◈ Oracle PrepSign Up Free
A-LevelMathematicsLogarithmic and exponential functions9709/31 · May/June 2021 · Q25 Marks

Find the real root of the equation (2eˣ + e⁻ˣ) / (2 + eˣ) = 3, giving your answer correct to 3 decimal places. Your working should show clearly that the equation has only one real root.

  1. Find the real root of the equation (2eˣ + e⁻ˣ) / (2 + eˣ) = 3, giving your answer correct to 3 decimal places. Your working should show clearly that the equation has only one real root.[5]

Mark scheme

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

  1. [B1] Reduce to a 3-term quadratic u² +6u−1=0 OE
  2. [M1] Solve a 3-term quadratic for u
  3. [A1] Obtain root √10-3
  4. [A1] Obtain answer x = -1.818 only
  5. [B1] Reject -√10-3 correctly

Write your answer and get it marked

On Oracle Prep you can write or photograph your own answer and have it marked by AI against this mark scheme, with feedback on what you missed.

Sign up free →

About this question

Topic
Logarithmic and exponential functions, A-Level Mathematics (syllabus 9709). Structured question, 5 marks.
Source
A-Level Mathematics, May/June 2021 Paper 3 Variant 1, question 2.

Questions and mark schemes are from past examination papers and remain the copyright of the exam board. How Oracle Prep builds its content

Related Mathematics questions

Logarithmic and exponential functionsFeb/Mar 20244m

Use logarithms to solve the equation 3⁴ˣ⁺³ = 5²ˣ⁺⁷. Give your answer correct to 3 significant figures.

Logarithmic and exponential functionsFeb/Mar 20244m

The positive numbers p and q are such that ln(p/q) = a and ln(q²p) = b. Express ln(p⁷q) in terms of a and b.

Logarithmic and exponential functionsFeb/Mar 20253m

Solve the equation ln(3x+1) – ln(x - 5) = ln 7.

AlgebraFeb/Mar 20245m

It is given that the coefficient of x³ in the expansion of (2+ax)⁴(5-ax) is 432. Find the value of the constant a.

QuadraticsFeb/Mar 20246m

The straight line y = x+5 meets the curve 2x² + 3y² = k at a single point P.

SeriesFeb/Mar 20248m

An arithmetic progression is such that its first term is 6 and its tenth term is 19.5. Find the sum of the first 100 …

← Browse all past-paper questions

© 2026 Oracle Prep