| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2010 |
| Session | June |
| Marks | 8 |
| Topic | Curve Sketching |
| Type | Sketch rational with quadratic numerator |
| Difficulty | Standard +0.8 This question requires algebraic division to identify the oblique asymptote, differentiation using the quotient rule, completing the square or sign analysis to prove the derivative inequality, and synthesizing multiple features (asymptotes, intercepts, monotonicity) into a coherent sketch. While the individual techniques are A-level standard, the proof requirement and the need to coordinate several analytical steps elevate this above routine curve sketching. |
| 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 sums |
**(i)**
$\frac{dy}{dx} = \frac{(x+3)(2x+5)-(x^2+5x-6)\cdot 1}{(x+3)^2}$ **M1** Use of quotient rule
$= \frac{x^2+6x+21}{(x+3)^2} = 1 + \frac{12}{(x+3)^2} > 1$ **M1** Division **A1** Fully explained **cso** [3]
**(ii)** Vertical asymptote $x = -3$ **B1**
Long-division: $y = \frac{x^2+5x-6}{x+3} = x+2-\frac{12}{x+3}$ **M1** This may happen first, then differentiation.
leading to oblique asymptote $y = x+2$ **A1**
Factorising numerator: $y = \frac{x^2+5x-6}{x+3} = \frac{(x+6)(x-1)}{x+3}$ **M1**
leading to $y=0$, $x=-6$, $1$ **A1**
When $x=0$, $y=-2$ **B1**
Graph of curve: **G1** No TPs
**G1** All correct, **ft** provided sensible oblique asymptote **[8]**5 A curve has equation $y = \frac { x ^ { 2 } + 5 x - 6 } { x + 3 }$ for $x \neq - 3$.\\
(i) Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } > 1$ at all points on the curve.\\
(ii) Sketch the curve, justifying all significant features.
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2010 Q5 [8]}}