| Exam Board | AQA |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2006 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Express cos²x or sin²x in terms of cos 2x |
| Difficulty | Moderate -0.8 This is a straightforward application of the double angle formula cos 2x = 2cos²x - 1, requiring simple rearrangement in part (a), followed by routine integration in part (b). Both parts are standard textbook exercises with no problem-solving insight required, making it easier than average. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\cos 2x = 2\cos^2 x - 1\) | B1B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\cos^2 x = \frac{1}{2}(\cos 2x + 1)\) | M1 | Attempt to express \(\cos^2 x\) in terms of \(\cos 2x\) |
| \(\frac{1}{2}\int_0^{\frac{\pi}{2}} \cos 2x + 1\ dx = \left[\frac{1}{4}\sin 2x + \frac{x}{2}\right]_0^{\frac{\pi}{2}}\) | A1, A1 | |
| \(= \frac{\pi}{4}\) | M1A1F | Use limits. Ft on integer \(a\) |
## Question 6:
### Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\cos 2x = 2\cos^2 x - 1$ | B1B1 | |
### Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\cos^2 x = \frac{1}{2}(\cos 2x + 1)$ | M1 | Attempt to express $\cos^2 x$ in terms of $\cos 2x$ |
| $\frac{1}{2}\int_0^{\frac{\pi}{2}} \cos 2x + 1\ dx = \left[\frac{1}{4}\sin 2x + \frac{x}{2}\right]_0^{\frac{\pi}{2}}$ | A1, A1 | |
| $= \frac{\pi}{4}$ | M1A1F | Use limits. Ft on integer $a$ |
---6
\begin{enumerate}[label=(\alph*)]
\item Express $\cos 2 x$ in the form $a \cos ^ { 2 } x + b$, where $a$ and $b$ are constants.
\item Hence show that $\int _ { 0 } ^ { \frac { \pi } { 2 } } \cos ^ { 2 } x \mathrm {~d} x = \frac { \pi } { a }$, where $a$ is an integer.
\end{enumerate}
\hfill \mbox{\textit{AQA C4 2006 Q6 [7]}}