| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2010 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trigonometric equations in context |
| Type | Show then solve substituted equation |
| Difficulty | Moderate -0.3 Part (a) is basic graph sketching requiring simple recall. Part (b)(i) is a straightforward algebraic manipulation using the Pythagorean identity sin²θ + cos²θ = 1 to reach cos θ = 1/2. Part (b)(ii) applies the result from (i) with a substitution (θ = 2x) and solving for x in the given interval. This is a standard C2 trigonometric equation with guided steps, slightly easier than average due to the scaffolding provided. |
| Spec | 1.05f Trigonometric function graphs: symmetries and periodicities1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05o Trigonometric equations: solve in given intervals |
| Answer | Marks |
|---|---|
| Correct cosine curve shape for \(0 \leq x \leq 2\pi\) | B1 |
| Intercepts: \((0,1)\), \(\left(\frac{\pi}{2}, 0\right)\), \(\left(\frac{3\pi}{2}, 0\right)\), \((2\pi, 1)\) | B1 |
| Answer | Marks |
|---|---|
| \(\sin^2\theta = 1 - \cos^2\theta\) used | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\cos\theta = \frac{1}{2}\) | A1 | Completion shown convincingly |
| Answer | Marks | Guidance |
|---|---|---|
| \(\cos 2x = \frac{1}{2}\) | M1 | Using result from (b)(i) with \(\theta = 2x\) |
| \(2x = \frac{\pi}{3}\) or \(2x = \frac{5\pi}{3}\) (and \(2x = \frac{7\pi}{3}\)) | M1 | Correct method for solving |
| \(x = \frac{\pi}{6} \approx 0.524\) | A1 | |
| \(x = \frac{5\pi}{6} \approx 2.62\) | A1 |
# Question 7:
## Part (a):
Correct cosine curve shape for $0 \leq x \leq 2\pi$ | B1 |
Intercepts: $(0,1)$, $\left(\frac{\pi}{2}, 0\right)$, $\left(\frac{3\pi}{2}, 0\right)$, $(2\pi, 1)$ | B1 |
## Part (b)(i):
$\sin^2\theta = 1 - \cos^2\theta$ used | M1 |
$1 - \cos^2\theta = \cos\theta(2-\cos\theta)$
$1 - \cos^2\theta = 2\cos\theta - \cos^2\theta$
$1 = 2\cos\theta$
$\cos\theta = \frac{1}{2}$ | A1 | Completion shown convincingly
## Part (b)(ii):
$\cos 2x = \frac{1}{2}$ | M1 | Using result from (b)(i) with $\theta = 2x$
$2x = \frac{\pi}{3}$ or $2x = \frac{5\pi}{3}$ (and $2x = \frac{7\pi}{3}$) | M1 | Correct method for solving
$x = \frac{\pi}{6} \approx 0.524$ | A1 |
$x = \frac{5\pi}{6} \approx 2.62$ | A1 |
---7
\begin{enumerate}[label=(\alph*)]
\item Sketch the graph of $y = \cos x$ in the interval $0 \leqslant x \leqslant 2 \pi$. State the values of the intercepts with the coordinate axes.
\item \begin{enumerate}[label=(\roman*)]
\item Given that
$$\sin ^ { 2 } \theta = \cos \theta ( 2 - \cos \theta )$$
prove that $\cos \theta = \frac { 1 } { 2 }$.
\item Hence solve the equation
$$\sin ^ { 2 } 2 x = \cos 2 x ( 2 - \cos 2 x )$$
in the interval $0 \leqslant x \leqslant \pi$, giving your answers in radians to three significant figures.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C2 2010 Q7 [8]}}