| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2008 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Expand then express in harmonic form |
| Difficulty | Standard +0.3 This is a standard C3 harmonic form question requiring compound angle expansion, then conversion to R sin(θ+α) form using standard techniques. Part (i) uses compound angle formulae (routine but multi-step), part (ii) is a textbook harmonic form conversion, and part (iii) is straightforward equation solving. Slightly above average due to the initial expansion step, but all techniques are standard C3 material with no novel insight required. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc |
| Answer | Marks | Guidance |
|---|---|---|
| (i) | Show at least correct \(\cos\theta\cos 60 + \sin\theta\sin 60\) or \(\cos\theta\cos 60 - \sin\theta\sin 60\) | B1 |
| Attempt expansion of both with exact numerical values attempted | M1 | and with \(\cos 60 \neq \sin 60\) |
| Obtain \(\frac{1}{2}\sqrt{3}\sin\theta + \frac{1}{2}\cos\theta\) | A1 | or exact equiv |
| (ii) | Attempt correct process for finding \(R\) | M1 |
| Attempt recognisable process for finding \(a\) | M1 | allowing \(\sin\)/cos muddles |
| Obtain \(\sqrt{7}\sin(\theta+70.9)\) | A1 | allow 2.65 for \(R\); allow 70.9 ± 0.1 for \(\alpha\) |
| (iii) | Attempt correct process to find any value of \(\theta +\) their \(a\) | M1 |
| Obtain any correct value for \(\theta + 70.9\) | A1 | −158, −22, 202, 338, … |
| Attempt correct process to find \(\theta +\) their \(a\) in 3rd quadrant | M1 | or several values including this |
| Obtain 131 | A1 | or greater accuracy and no other |
| [SC for solutions with no working shown: Correct answer only B4; 131 with other answers B2] |
(i) | Show at least correct $\cos\theta\cos 60 + \sin\theta\sin 60$ or $\cos\theta\cos 60 - \sin\theta\sin 60$ | B1 | |
| Attempt expansion of both with exact numerical values attempted | M1 | and with $\cos 60 \neq \sin 60$ |
| Obtain $\frac{1}{2}\sqrt{3}\sin\theta + \frac{1}{2}\cos\theta$ | A1 | or exact equiv |
(ii) | Attempt correct process for finding $R$ | M1 | whether exact or approx |
| Attempt recognisable process for finding $a$ | M1 | allowing $\sin$/cos muddles |
| Obtain $\sqrt{7}\sin(\theta+70.9)$ | A1 | allow 2.65 for $R$; allow 70.9 ± 0.1 for $\alpha$ |
(iii) | Attempt correct process to find any value of $\theta +$ their $a$ | M1 | |
| Obtain any correct value for $\theta + 70.9$ | A1 | −158, −22, 202, 338, … |
| Attempt correct process to find $\theta +$ their $a$ in 3rd quadrant | M1 | or several values including this |
| Obtain 131 | A1 | or greater accuracy and no other |
| [SC for solutions with no working shown: Correct answer only B4; 131 with other answers B2] | | |8 The expression $\mathrm { T } ( \theta )$ is defined for $\theta$ in degrees by
$$\mathrm { T } ( \theta ) = 3 \cos \left( \theta - 60 ^ { \circ } \right) + 2 \cos \left( \theta + 60 ^ { \circ } \right) .$$
(i) Express $\mathrm { T } ( \theta )$ in the form $A \sin \theta + B \cos \theta$, giving the exact values of the constants $A$ and $B$. [3]\\
(ii) Hence express $\mathrm { T } ( \theta )$ in the form $R \sin ( \theta + \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$.\\
(iii) Find the smallest positive value of $\theta$ such that $\mathrm { T } ( \theta ) + 1 = 0$.
\hfill \mbox{\textit{OCR C3 2008 Q8 [10]}}