| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2013 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Integration using harmonic form |
| Difficulty | Standard +0.3 This is a standard two-part harmonic form question requiring routine techniques: (i) uses the R cos(x-α) expansion formula with straightforward coefficient matching, and (ii) applies a standard substitution u = tan(x-α) to integrate sec²(x-α), which is a well-practiced method in P3. The exact answer verification adds minimal difficulty. Slightly easier than average due to its predictable structure. |
| Spec | 1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc1.08h Integration by substitution |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State \(R = 2\) | B1 | |
| Use trig formula to find \(\alpha\) | M1 | |
| Obtain \(\alpha = \frac{1}{6}\pi\) with no errors seen | A1 | [3 marks] |
| (ii) Substitute denominator of integrand and state integral \(k \tan(x - \alpha)\) | M1* | |
| State correct indefinite integral \(\frac{1}{4}\tan\left(x - \frac{1}{6}\pi\right)\) | A1 | |
| Substitute limits | M1 (dep*) | |
| Obtain the given answer correctly | A1 | [4 marks total] |
**(i)** State $R = 2$ | B1 |
Use trig formula to find $\alpha$ | M1 |
Obtain $\alpha = \frac{1}{6}\pi$ with no errors seen | A1 | [3 marks]
**(ii)** Substitute denominator of integrand and state integral $k \tan(x - \alpha)$ | M1* |
State correct indefinite integral $\frac{1}{4}\tan\left(x - \frac{1}{6}\pi\right)$ | A1 |
Substitute limits | M1 (dep*) |
Obtain the given answer correctly | A1 | [4 marks total]
---4 (i) Express $( \sqrt { } 3 ) \cos x + \sin x$ in the form $R \cos ( x - \alpha )$, where $R > 0$ and $0 < \alpha < \frac { 1 } { 2 } \pi$, giving the exact values of $R$ and $\alpha$.\\
(ii) Hence show that
$$\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 2 } \pi } \frac { 1 } { ( ( \sqrt { } 3 ) \cos x + \sin x ) ^ { 2 } } \mathrm {~d} x = \frac { 1 } { 4 } \sqrt { } 3$$
\hfill \mbox{\textit{CAIE P3 2013 Q4 [7]}}