| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2019 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Integration using reciprocal identities |
| Difficulty | Standard +0.3 This is a straightforward two-part question requiring standard double-angle identities (cos 2θ = 1-2sin²θ = 2cos²θ-1, sin 2θ = 2sinθcosθ) to simplify the expression to tan θ, followed by routine integration of tan θ and evaluation at given limits. While it requires multiple steps, the techniques are standard A-level fare with no novel insight needed, making it slightly easier than average. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05l Double angle formulae: and compound angle formulae1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use double angle formulae and express entire fraction in terms of \(\sin\theta\) and \(\cos\theta\) | M1 | |
| Obtain a correct expression | A1 | |
| Obtain the given answer | A1 | |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State integral of the form \(\pm\ln\cos\theta\) | M1* | |
| Use correct limits correctly and insert exact values for the trig ratios | DM1 | |
| Obtain a correct expression, e.g. \(-\ln\frac{1}{\sqrt{2}} + \ln\frac{\sqrt{3}}{2}\) | A1 | |
| Obtain the given answer following full and exact working | A1 | |
| Total: 4 |
## Question 3(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use double angle formulae and express entire fraction in terms of $\sin\theta$ and $\cos\theta$ | M1 | |
| Obtain a correct expression | A1 | |
| Obtain the given answer | A1 | |
| **Total: 3** | | |
## Question 3(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| State integral of the form $\pm\ln\cos\theta$ | M1* | |
| Use correct limits correctly and insert exact values for the trig ratios | DM1 | |
| Obtain a correct expression, e.g. $-\ln\frac{1}{\sqrt{2}} + \ln\frac{\sqrt{3}}{2}$ | A1 | |
| Obtain the given answer following full and exact working | A1 | |
| **Total: 4** | | |3 Let $f ( \theta ) = \frac { 1 - \cos 2 \theta + \sin 2 \theta } { 1 + \cos 2 \theta + \sin 2 \theta }$.\\
(i) Show that $\mathrm { f } ( \theta ) = \tan \theta$.\\
(ii) Hence show that $\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 4 } \pi } \mathrm { f } ( \theta ) \mathrm { d } \theta = \frac { 1 } { 2 } \ln \frac { 3 } { 2 }$.\\
\hfill \mbox{\textit{CAIE P3 2019 Q3 [7]}}