| Exam Board | Edexcel |
|---|---|
| Module | C34 (Core Mathematics 3 & 4) |
| Year | 2017 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Find value where max/min occurs |
| Difficulty | Standard +0.3 This is a standard harmonic form question with routine application of R-α form. Part (a) uses Pythagorean theorem and inverse tan (textbook procedure), part (b) is straightforward solving after substitution, and part (c) requires recognizing that y is minimized when the denominator is maximized. All steps are algorithmic 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 |
|---|---|---|
| \(R = 37\) | B1 | No working needed; condone \(R = \pm 37\) |
| \(\tan\alpha = \frac{12}{35} \Rightarrow \alpha =\) awrt 0.3303 | M1 A1 | M1: \(\tan\alpha = \pm\frac{12}{35}\) or \(\tan\alpha = \pm\frac{35}{12}\) with attempt to find \(\alpha\); A1: \(\alpha =\) awrt 0.3303; answers in degrees are A0 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\sin(x-\alpha) = \frac{37}{2R}\) \((= 0.5...)\) | M1 | Uses part (a) to solve equation \(\sin(x\pm\alpha) = \frac{37}{2\times theirR}\) |
| \(x = \arcsin\left(\frac{37}{2\times\text{their "37"}}\right) + \text{their "0.3303"}\) | M1 | Operations undone in correct order; accept \(\sin(x\pm\alpha)=k \Rightarrow x = \arcsin k \pm \alpha\) |
| \(x =\) awrt 0.854 or awrt 2.95 | A1 | One correct answer; allow \(0.272\pi\) or \(0.938\pi\) |
| \(x =\) awrt 0.854 and 2.95 | A1 | Both values (no extra values in range); allow \(0.272\pi, 0.938\pi\) |
| Answer | Marks |
|---|---|
| Find \(y = \frac{7000}{31+(\pm''R'')^2} = 5\) | M1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(x - \alpha = \frac{\pi}{2} \Rightarrow x = 1.90\) | M1 A1 | M1: uses \(x - \text{their } \alpha = (2n+1)\frac{\pi}{2}\); A1: awrt 1.90, condone 1.9 |
# Question 11:
## Part (a)
| $R = 37$ | B1 | No working needed; condone $R = \pm 37$ |
| $\tan\alpha = \frac{12}{35} \Rightarrow \alpha =$ awrt 0.3303 | M1 A1 | M1: $\tan\alpha = \pm\frac{12}{35}$ or $\tan\alpha = \pm\frac{35}{12}$ with attempt to find $\alpha$; A1: $\alpha =$ awrt 0.3303; answers in degrees are A0 |
## Part (b)
| $\sin(x-\alpha) = \frac{37}{2R}$ $(= 0.5...)$ | M1 | Uses part (a) to solve equation $\sin(x\pm\alpha) = \frac{37}{2\times theirR}$ |
| $x = \arcsin\left(\frac{37}{2\times\text{their "37"}}\right) + \text{their "0.3303"}$ | M1 | Operations undone in correct order; accept $\sin(x\pm\alpha)=k \Rightarrow x = \arcsin k \pm \alpha$ |
| $x =$ awrt 0.854 or awrt 2.95 | A1 | One correct answer; allow $0.272\pi$ or $0.938\pi$ |
| $x =$ awrt 0.854 and 2.95 | A1 | Both values (no extra values in range); allow $0.272\pi, 0.938\pi$ |
## Part (c)(i)
| Find $y = \frac{7000}{31+(\pm''R'')^2} = 5$ | M1 A1 | |
## Part (c)(ii)
| $x - \alpha = \frac{\pi}{2} \Rightarrow x = 1.90$ | M1 A1 | M1: uses $x - \text{their } \alpha = (2n+1)\frac{\pi}{2}$; A1: awrt 1.90, condone 1.9 |
---\begin{enumerate}
\item (a) Express $35 \sin x - 12 \cos x$ in the form $R \sin ( x - \alpha )$, where $R > 0$ and $0 < \alpha < \frac { \pi } { 2 }$
\end{enumerate}
Give the exact value of $R$, and give the value of $\alpha$, in radians, to 4 significant figures.\\
(b) Hence solve, for $0 \leqslant x < 2 \pi$,
$$70 \sin x - 24 \cos x = 37$$
(Solutions based entirely on graphical or numerical methods are not acceptable.)
$$y = \frac { 7000 } { 31 + ( 35 \sin x - 12 \cos x ) ^ { 2 } } , \quad x > 0$$
(c) Use your answer to part (a) to calculate\\
(i) the minimum value of $y$,\\
(ii) the smallest value of $x , x > 0$, at which this minimum value occurs.
\hfill \mbox{\textit{Edexcel C34 2017 Q11 [11]}}