| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Deduce related integral from numerical approximation |
| Difficulty | Standard +0.3 This is a slightly above-average C2 question. Part (a) is trivial recall. Part (b) is standard trapezium rule application with exact trigonometric values. Part (c) requires the insight that sin²x + cos²x = 1, making it a rectangle minus part (b), which elevates it slightly above routine but remains accessible for C2 students who recognize the identity. |
| Spec | 1.05g Exact trigonometric values: for standard angles1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Notes |
| \(\frac{1}{2}\sqrt{3}\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Notes |
| \(x\): \(0\), \(\frac{\pi}{6}\), \(\frac{\pi}{3}\) | M1 | |
| \(\cos^2 x\): \(1\), \(\frac{3}{4}\), \(\frac{1}{4}\) | A1 | |
| area \(\approx \frac{1}{2} \times \frac{\pi}{6} \times [1 + \frac{1}{4} + 2(\frac{3}{4})]\) | B1 M1 | |
| \(= 0.720\) (3sf) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Notes |
| area of \(S = \int_0^{\frac{\pi}{3}} \sin^2 x \, dx = \int_0^{\frac{\pi}{3}} (1 - \cos^2 x) \, dx\) | M1 | |
| \(= \frac{\pi}{3} - 0.71995 = 0.327\) (3sf) | M1 A1 | (9) |
## Question 5:
**(a)**
| Answer/Working | Marks | Notes |
|---|---|---|
| $\frac{1}{2}\sqrt{3}$ | B1 | |
**(b)**
| Answer/Working | Marks | Notes |
|---|---|---|
| $x$: $0$, $\frac{\pi}{6}$, $\frac{\pi}{3}$ | M1 | |
| $\cos^2 x$: $1$, $\frac{3}{4}$, $\frac{1}{4}$ | A1 | |
| area $\approx \frac{1}{2} \times \frac{\pi}{6} \times [1 + \frac{1}{4} + 2(\frac{3}{4})]$ | B1 M1 | |
| $= 0.720$ (3sf) | A1 | |
**(c)**
| Answer/Working | Marks | Notes |
|---|---|---|
| area of $S = \int_0^{\frac{\pi}{3}} \sin^2 x \, dx = \int_0^{\frac{\pi}{3}} (1 - \cos^2 x) \, dx$ | M1 | |
| $= \frac{\pi}{3} - 0.71995 = 0.327$ (3sf) | M1 A1 | **(9)** |
---\begin{enumerate}
\item (a) Write down the exact value of $\cos \frac { \pi } { 6 }$.
\end{enumerate}
The finite region $R$ is bounded by the curve $y = \cos ^ { 2 } x$, where $x$ is measured in radians, the positive coordinate axes and the line $x = \frac { \pi } { 3 }$.\\
(b) Use the trapezium rule with three equally-spaced ordinates to estimate the area of $R$, giving your answer to 3 significant figures.
The finite region $S$ is bounded by the curve $y = \sin ^ { 2 } x$, where $x$ is measured in radians, the positive coordinate axes and the line $x = \frac { \pi } { 3 }$.\\
(c) Using your answer to part (b), find an estimate for the area of $S$.\\
\hfill \mbox{\textit{Edexcel C2 Q5 [9]}}