| Exam Board | OCR |
|---|---|
| Module | H240/01 (Pure Mathematics) |
| Year | 2019 |
| Session | June |
| Marks | 11 |
| Paper | 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(x-α) formula requiring basic trigonometry (R=√58, tan α=7/3). Parts (b-d) follow directly from the harmonic form with no novel insight needed—just applying known results about amplitude, phase shift, and horizontal stretch. Slightly above average due to the multi-part nature and requiring careful ordering of transformations, but all techniques are textbook standard. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(R^2 = 9 + 49\) | M1 | Attempt correct process to find \(R\) |
| \(R\cos\alpha = 3\), \(R\sin\alpha = 7\), hence \(\tan\alpha = \frac{7}{3}\) | M1 | Attempt correct process to find \(\tan\alpha\) (or equiv with \(\sin\alpha\) or \(\cos\alpha\)). M0 for \(\tan\alpha = \frac{3}{7}\). Allow M1 even if evaluated in degrees |
| \(\sqrt{58}\cos(3x - 1.17)\) | A1 | Obtain \(\sqrt{58}\cos(3x - 1.17)\). Allow \(R = 7.62\) or better. \(\alpha\) must be in radians. If \(R\) and \(\alpha\) are correct then no need to see them substituted back |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Stretch in the \(y\) direction by sf \(\sqrt{58}\) | B1FT | Follow through their \(R\) (numerical or just '\(R\)'). Given at any point in the sequence of transformations. Allow BOD if no 'scale factor'. B1 for 'stretch in \(y\)-direction by \(\sqrt{58}\)'. Must be 'parallel to \(y\)-axis', 'in \(y\) direction', '\(x\)-axis invariant' or equiv. B0 for 'along / in / on / to \(y\)-axis', 'parallel to \(y\)' etc |
| Translation in the \(x\) direction by \(1.17\) | M1 | Translation by \(\pm\) their \(\alpha\) and stretch by (sf) \(3\) or \(\frac{1}{3}\), in either order, both in the \(x\) direction. Allow informal language eg 'shift', 'move', 'compression', 'squash'. Allow translation by \(\pm\frac{1}{3}\) (their \(\alpha\)) |
| Stretch in the \(x\) direction by sf \(\frac{1}{3}\) | A1FT | Translation by their \(\alpha\) (numerical, inc in degrees, or just '\(\alpha\)'). Must be in positive \(x\)-direction. Must use correct language (see B1FT) |
| A1 | Stretch by sf \(\frac{1}{3}\). A0A1 is possible. For A1A1 stretch must follow translation, unless using \(\frac{1}{3}\) (their \(\alpha\)). Must mention 'scale factor', 'factor' or 'sf' |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Greatest value is \(\sqrt{58}\) | B1FT | FT their \(R\). \(R\) must be numerical. Allow no method shown |
| when \(x = 0.389\) | B1 | Obtain \(0.389\). Must be in radians. 'Determine' so some method needed. eg \(3x - 1.17 = 0\) oe (minimum of \(x = \frac{1.17}{3}\)). Allow \(0.39\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Least value is \(-\sqrt{58}\) | B1FT | FT their \(R\). \(R\) must be numerical. Allow no method shown |
| when \(x = 1.44\) | B1 | Obtain \(1.44\). Must be in radians. 'Determine' so some method needed. eg \(3x - 1.17 = \pi\), or equiv |
# Question 9:
## Part (a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $R^2 = 9 + 49$ | M1 | Attempt correct process to find $R$ |
| $R\cos\alpha = 3$, $R\sin\alpha = 7$, hence $\tan\alpha = \frac{7}{3}$ | M1 | Attempt correct process to find $\tan\alpha$ (or equiv with $\sin\alpha$ or $\cos\alpha$). M0 for $\tan\alpha = \frac{3}{7}$. Allow M1 even if evaluated in degrees |
| $\sqrt{58}\cos(3x - 1.17)$ | A1 | Obtain $\sqrt{58}\cos(3x - 1.17)$. Allow $R = 7.62$ or better. $\alpha$ must be in radians. If $R$ and $\alpha$ are correct then no need to see them substituted back |
## Part (b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Stretch in the $y$ direction by sf $\sqrt{58}$ | B1FT | Follow through their $R$ (numerical or just '$R$'). Given at any point in the sequence of transformations. Allow BOD if no 'scale factor'. B1 for 'stretch in $y$-direction by $\sqrt{58}$'. Must be 'parallel to $y$-axis', 'in $y$ direction', '$x$-axis invariant' or equiv. B0 for 'along / in / on / to $y$-axis', 'parallel to $y$' etc |
| Translation in the $x$ direction by $1.17$ | M1 | Translation by $\pm$ their $\alpha$ and stretch by (sf) $3$ or $\frac{1}{3}$, in either order, both in the $x$ direction. Allow informal language eg 'shift', 'move', 'compression', 'squash'. Allow translation by $\pm\frac{1}{3}$ (their $\alpha$) |
| Stretch in the $x$ direction by sf $\frac{1}{3}$ | A1FT | Translation by their $\alpha$ (numerical, inc in degrees, or just '$\alpha$'). Must be in positive $x$-direction. Must use correct language (see B1FT) |
| | A1 | Stretch by sf $\frac{1}{3}$. A0A1 is possible. For A1A1 stretch must follow translation, unless using $\frac{1}{3}$ (their $\alpha$). Must mention 'scale factor', 'factor' or 'sf' |
## Part (c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Greatest value is $\sqrt{58}$ | B1FT | FT their $R$. $R$ must be numerical. Allow no method shown |
| when $x = 0.389$ | B1 | Obtain $0.389$. Must be in radians. 'Determine' so some method needed. eg $3x - 1.17 = 0$ oe (minimum of $x = \frac{1.17}{3}$). Allow $0.39$ |
## Part (d):
| Answer | Mark | Guidance |
|--------|------|----------|
| Least value is $-\sqrt{58}$ | B1FT | FT their $R$. $R$ must be numerical. Allow no method shown |
| when $x = 1.44$ | B1 | Obtain $1.44$. Must be in radians. 'Determine' so some method needed. eg $3x - 1.17 = \pi$, or equiv |
---9
\begin{enumerate}[label=(\alph*)]
\item Express $3 \cos 3 x + 7 \sin 3 x$ in the form $R \cos ( 3 x - \alpha )$, where $R > 0$ and $0 < \alpha < \frac { 1 } { 2 } \pi$.
\item Give full details of a sequence of three transformations needed to transform the curve $y = \cos x$ to the curve $y = 3 \cos 3 x + 7 \sin 3 x$.
\item Determine the greatest value of $3 \cos 3 x + 7 \sin 3 x$ as $x$ varies and give the smallest positive value of $x$ for which it occurs.
\item Determine the least value of $3 \cos 3 x + 7 \sin 3 x$ as $x$ varies and give the smallest positive value of $x$ for which it occurs.
\end{enumerate}
\hfill \mbox{\textit{OCR H240/01 2019 Q9 [11]}}