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WMA12/01 Jun 2021, Question 2: Solving Exponential and Log Equations

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 8 marks

The question, in short

Sketch two related exponential curves on copies of the given y = 4ˣ graph, labelling axis intercepts, then use algebra to find where y = 2ˣ and y = 4ˣ − 6 intersect.

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

Part (ai) · 1 mark

On a copy of the graph, sketch y = 2ˣ, showing its shape relative to y = 4ˣ.

  1. 1Both curves pass through (0, 1) since 2⁰ = 4⁰ = 1
  2. 2For x > 0, 2ˣ < 4ˣ so y = 2ˣ is below and less steep; for x < 0 it is above

AnswerSketch: y = 2ˣ is an increasing exponential through (0, 1), the same y-intercept as y = 4ˣ, above y = 4ˣ for x < 0 and below it (less steep) for x > 0, with the x-axis as asymptote.

Part (aii) · 3 marks

On a copy of the graph, sketch y = 4ˣ − 6, labelling coordinates where it meets the axes.

  1. 1Translate y = 4ˣ down by 6 units
  2. 2x = 0: y = 1 − 6 = −5, so (0, −5)
  3. 3y = 0: 4ˣ = 6, so x = log₄ 6 ≈ 1.29

AnswerSketch: y = 4ˣ − 6 is y = 4ˣ moved down 6, with asymptote y = −6, crossing the y-axis at (0, −5) and the x-axis at (log₄ 6, 0) ≈ (1.29, 0).

Part (b) · 4 marks

Using algebra, find the exact coordinates of the intersection point P of y = 2ˣ and y = 4ˣ − 6.

  1. 12ˣ = 4ˣ − 6, and 4ˣ = (2ˣ)²
  2. 2Let u = 2ˣ: u² − u − 6 = 0
  3. 3(u − 3)(u + 2) = 0, so u = 3 or u = −2
  4. 42ˣ cannot be negative, so 2ˣ = 3
  5. 5x = log₂ 3 and y = 3, so P = (log₂ 3, 3)

AnswerP = (log₂ 3, 3)

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Also searched as: WMA12 June 2021 · WMA12/01 · wma12 jun21 · IAS Maths Pure Mathematics 2 June 2021 · question 2.
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