4024/21 Oct/Nov 2021, Question 7: Exponential Functions
Cambridge O Level Mathematics 4024 · Paper 21 (Calculator) · Calculator · 11 marks
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The question, in short
Draw the curve y = 2ˣ from x = 0 to 4 on the grid, then use the given curve y = 1/(2x²) + 3x to estimate a gradient, a solution, and two solutions.
Summarised in our own words. Use the official paper for the exact wording and the diagram.
Part (a) · 4 marks
Plot y = 2ˣ for x from 0 to 4 on the grid provided.
- 1Table: x = 0, 1, 2, 3, 4 gives y = 1, 2, 4, 8, 16
- 2Plot (0, 1), (1, 2), (2, 4), (3, 8), (4, 16)
- 3Join with a smooth curve
AnswerCorrect smooth curve
Part (bi) · 2 marks
Draw a tangent at x = 0.5 and use it to estimate the curve's gradient there.
- 1At x = 0.5: y = 1/(2 × 0.25) + 3 × 0.5 = 2 + 1.5 = 3.5
- 2Draw a ruled tangent touching the curve at (0.5, 3.5)
- 3Pick two points on the tangent and find change in y ÷ change in x
- 4The tangent slopes down steeply: gradient ≈ −5
Answer−5.5 to −2.8
Part (bii) · 1 mark
Read off the x-value where the curve y = 1/(2x²) + 3x equals 2.
- 1Draw y = 2 and read where it meets the curve
- 2x ≈ −0.4
Answer−0.45 to −0.35
Part (biii) · 4 marks
Draw a suitable straight line and use it to estimate the two solutions of 1/(2x²) = 7 − 4x.
- 11/(2x²) = 7 − 4x → 1/(2x²) + 3x = 7 − x
- 2Draw the line y = 7 − x on the grid
- 3Read the x-values where the line meets the curve
- 4x ≈ −0.25, x ≈ 0.29, x ≈ 1.71
Answer−0.2 to −0.3, 0.2 to 0.4, 1.6 to 1.8
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