| Exam Board | AQA |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2010 |
| Session | January |
| Marks | 10 |
| 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 cos(x-α) method with multiple straightforward parts. Part (a) uses the standard formula R=√(1²+3²) and α=arctan(3/1), part (b) simply reads off the minimum value -R and solves R cos(x-α)=-R, and part (c) solves a basic equation using the harmonic form. All steps are textbook procedures 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 |
|---|---|---|
| \(R=\sqrt{10}\) | B1 | Accept \(R=3.16\) or better |
| \(\tan\alpha=3\) | M1 | OE |
| \(\alpha=1.249\), ignore extra out of range | A1 (3 marks) | AWRT 1.25; SC \(\alpha=0.322\) B1; radians only |
| Answer | Marks | Guidance |
|---|---|---|
| minimum value \(=-\sqrt{10}\) | B1F (1 mark) | F on \(R\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\cos(x-\alpha)=-1\) | M1 | |
| \(x=4.391\) | A1F (2 marks) | AWRT 4.39; \(51.56°\) or \(\ldots .57°\) or better |
| Answer | Marks | Guidance |
|---|---|---|
| \(\cos(x-\alpha)=\frac{2}{\sqrt{10}}\) | M1 | |
| \(x-\alpha=\pm0.886\quad 5.397\), ignore extra out of range | A1 | Two values, accept 2dp and condone 5.4; condone use of degrees |
| \(x=0.36296\ldots\quad 2.13512\ldots\) | A1F | F on \(x-\alpha\), either value. AWRT |
| \(x=0.363\qquad 2.135\) | A1 (4 marks) | CSO, 3dp or better |
| Answer | Marks | Guidance |
|---|---|---|
| \(10\sin^2 x - 12\sin x + 3 = 0\) | M1 | Or equivalent quadratic using \(\cos x\) (ie \(\sin^2 x+\cos^2 x=1\) used) or equivalent using \(\cos x\) |
| \(\sin x=\) two numerical answers, \(-1\leq\text{ans}\leq1\) | A1F | |
| \(x=\) one correct answer | A1F | |
| \(x=0.363\qquad 2.135\) | A1 | CSO, 3dp or better |
## Question 2:
### Part (a)
| $R=\sqrt{10}$ | B1 | Accept $R=3.16$ or better |
| $\tan\alpha=3$ | M1 | OE |
| $\alpha=1.249$, ignore extra out of range | A1 (3 marks) | AWRT 1.25; SC $\alpha=0.322$ B1; radians only |
### Part (b)(i)
| minimum value $=-\sqrt{10}$ | B1F (1 mark) | F on $R$ |
### Part (b)(ii)
| $\cos(x-\alpha)=-1$ | M1 | |
| $x=4.391$ | A1F (2 marks) | AWRT 4.39; $51.56°$ or $\ldots .57°$ or better |
### Part (c)
| $\cos(x-\alpha)=\frac{2}{\sqrt{10}}$ | M1 | |
| $x-\alpha=\pm0.886\quad 5.397$, ignore extra out of range | A1 | Two values, accept 2dp and condone 5.4; condone use of degrees |
| $x=0.36296\ldots\quad 2.13512\ldots$ | A1F | F on $x-\alpha$, either value. AWRT |
| $x=0.363\qquad 2.135$ | A1 (4 marks) | CSO, 3dp or better |
**Alternative (c):**
| $10\sin^2 x - 12\sin x + 3 = 0$ | M1 | Or equivalent quadratic using $\cos x$ (ie $\sin^2 x+\cos^2 x=1$ used) or equivalent using $\cos x$ |
| $\sin x=$ two numerical answers, $-1\leq\text{ans}\leq1$ | A1F | |
| $x=$ one correct answer | A1F | |
| $x=0.363\qquad 2.135$ | A1 | CSO, 3dp or better |
**Total: 10 marks**
---2
\begin{enumerate}[label=(\alph*)]
\item Express $\cos x + 3 \sin x$ in the form $R \cos ( x - \alpha )$, where $R > 0$ and $0 < \alpha < \frac { \pi } { 2 }$. Give your value of $\alpha$, in radians, to three decimal places.
\item \begin{enumerate}[label=(\roman*)]
\item Hence write down the minimum value of $\cos x + 3 \sin x$.
\item Find the value of $x$ in the interval $0 \leqslant x \leqslant 2 \pi$ at which this minimum occurs, giving your answer, in radians, to three decimal places.
\end{enumerate}\item Solve the equation $\cos x + 3 \sin x = 2$ in the interval $0 \leqslant x \leqslant 2 \pi$, giving all solutions, in radians, to three decimal places.
\end{enumerate}
\hfill \mbox{\textit{AQA C4 2010 Q2 [10]}}