O-LevelAdditional MathematicsDifferentiation and integration4037/11 · May/June 2010 · Q1111 Marks
Answer only one of the following two alternatives. EITHER A curve has equation y = ln x / x², where x > 0.
- (i)Find the exact coordinates of the stationary point of the curve.[6]
- (ii)Show that d²y/dx² can be written in the form (a ln x + b) / x⁴, where a and b are integers.[3]
- (iii)Hence, or otherwise, determine the nature of the stationary point of the curve.[2]
Mark scheme
Marking points from the official mark scheme (May/June 2010). The mark or mark type is shown in brackets.
- [6] (i) B3,2,1,0 for dy/dx = (x^2(1/x) - 2x ln x) / x^4 = (1 - 2lnx) / x^3 (-1 each error); M1 for dy/dx = 0; A1 for ln x = 1/2 => x = e^(1/2); A1 for y = 1/(2e)
- [3] (ii) M1 for attempt at 2nd derivative; A1 for a = 6; A1 for b = -5 (or equivalent form)
- [2] (iii) M1 for a correct method (substitute x into d²y/dx² and determine sign); A1 for max (must be from correct working only)
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