A-LevelFurther MathematicsIntegration9231/12 · May/June 2018 · Q69 Marks

The curve C has equation y = (x² + b)/(x + b), where b is a positive constant.

(i)Find the equations of the asymptotes of C.[3]
(ii)Show that C does not intersect the x-axis.[1]
(iii)Justifying your answer, find the number of stationary points on C.[2]
(iv)Sketch C. Your sketch should indicate the coordinates of any points of intersection with the y-axis. You do not need to find the coordinates of any stationary points.[3]

Mark scheme

Marking points from the official mark scheme (May/June 2018). The mark or mark type is shown in brackets.

B1 (i) Vertical asymptote is x = -b.
M1 (i) By inspection or long division. (x²+b)/(x+b) = x-b + (b²+b)/(x+b)
A1 (i) Thus the oblique asymptote is y = x-b
B1 (ii) Must refer to b > 0 OE. If y=0 then x²+b=0 which has no real root.
M1 (iii) Find dy/dx and set = 0. dy/dx = (x²+2bx-b)/(x+b)²
A1 (iii) Use discriminant or (x+b)² to show two stationary points. (D = 4b(b+1) > 0 since b>0)
B1 (iv) Intersection (0,1) given and asymptotes drawn.
B1 (iv) Each branch correct. (one branch, e.g., for x > -b)
B1 (iv) Each branch correct. (second branch, e.g., for x < -b)

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Topic: Integration, A-Level Further Mathematics (syllabus 9231). Structured question, 9 marks.
Source: A-Level Further Mathematics, May/June 2018 Paper 1 Variant 2, question 6.

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