O-LevelAdditional MathematicsDifferentiation and integration4037/21 · May/June 2019 · Q1213 Marks

Given that ∫₀ᵃ e^(2x) dx = 50, find the exact value of a. You must show all your working.

(a)Given that ∫₀ᵃ e^(2x) dx = 50, find the exact value of a. You must show all your working.[4]
(b)(i)Find the equation of the curve.[4]
(b)(ii)Find ∫ y dx and hence evaluate ∫_π/2^π y dx.[5]

Mark scheme

Marking points from the official mark scheme (May/June 2019). The mark or mark type is shown in brackets.

[B1] e^(2x)/2 seen
[M1] e^(2a)/2 - e^0/2 = 50 (Uses limits correctly for their integral and sets = 50)
[M1] 2a = ln101 oe (Using their integral)
[A1] a = ln101 / 2 or ln√101 final answer (Allow any exact equivalent)
[B2] [y =] 3x - (2/5)sin 5x [+c] (B1 for -k sin 5x where k > 0)
[M1] (8π/2) = 3(π/5) - (2/5)sin(5π/5) + c
[A1] y = 3x - (2/5)sin 5x + π
[B3] ∫ y dx = 3x²/2 - (2/25)cos 5x + πx [+c] (B2 for -(2/25)cos 5x oe nfww and B1FT for 3x²/2 + ... + πx [+c])
[M1] their F(π) - their F(π/2)
[A1] 16[.0] or 15.95 to 15.96 or 13π²/8 - 2/25

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Topic: Differentiation and integration, O-Level Additional Mathematics (syllabus 4037). Structured question, 13 marks.
Source: O-Level Additional Mathematics, May/June 2019 Paper 2 Variant 1, question 12.

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