(x\leq0\); local min in third quadrant |
| Maximum at origin \((0,0)\) | B1 | Does not need to be labelled |
| Passes through \((-7, 0)\) | B1 | Allow just \(x\)-value; condone slip of coords reversed if sketch is correct; do not allow \(7\) instead of \(-7\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Reflection in \(y\)-axis | B1 | Positive cubic in quadrants 1, 3, 4 only; local max on \(x\)-axis where \(x\leq0\); local min in quadrant 4; do not accept local min on \(y\)-axis |
| Touches at \((-2, 0)\) and passes through \((5, 0)\) | B1 | Allow just \(x\)-values; do not allow \(2\) instead of \(-2\) or \(-5\) instead of \(5\) |
| Passes through \((0, -3)\) | B1 | Allow just \(-3\); do not allow \(3\) instead of \(-3\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(x=0 \Rightarrow y=k\cos\!\left(\frac{\pi}{6}\right)=\sqrt{3}\); \(k\cdot\frac{\sqrt{3}}{2}=\sqrt{3} \Rightarrow k=2\) | B1 | \(k=2\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\left(p=\right)\dfrac{\pi}{3}\) or \(\left(q=\right)\dfrac{4\pi}{3}\) | B1 | Award for sight of either value; allow awrt 1.05 or awrt 4.19; allow in degrees (60 or 240) |
| \(\left(p=\right)\dfrac{\pi}{3}\) and \(\left(q=\right)\dfrac{4\pi}{3}\) | B1 | Both correct; ignore labelling; condone \(\left(0,\frac{\pi}{3}\right),\left(0,\frac{4\pi}{3}\right)\); allow exact equivalents |
# Question 5(i)(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Horizontal translation $\leftarrow$ | B1 | Negative cubic in quadrants 2, 3, 4 only; local max on $x$-axis where $x\leq0$; local min in third quadrant |
| Maximum at origin $(0,0)$ | B1 | Does not need to be labelled |
| Passes through $(-7, 0)$ | B1 | Allow just $x$-value; condone slip of coords reversed if sketch is correct; do not allow $7$ instead of $-7$ |
---
# Question 5(i)(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Reflection in $y$-axis | B1 | Positive cubic in quadrants 1, 3, 4 only; local max on $x$-axis where $x\leq0$; local min in quadrant 4; do not accept local min on $y$-axis |
| Touches at $(-2, 0)$ and passes through $(5, 0)$ | B1 | Allow just $x$-values; do not allow $2$ instead of $-2$ or $-5$ instead of $5$ |
| Passes through $(0, -3)$ | B1 | Allow just $-3$; do not allow $3$ instead of $-3$ |
---
# Question 5(ii)(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x=0 \Rightarrow y=k\cos\!\left(\frac{\pi}{6}\right)=\sqrt{3}$; $k\cdot\frac{\sqrt{3}}{2}=\sqrt{3} \Rightarrow k=2$ | B1 | $k=2$ |
---
# Question 5(ii)(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\left(p=\right)\dfrac{\pi}{3}$ **or** $\left(q=\right)\dfrac{4\pi}{3}$ | B1 | Award for sight of either value; allow awrt 1.05 or awrt 4.19; allow in degrees (60 or 240) |
| $\left(p=\right)\dfrac{\pi}{3}$ **and** $\left(q=\right)\dfrac{4\pi}{3}$ | B1 | Both correct; ignore labelling; condone $\left(0,\frac{\pi}{3}\right),\left(0,\frac{4\pi}{3}\right)$; allow exact equivalents |
**(9 marks)**5. (i)
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{dfb4b2bc-4bc8-4e5b-9b13-ffe4fbde1b4f-14_572_1025_212_463}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
Figure 2 shows a sketch of the curve with equation $y = \mathrm { f } ( x )$.\\
The curve passes through the points $( - 5,0 )$ and $( 0 , - 3 )$ and touches the $x$-axis at the point $( 2,0 )$.
On separate diagrams sketch the curve with equation
\begin{enumerate}[label=(\alph*)]
\item $y = \mathrm { f } ( x + 2 )$
\item $y = \mathrm { f } ( - x )$
On each diagram, show clearly the coordinates of all the points where the curve cuts or touches the coordinate axes.\\
(ii)
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{dfb4b2bc-4bc8-4e5b-9b13-ffe4fbde1b4f-14_415_814_1548_571}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}
Figure 3 shows a sketch of the curve with equation
$$y = k \cos \left( x + \frac { \pi } { 6 } \right) \quad 0 \leqslant x \leqslant 2 \pi$$
where $k$ is a constant.\\
The curve meets the $y$-axis at the point $( 0 , \sqrt { 3 } )$ and passes through the points $( p , 0 )$ and ( $q , 0$ ).
Find\\
(a) the value of $k$,\\
(b) the exact value of $p$ and the exact value of $q$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel P1 2020 Q5 [9]}}