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

OR: A curve has the equation y = Ae²ˣ + Be⁻ˣ where x ≥ 0. At the point where x = 0, y = 50 and dy/dx = -20.

(i)Show that A = 10 and find the value of B.[5]
(ii)Using the values of A and B found in part (i), find the coordinates of the stationary point on the curve.[4]
(iii)Determine the nature of the stationary point, giving a reason for your answer.[2]

Mark scheme

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

[B1] (i) 50 = A + B
[M1] (i) dy/dx = 2Ae^(2x) - Be^(-x) (M1 for attempt to differentiate)
[A1] (i) -20 = 2A - B (A1 all correct)
[DM1] (i) leads to A = 10 and B = 40 (DM1 for attempt to solve equations.)
[A1] (i) A=10 and B=40
[M1] (ii) dy/dx = 20e^(2x) - 40e^(-x), 20e^(2x) = 40e^(-x) (M1 for equating to zero and attempt at solution)
[M1] (ii) e^(3x) = 2 (M1 for dealing with exponentials)
[M1] (ii) x = (1/3)ln 2 or 0.231 (M1 for attempt to obtain x)
[A1] (ii) y = 47.6 (A1 for both)
[M1] (iii) d²y/dx² = 40e^(2x) + 40e^(-x) (M1 for attempt at second derivative or other valid method)
[A1] (iii) Always +ve, so min (A1 for a correct conclusion)

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About this question

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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