A-LevelFurther MathematicsRational functions and graphs9231/12 · Oct/Nov 2025 · Q714 Marks

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

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

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2025). The mark or mark type is shown in brackets.

[B1] Vertical asymptote. (x = -1)
[M1] Oblique asymptote. (y = x)
[A1]
[M1] Sets dy/dx = 0.
[A1 A1] 1 mark for each correct coordinate. SCB1 if just the correct x values listed
[B1FT] Axes and asymptotes labelled.
[B1] Left branch correct with all approaches to the asymptotes correct.
[B1] Right branch correct with a smooth minimum point.
[B1FT] Symmetrical about x = 0 and follow through from part (c) right branch
[B1] Correct shape of a curve and smooth at (0,1) from a correct graph in part (c)
[M1] Finds critical point. (x²-2x-2=0 OR x=1+√3)
[A1] For selecting just the positive value of x.
[A1] (-1-√3 < x < 1+√3)

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Topic: Rational functions and graphs, A-Level Further Mathematics (syllabus 9231). Structured question, 14 marks.
Source: A-Level Further Mathematics, Oct/Nov 2025 Paper 1 Variant 2, question 7.

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