| Exam Board | Edexcel |
|---|---|
| Module | C34 (Core Mathematics 3 & 4) |
| Year | 2016 |
| Session | January |
| Marks | 12 |
| 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 with routine techniques: converting to R sin(θ+α) using Pythagorean identity and arctan, solving a basic trigonometric equation, and finding max/min values using the range of sine. All steps are textbook 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 |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(R = \sqrt{34}\) | B1 | Condone \(\pm\sqrt{34}\) |
| \(\tan\alpha = \dfrac{5}{3}\) | M1 | Or \(\tan\alpha=\pm\dfrac{5}{3}\) or \(\pm\dfrac{3}{5}\); implied by awrt 1.0 rads or 59° |
| \(\alpha = 1.03\) | A1 | awrt 1.03; also accept \(\sqrt{34}\sin(2x+1.03)\); degrees answer (59.04) is A0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\sin(2x+\text{"1.03"}) = \dfrac{4}{\text{"}\sqrt{34}\text{"}}\) \((=0.68599...)\) | M1 | Uses part (a) to solve equation |
| \(2x+\text{"1.03"}=2\pi+\arcsin\left(\dfrac{4}{\text{"}\sqrt{34}\text{"}}\right)\Rightarrow x=...\) | M1 | Attempt at one solution in range; acceptable to find \(-0.14\) and add \(\pi\) |
| Either \(x=\) awrt 3.0 or awrt 0.68 | A1 | In degrees accept awrt 38.8 or 172.1; condone 3 for 3.0 |
| \(2x+\text{"1.03"}=\pi-\arcsin\left(\dfrac{4}{\text{"}\sqrt{34}\text{"}}\right)\Rightarrow x=...\) | M1 | Attempt at second solution |
| Both \(x=\) awrt 3.0 and awrt 0.68 | A1 | In degrees awrt 38.8 and awrt 172.1; condone 3 for 3.0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Greatest value: \(4(\sqrt{34})^2+3 = 139\) | M1 A1 | Attempts \(4(R)^2+3\); cao |
| Least value: \(4(0)^2+3 = 3\) | M1 A1 | Uses 0 for minimum value; accept \(4(0)^2+3\) |
# Question 10(a) - R-alpha Form
| Answer/Working | Mark | Guidance |
|---|---|---|
| $R = \sqrt{34}$ | B1 | Condone $\pm\sqrt{34}$ |
| $\tan\alpha = \dfrac{5}{3}$ | M1 | Or $\tan\alpha=\pm\dfrac{5}{3}$ or $\pm\dfrac{3}{5}$; implied by awrt 1.0 rads or 59° |
| $\alpha = 1.03$ | A1 | awrt 1.03; also accept $\sqrt{34}\sin(2x+1.03)$; degrees answer (59.04) is A0 |
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# Question 10(b) - Solving Equation
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\sin(2x+\text{"1.03"}) = \dfrac{4}{\text{"}\sqrt{34}\text{"}}$ $(=0.68599...)$ | M1 | Uses part (a) to solve equation |
| $2x+\text{"1.03"}=2\pi+\arcsin\left(\dfrac{4}{\text{"}\sqrt{34}\text{"}}\right)\Rightarrow x=...$ | M1 | Attempt at one solution in range; acceptable to find $-0.14$ and add $\pi$ |
| Either $x=$ awrt 3.0 **or** awrt 0.68 | A1 | In degrees accept awrt 38.8 or 172.1; condone 3 for 3.0 |
| $2x+\text{"1.03"}=\pi-\arcsin\left(\dfrac{4}{\text{"}\sqrt{34}\text{"}}\right)\Rightarrow x=...$ | M1 | Attempt at second solution |
| Both $x=$ awrt 3.0 **and** awrt 0.68 | A1 | In degrees awrt 38.8 and awrt 172.1; condone 3 for 3.0 |
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# Question 10(c) - Greatest/Least Values
| Answer/Working | Mark | Guidance |
|---|---|---|
| Greatest value: $4(\sqrt{34})^2+3 = 139$ | M1 A1 | Attempts $4(R)^2+3$; cao |
| Least value: $4(0)^2+3 = 3$ | M1 A1 | Uses 0 for minimum value; accept $4(0)^2+3$ |10. (a) Express $3 \sin 2 x + 5 \cos 2 x$ in the form $R \sin ( 2 x + \alpha )$, where $R > 0$ and $0 < \alpha < \frac { \pi } { 2 }$ Give the exact value of $R$ and give the value of $\alpha$ to 3 significant figures.\\
(b) Solve, for $0 < x < \pi$,
$$3 \sin 2 x + 5 \cos 2 x = 4$$
(Solutions based entirely on graphical or numerical methods are not acceptable.)
$$g ( x ) = 4 ( 3 \sin 2 x + 5 \cos 2 x ) ^ { 2 } + 3$$
(c) Using your answer to part (a) and showing your working,
\begin{enumerate}[label=(\roman*)]
\item find the greatest value of $\mathrm { g } ( x )$,
\item find the least value of $\mathrm { g } ( x )$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C34 2016 Q10 [12]}}