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A-LevelMathematicsDifferentiation9709/22 · May/June 2021 · Q68 Marks

The diagram shows the curve with equation y = (ln x)² – 2 ln x. The curve crosses the x-axis at the points A and B, and has a minimum point M.

  1. (a)Find the exact value of the gradient of the curve at each of the points A and B.[6]
  2. (b)Find the exact x-coordinate of M.[2]

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Mark scheme

Marking points from the official mark scheme (May/June 2021). The mark or mark type is shown in brackets.

  1. [B1] Obtain x = 1.
  2. [B1] Obtain x = e²
  3. [*M1] Differentiate to obtain at least one correct term
  4. [A1] Obtain correct first derivative (2ln x)/x - 2/x
  5. [DM1] Substitute at least one of their x-values corresponding to y = 0 to find gradient
  6. [A1] Obtain gradient -2 [at A] and gradient 2e⁻² [at B]
  7. [M1] Equate first derivative to zero
  8. [A1] Obtain x = e

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About this question

Topic
Differentiation, A-Level Mathematics (syllabus 9709). Structured question, 8 marks.
Source
A-Level Mathematics, May/June 2021 Paper 2 Variant 2, question 6.

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