| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2013 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Over/underestimate justification with graph |
| Difficulty | Moderate -0.3 Part (a) requires expanding and integrating a standard trigonometric expression using double angle formulas—routine A-level technique. Part (b)(i) is straightforward trapezium rule application with calculator work. Part (b)(ii) requires understanding concavity but is a standard textbook question type. Overall slightly easier than average due to mechanical nature and clear structure, though the reasoning element in (b)(ii) adds minor challenge. |
| Spec | 1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.08b Integrate x^n: where n != -1 and sums1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Expand brackets and use \(\sin^2 x + \cos^2 x = 1\) | M1 | |
| Obtain \(1 - \sin 2x\) | A1 | |
| Integrate and obtain term of form \(\pm k \cos 2x\), where \(k = \frac{1}{2}, 1\) or \(2\) | M1 | |
| State correct integral \(x + \frac{\cos 2x}{2} (+c)\) | A1 | [4] |
| (b) (i) State or imply correct ordinates \(1.4142\ldots, 1.0823\ldots, 1\) | B1 | |
| Use correct formula, or equivalent, correctly with \(h = \frac{\pi}{8}\) and three ordinates | M1 | |
| Obtain answer 0.899 with no errors seen | A1 | [3] |
| (ii) Make a recognisable sketch of \(y = \cosec x\) for \(0 < x \leq \frac{1}{2}\pi\) | B1 | |
| Justify statement that the trapezium rule gives an over-estimate | B1 | [2] |
(a) Expand brackets and use $\sin^2 x + \cos^2 x = 1$ | M1 |
Obtain $1 - \sin 2x$ | A1 |
Integrate and obtain term of form $\pm k \cos 2x$, where $k = \frac{1}{2}, 1$ or $2$ | M1 |
State correct integral $x + \frac{\cos 2x}{2} (+c)$ | A1 | [4]
(b) (i) State or imply correct ordinates $1.4142\ldots, 1.0823\ldots, 1$ | B1 |
Use correct formula, or equivalent, correctly with $h = \frac{\pi}{8}$ and three ordinates | M1 |
Obtain answer 0.899 with no errors seen | A1 | [3]
(ii) Make a recognisable sketch of $y = \cosec x$ for $0 < x \leq \frac{1}{2}\pi$ | B1 |
Justify statement that the trapezium rule gives an over-estimate | B1 | [2]6
\begin{enumerate}[label=(\alph*)]
\item Find $\int ( \sin x - \cos x ) ^ { 2 } \mathrm {~d} x$.
\item \begin{enumerate}[label=(\roman*)]
\item Use the trapezium rule with 2 intervals to estimate the value of
$$\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 2 } \pi } \operatorname { cosec } x d x$$
giving your answer correct to 3 decimal places.
\item Using a sketch of the graph of $y = \operatorname { cosec } x$ for $0 < x \leqslant \frac { 1 } { 2 } \pi$, explain whether the trapezium rule gives an under-estimate or an over-estimate of the true value of the integral in part (i).
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2013 Q6 [9]}}