O-LevelAdditional MathematicsDifferentiation and integration4037/12 · May/June 2018 · Q1110 Marks

The diagram shows the graph of the curve y= (e^(4x)+3)/8. The curve meets the y-axis at the point A. The normal to the curve at A meets the x-axis at the point B. Find the area of the shaded region enclosed by the curve, the line AB and the line through B parallel to the y-axis. Give your answer in the form e/a, where a is a constant. You must show all your working.

The diagram shows the graph of the curve y= (e^(4x)+3)/8. The curve meets the y-axis at the point A. The normal to the curve at A meets the x-axis at the point B. Find the area of the shaded region enclosed by the curve, the line AB and the line through B parallel to the y-axis. Give your answer in the form e/a, where a is a constant. You must show all your working.[10]

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Mark scheme

Marking points from the official mark scheme (May/June 2018). The mark or mark type is shown in brackets.

B1 When x = 0, y = 1/2 For y = 1/2
B1 dy/dx = 1/2 e^(4x)
B1 FT on their dy/dx, must be numeric dy/dx = 1/2, Gradient of normal = -2
M1 For an attempt at a normal equation passing through their (0, 1/2) and a substitution of y = 0 either: Normal y - 1/2 = -2x or: Gradient of normal = OA/OB
A1 When y=0, x=1/4
M1 For attempt to integrate to obtain ∫ (e^(4x)+3)/8 dx ke^(4x) + (3/8)x, k₁ ≠ 1/8, k₁ ≠ 1/2
A1 For correct integration [1/32 e^(4x) + 3x/8] from 0 to 1/4
M1dep Use of limits
B1 FT on their x = 1/4 For area of triangle = 1/16
A1 final answer in correct form e/32

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