| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2010 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Solve equation directly given harmonic form |
| Difficulty | Standard +0.3 This is a standard C3 question on the R-formula (harmonic form) followed by routine applications. Part (i) is textbook R cos(x-α) conversion. Part (ii)(a) requires finding where denominator equals zero (standard technique). Part (ii)(b) involves solving T(3x) = value, which requires working backwards through the R-form and solving a trigonometric equation—straightforward but multi-step. Slightly above average due to the composite argument 3x, but still a familiar question type with well-practiced techniques. |
| 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 |
|---|---|---|
| Obtain \(R = 3\sqrt{2}\) or \(R = \sqrt{18}\) or \(R = 4.24\) | B1 | or equiv |
| Attempt to find value of \(\alpha\) | M1 | condone sin/cos muddles and degrees |
| Obtain \(\frac{1}{4}\pi\) or \(0.785\) | A1 3 | in radians now |
| Answer | Marks | Guidance |
|---|---|---|
| a Equate \(x - \alpha\) to \(\frac{1}{4}\pi\) or attempt solution of \(3\cos x + 3\sin x = 0\) | M1 | condone degrees here |
| Obtain \(\frac{3}{4}\pi\) | A1 2 | or ..., \(-\frac{5}{4}\pi, \frac{3}{4}\pi, \frac{7}{4}\pi, ...\); in radians now |
| b Attempt correct process to find value of \(3x - \alpha\) | *M1 | with attempt at rearranging \(T(3x) = \frac{5}{6}\sqrt{6}\) |
| Obtain at least one correct exact value of \(3x - \alpha\) | A1 | \(\pm \frac{1}{3}\pi, \pm \frac{11}{6}\pi, ...\) |
| Attempt at least one positive value of \(x\) | M1 | dep *M |
| Obtain \(\frac{1}{6}\pi\) | A1 4 | — |
## (i)
Obtain $R = 3\sqrt{2}$ or $R = \sqrt{18}$ or $R = 4.24$ | B1 | or equiv
Attempt to find value of $\alpha$ | M1 | condone sin/cos muddles and degrees
Obtain $\frac{1}{4}\pi$ or $0.785$ | A1 3 | in radians now
## (ii)
a Equate $x - \alpha$ to $\frac{1}{4}\pi$ or attempt solution of $3\cos x + 3\sin x = 0$ | M1 | condone degrees here
Obtain $\frac{3}{4}\pi$ | A1 2 | or ..., $-\frac{5}{4}\pi, \frac{3}{4}\pi, \frac{7}{4}\pi, ...$; in radians now
b Attempt correct process to find value of $3x - \alpha$ | *M1 | with attempt at rearranging $T(3x) = \frac{5}{6}\sqrt{6}$
Obtain at least one correct exact value of $3x - \alpha$ | A1 | $\pm \frac{1}{3}\pi, \pm \frac{11}{6}\pi, ...$
Attempt at least one positive value of $x$ | M1 | dep *M
Obtain $\frac{1}{6}\pi$ | A1 4 | —
---\begin{enumerate}[label=(\roman*)]
\item Express $3 \cos x + 3 \sin x$ in the form $R \cos(x - \alpha)$, where $R > 0$ and $0 < \alpha < \frac{1}{2}\pi$. [3]
\item The expression T$(x)$ is defined by T$(x) = \frac{8}{3 \cos x + 3 \sin x}$.
\begin{enumerate}[label=(\alph*)]
\item Determine a value of $x$ for which T$(x)$ is not defined. [2]
\item Find the smallest positive value of $x$ satisfying T$(3x) = \frac{8}{3}\sqrt{6}$, giving your answer in an exact form. [4]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{OCR C3 2010 Q8 [9]}}