| Exam Board | Edexcel |
|---|---|
| Module | Paper 2 (Paper 2) |
| Session | Specimen |
| Marks | 3 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Quadratic trigonometric equations |
| Type | Identify error in student working |
| Difficulty | Easy -1.2 This is a pedagogical question about common student errors in solving a basic trig equation, requiring only identification and explanation of mistakes rather than solving a challenging problem. The actual equation cos θ = 2sin θ is straightforward (divide by cos θ to get tan θ = 2), making this significantly easier than average A-level questions. |
| Spec | 1.05o Trigonometric equations: solve in given intervals |
| \(\underline { \text { Student } A }\) |
| \(\cos \theta = 2 \sin \theta\) |
| \(\tan \theta = 2\) |
| \(\theta = 63.4 ^ { \circ }\) |
| Answer | Marks | Guidance |
|---|---|---|
| It should be \(\tan\theta = \frac{\sin\theta}{\cos\theta}\) | B1 | 2.3 |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Shows \(\cos(-26.6°) \neq 2\sin(-26.6°)\), so cannot be a solution | B1 | 2.4 |
| (ii) Explains that the incorrect answer was introduced by squaring | B1 | 2.4 |
# Question 2
## 2(a)
Identifies an error for student A: They use $\tan\theta = \frac{\sin\theta}{\sin\theta}$
It should be $\tan\theta = \frac{\sin\theta}{\cos\theta}$ | B1 | 2.3
(1 mark)
## 2(b)
(i) Shows $\cos(-26.6°) \neq 2\sin(-26.6°)$, so cannot be a solution | B1 | 2.4
(ii) Explains that the incorrect answer was introduced by squaring | B1 | 2.4
(2 marks)
**Total: 3 marks**
---
## Notes:
### (a)
B1: Accept a response of the type 'They use $\tan\theta = \frac{\cos\theta}{\sin\theta}$. This is incorrect as $\tan\theta = \frac{\sin\theta}{\cos\theta}$'
It can be implied by a response such as 'They should get $\tan\theta = \frac{1}{2}$ not $\tan\theta = 2$'
Accept also statements such as 'it should be $\cot\theta = 2$'
### (b)
B1: Accept a response where the candidate shows that $-26.6°$ is not a solution of $\cos\theta = 2\sin\theta$. This can be shown by, for example, finding both $\cos(-26.6°)$ and $2\sin(-26.6°)$ and stating that they are not equal. An acceptable alternative is to state that $\cos(-26.6°)$ is positive and $2\sin(-26.6°)$ is negative and stating that they therefore cannot be equal.
B1: Explains that the incorrect answer was introduced by squaring. Accept an example showing this. For example $x = 5$ squared gives $x^2 = 25$ which has answers $\pm 5$2. Some A level students were given the following question.
Solve, for $- 90 ^ { \circ } < \theta < 90 ^ { \circ }$, the equation
$$\cos \theta = 2 \sin \theta$$
The attempts of two of the students are shown below.
\begin{center}
\begin{tabular}{ | c | }
\hline
$\underline { \text { Student } A }$ \\
$\cos \theta = 2 \sin \theta$ \\
$\tan \theta = 2$ \\
$\theta = 63.4 ^ { \circ }$ \\
\end{tabular}
\end{center}
Student $B$
$$\begin{aligned}
\cos \theta & = 2 \sin \theta \\
\cos ^ { 2 } \theta & = 4 \sin ^ { 2 } \theta \\
1 - \sin ^ { 2 } \theta & = 4 \sin ^ { 2 } \theta \\
\sin ^ { 2 } \theta & = \frac { 1 } { 5 } \\
\sin \theta & = \pm \frac { 1 } { \sqrt { 5 } } \\
\theta & = \pm 26.6 ^ { \circ }
\end{aligned}$$
\begin{enumerate}[label=(\alph*)]
\item Identify an error made by student $A$.
Student $B$ gives $\theta = - 26.6 ^ { \circ }$ as one of the answers to $\cos \theta = 2 \sin \theta$.
\item \begin{enumerate}[label=(\roman*)]
\item Explain why this answer is incorrect.
\item Explain how this incorrect answer arose.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{Edexcel Paper 2 Q2 [3]}}