| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2023 |
| Session | September |
| Marks | 7 |
| Topic | Quadratic trigonometric equations |
| Type | Identify error in student working |
| Difficulty | Moderate -0.3 This is a standard A-level question testing understanding of common algebraic errors in trigonometric equations. Part (a) requires identifying that dividing by cos²θ loses solutions where cosθ=0, and that √(sin²θ) should give ±1/2. Part (b) is routine calculation. The question is slightly easier than average because the errors are clearly signposted and the working is provided, making it more about recognizing standard pitfalls than solving from scratch. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05o Trigonometric equations: solve in given intervals |
8.
Martin tried to find all the solutions of $4 \sin ^ { 2 } \theta \cos ^ { 2 } \theta - \cos ^ { 2 } \theta = 0$ for $0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }$\\
His working is shown below:
$$\begin{aligned}
& 4 \sin ^ { 2 } \theta \cos ^ { 2 } \theta - \cos ^ { 2 } \theta = 0 \\
& \Rightarrow 4 \sin ^ { 2 } \theta \cos ^ { 2 } \theta = \cos ^ { 2 } \theta \\
& \Rightarrow 4 \sin ^ { 2 } \theta = 1 \\
& \Rightarrow \sin ^ { 2 } \theta = \frac { 1 } { 4 } \\
& \Rightarrow \sin \theta = \frac { 1 } { 2 } \\
& \Rightarrow \theta = 30 ^ { \circ } , 150 ^ { \circ }
\end{aligned}$$
Martin did not find all the correct solutions because he made two errors.
\begin{enumerate}[label=(\alph*)]
\item Identify the two errors and explain the consequence of each error.\\[0pt]
[4 marks]
\item Find all the solutions that Martin did not find.
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2023 Q8 [7]}}