| Exam Board | AQA |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2010 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric differentiation |
| Type | Find tangent equation at parameter |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question testing standard C4 parametric differentiation. Part (a) involves routine double-angle formula recall and substitution. Part (b) requires the chain rule dy/dx = (dy/dθ)/(dx/dθ) and finding a tangent equation using given values—all standard textbook techniques with no novel problem-solving required. Slightly easier than average due to the scaffolded structure and explicit guidance. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| \(\sin2\theta=2\sin\theta\cos\theta\) | B1 | |
| \(\cos2\theta=\cos^2\theta-\sin^2\theta\) | B1 (2 marks) | OE; condone use of \(x\) etc, but variable must be consistent |
| Answer | Marks | Guidance |
|---|---|---|
| \(\sin\theta=\frac{4}{5}\Rightarrow\sin2\theta=2\times\frac{4}{5}\times\frac{3}{5}=\frac{24}{25}\) | B1 (1 mark) | AG; Use of \(106.26°\ldots\) B0 |
| \(\cos2\theta=\frac{9}{25}-\frac{16}{25}=-\frac{7}{25}\) | B1 (2 marks) | \(-0.28\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{dx}{d\theta}=6\cos2\theta\quad,\quad\frac{dy}{d\theta}=-8\sin2\theta\) | M1 | Attempt both derivatives, ie \(p\cos2\theta\), \(q\sin2\theta\) |
| A1 | Both correct | |
| \(\frac{dy}{dx}=-\frac{4\sin2\theta}{3\cos2\theta}\) ISW | A1 (3 marks) | CSO, OE |
| Answer | Marks | Guidance |
|---|---|---|
| \(P\left(\frac{72}{25},-\frac{28}{25}\right)\) | B1F | \((2.88,-1.12)\) |
| Gradient \(=-\frac{4}{3}\times-\frac{24}{7}\) | M1 | Their \(\frac{q\sin2\theta}{p\cos2\theta}\) or \(\frac{p\cos2\theta}{q\sin2\theta}\); must be working with rational numbers |
| Tangent \(\quad y+\frac{28}{25}=\frac{32}{7}\left(x-\frac{72}{25}\right)\) ISW | A1 (3 marks) | Any correct form; \(7y=32x-100\); Fractions in simplest form; Equation required |
## Question 6:
### Part (a)(i)
| $\sin2\theta=2\sin\theta\cos\theta$ | B1 | |
| $\cos2\theta=\cos^2\theta-\sin^2\theta$ | B1 (2 marks) | OE; condone use of $x$ etc, but variable must be consistent |
### Part (a)(ii)
| $\sin\theta=\frac{4}{5}\Rightarrow\sin2\theta=2\times\frac{4}{5}\times\frac{3}{5}=\frac{24}{25}$ | B1 (1 mark) | AG; Use of $106.26°\ldots$ B0 |
| $\cos2\theta=\frac{9}{25}-\frac{16}{25}=-\frac{7}{25}$ | B1 (2 marks) | $-0.28$ |
### Part (b)(i)
| $\frac{dx}{d\theta}=6\cos2\theta\quad,\quad\frac{dy}{d\theta}=-8\sin2\theta$ | M1 | Attempt both derivatives, ie $p\cos2\theta$, $q\sin2\theta$ |
| | A1 | Both correct |
| $\frac{dy}{dx}=-\frac{4\sin2\theta}{3\cos2\theta}$ ISW | A1 (3 marks) | CSO, OE |
### Part (b)(ii)
| $P\left(\frac{72}{25},-\frac{28}{25}\right)$ | B1F | $(2.88,-1.12)$ |
| Gradient $=-\frac{4}{3}\times-\frac{24}{7}$ | M1 | Their $\frac{q\sin2\theta}{p\cos2\theta}$ **or** $\frac{p\cos2\theta}{q\sin2\theta}$; must be working with rational numbers |
| Tangent $\quad y+\frac{28}{25}=\frac{32}{7}\left(x-\frac{72}{25}\right)$ ISW | A1 (3 marks) | Any correct form; $7y=32x-100$; Fractions in simplest form; Equation required |
**Total: 10 marks**6
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Express $\sin 2 \theta$ and $\cos 2 \theta$ in terms of $\sin \theta$ and $\cos \theta$.
\item Given that $0 < \theta < \frac { \pi } { 2 }$ and $\cos \theta = \frac { 3 } { 5 }$, show that $\sin 2 \theta = \frac { 24 } { 25 }$ and find the value of $\cos 2 \theta$.
\end{enumerate}\item A curve has parametric equations
$$x = 3 \sin 2 \theta , \quad y = 4 \cos 2 \theta$$
\begin{enumerate}[label=(\roman*)]
\item Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $\theta$.
\item At the point $P$ on the curve, $\cos \theta = \frac { 3 } { 5 }$ and $0 < \theta < \frac { \pi } { 2 }$. Find an equation of the tangent to the curve at the point $P$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C4 2010 Q6 [10]}}