O-LevelAdditional MathematicsDifferentiation and integration4037/12 · Oct/Nov 2016 · Q57 Marks

Find the equation of the normal to the curve y = (1/2)ln(3x + 2) at the point P where x = -1/3.

(i)Find the equation of the normal to the curve y = (1/2)ln(3x + 2) at the point P where x = -1/3.[4]
(ii)The normal to the curve at the point P intersects the y-axis at the point Q. The curve y = (1/2)ln(3x + 2) intersects the y-axis at the point R . Find the area of the triangle PQR.[3]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2016). The mark or mark type is shown in brackets.

[1] (i) dy/dx = 3/(2(3x+2)) B1 for correct derivative of log function
[1] (i) When x = -1/3, y = 0, dy/dx = 3/2 B1 for y = 0
[2] (i) Equation of normal: y = -2/3 (x + 1/3) M1 A1 M1 for attempt at a gradient of a perpendicular from differentiation and the equation of the normal
[1] (ii) Q(0, -2/9) or (0, -0.22) or better B1 ft Follow through on their c from part (i)
[1] (ii) R(0, 1/2 ln 2) or (0, 0.35) or better B1
[1] (ii) Area of PQR = 1/2 × (1/2 ln 2 + 2/9) × 1/3 = 0.0948 B1 Allow 0.095

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Topic: Differentiation and integration, O-Level Additional Mathematics (syllabus 4037). Structured question, 7 marks.
Source: O-Level Additional Mathematics, Oct/Nov 2016 Paper 1 Variant 2, question 5.

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