Easy -1.2 This is a straightforward application of the trapezium rule with clearly specified parameters (3 strips, width 2). It requires only substituting x-values into a simple function (8/x) and applying the standard trapezium rule formula—purely procedural with no problem-solving or conceptual challenge beyond basic recall.
Attempt the 4 correct \(y\)-coordinates, and no others. M0 if other \(y\)-values also found (unless not used). Allow decimal equivs
M1
Attempt correct trapezium rule, any \(h\), to find area between \(x=5\) and \(x=11\). Correct structure required, including placing of \(y\)-values. The 'big brackets' must be seen, or implied by later working. Allow BOD for eg \(\frac{1}{2} \times 2 \times (\frac{8}{5} + \frac{8}{11}) + 2(\frac{8}{7} + \frac{8}{9})\). Could use more than 3 strips as long as of equal width. Using \(x\)-values is M0
M1
Use correct \(h\) (soi) for their \(y\)-values (must have equally spaced \(x\)-values), to find area between \(x=5\) and \(x=11\). Must be in attempt at trap rule, not Simpson's rule. Allow if \(\frac{1}{2}\) missing. Allow other than 3 strips, as long as \(h\) is consistent
\(= 6.39\)
A1
Obtain 6.39, or better. Allow answers in range \([6.390, 6.391]\) if \(>3\)sf. Answer only is 0/4
# Question 1:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{1}{2} \times 2 \times (\frac{8}{5} + 2(\frac{8}{7} + \frac{8}{9}) + \frac{8}{11})$ | M1 | Attempt the 4 correct $y$-coordinates, and no others. M0 if other $y$-values also found (unless not used). Allow decimal equivs |
| | M1 | Attempt correct trapezium rule, any $h$, to find area between $x=5$ and $x=11$. Correct structure required, including placing of $y$-values. The 'big brackets' must be seen, or implied by later working. Allow BOD for eg $\frac{1}{2} \times 2 \times (\frac{8}{5} + \frac{8}{11}) + 2(\frac{8}{7} + \frac{8}{9})$. Could use more than 3 strips as long as of equal width. Using $x$-values is M0 |
| | M1 | Use correct $h$ (soi) for their $y$-values (must have equally spaced $x$-values), to find area between $x=5$ and $x=11$. Must be in attempt at trap rule, not Simpson's rule. Allow if $\frac{1}{2}$ missing. Allow other than 3 strips, as long as $h$ is consistent |
| $= 6.39$ | A1 | Obtain 6.39, or better. Allow answers in range $[6.390, 6.391]$ if $>3$sf. Answer only is 0/4 |
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1 Use the trapezium rule, with 3 strips each of width 2 , to estimate the value of
$$\int _ { 5 } ^ { 11 } \frac { 8 } { x } \mathrm {~d} x .$$
\hfill \mbox{\textit{OCR C2 2013 Q1 [4]}}