| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2007 |
| 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 harmonic form question requiring routine application of the R sin(x + α) method with straightforward follow-up parts. Part (a) uses the standard formula R = √(a² + b²) and tan α = b/a, part (b) is immediate once (a) is done, and part (c) involves solving a simple equation in the transformed form. While multi-part, each step follows a well-practiced procedure 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 |
|---|---|---|
| (a) Complete method for \(R\): e.g. \(R\cos\alpha = 3\), \(R\sin\alpha = 2\), \(R = \sqrt{(3^2+2^2)}\) | M1 | |
| \(R = \sqrt{13}\) or 3.61 (or more accurate) | A1 | |
| Complete method for \(\tan\alpha = \frac{2}{3}\) | M1 | |
| [Allow \(\tan\alpha = \frac{3}{2}\)] | ||
| \(\alpha = 0.588\) (Allow 33.7°) | A1 | (4 marks) |
| (b) Greatest value \(= \left(\sqrt{13}\right) = 169\) | M1, A1 | (2 marks) |
| (c) \(\sin(x + 0.588) = \frac{1}{\sqrt{13}}\) (\(= 0.27735...\)) | M1 | |
| \(\sin(x +\) their \(\alpha) = \frac{1}{\text{their }R}\) | ||
| \((x + 0.588)\) \(= 0.281\) (03... or 16.1° | A1 | |
| \((x + 0.588)\) Must be \(\pi - 0.281\) or \(180° -\) their 16.1° | M1 | |
| or \((x + 0.588) = 2\pi +\) their 0.281 or 360° \(+\) their 16.1° | M1 | |
| \(x = 2.273\) or \(x = 5.976\) (awrt) Both (radians only) | A1 | (5 marks) |
| If 0.281 or 16.1° not seen, correct answers imply this A mark |
| Answer | Marks |
|---|---|
| - (i) Squaring to form quadratic in \(\sin x\) or \(\cos x\) | M1 |
| [\(13\cos^2 x - 4\cos x - 8 = 0\), \(13\sin^2 x - 6\sin x - 3 = 0\)] | |
| Correct values for \(\cos x = 0.953...\), \(-0.646\); or \(\sin x = 0.767, 2.27\) awrt | A1 |
| For any one value of \(\cos x\) or \(\sin x\), correct method for two values of | M1 |
| \(x = 2.273\) or \(x = 5.976\) (awrt) Both seen anywhere | A1 |
| Checking other values (0.307, 4.011 or 0.869, 3.449) and discarding | M1 |
| - (ii) Squaring and forming equation of form \(a\cos 2x + b\sin 2x = c\) | M1 |
| \(9\sin^2 x + 4\cos^2 x + 12\sin 2x = 1 \Rightarrow 12\sin 2x + 5\cos 2x = 11\) | |
| Setting up to solve using R formula e.g. \(\sqrt{13}\cos(2x-1.176) = 11\) | M1 |
| \((2x-1.176) = \cos^{-1}\left(\frac{11}{\sqrt{13}}\right) = 0.562(0...)\) (\(\alpha\)) | A1 |
| \((2x-1.176) = 2\pi - \alpha\), \(2\pi + \alpha\),........ | M1 |
| \(x = 2.273\) or \(x = 5.976\) (awrt) Both seen anywhere | A1 |
| Checking other values and discarding | M1 |
| (11 marks) |
**(a)** Complete method for $R$: e.g. $R\cos\alpha = 3$, $R\sin\alpha = 2$, $R = \sqrt{(3^2+2^2)}$ | M1 | |
$R = \sqrt{13}$ or 3.61 (or more accurate) | A1 | |
Complete method for $\tan\alpha = \frac{2}{3}$ | M1 | |
[Allow $\tan\alpha = \frac{3}{2}$] | | |
$\alpha = 0.588$ (Allow 33.7°) | A1 | (4 marks) |
**(b)** Greatest value $= \left(\sqrt{13}\right) = 169$ | M1, A1 | (2 marks) |
**(c)** $\sin(x + 0.588) = \frac{1}{\sqrt{13}}$ ($= 0.27735...$) | M1 | |
$\sin(x +$ their $\alpha) = \frac{1}{\text{their }R}$ | | |
$(x + 0.588)$ $= 0.281$ (03... or 16.1° | A1 | |
$(x + 0.588)$ Must be $\pi - 0.281$ or $180° -$ their 16.1° | M1 | |
or $(x + 0.588) = 2\pi +$ their 0.281 or 360° $+$ their 16.1° | M1 | |
$x = 2.273$ or $x = 5.976$ (awrt) Both (radians only) | A1 | (5 marks) |
If 0.281 or 16.1° not seen, correct answers imply this A mark | | |
**Notes:**
- (a) 1st M1 on Open for correct method for $R$, even if found second. 2nd M1 for correct method for $\tan\alpha$. No working at all: M1A1 for $\sqrt{13}$, M1A1 for 0.588 or 33.7°. N.B. $\cos\alpha = 3$, $\sin\alpha = 2$ used, can still score M1A1 for $R$, but loses the A mark for $\alpha$. $\cos\alpha = 3$, $\sin\alpha = 2$: apply the same marking.
- (b) M1 for realising $\sin(x + \alpha) = \pm1$, so finding $R^4$.
- (c) Working in mixed degrees/rads: first two marks available. Working consistently in degrees: Possible to score first 4 marks. [Degree answers, just for reference, Only are 130.2° and 342.4°]. Third M1 can be gained for candidate's 0.281 – candidate's 0.588 + 2π or equiv. in degrees. **One of the answers correct in radians or degrees implies the corresponding M mark.**
- **Alt. (c):**
- (i) Squaring to form quadratic in $\sin x$ or $\cos x$ | M1 |
[$13\cos^2 x - 4\cos x - 8 = 0$, $13\sin^2 x - 6\sin x - 3 = 0$] | |
Correct values for $\cos x = 0.953...$, $-0.646$; or $\sin x = 0.767, 2.27$ awrt | A1 |
For any one value of $\cos x$ or $\sin x$, correct method for two values of | M1 |
$x = 2.273$ or $x = 5.976$ (awrt) Both seen anywhere | A1 |
Checking other values (0.307, 4.011 or 0.869, 3.449) and discarding | M1 |
- (ii) Squaring and forming equation of form $a\cos 2x + b\sin 2x = c$ | M1 |
$9\sin^2 x + 4\cos^2 x + 12\sin 2x = 1 \Rightarrow 12\sin 2x + 5\cos 2x = 11$ | |
Setting up to solve using R formula e.g. $\sqrt{13}\cos(2x-1.176) = 11$ | M1 |
$(2x-1.176) = \cos^{-1}\left(\frac{11}{\sqrt{13}}\right) = 0.562(0...)$ ($\alpha$) | A1 |
$(2x-1.176) = 2\pi - \alpha$, $2\pi + \alpha$,........ | M1 |
$x = 2.273$ or $x = 5.976$ (awrt) Both seen anywhere | A1 |
Checking other values and discarding | M1 |
| (11 marks) |
---\begin{enumerate}
\item (a) Express $3 \sin x + 2 \cos x$ in the form $R \sin ( x + \alpha )$ where $R > 0$ and $0 < \alpha < \frac { \pi } { 2 }$.\\
(b) Hence find the greatest value of $( 3 \sin x + 2 \cos x ) ^ { 4 }$.\\
(c) Solve, for $0 < x < 2 \pi$, the equation
\end{enumerate}
$$3 \sin x + 2 \cos x = 1$$
giving your answers to 3 decimal places.\\
\hfill \mbox{\textit{Edexcel C3 2007 Q6 [11]}}