| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2012 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Implicit equations and differentiation |
| Type | Find dy/dx at a point |
| Difficulty | Moderate -0.3 Part (a) is a straightforward implicit differentiation exercise requiring product rule and collecting terms, then substituting a point. Parts (b)(i) and (b)(ii) are standard parametric differentiation applications testing dx/dt = 0 and dx/dt = dy/dt respectively. All techniques are routine C4 content with no novel problem-solving required, making this slightly easier than average. |
| Spec | 1.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{d}{dx}(xy) = x\frac{dy}{dx}+y\) | B1 | |
| \(\frac{d}{dx}(y^2) = 2y\frac{dy}{dx}\) | B1 | |
| Substitute \((-1,-1)\) for \((x,y)\) and attempt to solve for \(\frac{dy}{dx}\) | M1 | or solve then substitute |
| Obtain \(\frac{dy}{dx} = -1\) WWW | A1 | |
| Total: [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Tangent parallel to \(y\)-axis \(\to \frac{dx}{dt}=0\) or \(\frac{dy}{dx}\to\infty\) or \(\frac{dy}{dx}=\infty\) | M1 | Accept clear intention |
| Obtain \(t=0\) | A1 | |
| \((-1, 0)\) with no other possibilities | A1 | Accept \(x=-1\), \(y=0\) |
| Total: [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State or imply or use \(\frac{dy}{dt} = \frac{dx}{dt}\) | M1 | |
| Produce \(3t^2+1=4t\) oe | A1 | |
| \(t=\frac{1}{3}\) or \(1\) | A1 | |
| Total: [3] |
# Question 8:
## Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{d}{dx}(xy) = x\frac{dy}{dx}+y$ | B1 | |
| $\frac{d}{dx}(y^2) = 2y\frac{dy}{dx}$ | B1 | |
| Substitute $(-1,-1)$ for $(x,y)$ and attempt to solve for $\frac{dy}{dx}$ | M1 | or solve then substitute |
| Obtain $\frac{dy}{dx} = -1$ WWW | A1 | |
| **Total: [4]** | | |
## Part (b)(i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Tangent parallel to $y$-axis $\to \frac{dx}{dt}=0$ or $\frac{dy}{dx}\to\infty$ or $\frac{dy}{dx}=\infty$ | M1 | Accept clear intention |
| Obtain $t=0$ | A1 | |
| $(-1, 0)$ with no other possibilities | A1 | Accept $x=-1$, $y=0$ |
| **Total: [3]** | | |
## Part (b)(ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| State or imply or use $\frac{dy}{dt} = \frac{dx}{dt}$ | M1 | |
| Produce $3t^2+1=4t$ oe | A1 | |
| $t=\frac{1}{3}$ or $1$ | A1 | |
| **Total: [3]** | | |8
\begin{enumerate}[label=(\alph*)]
\item Find the gradient of the curve $x ^ { 2 } + x y + y ^ { 2 } = 3$ at the point $( - 1 , - 1 )$.
\item A curve $C$ has parametric equations
$$x = 2 t ^ { 2 } - 1 , y = t ^ { 3 } + t$$
\begin{enumerate}[label=(\roman*)]
\item Find the coordinates of the point on $C$ at which the tangent is parallel to the $y$-axis.
\item Find the values of $t$ for which $x$ and $y$ have the same rate of change with respect to $t$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR C4 2012 Q8 [10]}}