| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2002 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Area under curve using integration |
| Difficulty | Moderate -0.8 This is a straightforward integration question testing standard techniques. Part (a) requires direct integration of sin 2x and cos x with simple substitution of limits. Part (b) involves integrating 1/(x+1) to get ln(x+1) and solving ln(p+1) - ln(2) = 2, which is routine manipulation. All techniques are standard textbook exercises with no problem-solving insight required, making it easier than average but not trivial due to the multi-part nature and algebraic manipulation needed. |
| Spec | 1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08d Evaluate definite integrals: between limits |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Obtain indefinite integral \(-\frac{1}{2}\cos 2x + \sin x\) | B1 + B1 | |
| Use limits with attempted integral | M1 | |
| Obtain answer 2 correctly with no errors | A1 | 4 |
| (b)(i) Identify \(R\) with correct definite integral and attempt to integrate | M1 | |
| Obtain indefinite integral \(\ln(x+1)\) | B1 | |
| Obtain answer \(R = \ln(p+1) - \ln 2\) | A1 | 3 |
| (b)(ii) Use exponential method to solve an equation of the form \(\ln x = k\) | M1 | |
| Obtain answer \(p = 13.8\) | A1 | 2 |
**(a)** Obtain indefinite integral $-\frac{1}{2}\cos 2x + \sin x$ | B1 + B1 |
Use limits with attempted integral | M1 |
Obtain answer 2 correctly with no errors | A1 | 4 |
**(b)(i)** Identify $R$ with correct definite integral and attempt to integrate | M1 |
Obtain indefinite integral $\ln(x+1)$ | B1 |
Obtain answer $R = \ln(p+1) - \ln 2$ | A1 | 3 |
**(b)(ii)** Use exponential method to solve an equation of the form $\ln x = k$ | M1 |
Obtain answer $p = 13.8$ | A1 | 2 |
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\begin{enumerate}[label=(\alph*)]
\item Find the value of $\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } ( \sin 2 x + \cos x ) \mathrm { d } x$.
\item \\
\includegraphics[max width=\textwidth, alt={}, center]{9894d97f-3b7b-4dbe-b94a-2c8415442038-3_517_880_422_669}
The diagram shows part of the curve $y = \frac { 1 } { x + 1 }$. The shaded region $R$ is bounded by the curve and by the lines $x = 1 , y = 0$ and $x = p$.
\begin{enumerate}[label=(\roman*)]
\item Find, in terms of $p$, the area of $R$.
\item Hence find, correct to 1 decimal place, the value of $p$ for which the area of $R$ is equal to 2 .
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2002 Q6 [9]}}