| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2006 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trigonometric equations in context |
| Type | Solve shifted trig equation |
| Difficulty | Moderate -0.8 This is a straightforward C2 question testing basic transformations, a routine trig equation with a phase shift, and an algebraic identity proof. All parts require only standard techniques with no problem-solving insight—part (a) is pure recall, part (b) is a textbook exercise in solving shifted sine equations, and part (c) is simple algebraic expansion. Easier than average A-level content. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.05o Trigonometric equations: solve in given intervals1.05p Proof involving trig: functions and identities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Stretch (I) in \(y\)-direction (II), Scale factor 2 (III) | M1A1 | \(>1\) transformation is M0. M1 for (I) and either (II) or (III) or (III) |
| Total (a)(i) | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Reflection; in \(x\)-axis | M1; A1 | 'Reflection'/'reflect(ed)' (or in \(y\)-axis or \(y=0\) or \(x=0\)) |
| Total (a)(ii) | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Translation; \(\begin{bmatrix}30°\\0\end{bmatrix}\) | B1; B1 | Accept full equivalent in words provided linked to 'translation/move/shift' and positive \(x\)-direction. (Note: B0 B1 is possible) |
| Total (a)(iii) | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\{\theta - 30° =\} \sin^{-1}(0.7) = 44.4\ldots°\) | M1 | Inverse sine of 0.7 PI eg by sight of 44, 74 or better |
| \(\ldots\ldots\ldots = 180° - 44.4°\) | m1 | Valid method for 2nd angle |
| \(\theta = 74.4°,\ 165.6°\) | A1 | Condone \(>\)1dp accuracy |
| Total (b) | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\ldots = \cos^2 x + 2\cos x\sin x + \sin^2 x + \cos^2 x - 2\cos x\sin x + \sin^2 x\) | M1 | Award for either bracket expanded correctly |
| \(\ldots\ldots = 2\cos^2 x + 2\sin^2 x\) | A1 | OE |
| \(= 2(\cos^2 x + \sin^2 x) = 2(1)\) | M1 | \(\cos^2 x + \sin^2 x = 1\) stated or used |
| \(= 2\) | A1 | AG (be convinced) |
| Total (c) | 4 |
## Question 6(a)(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Stretch **(I)** in $y$-direction **(II)**, Scale factor 2 **(III)** | M1A1 | $>1$ transformation is M0. M1 for **(I)** and either **(II)** or **(III)** or **(III)** |
| **Total (a)(i)** | **2** | |
## Question 6(a)(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Reflection; in $x$-axis | M1; A1 | 'Reflection'/'reflect(ed)' (or in $y$-axis or $y=0$ or $x=0$) |
| **Total (a)(ii)** | **2** | |
## Question 6(a)(iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Translation; $\begin{bmatrix}30°\\0\end{bmatrix}$ | B1; B1 | Accept **full** equivalent in words provided linked to 'translation/move/shift' and **positive** $x$-direction. (Note: B0 B1 is possible) |
| **Total (a)(iii)** | **2** | |
## Question 6(b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\{\theta - 30° =\} \sin^{-1}(0.7) = 44.4\ldots°$ | M1 | Inverse sine of 0.7 PI eg by sight of 44, 74 or better |
| $\ldots\ldots\ldots = 180° - 44.4°$ | m1 | Valid method for 2nd angle |
| $\theta = 74.4°,\ 165.6°$ | A1 | Condone $>$1dp accuracy |
| **Total (b)** | **3** | |
## Question 6(c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\ldots = \cos^2 x + 2\cos x\sin x + \sin^2 x + \cos^2 x - 2\cos x\sin x + \sin^2 x$ | M1 | Award for either bracket expanded correctly |
| $\ldots\ldots = 2\cos^2 x + 2\sin^2 x$ | A1 | OE |
| $= 2(\cos^2 x + \sin^2 x) = 2(1)$ | M1 | $\cos^2 x + \sin^2 x = 1$ stated or used |
| $= 2$ | A1 | AG (be convinced) |
| **Total (c)** | **4** | |
---6
\begin{enumerate}[label=(\alph*)]
\item Describe the geometrical transformation that maps the curve with equation $y = \sin x$ onto the curve with equation:
\begin{enumerate}[label=(\roman*)]
\item $y = 2 \sin x$;
\item $y = - \sin x$;
\item $y = \sin \left( x - 30 ^ { \circ } \right)$.
\end{enumerate}\item Solve the equation $\sin \left( \theta - 30 ^ { \circ } \right) = 0.7$, giving your answers to the nearest $0.1 ^ { \circ }$ in the interval $0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }$.
\item Prove that $( \cos x + \sin x ) ^ { 2 } + ( \cos x - \sin x ) ^ { 2 } = 2$.
\end{enumerate}
\hfill \mbox{\textit{AQA C2 2006 Q6 [12]}}