| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2019 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Rational functions with parameters: finding parameter values from conditions |
| Difficulty | Standard +0.3 This is a slightly above-average A-level question requiring polynomial long division to find the oblique asymptote, then matching coefficients to find constants. The vertical asymptote follows immediately, and the sketch is routine once asymptotes are known. It's methodical rather than insightful, with clear signposting across 7 marks total. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.02y Partial fractions: decompose rational functions |
| Answer | Marks | Guidance |
|---|---|---|
| 4(i) | x2 +1=( ax+b )( 2x+1 )+c | M1 |
| Answer | Marks |
|---|---|
| 2 4 | A1 A1 |
| Answer | Marks |
|---|---|
| 4(ii) | 1 |
| Answer | Marks |
|---|---|
| 2 | B1 FT |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
| Answer | Marks | Guidance |
|---|---|---|
| 4(iii) | B1 | Intersection (0,-4) given and asymptotes drawn. |
| B1 | Left branch correct. | |
| B1 FT | Right branch correct. |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 4:
--- 4(i) ---
4(i) | x2 +1=( ax+b )( 2x+1 )+c | M1 | Uses that 2x+1 is the quotient.
1 1
⇒a= , b=−
2 4 | A1 A1
3
--- 4(ii) ---
4(ii) | 1
x=
2 | B1 FT
1
Question | Answer | Marks | Guidance
--- 4(iii) ---
4(iii) | B1 | Intersection (0,-4) given and asymptotes drawn.
B1 | Left branch correct.
B1 FT | Right branch correct.
Deduct at most one mark for poor forms at infinity.
3
Question | Answer | Marks | GuidanceThe line $y = 2x + 1$ is an asymptote of the curve $C$ with equation
$$y = \frac{x^2 + 1}{ax + b}.$$
\begin{enumerate}[label=(\roman*)]
\item Find the values of the constants $a$ and $b$. [3]
\item State the equation of the other asymptote of $C$. [1]
\item Sketch $C$. [Your sketch should indicate the coordinates of any points of intersection with the $y$-axis. You do not need to find the coordinates of any stationary points.] [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE FP1 2019 Q4 [7]}}