WMA11/01 Oct 2024, Question 7: Trig Graphs: Shifts and Counting Solutions

Part (ai) · 2 marks

1. 1Minima of y = −4cos x are where cos x = 1: x = 0, ±2π, …, with y = −4
2. 2P is the minimum to the left of the y-axis, so P = (−2π, −4)

AnswerP = (−2π, −4)

Part (aii) · 1 mark

1. 1Zeros are where cos x = 0: x = π/2, 3π/2, …
2. 2Q follows the maximum at x = π, so Q = (3π/2, 0)

AnswerQ = (3π/2, 0)

Part (bi) · 1 mark

1. 1Maximum of −4cos x is 4, so maximum of C₂ is 4 + k
2. 24 + k = 11, so k = 7

Answerk = 7

Part (bii) · 2 marks

1. 1Minima of C₂ are at x = 0, ±2π, … with y = −4 + 7 = 3
2. 2The smallest positive x is 2π, so the point is (2π, 3)

Answer(2π, 3)

Part (c) · 2 marks

1. 1Draw the line y = 5 − (10/π)x: it passes through (0, 5) and (π/2, 0)
2. 2For x < 0 the line is above 5, higher than the curve's maximum of 4; for large x it is below −4
3. 3The line crosses the curve only once, at (π/2, 0)
4. 4So the equation has exactly 1 solution

Answer1 solution, because the line y = 5 − (10/π)x crosses the curve y = −4cos x exactly once

More past-paper questions on trig graphs: shifts and counting solutions

• WMA11/01 Jun 2025 Q4 · 6 marks
• WMA11/01 Jun 2024 Q11 · 6 marks
• WMA11/01 Jan 2023 Q9 · 4 marks
• WMA11/01 Jan 2022 Q9 · 4 marks
• WMA11/01 Jun 2022 Q9 · 8 marks
• WMA11/01 Jun 2021 Q9 · 7 marks