Consider these simultaneous equations, where c is a constant:
y = 3sin x + 2 y = x + c
Which of the following statements is/are true?
1 For some value of c: there is exactly one solution with 0 ≤ x ≤ π and there is at least one solution with −π < x < 0.
2 For some value of c: there is exactly one solution with 0 ≤ x ≤ π and there are no solutions with −π < x < 0.
3 For some value of c: there is exactly one solution with 0 ≤ x ≤ π and there are no solutions with x > π.
- Anone
- B1 only
- C2 only
- D3 only
- E1 and 2 only
- F1 and 3 only
- G2 and 3 only
- H1, 2 and 3
Show the answer and worked solution
answer · H
- Anone
- B1 only
- C2 only
- D3 only
- E1 and 2 only
- F1 and 3 only
- G2 and 3 only
- H1, 2 and 3
Rearrange to c = g(x) where g(x) = 3sin x + 2 − x, so each statement is about how many times the horizontal line y = c meets the graph of g. Since g'(x) = 3cos x − 1, the turning points are at cos x = 13, i.e. x ≈ ± 1.23. On [0, π], g rises from 2 to about 3.60 then falls to 2 − π ≈ −1.14, so there is exactly one solution there whenever 2 − π ≤ c < 2. On (−π, 0), g dips to a minimum of about 0.40, so choosing c = 1 gives a solution there (statement 1) while c = 0 gives none (statement 2). For x > π the values of g stay below 2−π, so any c ≥ 2−π — c = 0 again — gives no solution there (statement 3). All three hold.