| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2011 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Express and solve equation |
| Difficulty | Standard +0.3 This is a standard C3 harmonic form question requiring routine application of the R sin(θ + α) method and solving a resulting trigonometric equation. While it involves multiple steps (finding R and α, then solving in two quadrants), the techniques are well-practiced and follow a predictable template with no novel insight required, making it slightly easier than average. |
| 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 \(R = 25\) | B1 | allow \(\sqrt{625}\) or value rounding to 25 |
| Attempt to find value of \(\alpha\) | M1 | implied by correct answer or its complement; allow sin/cos muddles; allow use of radians for this mark; condone \(\sin\alpha = 7\), \(\cos\alpha = 24\) in the working |
| Obtain \(16.3°\) | A1 | 3 or greater accuracy 16.260...; must be degrees now; allow 16° here |
| (ii) Show correct process for finding one answer | M1 | even if leading to answer outside 0 to 360 or greater accuracy 12.425... or anything rounding to 12.4 |
| Obtain \((28.69 - 16.26\) and hence\() 12.4°\) | A1 | |
| Show correct process for finding second answer | M1 | even if further incorrect answers produced |
| Obtain \((151.31 - 16.26\) and hence\() 135°\) or \(135.1°\) | A1 | 4 or greater accuracy 135.054...; and no other between 0 and 360 |
**(i)** Obtain $R = 25$ | B1 | allow $\sqrt{625}$ or value rounding to 25
Attempt to find value of $\alpha$ | M1 | implied by correct answer or its complement; allow sin/cos muddles; allow use of radians for this mark; condone $\sin\alpha = 7$, $\cos\alpha = 24$ in the working
Obtain $16.3°$ | A1 | 3 or greater accuracy 16.260...; must be degrees now; allow 16° here
**(ii)** Show correct process for finding one answer | M1 | even if leading to answer outside 0 to 360 or greater accuracy 12.425... or anything rounding to 12.4
Obtain $(28.69 - 16.26$ and hence$) 12.4°$ | A1 |
Show correct process for finding second answer | M1 | even if further incorrect answers produced
Obtain $(151.31 - 16.26$ and hence$) 135°$ or $135.1°$ | A1 | 4 or greater accuracy 135.054...; and no other between 0 and 360
[SC: No working shown and 2 correct angles stated - B1 only in part (ii)]
---4 (i) Express $24 \sin \theta + 7 \cos \theta$ in the form $R \sin ( \theta + \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$.\\
(ii) Hence solve the equation $24 \sin \theta + 7 \cos \theta = 12$ for $0 ^ { \circ } < \theta < 360 ^ { \circ }$.
\hfill \mbox{\textit{OCR C3 2011 Q4 [7]}}