| Exam Board | Edexcel |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2021 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Functions |
| Type | Find intersection of exponential curves |
| Difficulty | Moderate -0.3 Part (a) requires sketching standard exponential transformations with routine intercept calculations. Part (b) involves solving 2^x = 4^x - 6, which requires recognizing 4^x = (2^x)^2 to form a quadratic, then using the quadratic formula—a standard technique for P2 exponential equations with no novel insight required. Slightly easier than average due to straightforward algebraic manipulation. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.06a Exponential function: a^x and e^x graphs and properties1.06g Equations with exponentials: solve a^x = b |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Correct shape and position for \(y = 2^x\); crosses \(y\)-axis with same intersection as \(y = 4^x\) but with gentler slope | B1 | Above \(y=4^x\) to left of \(y\)-axis, below \(y=4^x\) to right; be tolerant of "wobbles" if correct shapes intended |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Graph of same shape as \(y = 4^x\) but translated down | B1 | Always below \(y = 4^x\) |
| \(y\) intercept at \(-5\) | B1 | |
| \(x\) intercept at \(\log_4 6\) | B1 | Accept equivalents such as \(\log 6 / \log 4\) or awrt 1.29 |
| (4) | Ignore references to asymptotes |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(2^x = 4^x - 6\) or \(y = (2^x)^2 - 6 = y^2 - 6\) | M1 | Sets equations equal in \(x\) only, or writes \(4^x\) in terms of \(2^x\) |
| \(\Rightarrow 2^{2x} - 2^x - 6 = 0 \Rightarrow (2^x - 3)(2^x + 2) = 0\) or \((y-3)(y+2) = 0\) | M1 | Solves to find value for \(2^x\) or \(y\) |
| \(y = 2^x = 3\) | A1 | Correct \(y\) (or \(2^x\)) value; may have second solution |
| \(x = \log_2 3\) | A1 | Must reject second "solution"; accept \(\log 3/\log 2\) or \(2\log_4 3\) |
| (4) |
## Question 2:
### Part (a)(i)
| Answer | Mark | Guidance |
|--------|------|----------|
| Correct shape and position for $y = 2^x$; crosses $y$-axis with same intersection as $y = 4^x$ but with gentler slope | **B1** | Above $y=4^x$ to left of $y$-axis, below $y=4^x$ to right; be tolerant of "wobbles" if correct shapes intended |
### Part (a)(ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| Graph of same shape as $y = 4^x$ but translated down | **B1** | Always below $y = 4^x$ |
| $y$ intercept at $-5$ | **B1** | |
| $x$ intercept at $\log_4 6$ | **B1** | Accept equivalents such as $\log 6 / \log 4$ or awrt 1.29 |
| | **(4)** | Ignore references to asymptotes |
### Part (b)
| Answer | Mark | Guidance |
|--------|------|----------|
| $2^x = 4^x - 6$ or $y = (2^x)^2 - 6 = y^2 - 6$ | **M1** | Sets equations equal in $x$ only, or writes $4^x$ in terms of $2^x$ |
| $\Rightarrow 2^{2x} - 2^x - 6 = 0 \Rightarrow (2^x - 3)(2^x + 2) = 0$ or $(y-3)(y+2) = 0$ | **M1** | Solves to find value for $2^x$ or $y$ |
| $y = 2^x = 3$ | **A1** | Correct $y$ (or $2^x$) value; may have second solution |
| $x = \log_2 3$ | **A1** | Must reject second "solution"; accept $\log 3/\log 2$ or $2\log_4 3$ |
| | **(4)** | |
---2.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{515f245f-9c5b-4263-ab2c-0a4f96f3bff0-04_1001_1481_267_221}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows a sketch of the curve with equation $y = 4 ^ { x }$
A copy of Figure 1, labelled Diagram 1, is shown on the next page.
\begin{enumerate}[label=(\alph*)]
\item On Diagram 1, sketch the curve with equation
\begin{enumerate}[label=(\roman*)]
\item $y = 2 ^ { x }$
\item $y = 4 ^ { x } - 6$
Label clearly the coordinates of any points of intersection with the coordinate axes.
The curve with equation $y = 2 ^ { x }$ meets the curve with equation $y = 4 ^ { x } - 6$ at the point $P$.
\end{enumerate}\item Using algebra, find the exact coordinates of $P$.
\begin{center}
\includegraphics[max width=\textwidth, alt={}]{515f245f-9c5b-4263-ab2c-0a4f96f3bff0-05_1009_1490_264_219}
\end{center}
\section*{Diagram 1}
\end{enumerate}
\hfill \mbox{\textit{Edexcel P2 2021 Q2 [8]}}