| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2006 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Transformations of trigonometric graphs |
| Difficulty | Standard +0.3 This is a standard harmonic form question requiring the R cos(x-α) transformation (using R²=a²+b², tan α=b/a), describing transformations (stretch and translation), and solving a trigonometric equation. While it involves multiple parts and techniques, these are well-practiced C3 procedures with no novel insight required, making it slightly easier than average. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)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) State \(R = 13\) | B1 | or equiv |
| State at least one equation of form \(R\cos\alpha = k\), \(R\sin\alpha = k'\), \(\tan\alpha = k''\) | M1 | or equiv; allow \(\sin / \cos\) muddles; implied by correct \(\alpha\) |
| Obtain \(67.4\) | A1 3 | allow 67 or greater accuracy |
| (ii) Refer to translation and stretch | M1 | in either order; allow here equiv terms such as 'move', 'shift'; with both transformations involving constants |
| State translation in positive \(x\) direction by \(67.4\) | A1√ | or equiv; following their \(\alpha\); using correct terminology now |
| State stretch in \(y\) direction by factor \(13\) | A1√ 3 | or equiv; following their \(R\); using correct terminology now |
| (iii) Attempt value of \(\cos^{-1}(2 + R)\) | M1 | |
| Obtain \(81.15\) | A1√ | following their \(R\); accept 81 |
| Obtain \(148.5\) as one solution | A1 | accept 148.5 or 148.6 or value rounding to either of these |
| Add their \(\alpha\) value to second value correctly attempted | M1 | |
| Obtain \(346.2\) | A1 5 | accept 346.2 or 346.3 or value rounding to either of these; and no other solutions |
**(i)** State $R = 13$ | B1 | or equiv
State at least one equation of form $R\cos\alpha = k$, $R\sin\alpha = k'$, $\tan\alpha = k''$ | M1 | or equiv; allow $\sin / \cos$ muddles; implied by correct $\alpha$
Obtain $67.4$ | A1 3 | allow 67 or greater accuracy
**(ii)** Refer to translation and stretch | M1 | in either order; allow here equiv terms such as 'move', 'shift'; with both transformations involving constants
State translation in positive $x$ direction by $67.4$ | A1√ | or equiv; following their $\alpha$; using correct terminology now
State stretch in $y$ direction by factor $13$ | A1√ 3 | or equiv; following their $R$; using correct terminology now
**(iii)** Attempt value of $\cos^{-1}(2 + R)$ | M1 |
Obtain $81.15$ | A1√ | following their $R$; accept 81
Obtain $148.5$ as one solution | A1 | accept 148.5 or 148.6 or value rounding to either of these
Add their $\alpha$ value to second value correctly attempted | M1 |
Obtain $346.2$ | A1 5 | accept 346.2 or 346.3 or value rounding to either of these; and no other solutions8 (i) Express $5 \cos x + 12 \sin x$ in the form $R \cos ( x - \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$.\\
(ii) Hence give details of a pair of transformations which transforms the curve $y = \cos x$ to the curve $y = 5 \cos x + 12 \sin x$.\\
(iii) Solve, for $0 ^ { \circ } < x < 360 ^ { \circ }$, the equation $5 \cos x + 12 \sin x = 2$, giving your answers correct to the nearest $0.1 ^ { \circ }$.
\hfill \mbox{\textit{OCR C3 2006 Q8 [11]}}