A-LevelFurther MathematicsRational functions and graphs9231/12 · Oct/Nov 2023 · Q716 Marks

The curve C has equation y = f(x), where f(x) = (x²)/(x+1).

(a)Find the equations of the asymptotes of C.[3]
(b)Find the coordinates of any stationary points on C.[2]
(c)Sketch C.[3]
(d)Find the coordinates of any stationary points on the curve with equation y = 1/(f(x)).[2]
(e)Sketch the curve with equation y = 1/(f(x)) and find, in exact form, the set of values for which 1/(f(x)) > f(x).[6]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2023). The mark or mark type is shown in brackets.

[B1] Vertical asymptote `x=-1`.
[M1 A1] Oblique asymptote `y=x-1` (from `(x²)/(x+1) = x-1 + 1/(x+1)`).
[M1] Sets `dy/dx = (x²+2x)/((x+1)²) = 0`.
[A1] Coordinates `(0,0)` and `(-2,-4)`.
[B1] Sketch C: Axes and asymptotes.
[B1] Sketch C: Left branch correct.
[B1] Sketch C: Right branch correct.
[M1] Sets `d/dx ((x+1)/(x²)) = -(x+2)/(x³) = 0`.
[B1] x-coordinate of stationary point `x=-2`.
[B1] y-coordinate of stationary point `y=-1/4`.
[B1] Sketch `y = 1/(f(x))`: Left branch correct.
[B1] Sketch `y = 1/(f(x))`: Right branch correct.
[M2 A1] Finds critical points `x=-1`, `x=0`, `x=(1±√5)/2` (from `f(x) = ± 1`). Solves `(x+1)/(x²) > (x²)/(x+1)` by checking intervals.
[B1] Solution set `x<-1, (1-√5)/2<x<(1+√5)/2, x≠0`. Condone missing `x≠0`.

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