O-LevelAdditional MathematicsDifferentiation and integration4037/22 · Oct/Nov 2019 · Q1011 Marks
Given that y = ln x / x², find dy/dx.
- (i)Given that y = ln x / x², find dy/dx.[3]
- (ii)Find the coordinates of the stationary point on the curve y = ln x / x².[3]
- (iii)Using your answer to part (i), find ∫ ln x / x³ dx.[3]
- (iv)Hence evaluate ∫ from 1 to 2 (ln x / x³) dx.[2]
Mark scheme
Marking points from the official mark scheme (Oct/Nov 2019). The mark or mark type is shown in brackets.
- M1 Quotient rule or product rule
- A2/1/0 (x - 2x ln x) / x⁴ or (x - ln x . 2x) / x⁴ oe isw Minus one each error. Allow unsimplified.
- M1 x - 2xlnx = 0 Set dy/dx = 0 and attempt to solve. Must have two terms and obtain lnx = k only.
- A1 x = 1.65 awrt or √e
- A1 y = 0.184 awrt or 1/(2e)
- M1 Integrate their derivative from (i) which must have two terms. Condone omission of dx.
- A1 -1/(2x²) Find ∫ 1/x³ dx
- A1 ∫ ln x / x³ dx = -1/(2x²) - (ln x)/(2x²) + (C) oe Rearrange and complete
- M1 dep Insert limits and subtract correctly Must be inserting into two terms in x from (iii). Values explicitly seen if expression is incorrect.
- A1 3ln2/16 - 1/8 or 0.101 awrt
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