O-LevelAdditional MathematicsDifferentiation and integration4037/22 · May/June 2014 · Q129 Marks

A curve has equation y = x³ – 9x² + 24x.

(i)Find the set of values of x for which dy/dx ≥ 0.[4]
(ii)The normal to the curve at the point on the curve where x = 3 cuts the y-axis at the point P. Find the equation of the normal and the coordinates of P.[5]

Mark scheme

Marking points from the official mark scheme (May/June 2014). The mark or mark type is shown in brackets.

[B1] dy/dx = 3x²-18x+24 attempt at differentiation resulting in quadratic expression with two terms correct; allow = or < or<or > or ≥ 0 omitted here.
[M1] Solving their 3x² – 18x + 24 = 0 by factorising or quadratic formula or completing the square
[A1] Critical values 2 and 4
[A1] x ≤ 2, x ≥ 4 A0 if spurious attempt to combine; mark final answer
[M1] Evaluating their dy/dx at x = 3
[M1] Use of m₁m₂ = -1 to get m_normal = -1 / their(-3)
[B1] y = 18 soi
[A1ft] y-their-18 = (their)(-1/3)(x-3) or y = their (-1/3)x+c and c = their17 isw ft their m provided a genuine attempt at m_normal ; no ft if m = their m_tangent
[B1] P(0, 17) cao

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