| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2002 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indefinite & Definite Integrals |
| Type | Trapezium rule estimation |
| Difficulty | Moderate -0.3 This is a multi-part question with standard techniques: part (a)(i) is routine integration of cos 2x, part (a)(ii) requires the standard identity cos 2x = 1 - 2sin²x, part (b)(i) is straightforward trapezium rule application with 2 intervals, and part (b)(ii) requires recognizing that sec x is convex. All components are textbook exercises requiring recall and direct application rather than problem-solving, making it slightly easier than average. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| (a) (i) State indefinite integral \(k \sin 2x\) and use limits | M1 | |
| Obtain given answer correctly | A1 | 2 |
| (ii) Use double-angle formula to convert integrand to the form \(a + b\cos 2x\), where \(ab \neq 0\) | M1* | |
| Integrate and use limits (both terms) | M1(dep*) | |
| Obtain answer \(\frac{1}{4}(x - 2)\), or equivalent | A1 | 3 |
| (b) (i) Show or imply correct ordinates \(1, 1.08239..., \sqrt{2}(1.41421...)\) | B1 | |
| Use correct formula, or equivalent, with \(h = \pi/8\) and three ordinates | M1 | |
| Obtain correct answer \(0.90\) with no errors seen | A1 | 3 |
| (ii) Make a correct relevant sketch of \(y = \sec x\) | B1* | |
| State that the rule gives an over-estimate | B1(dep*) | 2 |
**(a) (i)** State indefinite integral $k \sin 2x$ and use limits | M1 |
Obtain given answer correctly | A1 | 2
**(ii)** Use double-angle formula to convert integrand to the form $a + b\cos 2x$, where $ab \neq 0$ | M1* |
Integrate and use limits (both terms) | M1(dep*) |
Obtain answer $\frac{1}{4}(x - 2)$, or equivalent | A1 | 3
**(b) (i)** Show or imply correct ordinates $1, 1.08239..., \sqrt{2}(1.41421...)$ | B1 |
Use correct formula, or equivalent, with $h = \pi/8$ and three ordinates | M1 |
Obtain correct answer $0.90$ with no errors seen | A1 | 3
**(ii)** Make a correct relevant sketch of $y = \sec x$ | B1* |
State that the rule gives an over-estimate | B1(dep*) | 2
---6
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Show that $\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \cos 2 x \mathrm {~d} x = \frac { 1 } { 2 }$.
\item By using an appropriate trigonometrical identity, find the exact value of $\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sin ^ { 2 } x \mathrm {~d} x$.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Use the trapezium rule with 2 intervals to estimate the value of $\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sec x d x$, giving your answer correct to 2 significant figures.
\item Determine, by sketching the appropriate part of the graph of $y = \sec x$, whether the trapezium rule gives an under-estimate or an over-estimate of the true value.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2002 Q6 [10]}}