| Exam Board | OCR MEI |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2010 |
| Session | June |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Conic sections |
| Type | Conic translation and transformation |
| Difficulty | Challenging +1.2 This is an extended investigation question requiring sketching and analysis of generalized conic sections across multiple cases. While it covers many parts, each individual step involves standard techniques (recognizing circles/lines, sketching curves, identifying asymptotes, considering limiting behavior). The conceptual demand is moderate—students must understand how exponents affect curve shape—but no deep proof or novel insight is required. The systematic structure guides students through the investigation, making it more accessible than its length suggests. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.02w Graph transformations: simple transformations of f(x) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| (A) Circle | B1 | |
| Sketch of circle, centre \((0,0)\) | G1 | |
| Sketch of "squarer" circle on same axes | G1 | |
| (C) Square | B1 | |
| (D) \(-1\leq x\leq 1\), \(-1\leq y\leq 1\) | B1, B1 | Give B1B0 for not all non-strict or unclear |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| (A) Odd roots exist for all real numbers | B1 | Any equivalent explanation |
| (B) Line | B1 | Sketch insufficient |
| Sketch of line | G1 | |
| Asymptote: \(x+y=0\) | B1 | |
| Sketch with asymptote | G1 | Line \(x+y=0\) outside unit square |
| Lines \(y=1\) and \(x=1\) on unit square | G1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Sketch for \(0\leq x, y\leq 1\) | G1 | G0 if curve beyond \((1,0)\) or \((0,1)\) |
| \(0\leq x, y\leq 1\) | B1 | Accept strict, or indication on graph |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| (A) Sketch (ft from (iii) on shape) | G2ft | Give G1 for partial attempt; ft from (iii) on shape |
| (B) Limit is a "plus sign" where \(x\to 0\) for \(-1\leq y\leq 1\) and vice versa | B1, B1 |
# Question 5:
## Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| (A) Circle | B1 | |
| Sketch of circle, centre $(0,0)$ | G1 | |
| Sketch of "squarer" circle on same axes | G1 | |
| (C) Square | B1 | |
| (D) $-1\leq x\leq 1$, $-1\leq y\leq 1$ | B1, B1 | Give B1B0 for not all non-strict or unclear |
## Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| (A) Odd roots exist for all real numbers | B1 | Any equivalent explanation |
| (B) Line | B1 | Sketch insufficient |
| Sketch of line | G1 | |
| Asymptote: $x+y=0$ | B1 | |
| Sketch with asymptote | G1 | Line $x+y=0$ outside unit square |
| Lines $y=1$ and $x=1$ on unit square | G1 | |
## Part (iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Sketch for $0\leq x, y\leq 1$ | G1 | G0 if curve beyond $(1,0)$ or $(0,1)$ |
| $0\leq x, y\leq 1$ | B1 | Accept strict, or indication on graph |
## Part (iv):
| Answer/Working | Mark | Guidance |
|---|---|---|
| (A) Sketch (ft from (iii) on shape) | G2ft | Give G1 for partial attempt; ft from (iii) on shape |
| (B) Limit is a "plus sign" where $x\to 0$ for $-1\leq y\leq 1$ and vice versa | B1, B1 | |5 In parts (i), (ii), (iii) of this question you are required to investigate curves with the equation
$$x ^ { k } + y ^ { k } = 1$$
for various positive values of $k$.
\begin{enumerate}[label=(\roman*)]
\item Firstly consider cases in which $k$ is a positive even integer.\\
(A) State the shape of the curve when $k = 2$.\\
(B) Sketch, on the same axes, the curves for $k = 2$ and $k = 4$.\\
(C) Describe the shape that the curve tends to as $k$ becomes very large.\\
(D) State the range of possible values of $x$ and $y$.
\item Now consider cases in which $k$ is a positive odd integer.\\
(A) Explain why $x$ and $y$ may take any value.\\
(B) State the shape of the curve when $k = 1$.\\
(C) Sketch the curve for $k = 3$. State the equation of the asymptote of this curve.\\
(D) Sketch the shape that the curve tends to as $k$ becomes very large.
\item Now let $k = \frac { 1 } { 2 }$.
Sketch the curve, indicating the range of possible values of $x$ and $y$.
\item Now consider the modified equation $| x | ^ { k } + | y | ^ { k } = 1$.\\
(A) Sketch the curve for $k = \frac { 1 } { 2 }$.\\
(B) Investigate the shape of the curve for $k = \frac { 1 } { n }$ as the positive integer $n$ becomes very large.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI FP2 2010 Q5 [18]}}