| Exam Board | AQA |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Year | 2019 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard trigonometric equations |
| Type | Deduce related solution |
| Difficulty | Moderate -0.3 This is a standard AS-level trigonometric equation question requiring substitution of the Pythagorean identity, solving a quadratic, and applying a double angle transformation. Part (a)(i) is verification only (2 marks), part (a)(ii) is routine solving (2 marks), and part (b) is a straightforward extension using the 'hence' structure. While it requires multiple techniques, all are standard AS procedures with no novel insight needed, making it slightly easier than average. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| (a)(i) | Uses identity to replace sin2θ with | |
| (1 – cos2θ ) or uses sinθ = | 1.2 | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | 1.1a | A1 |
| Answer | Marks |
|---|---|
| (a)(ii) | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | 1.1b | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| lose second B1 | 1.1b | B1 |
| Answer | Marks |
|---|---|
| 6(b) | Writes down a set that is half the |
| Answer | Marks | Guidance |
|---|---|---|
| part (a) Accept 36° or 144° | 2.2a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| Condone AWRT integer values | 1.1b | A1F |
| Total | 6 | |
| Q | Marking Instructions | AO |
Question 6:
--- 6
(a)(i) ---
6
(a)(i) | Uses identity to replace sin2θ with
(1 – cos2θ ) or uses sinθ = | 1.2 | M1 | 6(1 – cos2θ ) + 5cosθ = 7
6cos2θ – 5cosθ +1 = 0
(2cosθ – 1)(3cosθ – 1) = 0
cosθ =
1
√3
Solves quadratic equation to get
2
one solution cosθ = .
1
Or verifying using cosθ =
2 | 1.1a | A1
--- 6
(a)(ii) ---
6
(a)(ii) | 1
States any two correct solutions
2 | 1.1b | B1 | 2
θ = 60°, 300°,
Or cosθ =
1
3
θ =71°,289°
States two additional correct
solutions. Condone answers of
70.5 and 289.5 or greater
accuracy. Ignore any additional
answers outside the range but any
additional answers inside range
lose second B1 | 1.1b | B1
--- 6(b) ---
6(b) | Writes down a set that is half the
values given as their solutions in
part (a) Accept 36° or 144° | 2.2a | M1 | θ = 30°, 150°, 35°, 145°
210°, 330°, 215°, 325°
Writes down an additional set that
is 180° more than the first set.
Condone AWRT integer values | 1.1b | A1F
Total | 6
Q | Marking Instructions | AO | Marks | Typical Solution\begin{enumerate}[label=(\alph*)]
\item
\begin{enumerate}[label=(\roman*)]
\item Show that $\cos \theta = \frac{1}{2}$ is one solution of the equation
$$6\sin^2 \theta + 5\cos \theta = 7$$
[2 marks]
\item Find all the values of $\theta$ that solve the equation
$$6\sin^2 \theta + 5\cos \theta = 7$$
for $0° \leq \theta \leq 360°$
Give your answers to the nearest degree.
[2 marks]
\end{enumerate}
\item Hence, find all the solutions of the equation
$$6\sin^2 2\theta + 5\cos 2\theta = 7$$
for $0° \leq \theta \leq 360°$
Give your answers to the nearest degree.
[2 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA AS Paper 1 2019 Q6 [6]}}