| Exam Board | AQA |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Year | 2020 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard trigonometric equations |
| Type | Convert to quadratic in sin/cos |
| Difficulty | Moderate -0.3 This is a standard AS-level trigonometry question testing the Pythagorean identity and solving quadratic equations. Part (a) requires identifying two common algebraic errors (dividing by cos θ loses solutions, and missing solutions in the given range), while part (b) requires correctly solving 2cos²θ - cos θ = 0 by factoring. The question is slightly easier than average because it's scaffolded with worked solutions to critique, making the errors explicit rather than requiring independent problem-solving from scratch. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05o Trigonometric equations: solve in given intervals |
| Answer | Marks |
|---|---|
| 3(a) | Explains that Jia should not have |
| Answer | Marks | Guidance |
|---|---|---|
| Or obtains solution cosθ = 0 | 2.3 | E1 |
| Answer | Marks | Guidance |
|---|---|---|
| OE | 2.3 | E1 |
| Subtotal | 2 |
| Answer | Marks |
|---|---|
| 3(b) | Obtains any two correct |
| Answer | Marks | Guidance |
|---|---|---|
| Accept answer written in part (a) | 1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| and no extra ones | 1.1b | A1 |
| Subtotal | 2 | |
| Question Total | 4 | |
| Q | Marking Instructions | AO |
Question 3:
--- 3(a) ---
3(a) | Explains that Jia should not have
cancelled by cosθ OE
Or obtains solution cosθ = 0 | 2.3 | E1 | Jia should not have cancelled by cosθ
Jia has forgotten a second solution to
cosθ = 0.5
Explains that Jia has omitted the
other solution to cosθ = 0.5
OE | 2.3 | E1
Subtotal | 2
--- 3(b) ---
3(b) | Obtains any two correct
solutions.
Accept answer written in part (a) | 1.1a | M1 | θ = ±60°
and ±90°
Obtains four correct solutions
and no extra ones | 1.1b | A1
Subtotal | 2
Question Total | 4
Q | Marking Instructions | AO | Marks | Typical SolutionJia has to solve the equation
$$2 - 2\sin^2 \theta = \cos \theta$$
where $-180° \leq \theta \leq 180°$
Jia's working is as follows:
$$2 - 2(1 - \cos^2 \theta) = \cos \theta$$
$$2 - 2 + 2\cos^2 \theta = \cos \theta$$
$$2\cos^2 \theta = \cos \theta$$
$$2\cos \theta = 1$$
$$\cos \theta = 0.5$$
$$\theta = 60°$$
Jia's teacher tells her that her solution is incomplete.
\begin{enumerate}[label=(\alph*)]
\item Explain the two errors that Jia has made. [2 marks]
\item Write down all the values of $\theta$ that satisfy the equation
$$2 - 2\sin^2 \theta = \cos \theta$$
where $-180° \leq \theta \leq 180°$ [2 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA AS Paper 1 2020 Q3 [4]}}