| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2004 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Integration using harmonic form |
| Difficulty | Standard +0.3 This is a structured multi-part question that guides students through standard harmonic form techniques and integration. Part (i) is routine harmonic form conversion, part (ii) follows directly by squaring, part (iii) is a standard quotient rule exercise, and part (iv) combines these results in a straightforward substitution. While it requires multiple techniques, each step is well-signposted with no novel insight required, making it slightly easier than average. |
| Spec | 1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) | State answer \(R = \sqrt{2}\) | B1 |
| Use trigonometric formulae to find \(\alpha\) | M1 | |
| Obtain answer \(\alpha = \frac{1}{4}\pi\) (NOT \(45°\), unless \(45° = \frac{\pi}{4}^c\) somewhere, later) | A1 | [3] |
| (ii) | Use \(\cos\theta + \sin\theta = \sqrt{2}\cos(\theta - \frac{1}{4}\pi)\) to justify the given answer | B1 |
| (iii) | Differentiate using the quotient or product rule | M1 |
| Obtain derivative in any correct form | A1 | |
| Obtain the given answer correctly | A1 | [3] |
| (iv) | Convert integrand to give \(\int \frac{1}{2}\sec^2(\theta - \frac{\pi}{4})\,d\theta\) | B1 |
| Integrate, to obtain function \(\frac{1}{2}\tan(\theta - \frac{\pi}{4})\) | M1 | |
| Substitute (correct) limits correctly, to obtain given result | A1 | [3] |
# Question 8:
**(i)** | State answer $R = \sqrt{2}$ | B1 | |
Use trigonometric formulae to find $\alpha$ | M1 | |
Obtain answer $\alpha = \frac{1}{4}\pi$ (NOT $45°$, unless $45° = \frac{\pi}{4}^c$ somewhere, later) | A1 | **[3]**
**(ii)** | Use $\cos\theta + \sin\theta = \sqrt{2}\cos(\theta - \frac{1}{4}\pi)$ to justify the given answer | B1 | **[1]**
**(iii)** | Differentiate using the quotient or product rule | M1 | |
Obtain derivative in any correct form | A1 | |
Obtain the given answer correctly | A1 | **[3]**
**(iv)** | Convert integrand to give $\int \frac{1}{2}\sec^2(\theta - \frac{\pi}{4})\,d\theta$ | B1 | |
Integrate, to obtain function $\frac{1}{2}\tan(\theta - \frac{\pi}{4})$ | M1 | |
Substitute (correct) limits correctly, to obtain given result | A1 | **[3]**8 (i) Express $\cos \theta + \sin \theta$ in the form $R \cos ( \theta - \alpha )$, where $R > 0$ and $0 < \alpha < \frac { 1 } { 2 } \pi$, giving the exact values of $R$ and $\alpha$.\\
(ii) Hence show that
$$\frac { 1 } { ( \cos \theta + \sin \theta ) ^ { 2 } } = \frac { 1 } { 2 } \sec ^ { 2 } \left( \theta - \frac { 1 } { 4 } \pi \right)$$
(iii) By differentiating $\frac { \sin x } { \cos x }$, show that if $y = \tan x$ then $\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec ^ { 2 } x$.\\
(iv) Using the results of parts (ii) and (iii), show that
$$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { 1 } { ( \cos \theta + \sin \theta ) ^ { 2 } } \mathrm {~d} \theta = 1$$
\hfill \mbox{\textit{CAIE P2 2004 Q8 [10]}}