WMA12/01 Jun 2024, Question 6: Exponentials, Logarithms, e and ln

Part (a) · 3 marks

Sketch the curve y = aˣ + 4, marking the y-axis crossing coordinates and the asymptote equation.

1. 1y = aˣ is an increasing curve through (0, 1) with asymptote y = 0
2. 2Adding 4 moves it up 4: it crosses the y-axis at (0, 1 + 4) = (0, 5)
3. 3The asymptote moves up to y = 4; the curve stays above it in quadrants 1 and 2

AnswerAn increasing exponential curve in quadrants 1 and 2 only, crossing the y-axis at (0, 5) and approaching the horizontal asymptote y = 4 as x → −∞

Part (b) · 3 marks

Apply the trapezium rule to all tabulated y values to estimate the integral of 2ˣ − 2x from 2 to 3.5, to 2 decimal places.

1. 1h = 0.3
2. 2Area ≈ (0.3/2)[0 + 4.3137 + 2(0.3246 + 0.8629 + 1.6643 + 2.7896)]
3. 3= 0.15 × [4.3137 + 11.2828] = 0.15 × 15.5965
4. 4≈ 2.34

Answer2.34

Part (ci) · 2 marks

Using the part (b) result, estimate the integral of 2ˣ + 2x from 2 to 3.5, showing the method.

1. 12ˣ + 2x = (2ˣ − 2x) + 4x
2. 2∫₂^3.5 4x dx = [2x²]₂^3.5 = 24.5 − 8 = 16.5
3. 3Estimate = 2.34 + 16.5 = 18.84

Answer18.84

Part (cii) · 1 mark

Using earlier results, estimate the integral of 2ˣ⁺¹ − 4x from 2 to 3.5.

1. 12ˣ⁺¹ − 4x = 2(2ˣ − 2x)
2. 2Estimate = 2 × 2.34 = 4.68

Answer4.68

More past-paper questions on exponentials, logarithms, e and ln

• WMA12/01 Jan 2024 Q4 · 9 marks
• WMA12/01 Jun 2022 Q9 · 9 marks
• WMA12/01 Jun 2021 Q2 · 8 marks
• WMA12/01 Jan 2020 Q9 · 7 marks
• WMA12/01 Oct 2019 Q5 · 7 marks
• 0606/11 May/Jun 2026 Q8 · 8 marks