| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2017 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Solve reciprocal trig equation |
| Difficulty | Standard +0.3 This is a standard C3 harmonic form question with routine steps: (a) uses the R cos(θ+α) formula with straightforward calculation, (b) requires algebraic manipulation of reciprocal trig identities (dividing by sin 2x), and (c) applies part (a) to solve the equation. While it has multiple parts and requires careful algebra with reciprocal trig functions, all techniques are standard textbook exercises with no novel insight required. Slightly easier than average due to the guided structure. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(R = \sqrt{29}\) | B1 | Condone \(R = \pm\sqrt{29}\); do not allow decimals (e.g. 5.39); ISW after \(\sqrt{29}\) |
| \(\tan\alpha = \frac{2}{5} \Rightarrow \alpha =\) awrt \(0.381\) | M1 A1 | Accept \(\tan\alpha = \pm\frac{2}{5}\), \(\tan\alpha = \pm\frac{5}{2}\); also accept using \(\sin\alpha = \pm\frac{2}{R}\) or \(\cos\alpha = \pm\frac{5}{R}\); degree equivalent awrt \(21.8°\) is A0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(5\cot 2x - 3\text{cosec}\, 2x = 2 \Rightarrow 5\frac{\cos 2x}{\sin 2x} - \frac{3}{\sin 2x} = 2\) | M1 | Replaces \(\cot 2x\) by \(\frac{\cos 2x}{\sin 2x}\) and \(\text{cosec}\, 2x\) by \(\frac{1}{\sin 2x}\); do not be concerned by coefficients |
| \(\Rightarrow 5\cos 2x - 2\sin 2x = 3\) | A1 | cso; notation must be correct; cannot mix variables |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(5\cos 2x - 2\sin 2x = 3 \Rightarrow \cos(2x + 0.381) = \frac{3}{\sqrt{29}}\) | M1 | Must use their \(R\) and \(\alpha\) from (a) and their \(c\) from (b) |
| \(2x + 0.381 = \arccos\left(\frac{3}{\sqrt{29}}\right) \Rightarrow x = \ldots\) | dM1 | Score for dealing with cos, \(\alpha\) and the 2 correctly in that order to reach \(x\); implied by one correct answer |
| \(x =\) awrt \(0.30\), \(2.46\) | A1 A1 | One solution correct (usually \(x = 0.3/0.30\) or \(x = 2.46\)); both correct with no extra values in range; in degrees accept awrt 1dp \(17.2°\), \(141.(0)°\) |
## Question 4:
### Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $R = \sqrt{29}$ | B1 | Condone $R = \pm\sqrt{29}$; do not allow decimals (e.g. 5.39); ISW after $\sqrt{29}$ |
| $\tan\alpha = \frac{2}{5} \Rightarrow \alpha =$ awrt $0.381$ | M1 A1 | Accept $\tan\alpha = \pm\frac{2}{5}$, $\tan\alpha = \pm\frac{5}{2}$; also accept using $\sin\alpha = \pm\frac{2}{R}$ or $\cos\alpha = \pm\frac{5}{R}$; degree equivalent awrt $21.8°$ is A0 |
### Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $5\cot 2x - 3\text{cosec}\, 2x = 2 \Rightarrow 5\frac{\cos 2x}{\sin 2x} - \frac{3}{\sin 2x} = 2$ | M1 | Replaces $\cot 2x$ by $\frac{\cos 2x}{\sin 2x}$ **and** $\text{cosec}\, 2x$ by $\frac{1}{\sin 2x}$; do not be concerned by coefficients |
| $\Rightarrow 5\cos 2x - 2\sin 2x = 3$ | A1 | cso; notation must be correct; cannot mix variables |
### Part (c)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $5\cos 2x - 2\sin 2x = 3 \Rightarrow \cos(2x + 0.381) = \frac{3}{\sqrt{29}}$ | M1 | Must use their $R$ and $\alpha$ from (a) and their $c$ from (b) |
| $2x + 0.381 = \arccos\left(\frac{3}{\sqrt{29}}\right) \Rightarrow x = \ldots$ | dM1 | Score for dealing with cos, $\alpha$ and the 2 **correctly** in that order to reach $x$; implied by one correct answer |
| $x =$ awrt $0.30$, $2.46$ | A1 A1 | One solution correct (usually $x = 0.3/0.30$ or $x = 2.46$); both correct with no extra values in range; in degrees accept awrt 1dp $17.2°$, $141.(0)°$ |\begin{enumerate}
\item (a) Write $5 \cos \theta - 2 \sin \theta$ in the form $R \cos ( \theta + \alpha )$, where $R$ and $\alpha$ are constants, $R > 0$ and $0 \leqslant \alpha < \frac { \pi } { 2 }$
\end{enumerate}
Give the exact value of $R$ and give the value of $\alpha$ in radians to 3 decimal places.\\
(b) Show that the equation
$$5 \cot 2 x - 3 \operatorname { cosec } 2 x = 2$$
can be rewritten in the form
$$5 \cos 2 x - 2 \sin 2 x = c$$
where $c$ is a positive constant to be determined.\\
(c) Hence or otherwise, solve, for $0 \leqslant x < \pi$,
$$5 \cot 2 x - 3 \operatorname { cosec } 2 x = 2$$
giving your answers to 2 decimal places.\\
(Solutions based entirely on graphical or numerical methods are not acceptable.)
\hfill \mbox{\textit{Edexcel C3 2017 Q4 [9]}}