| Exam Board | Edexcel |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2020 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Transformations of trigonometric graphs |
| Difficulty | Standard +0.3 This is a standard harmonic form question with routine transformations. Part (a) uses the standard R cos(θ+α) formula with straightforward Pythagorean calculation and arctan. Parts (b) require identifying stretch factor R and translation -α from the harmonic form. Part (c) involves finding range by substituting the known range of f(θ), which is [-R, R]. All techniques are textbook exercises with no novel problem-solving required, making this slightly easier than average. |
| Spec | 1.02u Functions: definition and vocabulary (domain, range, mapping)1.02w Graph transformations: simple transformations of f(x)1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc |
| Leave blank | Q9 |
| END |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(R = \sqrt{41}\) | B1 | |
| \(\tan\alpha = \frac{4}{5} \Rightarrow \alpha = \text{awrt } 0.675\) | M1 A1 | (3) |
| Answer | Marks |
|---|---|
| Do not award for \(y\) is translated/transformed by "\(\sqrt{41}\)" | B1 ft |
| Answer | Marks | Guidance |
|---|---|---|
| If there are no labels score in the order given but do allow these to be written in any order as long as the candidate clearly states which one they are answering. For example it is fine to write ....translation is.....................stretch is ....................If the candidate does not label correctly, or states which one they are doing, but otherwise gets both completely correct then award SC B1 B0 | B1 ft | (2) |
| (c) Score for either end achieved by a correct method. Look for \(\frac{90}{4}\) (implied by 22.5), \(\frac{90}{4+(\sqrt{41})^2}\), \(g...22.5\) or \(g...2\) etc | M1 | |
| See scheme but allow 22.5 to be written as \(\frac{90}{4}\); Accept equivalent ways of writing the interval such as \([2, 22.5]\); Condone \(2 \leq g(x) \leq 22.5\) or \(2 \leq y \leq 22.5\) | A1 | (2) |
**(a)** $R = \sqrt{41}$ | B1 |
$\tan\alpha = \frac{4}{5} \Rightarrow \alpha = \text{awrt } 0.675$ | M1 A1 | (3)
**(b)(i)** Fully describes the stretch. Follow through on their $R$. **Requires the size and the direction. Allow** responses such as:
- stretch in the $y$ direction by "$\sqrt{41}$"
- multiplies all the $y$ coordinates/values by "$\sqrt{41}$"
- stretch in $\uparrow$ direction by "$\sqrt{41}$"
- vertical stretch by "$\sqrt{41}$"
- Scale Factor "$\sqrt{41}$" in just the $y$ direction
**Do not award** for $y$ is translated/transformed by "$\sqrt{41}$" | B1 ft |
**(ii)** Fully describes the translation. **Requires the size and the direction.** Follow through on their 0.675 or $\alpha = \text{awrt } 38.7°$ or $\arctan\frac{4}{5}$
**Allow** responses such as:
- translates left by 0.675
- horizontal by $-0.675$
- condone "transforms" left by 0.675. (question asks for the translation)
- moves $\leftarrow$ by $38.7°$
- $x$ values move back by 0.675
- shifts in the negative $x$ direction by $\arctan\frac{4}{5}$
- $\begin{pmatrix}-0.675\\0\end{pmatrix}$
**Do not award** for translates left by $-0.675$ (double negative...wrong direction); horizontal shift of 0.675 (no direction)
If there are no labels score in the order given but do allow these to be written in any order as long as the candidate clearly states which one they are answering. For example it is fine to write ....translation is.....................stretch is ....................If the candidate does not label correctly, or states which one they are doing, but otherwise gets both completely correct then award SC B1 B0 | B1 ft | (2)
**(c)** Score for either end achieved by a correct method. Look for $\frac{90}{4}$ (implied by 22.5), $\frac{90}{4+(\sqrt{41})^2}$, $g...22.5$ or $g...2$ etc | M1 |
See scheme but allow 22.5 to be written as $\frac{90}{4}$; Accept equivalent ways of writing the interval such as $[2, 22.5]$; Condone $2 \leq g(x) \leq 22.5$ or $2 \leq y \leq 22.5$ | A1 | (2)
**Total: 7 marks**9.
$$\mathrm { f } ( \theta ) = 5 \cos \theta - 4 \sin \theta \quad \theta \in \mathbb { R }$$
\begin{enumerate}[label=(\alph*)]
\item Express $\mathrm { f } ( \theta )$ in the form $R \cos ( \theta + \alpha )$, where $R$ and $\alpha$ are constants, $R > 0$ and $0 < \alpha < \frac { \pi } { 2 }$. Give the exact value of $R$ and give the value of $\alpha$, in radians, to 3 decimal places.
The curve with equation $y = \cos \theta$ is transformed onto the curve with equation $y = \mathrm { f } ( \theta )$ by a sequence of two transformations.
Given that the first transformation is a stretch and the second a translation,
\item \begin{enumerate}[label=(\roman*)]
\item describe fully the transformation that is a stretch,
\item describe fully the transformation that is a translation.
Given
$$g ( \theta ) = \frac { 90 } { 4 + ( f ( \theta ) ) ^ { 2 } } \quad \theta \in \mathbb { R }$$
\end{enumerate}\item find the range of g.
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\hfill \mbox{\textit{Edexcel P3 2020 Q9 [7]}}