| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2014 |
| Session | June |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Proving Excluded Range of Rational Function |
| Difficulty | Challenging +1.2 This is a multi-part question requiring understanding of asymptotes and polynomial division, followed by a range restriction proof. Parts (i) and (ii) are straightforward identification from asymptote conditions (vertical asymptote gives d, oblique asymptote gives a and b, y-intercept gives c). Part (iii) requires rearranging to find when the equation has real solutions, leading to a discriminant condition—a standard technique but requiring careful algebraic manipulation. This is moderately above average for A-level, typical of Further Maths content but not requiring exceptional insight. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.02y Partial fractions: decompose rational functions |
The curve $C$ has equation
$$y = \frac { a x ^ { 2 } + b x + c } { x + d }$$
where $a , b , c$ and $d$ are constants. The curve cuts the $y$-axis at $( 0 , - 2 )$ and has asymptotes $x = 2$ and $y = x + 1$.\\
(i) Write down the value of $d$.\\
(ii) Determine the values of $a , b$ and $c$.\\
(iii) Show that, at all points on $C$, either $y \leqslant 3 - 2 \sqrt { 6 }$ or $y \geqslant 3 + 2 \sqrt { 6 }$.
\hfill \mbox{\textit{CAIE FP1 2014 Q12 OR}}