O-LevelAdditional MathematicsDifferentiation and integration4037/12 · Oct/Nov 2010 · Q1211 Marks

EITHER: A curve is such that dy/dx = 4x² – 9. The curve passes through the point (3, 1).

(i)Find the equation of the curve.[4]
(ii)The curve has stationary points at A and B. Find the coordinates of A and of B.[3]
(iii)Find the equation of the perpendicular bisector of the line AB.[4]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2010). The mark or mark type is shown in brackets.

[M1, A1] (i) y = 4x³/3 - 9x (+c) (M1 for attempt to integrate)
[M1, A1] (i) when x = 3, y = 1, so c = -8 (M1 for attempt to find c)
[M1] (ii) 4x² - 9 = 0, leads to x = ±1.5 (M1 for attempt to solve dy/dx = 0)
[A1, A1] (ii) Points (1.5, -17), (-1.5, 1) (A1 for each pair)
[M1] (iii) Midpoint AB: (0, -8) (M1 for attempt to find midpoint)
[M1] (iii) Gradient of AB = -6, perp grad = 1/6 (M1 for attempt to find grad of perp)
[M1, A1] (iii) Equation: x - 6y = 48 (M1 must be working with perp)

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Topic: Differentiation and integration, O-Level Additional Mathematics (syllabus 4037). Structured question, 11 marks.
Source: O-Level Additional Mathematics, Oct/Nov 2010 Paper 1 Variant 2, question 12.

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