| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2012 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Implicit equations and differentiation |
| Type | Iterative method for special point |
| Difficulty | Standard +0.3 This is a straightforward multi-part question combining implicit differentiation with an iterative method. Part (i) is routine differentiation, part (ii) requires algebraic manipulation of dy/dx = 1/4, and part (iii) is a standard iterative calculation with a given starting value. All techniques are standard C3 material with no novel insight required, making it slightly easier than average. |
| Spec | 1.07s Parametric and implicit differentiation1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Obtain rational expression of form \(\frac{f(y)}{y^3+2y}\) | M1 | Where \(f(y)\) is not constant; ignore how expression is labelled |
| Obtain \(\frac{3y^2+2}{y^3+2y}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Recognise that \(\frac{dy}{dx} = 1 \div \frac{dx}{dy}\) for rational expression of form \(\frac{f(y)}{y^3+2y}\) | M1 | May be implied |
| Obtain \(\frac{y^3+2y}{3y^2+2} = 4\) or \(\frac{3y^2+2}{y^3+2y} = \frac{1}{4}\) | A1ft | Following their rational expression from (i) |
| Confirm \(y = \frac{12y^2+8}{y^2+2}\) | A1 | AG; following correct work and with at least one step between \(\frac{y^3+2y}{3y^2+2}=4\) or equiv and answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Obtain correct first iterate \(11.89\) | B1 | Or greater accuracy; having started with 12 |
| Attempt iteration process to produce at least 3 iterates in all | M1 | Implied by plausible sequence of values; having started anywhere; if formula clearly not based on equation from part (ii), award M0 |
| Obtain at least 2 more correct iterates | A1 | Showing at least 3 decimal places |
| Obtain \(11.888\) for \(y\) | A1 | Answer needed to exactly 3 decimal places |
| Obtain \(7.441\) for \(x\) | A1 | Answer needed to exactly 3 decimal places; award final A0 if not clear which is \(x\) and which is \(y\); \([12 \to 11.89041 \to 11.88841 \to 11.88837]\) |
# Question 6(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Obtain rational expression of form $\frac{f(y)}{y^3+2y}$ | M1 | Where $f(y)$ is not constant; ignore how expression is labelled |
| Obtain $\frac{3y^2+2}{y^3+2y}$ | A1 | |
---
# Question 6(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Recognise that $\frac{dy}{dx} = 1 \div \frac{dx}{dy}$ for rational expression of form $\frac{f(y)}{y^3+2y}$ | M1 | May be implied |
| Obtain $\frac{y^3+2y}{3y^2+2} = 4$ or $\frac{3y^2+2}{y^3+2y} = \frac{1}{4}$ | A1ft | Following their rational expression from (i) |
| Confirm $y = \frac{12y^2+8}{y^2+2}$ | A1 | AG; following correct work and with at least one step between $\frac{y^3+2y}{3y^2+2}=4$ or equiv and answer |
---
# Question 6(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Obtain correct first iterate $11.89$ | B1 | Or greater accuracy; having started with 12 |
| Attempt iteration process to produce at least 3 iterates in all | M1 | Implied by plausible sequence of values; having started anywhere; if formula clearly not based on equation from part (ii), award M0 |
| Obtain at least 2 more correct iterates | A1 | Showing at least 3 decimal places |
| Obtain $11.888$ for $y$ | A1 | Answer needed to exactly 3 decimal places |
| Obtain $7.441$ for $x$ | A1 | Answer needed to exactly 3 decimal places; award final A0 if not clear which is $x$ and which is $y$; $[12 \to 11.89041 \to 11.88841 \to 11.88837]$ |6\\
\includegraphics[max width=\textwidth, alt={}, center]{89e54367-bb83-483a-add5-0527b71a5cac-4_476_709_251_683}
The diagram shows the curve with equation $x = \ln \left( y ^ { 3 } + 2 y \right)$. At the point $P$ on the curve, the gradient is 4 and it is given that $P$ is close to the point with coordinates (7.5,12).\\
(i) Find $\frac { \mathrm { d } x } { \mathrm {~d} y }$ in terms of $y$.\\
(ii) Show that the $y$-coordinate of $P$ satisfies the equation
$$y = \frac { 12 y ^ { 2 } + 8 } { y ^ { 2 } + 2 }$$
(iii) By first using an iterative process based on the equation in part (ii), find the coordinates of $P$, giving each coordinate correct to 3 decimal places.
\hfill \mbox{\textit{OCR C3 2012 Q6 [10]}}