O-LevelAdditional MathematicsDifferentiation and integration4037/21 · Oct/Nov 2011 · Q1211 Marks
Answer only one of the following two alternatives. OR A curve has equation y = xe^x. The curve has a stationary point at P.
- (i)Find, in terms of e, the coordinates of P and determine the nature of this stationary point.[5]
- (ii)The normal to the curve at the point Q (1, e) meets the x-axis at R and the y-axis at S. Find, in terms of e, the area of triangle ORS, where O is the origin.[6]
Mark scheme
Marking points from the official mark scheme (Oct/Nov 2011). The mark or mark type is shown in brackets.
- [1] Use product rule
- [1] (x + 1)e^x
- [1] Solve dy/dx = 0
- [1] (-1, -1/e)
- [1] Shows minimum
- [1] Gradient tangent = 2e
- [1] Use m₁m₂ = -1 in equation of line (y - e = -1/(2e) (x - 1))
- [1] R(1 + 2e², 0)
- [1] S(0, (1 + 2e²)/(2e))
- [2] Area of triangle = (1 + 2e²)² / (4e)
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