| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2005 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Express and solve equation |
| Difficulty | Standard +0.3 This is a standard two-part harmonic form question requiring routine application of the R sin(θ + α) method followed by solving a trigonometric equation. While it involves multiple steps (finding R and α, then solving), both parts follow textbook procedures with no novel insight required, making it slightly easier than the average A-level question. |
| Spec | 1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Obtain \(R = \sqrt{13}\), or 3.6 or 3.61 or greater accuracy | B1 | |
| Attempt recognisable process for finding \(\alpha\) | M1 | [allow sine/cosine muddles] |
| Obtain \(\alpha = 33.7\) | A1 | Total: 3 marks [or greater accuracy] |
| (ii) Attempt to find at least one value of \(\theta + \alpha\) | *M1 | [following their \(R\)] |
| Obtain value rounding to 76 or 104 | A1∨ | |
| Subtract their \(\alpha\) from at least one value | M1 | [dependent on *M] |
| Obtain one value rounding to 42 or 43, or to 70 | A1 | |
| Obtain other value 42.4 or 70.2 | A1 | Total: 5 marks [or greater accuracy; no other answers between 0 and 360; ignore answers outside 0 to 360] |
**(i)** Obtain $R = \sqrt{13}$, or 3.6 or 3.61 or greater accuracy | B1 |
Attempt recognisable process for finding $\alpha$ | M1 | [allow sine/cosine muddles]
Obtain $\alpha = 33.7$ | A1 | Total: 3 marks [or greater accuracy]
**(ii)** Attempt to find at least one value of $\theta + \alpha$ | *M1 | [following their $R$]
Obtain value rounding to 76 or 104 | A1∨ |
Subtract their $\alpha$ from at least one value | M1 | [dependent on *M]
Obtain one value rounding to 42 or 43, or to 70 | A1 |
Obtain other value 42.4 or 70.2 | A1 | Total: 5 marks [or greater accuracy; no other answers between 0 and 360; ignore answers outside 0 to 360]5 (i) Express $3 \sin \theta + 2 \cos \theta$ in the form $R \sin ( \theta + \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$.\\
(ii) Hence solve the equation $3 \sin \theta + 2 \cos \theta = \frac { 7 } { 2 }$, giving all solutions for which $0 ^ { \circ } < \theta < 360 ^ { \circ }$.
\hfill \mbox{\textit{OCR C3 2005 Q5 [8]}}