| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2012 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Prove identity then solve equation only (no integral) |
| Difficulty | Standard +0.3 This is a standard C3 question testing double angle formulae and transformations. Part (i) requires routine application of cos(2θ) = 1-2sin²θ twice, part (ii) is straightforward substitution, and part (iii) involves algebraic manipulation and finding max/min of cosine. The transformations question is textbook material. All parts follow predictable patterns with no novel insight required, making it slightly easier than average. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.02z Models in context: use functions in modelling1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05l Double angle formulae: and compound angle formulae |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State \(\cos 4\theta = 1 - 2\sin^2 2\theta\) | B1 | |
| State or clearly imply \(\sin 2\theta = 2\sin\theta\cos\theta\) | B1 | possibly substituted in incorrect expression |
| Obtain \(1 - 8\sin^2\theta\cos^2\theta\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Produce expression involving \(\cos\frac{4}{24}\pi\) as only trigonometrical ratio | M1 | |
| Obtain \(\frac{1}{8} - \frac{1}{16}\sqrt{3}\) | A1 | or exact equiv (including, eg \(\frac{1 - \frac{1}{2}\sqrt{3}}{8}\)) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use \(2\cos^2 2\theta = 1 + \cos 4\theta\) | B1 | or use \(2\cos^2 2\theta = 2 - 8\sin^2\theta\cos^2\theta\) |
| Attempt to express in terms of \(\cos 4\theta\) | M1 | or unsimplified equiv |
| Obtain \(\frac{2}{3} + \frac{4}{3}\cos 4\theta\) | A1 | |
| Substitute at least one of \(-1\) and \(1\) for \(\cos 4\theta\) in expression where \(\cos 4\theta\) is only trigonometrical ratio | M1 | or at least one of \(\theta = \frac{1}{4}\pi\) and \(\theta = 0\) |
| Obtain \(2\) and \(-\frac{2}{3}\) | A1 |
## Question 8:
### Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State $\cos 4\theta = 1 - 2\sin^2 2\theta$ | B1 | |
| State or clearly imply $\sin 2\theta = 2\sin\theta\cos\theta$ | B1 | possibly substituted in incorrect expression |
| Obtain $1 - 8\sin^2\theta\cos^2\theta$ | B1 | |
### Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Produce expression involving $\cos\frac{4}{24}\pi$ as only trigonometrical ratio | M1 | |
| Obtain $\frac{1}{8} - \frac{1}{16}\sqrt{3}$ | A1 | or exact equiv (including, eg $\frac{1 - \frac{1}{2}\sqrt{3}}{8}$) |
### Part (iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use $2\cos^2 2\theta = 1 + \cos 4\theta$ | B1 | or use $2\cos^2 2\theta = 2 - 8\sin^2\theta\cos^2\theta$ |
| Attempt to express in terms of $\cos 4\theta$ | M1 | or unsimplified equiv |
| Obtain $\frac{2}{3} + \frac{4}{3}\cos 4\theta$ | A1 | |
| Substitute at least one of $-1$ and $1$ for $\cos 4\theta$ in expression where $\cos 4\theta$ is only trigonometrical ratio | M1 | or at least one of $\theta = \frac{1}{4}\pi$ and $\theta = 0$ |
| Obtain $2$ and $-\frac{2}{3}$ | A1 | |
---8 (i) Express $\cos 4 \theta$ in terms of $\sin 2 \theta$ and hence show that $\cos 4 \theta$ can be expressed in the form $1 - k \sin ^ { 2 } \theta \cos ^ { 2 } \theta$, where $k$ is a constant to be determined.\\
(ii) Hence find the exact value of $\sin ^ { 2 } \left( \frac { 1 } { 24 } \pi \right) \cos ^ { 2 } \left( \frac { 1 } { 24 } \pi \right)$.\\
(iii) By expressing $2 \cos ^ { 2 } 2 \theta - \frac { 8 } { 3 } \sin ^ { 2 } \theta \cos ^ { 2 } \theta$ in terms of $\cos 4 \theta$, find the greatest and least possible values of
$$2 \cos ^ { 2 } 2 \theta - \frac { 8 } { 3 } \sin ^ { 2 } \theta \cos ^ { 2 } \theta$$
as $\theta$ varies.\\
\includegraphics[max width=\textwidth, alt={}, center]{89e54367-bb83-483a-add5-0527b71a5cac-5_606_926_267_552}
The function f is defined for all real values of $x$ by
$$\mathrm { f } ( x ) = k \left( x ^ { 2 } + 4 x \right) ,$$
where $k$ is a positive constant. The diagram shows the curve with equation $y = \mathrm { f } ( x )$.\\
(i) The curve $y = x ^ { 2 }$ can be transformed to the curve $y = \mathrm { f } ( x )$ by the following sequence of transformations: a translation parallel to the $x$-axis,\\
a translation parallel to the $y$-axis,\\
a stretch. a translation parallel to the $x$-axis, a translation parallel to the $y$-axis, a stretch.\\
Give details, in terms of $k$ where appropriate, of these transformations.\\
(ii) Find the range of f in terms of $k$.\\
(iii) It is given that there are three distinct values of $x$ which satisfy the equation $| \mathrm { f } ( x ) | = 20$. Find the value of $k$ and determine exactly the three values of $x$ which satisfy the equation in this case.
\hfill \mbox{\textit{OCR C3 2012 Q8 [10]}}