Edexcel P1 2019 June — Question 9 7 marks

Exam BoardEdexcel
ModuleP1 (Pure Mathematics 1)
Year2019
SessionJune
Marks7
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicTrig Graphs & Exact Values
TypeSketch two trig curves and count intersections/solutions
DifficultyModerate -0.3 This is a multi-part question requiring basic transformations of sin (vertical stretch/translation), sketching tan with asymptotes, and counting intersections by recognizing periodicity. Part (a) is trivial recall, parts (b) are standard P1 graph sketching, and part (c) requires understanding that intersections repeat every 360° to scale up from the visible range. While multi-step, each component is routine for P1 level with no novel problem-solving required.
Spec1.02q Use intersection points: of graphs to solve equations1.05f Trigonometric function graphs: symmetries and periodicities1.05g Exact trigonometric values: for standard angles1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^2
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5eee32af-9b0e-428c-bbc6-1feef44e0e1e-24_741_806_255_577} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a plot of the curve with equation \(y = \sin \theta , \quad 0 \leqslant \theta \leqslant 360 ^ { \circ }\)
  1. State the coordinates of the minimum point on the curve with equation $$y = 4 \sin \theta , \quad 0 \leqslant \theta \leqslant 360 ^ { \circ }$$ A copy of Figure 3, called Diagram 1, is shown on the next page.
  2. On Diagram 1, sketch and label the curves
    1. \(y = 1 + \sin \theta , \quad 0 \leqslant \theta \leqslant 360 ^ { \circ }\)
    2. \(y = \tan \theta , \quad 0 \leqslant \theta \leqslant 360 ^ { \circ }\)
  3. Hence find the number of solutions of the equation
    1. \(\tan \theta = 1 + \sin \theta\) that lie in the region \(0 \leqslant \theta \leqslant 2160 ^ { \circ }\)
    2. \(\tan \theta = 1 + \sin \theta\) that lie in the region \(0 \leqslant \theta \leqslant 1980 ^ { \circ }\)
      \includegraphics[max width=\textwidth, alt={}]{5eee32af-9b0e-428c-bbc6-1feef44e0e1e-25_746_808_577_575}
      \section*{Diagram 1}
Question 9:
Part (a)
AnswerMarks Guidance
Answer/WorkingMark Guidance
\((270°, -4)\)B1 B1 (2) Either coordinate correct scores B1. Condone \(\frac{3\pi}{2}=270°\). Condone swapped coordinates \((-4, 270)\) for first B1 only. Do not accept multiple answers unless one point chosen.
Part (b)
AnswerMarks Guidance
Answer/WorkingMark Guidance
Graph of \(y = 1 + \sin\theta\)B1 Curve through \((0,1)\), \((90°,2)\), \((180°,1)\), \((270°,0)\), \((360°,1)\) with acceptable curvature. No straight lines.
Graph of \(y = \tan\theta\)B1 (2) Acceptable curvature. Must go beyond \(y=1\) and \(y=-1\). First quadrant \((0,0)\to(90°,\infty)\); second/third quadrants \((90°,-\infty)\to(270°,\infty)\) through \((180°,0)\); fourth quadrant \((270°,-\infty)\to(0,0)\)
Part (c)(i)
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(\frac{2160}{360}=6\) or \(\frac{2160}{180}=12\)M1 For this calculation or multiplying intersections in (b) by 6. Sight of 6 or 12 implies this mark.
\(6 \times 2 = 12\)A1 12 scores both marks
Part (c)(ii)
AnswerMarks Guidance
Answer/WorkingMark Guidance
11B1ft (3) (7 marks) Follow through on \(n\) less than answer to (c)(i), where \(n\) is number of solutions in range \(180° < \theta \leqslant 360°\) 
## Question 9:

### Part (a)

| Answer/Working | Mark | Guidance |
|---|---|---|
| $(270°, -4)$ | B1 B1 **(2)** | Either coordinate correct scores B1. Condone $\frac{3\pi}{2}=270°$. Condone swapped coordinates $(-4, 270)$ for first B1 only. Do not accept multiple answers unless one point chosen. |

### Part (b)

| Answer/Working | Mark | Guidance |
|---|---|---|
| Graph of $y = 1 + \sin\theta$ | B1 | Curve through $(0,1)$, $(90°,2)$, $(180°,1)$, $(270°,0)$, $(360°,1)$ with acceptable curvature. No straight lines. |
| Graph of $y = \tan\theta$ | B1 **(2)** | Acceptable curvature. Must go beyond $y=1$ and $y=-1$. First quadrant $(0,0)\to(90°,\infty)$; second/third quadrants $(90°,-\infty)\to(270°,\infty)$ through $(180°,0)$; fourth quadrant $(270°,-\infty)\to(0,0)$ |

### Part (c)(i)

| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{2160}{360}=6$ or $\frac{2160}{180}=12$ | M1 | For this calculation or multiplying intersections in (b) by 6. Sight of 6 or 12 implies this mark. |
| $6 \times 2 = 12$ | A1 | 12 scores both marks |

### Part (c)(ii)

| Answer/Working | Mark | Guidance |
|---|---|---|
| 11 | B1ft **(3) (7 marks)** | Follow through on $n$ less than answer to (c)(i), where $n$ is number of solutions in range $180° < \theta \leqslant 360°$ |

---
9.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{5eee32af-9b0e-428c-bbc6-1feef44e0e1e-24_741_806_255_577}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}

Figure 3 shows a plot of the curve with equation $y = \sin \theta , \quad 0 \leqslant \theta \leqslant 360 ^ { \circ }$
\begin{enumerate}[label=(\alph*)]
\item State the coordinates of the minimum point on the curve with equation

$$y = 4 \sin \theta , \quad 0 \leqslant \theta \leqslant 360 ^ { \circ }$$

A copy of Figure 3, called Diagram 1, is shown on the next page.
\item On Diagram 1, sketch and label the curves
\begin{enumerate}[label=(\roman*)]
\item $y = 1 + \sin \theta , \quad 0 \leqslant \theta \leqslant 360 ^ { \circ }$
\item $y = \tan \theta , \quad 0 \leqslant \theta \leqslant 360 ^ { \circ }$
\end{enumerate}\item Hence find the number of solutions of the equation
\begin{enumerate}[label=(\roman*)]
\item $\tan \theta = 1 + \sin \theta$ that lie in the region $0 \leqslant \theta \leqslant 2160 ^ { \circ }$
\item $\tan \theta = 1 + \sin \theta$ that lie in the region $0 \leqslant \theta \leqslant 1980 ^ { \circ }$

\begin{center}

\end{center}

\begin{center}
\includegraphics[max width=\textwidth, alt={}]{5eee32af-9b0e-428c-bbc6-1feef44e0e1e-25_746_808_577_575}
\end{center}

\section*{Diagram 1}
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{Edexcel P1 2019 Q9 [7]}}
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