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4PM1/01 Jun 2019, Question 8: Solving Exponential and Log Equations

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 10 marks

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The question, in short

Solve a quadratic in p, use it to solve an exponential equation in 5^x, then find where two exponential curves intersect, giving values to 3 significant figures.

Summarised in our own words. Use the official paper for the exact wording and any diagrams.

Part (a) · 2 marks

Solve the quadratic equation 5p² − 9p + 4 = 0 for p.

  1. 1(5p − 4)(p − 1) = 0
  2. 2p = 4/5 or p = 1

Answerp = 4/5, p = 1

Part (b) · 4 marks

Using the result from part a, solve 5^(2x + 1) − 9(5^x) + 4 = 0 for x, to 3 significant figures where appropriate.

  1. 15^(2x + 1) = 5 × (5^x)², so 5(5^x)² − 9(5^x) + 4 = 0
  2. 25^x = 1 → x = 0
  3. 35^x = 4/5 → x ln 5 = ln(4/5)
  4. 4x = −0.1386… = −0.139

Answerx = 0, x = −0.139

Part (c) · 4 marks

Find the coordinates of the two intersection points of y = 5^(2x + 1) + 5^x and y = 2(5^(x + 1)) − 4, to 3 significant figures where appropriate.

  1. 15^(2x + 1) + 5^x = 10(5^x) − 4
  2. 25^(2x + 1) − 9(5^x) + 4 = 0, the equation from (b)
  3. 3x = 0: y = 5 + 1 = 6 → (0, 6)
  4. 45^x = 4/5: y = 10(4/5) − 4 = 4 → (−0.139, 4)

Answer(0, 6) and (−0.139, 4)

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