| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2006 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Trapezium rule then exact integration comparison |
| Difficulty | Moderate -0.3 This is a straightforward application of the trapezium rule with calculator work. Part (a) requires evaluating sec x at given points (basic calculator use), part (b) is a standard trapezium rule application following a formula, and part (c) is a routine percentage error calculation. While it involves multiple steps, each component is mechanical and requires no problem-solving insight—slightly easier than average due to its purely procedural nature. |
| Spec | 1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.09f Trapezium rule: numerical integration |
| \(x\) | 0 | \(\frac { \pi } { 16 }\) | \(\frac { \pi } { 8 }\) | \(\frac { 3 \pi } { 16 }\) | \(\frac { \pi } { 4 }\) |
| \(y\) | 1 | 1.20269 |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| \(x\): \(0\), \(\frac{\pi}{16}\), \(\frac{\pi}{8}\), \(\frac{3\pi}{16}\), \(\frac{\pi}{4}\); \(y\): \(1\), 1.01959, 1.08239, \(1.20269\), 1.41421 | M1, A1 | M1 for one correct, A1 for all correct. (2) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| Integral \(= \frac{1}{2} \times \frac{\pi}{16} \times \{1 + 1.4142 + 2(1.01959 + \ldots + 1.20269)\}\) | M1, A1\(\sqrt{}\) | |
| \(= \frac{\pi}{32} \times 9.02355 = 0.8859\) | A1 cao | (3) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| Percentage error \(= \frac{approx - 0.88137}{0.88137} \times 100 = 0.51\%\) (allow \(0.5\%\) to \(0.54\%\) for A1) | M1, A1 | (2) [7] |
| M1 gained for \((\pm)\frac{approx - \ln(1+\sqrt{2})}{\ln(1+\sqrt{2})}\) |
## Question 2:
### Part (a):
| Working/Answer | Marks | Guidance |
|---|---|---|
| $x$: $0$, $\frac{\pi}{16}$, $\frac{\pi}{8}$, $\frac{3\pi}{16}$, $\frac{\pi}{4}$; $y$: $1$, **1.01959**, **1.08239**, $1.20269$, **1.41421** | M1, A1 | M1 for one correct, A1 for all correct. **(2)** |
### Part (b):
| Working/Answer | Marks | Guidance |
|---|---|---|
| Integral $= \frac{1}{2} \times \frac{\pi}{16} \times \{1 + 1.4142 + 2(1.01959 + \ldots + 1.20269)\}$ | M1, A1$\sqrt{}$ | |
| $= \frac{\pi}{32} \times 9.02355 = 0.8859$ | A1 cao | **(3)** |
### Part (c):
| Working/Answer | Marks | Guidance |
|---|---|---|
| Percentage error $= \frac{approx - 0.88137}{0.88137} \times 100 = 0.51\%$ (allow $0.5\%$ to $0.54\%$ for A1) | M1, A1 | **(2) [7]** |
| M1 gained for $(\pm)\frac{approx - \ln(1+\sqrt{2})}{\ln(1+\sqrt{2})}$ | | |
---2. (a) Given that $y = \sec x$, complete the table with the values of $y$ corresponding to $x = \frac { \pi } { 16 } , \frac { \pi } { 8 }$ and $\frac { \pi } { 4 }$.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$x$ & 0 & $\frac { \pi } { 16 }$ & $\frac { \pi } { 8 }$ & $\frac { 3 \pi } { 16 }$ & $\frac { \pi } { 4 }$ \\
\hline
$y$ & 1 & & & 1.20269 & \\
\hline
\end{tabular}
\end{center}
(b) Use the trapezium rule, with all the values for $y$ in the completed table, to obtain an estimate for $\int _ { 0 } ^ { \frac { \pi } { 4 } } \sec x \mathrm {~d} x$. Show all the steps of your working, and give your answer to 4 decimal places.
The exact value of $\int _ { 0 } ^ { \frac { \pi } { 4 } } \sec x \mathrm {~d} x$ is $\ln ( 1 + \sqrt { } 2 )$.\\
(c) Calculate the \% error in using the estimate you obtained in part (b).\\
\hfill \mbox{\textit{Edexcel C4 2006 Q2 [7]}}