| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2013 |
| Session | June |
| Marks | 2 |
| Topic | Integration with Partial Fractions |
| Type | Rational function curve sketching |
| Difficulty | Standard +0.3 This is a straightforward rational function analysis question requiring differentiation using the quotient rule, sign analysis of the derivative, identification of asymptotes (x = ±2, y = 0), and intercepts. While it involves multiple steps, each is a standard technique with no novel insight required, making it slightly easier than average for A-level standard. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.07i Differentiate x^n: for rational n and sums |
**(i)** $\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{(x^2-4)\cdot1-(x+1)\cdot2x}{(x^2-4)^2}$ Use of quotient rule; correct unsimplified **M1 A1**
$= -\dfrac{(x^2+2x+4)}{(x^2-4)^2} = -\dfrac{(x+1)^2+3}{(x^2-4)^2}$ or clear explanation this is $< 0$ **E1**
**ALT:** $y = \dfrac{^3\!/_4}{x-2} + \dfrac{^1\!/_4}{x+2} \Rightarrow \dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{-\,^3\!/_4}{(x-2)^2} + \dfrac{-\,^1\!/_4}{(x+2)^2} < 0$
**[3]**
**(ii)** Asymptotes $y = 0$ stated or clear from graph **B1**; $x = \pm 2$ stated or clear from graph **B1**
Crossing-points $(0, -\frac{1}{4})$ and $(-1, 0)$ noted or clearly shown on graph **B1 B1**
3 regions **M1**; All correct (incl. no TPs) **A1**
[Graph showing 3 regions with correct asymptotes and intercepts]
**[6]**3 The curve $C$ has equation $y = \frac { x + 1 } { x ^ { 2 } - 4 }$.\\
(i) Show that the gradient of $C$ is always negative.\\
(ii) Sketch $C$, showing all significant features.
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2013 Q3 [2]}}