| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2014 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Over/underestimate justification with graph |
| Difficulty | Standard +0.3 This is a straightforward trapezium rule application with standard interval calculation, followed by a routine concavity analysis using a sketch. The cosecant function is A-level standard, and determining over/underestimate from concavity is a textbook exercise, making this slightly easier than average overall. |
| Spec | 1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State or imply ordinates 2, 1.1547…, 1, 1.1547… | B1 | |
| Use correct formula, or equivalent, with \(h = \frac{1}{6}\pi\) and four ordinates | M1 | |
| Obtain answer 1.95 | A1 | [3] |
| (ii) Make recognisable sketch of \(y = \cos x\) for the given interval | B1 | |
| Justify a statement that the estimate will be an overestimate | B1 | [2] |
**(i)** State or imply ordinates 2, 1.1547…, 1, 1.1547… | B1 |
Use correct formula, or equivalent, with $h = \frac{1}{6}\pi$ and four ordinates | M1 |
Obtain answer 1.95 | A1 | [3]
**(ii)** Make recognisable sketch of $y = \cos x$ for the given interval | B1 |
Justify a statement that the estimate will be an overestimate | B1 | [2]2 (i) Use the trapezium rule with 3 intervals to estimate the value of
$$\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 2 } { 3 } \pi } \operatorname { cosec } x d x$$
giving your answer correct to 2 decimal places.\\
(ii) Using a sketch of the graph of $y = \operatorname { cosec } x$, explain whether the trapezium rule gives an overestimate or an underestimate of the true value of the integral in part (i).
\hfill \mbox{\textit{CAIE P3 2014 Q2 [5]}}