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WMA12/01 Jan 2024, Question 4: Exponentials, Logarithms, e and ln

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 9 marks

The question, in short

Sketch y = a⁻ˣ + 4 for a > 1, showing the y-axis crossing and asymptote; then use the given table with the trapezium rule to estimate integrals of 3^(−x/2) + 4.

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

Part (a) · 3 marks

Draw the curve y = a⁻ˣ + 4 with a > 1, marking the y-intercept coordinates and the asymptote's equation.

  1. 1At x = 0: y = a⁰ + 4 = 5, so the curve meets the y-axis at (0, 5)
  2. 2As x → ∞, a⁻ˣ → 0, so y → 4: asymptote y = 4
  3. 3a⁻ˣ = (1/a)ˣ with a > 1, so the curve decreases from left to right and stays above y = 4

AnswerDecreasing exponential curve, entirely above the line y = 4, crossing the y-axis at (0, 5) and approaching the horizontal asymptote y = 4 as x → ∞ (rising steeply as x → −∞).

Part (b) · 3 marks

Apply the trapezium rule to all tabulated y values to estimate the integral from −4 to 8.5, giving two significant figures.

  1. 1h = (8.5 − (−4))/5 = 2.5
  2. 2Area ≈ ½ × 2.5 × [13 + 4.009 + 2(6.280 + 4.577 + 4.146 + 4.037)]
  3. 3= 1.25 × [17.009 + 38.08] = 1.25 × 55.089
  4. 4= 68.86… ≈ 69 (2 s.f.)

Answer69

Part (ci) · 2 marks

Use the part (b) result to estimate the integral of just 3^(−x/2) over the same limits.

  1. 1∫ 3^(−x/2) dx = ∫ (3^(−x/2) + 4) dx − ∫ 4 dx
  2. 2∫ from −4 to 8.5 of 4 dx = 4 × 12.5 = 50
  3. 3≈ 68.86 − 50 = 18.86 ≈ 19

Answer19

Part (cii) · 1 mark

Combine the part (b) integral with the related integral from −8.5 to 4 of 3^(x/2) + 4.

  1. 1y = 3^(x/2) + 4 on [−8.5, 4] is the reflection of y = 3^(−x/2) + 4 on [−4, 8.5] in the y-axis
  2. 2So both integrals have the same value
  3. 3Total ≈ 2 × 68.86 = 137.7 ≈ 138

Answer138

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