| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2015 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trig Graphs & Exact Values |
| Type | Sketch single reciprocal or inverse trig graph |
| Difficulty | Moderate -0.3 Part (a) requires knowing the domain and range of arcsin and applying a horizontal stretch - straightforward but tests understanding of inverse trig functions. Part (b) is a standard application of the chain rule for implicit differentiation. Both parts are routine C3 content with no novel problem-solving required, making this slightly easier than average. |
| Spec | 1.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates |
| Answer | Marks | Guidance |
|---|---|---|
| Curve passes through \(\left(-\frac{1}{3}, -\frac{\pi}{2}\right)\) and \(\left(\frac{1}{3}, \frac{\pi}{2}\right)\) | B1 | Correct end points stated |
| Correct shape: increasing curve through origin | B1 | Correct shape |
| End points correctly marked on sketch | B1 | Coordinates correctly placed |
| Answer | Marks | Guidance |
|---|---|---|
| \(x = \frac{1}{3}\sin y \Rightarrow \frac{dx}{dy} = \frac{1}{3}\cos y\) | B1 | Correct \(\frac{dx}{dy}\) |
| \(\frac{dy}{dx} = \frac{1}{\frac{1}{3}\cos y} = \frac{3}{\cos y}\) | B1 | Correct \(\frac{dy}{dx}\) in terms of \(y\) |
# Question 6:
## Part (a):
| Curve passes through $\left(-\frac{1}{3}, -\frac{\pi}{2}\right)$ and $\left(\frac{1}{3}, \frac{\pi}{2}\right)$ | B1 | Correct end points stated |
|---|---|---|
| Correct shape: increasing curve through origin | B1 | Correct shape |
| End points correctly marked on sketch | B1 | Coordinates correctly placed |
## Part (b):
| $x = \frac{1}{3}\sin y \Rightarrow \frac{dx}{dy} = \frac{1}{3}\cos y$ | B1 | Correct $\frac{dx}{dy}$ |
|---|---|---|
| $\frac{dy}{dx} = \frac{1}{\frac{1}{3}\cos y} = \frac{3}{\cos y}$ | B1 | Correct $\frac{dy}{dx}$ in terms of $y$ |
---6
\begin{enumerate}[label=(\alph*)]
\item Sketch, on the axes below, the curve with equation $y = \sin ^ { - 1 } ( 3 x )$, where $y$ is in radians.
State the exact values of the coordinates of the end points of the graph.
\item Given that $x = \frac { 1 } { 3 } \sin y$, write down $\frac { \mathrm { d } x } { \mathrm {~d} y }$ and hence find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $y$.
\section*{Answer space for question 6}
(a)\\
\includegraphics[max width=\textwidth, alt={}, center]{2df59047-3bfe-4b9c-a77f-142bc7506cbc-14_839_1451_813_324}
\end{enumerate}
\hfill \mbox{\textit{AQA C3 2015 Q6 [5]}}