O-LevelAdditional MathematicsDifferentiation and integration4037/12 · May/June 2021 · Q67 Marks

DO NOT USE A CALCULATOR IN THIS QUESTION. A curve has equation y = (3+√5)x²-8√5x+60.

(a)Find the x-coordinate of the stationary point on the curve, giving your answer in the form a+b√5, where a and b are integers.[4]
(b)Hence find the y-coordinate of this stationary point, giving your answer in the form c√5, where c is an integer.[3]

Mark scheme

Marking points from the official mark scheme (May/June 2021). The mark or mark type is shown in brackets.

[1] M1 Either For differentiation must have one correct term. Or for use of 'b² – 4ac = 0', so x = -b/2a at the stationary point.
[1] M1 Dep for equating their dy/dx to zero with attempt to solve and rationalisation using a two term factor, or rationalisation of x = -b/2a, using a two term factor with sufficient detail to imply no use of a calculator. Allow multiple equivalents. Allow one numerical slip or sign error.
[1] A1 for -5
[1] A1 for 3√5
[1] M1 For substitution of their x and simplification with sufficient detail to imply no use of a calculator. Allow one numerical slip or sign error in the expansion of (3+√5)(3√5-5)² or one sign error in the other terms.
[1] A1 for all non surd terms = 0
[1] A1 for 20√5

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