| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2021 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trig Graphs & Exact Values |
| Type | Sketch trig curve and straight line, count intersections |
| Difficulty | Standard +0.3 This is a multi-part question requiring sketching y=1/x+1, identifying asymptotes, and counting intersections with tan x. While it involves multiple steps, each component is straightforward: recognizing the horizontal asymptote y=1, sketching a standard reciprocal function, and visually counting intersections. Part (c) extends the pattern recognition to larger domains, which is routine once the periodic behavior is understood. This is slightly easier than average as it's primarily visual pattern recognition rather than algebraic manipulation or proof. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.02o Sketch reciprocal curves: y=a/x and y=a/x^21.02q Use intersection points: of graphs to solve equations1.05a Sine, cosine, tangent: definitions for all arguments1.05f Trigonometric function graphs: symmetries and periodicities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(x = \dfrac{3\pi}{2}\) | B1 | oe and no others. Do not accept in degrees. May be labelled on graph but must be an equation. If multiple answers given then \(x = \dfrac{3\pi}{2}\) must be identified (e.g. circled) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Correct shape of \(\dfrac{1}{x}\) type curve in Quadrant 1 | B1 | Must not cross either axis; acceptable curvature — do not penalise unless it is clear a minimum point was intended |
| Correct shape and position for both branches with asymptote at \(y = 1\), labelled as \(y = 1\) or stated in working | B1 | Do not penalise sketch unless turning points are clearly intended. Asymptote/dashed line does not need to be drawn on sketch |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(5\) (solutions) | B1 | 5 only |
| Number of solutions are the number of points of intersections between the graphs | B1 | Do not allow if they mention where the graphs cross the \(x\)-axis |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Number of solutions \(= 40\) | B1ft | Follow through from their sketch (e.g. number of intersections in first quadrant \(\times 20\)) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Number of solutions \(= 14\) | B1 |
## Question 9:
**Part (a):**
| Answer | Mark | Guidance |
|--------|------|----------|
| $x = \dfrac{3\pi}{2}$ | B1 | oe and no others. Do not accept in degrees. May be labelled on graph but must be an equation. If multiple answers given then $x = \dfrac{3\pi}{2}$ must be identified (e.g. circled) |
**Total: (1)**
---
**Part (b)(i):**
| Answer | Mark | Guidance |
|--------|------|----------|
| Correct shape of $\dfrac{1}{x}$ type curve in Quadrant 1 | B1 | Must not cross either axis; acceptable curvature — do not penalise unless it is clear a minimum point was intended |
| Correct shape and position for both branches with asymptote at $y = 1$, labelled as $y = 1$ or stated in working | B1 | Do not penalise sketch unless turning points are clearly intended. Asymptote/dashed line does not need to be drawn on sketch |
**Total: (4)**
---
**Part (b)(ii):**
| Answer | Mark | Guidance |
|--------|------|----------|
| $5$ (solutions) | B1 | 5 only |
| Number of solutions are the number of points of intersections between the graphs | B1 | Do not allow if they mention where the graphs cross the $x$-axis |
---
**Part (c)(i):**
| Answer | Mark | Guidance |
|--------|------|----------|
| Number of solutions $= 40$ | B1ft | Follow through from their sketch (e.g. number of intersections in first quadrant $\times 20$) |
---
**Part (c)(ii):**
| Answer | Mark | Guidance |
|--------|------|----------|
| Number of solutions $= 14$ | B1 | |
**Total: (2)**
**Question Total: (7 marks)**9.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{877d03f2-d62c-4060-bdd2-f0d5dfbe6203-30_707_1034_251_456}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{center}
\end{figure}
Figure 4 shows a sketch of the curve with equation
$$y = \tan x \quad - 2 \pi \leqslant x \leqslant 2 \pi$$
The line $l$, shown in Figure 4, is an asymptote to $y = \tan x$
\begin{enumerate}[label=(\alph*)]
\item State an equation for $l$.
A copy of Figure 4, labelled Diagram 1, is shown on the next page.
\item \begin{enumerate}[label=(\roman*)]
\item On Diagram 1, sketch the curve with equation
$$y = \frac { 1 } { x } + 1 \quad - 2 \pi \leqslant x \leqslant 2 \pi$$
stating the equation of the horizontal asymptote of this curve.
\item Hence, giving a reason, state the number of solutions of the equation
\end{enumerate}\item State the number of solutions of the equation $\tan x = \frac { 1 } { x } + 1$ in the region
\begin{enumerate}[label=(\roman*)]
\item $0 \leqslant x \leqslant 40 \pi$
\item $- 10 \pi \leqslant x \leqslant \frac { 5 } { 2 } \pi$
$$\begin{aligned}
& \qquad \tan x = \frac { 1 } { x } + 1 \\
& \text { in the region } - 2 \pi \leqslant x \leqslant 2 \pi
\end{aligned}$$"
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{877d03f2-d62c-4060-bdd2-f0d5dfbe6203-31_725_1047_1078_447}
\captionsetup{labelformat=empty}
\caption{Diagram 1}
\end{center}
\end{figure}
\begin{center}
\includegraphics[max width=\textwidth, alt={}]{877d03f2-d62c-4060-bdd2-f0d5dfbe6203-32_2644_1837_118_114}
\end{center}
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{Edexcel P1 2021 Q9 [7]}}