| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2017 |
| Session | June |
| Marks | 7 |
| Topic | Integration with Partial Fractions |
| Type | Rational function curve sketching |
| Difficulty | Standard +0.3 This is a standard curve sketching question requiring asymptote identification (vertical at x=-1, horizontal at y=1 via polynomial division), differentiation using quotient rule, and finding turning points. While it involves multiple steps, each technique is routine for Further Maths students and requires no novel insight—slightly above average difficulty only due to the algebraic manipulation involved. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives |
**Question 6(i)**
HA $y=1$   VA $x=-1$ **B2** (B1 for each)
**Question 6(ii)**
$y = \dfrac{x^2+1}{(x+1)^2}$ or $y = 1 - \dfrac{2x}{(x+1)^2}$ **M1A1** Attempted; correct unsimplified
$\Rightarrow \dfrac{\text{d}y}{\text{d}x} = \dfrac{(x+1)^2(2x)-(x^2+1).2(x+1)}{(x+1)^4}$ or $-\dfrac{(x+1)^2.2-2x.2(x+1)}{(x+1)^4} = \dfrac{2(x-1)}{(x+1)^3}$
$\Rightarrow \dfrac{\text{d}y}{\text{d}x} = 0$ when $x=1,\ y=\dfrac{1}{2}$ **A2** (A1 for each)
**Question 6(iii)**
[Graph sketch] **3 marks**
- **G1** for graph in 2 bits, separated by a (FT) vertical asymptote and all positive
- **G1** for $y$-intercept at $(0,1)$ and MIN. in (approx. FT) correct place
- **G1** for correct asymptotic behaviour
**Total: 7 marks**6 The curve $S$ has equation $y = \frac { x ^ { 2 } + 1 } { ( x + 1 ) ^ { 2 } }$.\\
(i) Write down the equations of the asymptotes of $S$.\\
(ii) Determine $\frac { \mathrm { d } y } { \mathrm {~d} x }$ and hence find the coordinates of any turning points of $S$.\\
(iii) Sketch $S$.
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2017 Q6 [7]}}