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A-LevelMathematicsLogarithmic and exponential functions9709/33 · May/June 2020 · Q36 Marks

Show that the equation ln(1 + e⁻ˣ) + 2x = 0 can be expressed as a quadratic equation in eˣ.

  1. (a)Show that the equation ln(1 + e⁻ˣ) + 2x = 0 can be expressed as a quadratic equation in eˣ.[2]
  2. (b)Hence solve the equation ln(1 + e⁻ˣ) + 2x = 0, giving your answer correct to 3 decimal places.[4]

Mark scheme

Marking points from the official mark scheme (May/June 2020). The mark or mark type is shown in brackets.

  1. [B1] Remove logarithms correctly and state 1 + e⁻ˣ = e⁻²ˣ, or equivalent
  2. [B1] Show equation is u² + u − 1 = 0, where u = eˣ, or equivalent
  3. [M1] Solve a 3-term quadratic for u
  4. [A1] Obtain root ½(-1+√5), or decimal in [0.61, 0.62]
  5. [M1] Use correct method for finding x from a positive root
  6. [A1] Obtain answer x = -0.481 only

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About this question

Topic
Logarithmic and exponential functions, A-Level Mathematics (syllabus 9709). Structured question, 6 marks.
Source
A-Level Mathematics, May/June 2020 Paper 3 Variant 3, question 3.

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