| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2010 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Stationary Points of Rational Functions |
| Difficulty | Standard +0.8 This FP1 question requires multiple techniques: finding asymptotes (polynomial division for oblique asymptote), differentiation using quotient rule, proving an inequality involving the derivative for all points, and sketching. The proof that dy/dx > 1 everywhere requires completing the square or analyzing a quadratic expression, which demands insight beyond routine calculation. Multi-part with conceptual depth typical of Further Maths. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.02n Sketch curves: simple equations including polynomials1.07i Differentiate x^n: for rational n and sums1.07j Differentiate exponentials: e^(kx) and a^(kx)1.07m Tangents and normals: gradient and equations |
| Answer | Marks | Guidance |
|---|---|---|
| (i) One asymptote is \(x = -1\) | B1 | |
| \(y = x - 4 - \frac{3}{(x+1)}\) | M1 | |
| Require \(y = x +\) non-zero constant. Other asymptote is \(y = x - 4\) | A1 | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| \(\mathbf{OR}\) \(x + k \approx (x^2 - 3x - 7)/(x+1)\) for large \(x \Rightarrow x^2 + (k+1)x + k \approx x^2 - 3x - 7\) for large \(x\) ⟹ \(k + 1 = -3 \Rightarrow k = -4\) ⟹ other asymptote is \(y = x - 4\) | M1 A1 | |
| \(\mathbf{OR}\) \((x^2 - 3x - 7)/(x+1) = mx + c \Rightarrow (m-1)x^2 + (m + c + 3)x + c + 7 = 0\) | M1 | |
| Put \(m - 1 = 0, m + c + 3 = 0\) to obtain other asymptote is \(y = x - 4\) | A1 | |
| (ii) Obtains any correct result for \(\frac{dy}{dx}\) | M1 | |
| \(\Rightarrow \ldots \Rightarrow \frac{dy}{dx} = [(x+1)^2 + 3]/[(x+1)^2] = 1 + \frac{3}{(x+1)^2}\) | A1 | [2] |
| Answer | Marks |
|---|---|
| (iii) Axes and asymptotes correctly placed | B1 |
| Right-hand branch | B1 |
| Left-hand branch | B1 |
| Deduct 1 overall for bad form(s) at infinity | [3] |
**(i)** One asymptote is $x = -1$ | B1 |
$y = x - 4 - \frac{3}{(x+1)}$ | M1 |
Require $y = x +$ non-zero constant. Other asymptote is $y = x - 4$ | A1 | [3]
**Alternatively for last two marks:**
$\mathbf{OR}$ $x + k \approx (x^2 - 3x - 7)/(x+1)$ for large $x \Rightarrow x^2 + (k+1)x + k \approx x^2 - 3x - 7$ for large $x$ ⟹ $k + 1 = -3 \Rightarrow k = -4$ ⟹ other asymptote is $y = x - 4$ | M1 A1 |
$\mathbf{OR}$ $(x^2 - 3x - 7)/(x+1) = mx + c \Rightarrow (m-1)x^2 + (m + c + 3)x + c + 7 = 0$ | M1 |
Put $m - 1 = 0, m + c + 3 = 0$ to obtain other asymptote is $y = x - 4$ | A1 |
**(ii)** Obtains any correct result for $\frac{dy}{dx}$ | M1 |
$\Rightarrow \ldots \Rightarrow \frac{dy}{dx} = [(x+1)^2 + 3]/[(x+1)^2] = 1 + \frac{3}{(x+1)^2}$ | A1 | [2]
No comment required.
**(iii)** Axes and asymptotes correctly placed | B1 |
Right-hand branch | B1 |
Left-hand branch | B1 |
Deduct 1 overall for bad form(s) at infinity | [3] |
---6 The curve $C$ has equation
$$y = \frac { x ^ { 2 } - 3 x - 7 } { x + 1 }$$
(i) Obtain the equations of the asymptotes of $C$.\\
(ii) Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } > 1$ at all points of $C$.\\
(iii) Draw a sketch of $C$.
\hfill \mbox{\textit{CAIE FP1 2010 Q6 [8]}}