| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2003 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Integration using reciprocal identities |
| Difficulty | Standard +0.3 This is a structured three-part question requiring standard reciprocal trig identities and integration techniques. Part (i) is routine algebraic manipulation of cotangent and cosecant. Part (ii) uses the standard result that ∫cot x dx = ln|sin x|. Part (iii) cleverly combines parts (i) and (ii) using the proven identity. While it requires careful execution across multiple steps, all techniques are standard P3 material with no novel insight required, making it slightly easier than average. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) EITHER Make relevant use of the correct sin 2A formula | M1 | Make relevant use of the correct cos 2A formula |
| OR Make relevant use of the tan 2A formula | M1 | Make relevant use of \(1 + \tan^2 A = \sec^2 A\) or \(\cos^2 A + \sin^2 A = 1\) |
| (ii) State or imply indefinite integral is ln sin x, or equivalent | B1 | Substitute correct limits correctly |
| (iii) EITHER State indefinite integral of cos 2x is of the form \(k \ln \sin 2x\) | M1 | State correct integral \(\frac{1}{2} \ln \sin 2x\) |
| OR State or obtain indefinite integral of cosec 2x is of the form \(k \ln \tan x\), or equivalent | M1 | State correct integral \(\frac{1}{2} \ln \tan x\), or equivalent |
**(i)** **EITHER** Make relevant use of the correct sin 2A formula | M1 | Make relevant use of the correct cos 2A formula | M1 | Derive the given result correctly | A1 |
**OR** Make relevant use of the tan 2A formula | M1 | Make relevant use of $1 + \tan^2 A = \sec^2 A$ or $\cos^2 A + \sin^2 A = 1$ | M1 | Derive the given result correctly | A1 | **[3]**
**(ii)** State or imply indefinite integral is ln sin x, or equivalent | B1 | Substitute correct limits correctly | M1 | Obtain given exact answer correctly | A1 | **[3]**
**(iii)** **EITHER** State indefinite integral of cos 2x is of the form $k \ln \sin 2x$ | M1 | State correct integral $\frac{1}{2} \ln \sin 2x$ | A1 | Substitute limits correctly throughout | M1 | Obtain answer $\frac{1}{4} \ln 3$, or equivalent | A1 |
**OR** State or obtain indefinite integral of cosec 2x is of the form $k \ln \tan x$, or equivalent | M1 | State correct integral $\frac{1}{2} \ln \tan x$, or equivalent | A1 | Substitute limits correctly | M1 | Obtain answer $\frac{1}{4} \ln 3$, or equivalent | A1 | **[4]**10 (i) Prove the identity
$$\cot x - \cot 2 x \equiv \operatorname { cosec } 2 x$$
(ii) Show that $\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 4 } \pi } \cot x \mathrm {~d} x = \frac { 1 } { 2 } \ln 2$.\\
(iii) Find the exact value of $\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 4 } \pi } \operatorname { cosec } 2 x \mathrm {~d} x$, giving your answer in the form $a \ln b$.
\hfill \mbox{\textit{CAIE P3 2003 Q10 [10]}}