O-LevelAdditional MathematicsDifferentiation and integration4037/11 · May/June 2011 · Q1010 Marks
(a) A curve is such that dy/dx = ae^(1-x) - 3x², where a is a constant. At the point (1, 4), the gradient of the curve is 2.
- (a)(i)Find the value of a.[1]
- (a)(ii)Find the equation of the curve.[5]
- (b)(i)Find ∫(7x + 8)¹/³ dx.[2]
- (b)(ii)Hence evaluate ∫⁸₀ (7x + 8)¹/³ dx.[2]
Mark scheme
Marking points from the official mark scheme (May/June 2011). The mark or mark type is shown in brackets.
- [1] B1 for 2 = a*e^(1-1) - 3(1)² => 2 = a - 3 => a = 5.
- [1] √B1 for integrating ae^(1-x) to -ae^(1-x) (using their a).
- [1] B1 for integrating -3x² to -x³.
- [1] B1 for adding constant of integration +c.
- [1] M1 for substituting (1,4) into their integrated expression to find c.
- [1] A1 for y = -5e^(1-x) - x³ + 10.
- [1] B1 for (7x+8)^(4/3).
- [1] B1 for (3/28) as coefficient.
- [1] M1 for substituting limits 8 and 0 into their integrated expression.
- [1] A1 for 180/7 or 25.7 (allow 3sf).
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