| Exam Board | Edexcel |
|---|---|
| Module | Paper 1 (Paper 1) |
| Year | 2019 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Small angle approximation |
| Type | Estimate root of equation |
| Difficulty | Moderate -0.3 This is a straightforward application of small angle approximation requiring students to sketch a line on a given graph, identify one intersection point, then substitute cos x ≈ 1 - x²/2 into the equation and solve a simple quadratic. While it involves multiple steps, each step is routine and the question explicitly guides students through the process with clear instructions. |
| Spec | 1.05e Small angle approximations: sin x ~ x, cos x ~ 1-x^2/2, tan x ~ x1.09a Sign change methods: locate roots1.09b Sign change methods: understand failure cases |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Draws \(y = 2x + \frac{1}{2}\) on Figure 1/Diagram 1 with correct gradient and intercept | B1 | Look for straight line with intercept \(\approx \frac{1}{2}\) and further point at \(\approx\left(\frac{1}{2}, 1\frac{1}{2}\right)\). Tolerance of 0.25 of a square. Must appear in quadrants 1, 2 and 3. |
| States there is only one intersection, therefore only one root | B1 | Requires a reason and minimal conclusion. Allowable linear graph must have intercept \(\pm\frac{1}{2}\) with one intersection with \(\cos x\) OR gradient \(\pm 2\) with one intersection with \(\cos x\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\cos x - 2x - \frac{1}{2} = 0 \Rightarrow 1 - \frac{x^2}{2} - 2x - \frac{1}{2} = 0\) | M1 | Attempts small angle approximation \(\cos x = 1 - \frac{x^2}{2}\) in given equation |
| Solves their \(x^2 + 4x - 1 = 0\) | dM1 | Proceeds to 3TQ in single variable and attempts to solve. Allow completion of square, formula or calculator. Do not allow factorisation unless equation does factorise. |
| \(x \approx 0.236\), accept \(-2 + \sqrt{5}\) | A1 | Do not allow if another root given and it is not obvious that 0.236 has been chosen |
## Question 2:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Draws $y = 2x + \frac{1}{2}$ on Figure 1/Diagram 1 with correct gradient and intercept | B1 | Look for straight line with intercept $\approx \frac{1}{2}$ and further point at $\approx\left(\frac{1}{2}, 1\frac{1}{2}\right)$. Tolerance of 0.25 of a square. Must appear in quadrants 1, 2 and 3. |
| States there is only one intersection, therefore only one root | B1 | Requires a reason and minimal conclusion. Allowable linear graph must have intercept $\pm\frac{1}{2}$ with one intersection with $\cos x$ OR gradient $\pm 2$ with one intersection with $\cos x$ |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\cos x - 2x - \frac{1}{2} = 0 \Rightarrow 1 - \frac{x^2}{2} - 2x - \frac{1}{2} = 0$ | M1 | Attempts small angle approximation $\cos x = 1 - \frac{x^2}{2}$ in given equation |
| Solves their $x^2 + 4x - 1 = 0$ | dM1 | Proceeds to 3TQ in single variable and attempts to solve. Allow completion of square, formula or calculator. Do not allow factorisation unless equation does factorise. |
| $x \approx 0.236$, accept $-2 + \sqrt{5}$ | A1 | Do not allow if another root given and it is not obvious that 0.236 has been chosen |
---2.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{91a2f26a-add2-4b58-997d-2ae229548217-04_670_1447_212_333}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows a plot of part of the curve with equation $y = \cos x$ where $x$ is measured in radians. Diagram 1, on the opposite page, is a copy of Figure 1.
\begin{enumerate}[label=(\alph*)]
\item Use Diagram 1 to show why the equation
$$\cos x - 2 x - \frac { 1 } { 2 } = 0$$
has only one real root, giving a reason for your answer.
Given that the root of the equation is $\alpha$, and that $\alpha$ is small,
\item use the small angle approximation for $\cos x$ to estimate the value of $\alpha$ to 3 decimal places.
\begin{center}
\includegraphics[max width=\textwidth, alt={}]{91a2f26a-add2-4b58-997d-2ae229548217-05_664_1452_246_333}
\end{center}
\section*{Diagram 1}
\end{enumerate}
\hfill \mbox{\textit{Edexcel Paper 1 2019 Q2 [5]}}