| Exam Board | OCR MEI |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2006 |
| Session | January |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration with Partial Fractions |
| Type | Rational function curve sketching |
| Difficulty | Challenging +1.2 This is a structured Further Maths curve sketching question requiring multiple techniques (asymptotes, calculus for stationary points, algebraic analysis of asymptote crossing), but each part is guided and uses standard methods. The polynomial division and differentiation are routine for FP2 level, though the multi-part nature and case analysis elevate it slightly above average difficulty. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.02n Sketch curves: simple equations including polynomials1.02p Interpret algebraic solutions: graphically1.02q Use intersection points: of graphs to solve equations1.07n Stationary points: find maxima, minima using derivatives |
5 A curve has equation $y = \frac { x ^ { 3 } - k ^ { 3 } } { x ^ { 2 } - 4 }$, where $k$ is a positive constant and $k \neq 2$.\\
(i) Find the equations of the three asymptotes.\\
(ii) Use your graphical calculator to obtain rough sketches of the curve in the two separate cases $k < 2$ and $k > 2$.\\
(iii) In the case $k < 2$, your sketch may not show clearly the shape of the curve near $x = 0$. Use calculus to show that the curve has a minimum point when $x = 0$.\\
(iv) In the case $k > 2$, your sketch may not show clearly how the curve approaches its asymptote as $x \rightarrow + \infty$. Show algebraically that the curve crosses this asymptote.\\
(v) Use the results of parts (iii) and (iv) to produce more accurate sketches of the curve in the two separate cases $k < 2$ and $k > 2$. These sketches should indicate where the curve crosses the axes, and should show clearly how the curve approaches its asymptotes. The presence of stationary points should be clearly shown, but there is no need to find their coordinates.
\hfill \mbox{\textit{OCR MEI FP2 2006 Q5 [18]}}