O-LevelAdditional MathematicsDifferentiation and integration4037/11 · May/June 2025 · Q68 Marks

A curve has equation y = ((x²-1)/(x²+1))⁴.

(a)Show that dy/dx can be written as Ax(x²-1)³ / (x²+1)⁵, where A is a positive integer to be found.[5]
(b)(i)Show that the curve has stationary points where x =-1, x = 0 and x = 1.[1]
(b)(ii)Use the first derivative test to determine which two stationary points have the same nature and state whether they are maximum or minimum points.[2]

Mark scheme

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

[M1] for an attempt at the quotient or product rule
[A1] fully correct
[M1] Dep M1 for dy/dx = k * ( (x²-1)/(x²+1) )³ * x * f(x)
[A1] fully correct
[A1] Correct completion to 16x(x²-1)³ / (x²+1)⁵
[B1] FT their 16x(x²-1)³ = 0 therefore x = 0, x²-1=0 → x = ±1 oe
[M1] Any two of: x -1<x<0, 0, 0<x<1, dy/dx positive, 0, negative. OR x 0<x<1, 1, x>1, dy/dx negative, 0, positive. OR x x<-1, -1, -1<x<0, dy/dx negative, 0, positive.
[A1] x = 1 and x = -1 are minimum points

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Topic: Differentiation and integration, O-Level Additional Mathematics (syllabus 4037). Structured question, 8 marks.
Source: O-Level Additional Mathematics, May/June 2025 Paper 1 Variant 1, question 6.

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