| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2012 |
| Session | June |
| Marks | 11 |
| 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 with routine techniques: converting to R sin(θ+α) using Pythagorean identity and tan α, solving a transformed equation, and finding range bounds. While multi-part, each step follows textbook methods 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 |
|---|---|---|
| Answer | Marks | Guidance |
| State \(R = 5\) | B1 | |
| Attempt to find value of \(\alpha\) | M1 | Implied by correct value or its complement |
| Obtain 53.1 | A1 | Allow \(\tan^{-1}\frac{4}{3}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Attempt to find at least one value of \(\theta + \alpha\) | M1 | (Should be \(-168.5\) or \(-11.5\) or \(191.5\) or …) |
| Obtain 1 correct value of \(\theta\) (\(-64.7\) or 138) | A1 | Allow \(\pm 0.1\) in answer and greater accuracy. Note that 138 needs to be obtained legitimately from positive value of \(\sin^{-1}(-\frac{1}{5})\) and not from \(180 - 41.6\) |
| Attempt correct process to find the second value | M1 | Involving a positive value of \(\sin^{-1}(-\frac{1}{5})\) and subtraction of their \(\alpha\) |
| Obtain second value of \(\theta\) (138 or \(-64.7\)) | A1 | Allow \(\pm 0.1\) in answer and greater accuracy; and no others between \(-180\) and \(180\); answers only: 0/4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use \(-1\) as minimum or \(1\) as maximum value of \(\sin(\theta + \alpha)\) | *M1 | |
| Relate \(-5k + c\) to \(-37\) and \(5k + c\) to \(43\) | A1 | As equations or inequalities |
| Attempt solution of pair of linear eqns | M1 | dep *M; must be equations now. SC: both \(k=8\) and \(c=3\) obtained with no working or unconvincing working, award B2 (i.e. max 2/4) |
| Obtain \(k = 8\) and \(c = 3\) | A1 | Note that alternative solutions may occur. If mathematically sound, all 4 marks available; if work is not fully convincing, apply SC |
# Question 8(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State $R = 5$ | B1 | |
| Attempt to find value of $\alpha$ | M1 | Implied by correct value or its complement |
| Obtain 53.1 | A1 | Allow $\tan^{-1}\frac{4}{3}$ |
---
# Question 8(ii)(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Attempt to find at least one value of $\theta + \alpha$ | M1 | (Should be $-168.5$ or $-11.5$ or $191.5$ or …) |
| Obtain 1 correct value of $\theta$ ($-64.7$ or 138) | A1 | Allow $\pm 0.1$ in answer and greater accuracy. Note that 138 needs to be obtained legitimately from positive value of $\sin^{-1}(-\frac{1}{5})$ and not from $180 - 41.6$ |
| Attempt correct process to find the second value | M1 | Involving a positive value of $\sin^{-1}(-\frac{1}{5})$ and subtraction of their $\alpha$ |
| Obtain second value of $\theta$ (138 or $-64.7$) | A1 | Allow $\pm 0.1$ in answer and greater accuracy; and no others between $-180$ and $180$; answers only: 0/4 |
---
# Question 8(ii)(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use $-1$ as minimum or $1$ as maximum value of $\sin(\theta + \alpha)$ | *M1 | |
| Relate $-5k + c$ to $-37$ and $5k + c$ to $43$ | A1 | As equations or inequalities |
| Attempt solution of pair of linear eqns | M1 | dep *M; must be equations now. **SC**: both $k=8$ and $c=3$ obtained with no working or unconvincing working, award B2 (i.e. max 2/4) |
| Obtain $k = 8$ and $c = 3$ | A1 | Note that alternative solutions may occur. If mathematically sound, all 4 marks available; if work is not fully convincing, apply SC |8 (i) Express $3 \sin \theta + 4 \cos \theta$ in the form $R \sin ( \theta + \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$.\\
(ii) Hence
\begin{enumerate}[label=(\alph*)]
\item solve the equation $3 \sin \theta + 4 \cos \theta + 1 = 0$, giving all solutions for which $- 180 ^ { \circ } < \theta < 180 ^ { \circ }$,
\item find the values of the positive constants $k$ and $c$ such that
$$- 37 \leqslant k ( 3 \sin \theta + 4 \cos \theta ) + c \leqslant 43$$
for all values of $\theta$.
\end{enumerate}
\hfill \mbox{\textit{OCR C3 2012 Q8 [11]}}