| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2003 |
| Session | November |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Definite integral with trigonometric functions |
| Difficulty | Moderate -0.3 This is a structured multi-part question that guides students through standard trigonometric integration techniques. Part (i) is routine differentiation using the quotient rule, part (ii) applies the result directly, and parts (iii-iv) require knowing standard identities (cot²x = cosec²x - 1 and 1-cos2x = 2sin²x) but the approach is straightforward once the identity is recalled. The question is slightly easier than average due to its scaffolded nature and reliance on standard techniques rather than problem-solving insight. |
| Spec | 1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08d Evaluate definite integrals: between limits |
| Answer | Marks |
|---|---|
| Attempt to differentiate using the quotient, product or chain rule | M1 |
| Obtain derivative in any correct form | A1 |
| Obtain the given answer correctly | A1 |
| Answer | Marks |
|---|---|
| State or imply the indefinite integral is \(-\cot x\) | B1 |
| Substitute limits and obtain given answer correctly | B1 |
| Answer | Marks |
|---|---|
| Use \(\cot^2 x = \csc^2 x - 1\) and attempt to integrate both terms, or equivalent | M1 |
| Substitute limits where necessary and obtain a correct unsimplified answer | A1 |
| Obtain final answer \(\sqrt{3} - \frac{1}{3}\pi\) | A1 |
| Answer | Marks |
|---|---|
| Use \(\cos 2A\) formula and reduce denominator to \(2\sin^2 x\) | B1 |
| Use given result and obtain answer of the form \(k\sqrt{3}\) | M1 |
| Obtain correct answer \(\frac{1}{2}\sqrt{3}\) | A1 |
### (i)
Attempt to differentiate using the quotient, product or chain rule | M1 |
Obtain derivative in any correct form | A1 |
Obtain the given answer correctly | A1 |
**Total: [3]**
### (ii)
State or imply the indefinite integral is $-\cot x$ | B1 |
Substitute limits and obtain given answer correctly | B1 |
**Total: [2]**
### (iii)
Use $\cot^2 x = \csc^2 x - 1$ and attempt to integrate both terms, or equivalent | M1 |
Substitute limits where necessary and obtain a correct unsimplified answer | A1 |
Obtain final answer $\sqrt{3} - \frac{1}{3}\pi$ | A1 |
**Total: [3]**
### (iv)
Use $\cos 2A$ formula and reduce denominator to $2\sin^2 x$ | B1 |
Use given result and obtain answer of the form $k\sqrt{3}$ | M1 |
Obtain correct answer $\frac{1}{2}\sqrt{3}$ | A1 |
**Total: [3]**\begin{enumerate}[label=(\roman*)]
\item By differentiating $\frac{\cos x}{\sin x}$, show that if $y = \cot x$ then $\frac{dy}{dx} = -\cosec^2 x$. [3]
\item Hence show that $\int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \cosec^2 x \, dx = \sqrt{3}$. [2]
\end{enumerate}
By using appropriate trigonometrical identities, find the exact value of
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{2}
\item $\int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \cot^2 x \, dx$, [3]
\item $\int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \frac{1}{1 - \cos 2x} \, dx$. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2003 Q7 [11]}}