Hint
Angles are radians. Count \(\displaystyle j=0\) through kn, including both endpoints.
Worked solution
(a)
The sine is zero when \(\displaystyle nx=j\pi\); the stated x interval gives \(\displaystyle 0\le j\le 12\).
Answer: \(\displaystyle x= \frac{j\pi }{3}, \quad j= 0, 1, \ldots , 12\)
(b)
Use \(\displaystyle nx=\pi\) for the first positive zero.
\(\displaystyle \frac{1}{3}\)
Answer: \(\displaystyle \frac{1}{3}\)
(c)
The upper endpoint is also a root.
\(4\)
Answer: \(4\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit complete valid alternatives and consequential reasoning; an unexplained final answer does not earn a method mark.
(a) · 2 marks
- 1 mark: Solve \(\displaystyle nx=j\pi\) and determine every permitted integer j from the given closed interval.
- 1 mark: Give all results requested in this part, with the correct signs, exact values, units, branch and boundary inclusion where applicable.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: An inverse-trig value gives only one branch.
(b) · 2 marks
- 1 mark: Use \(\displaystyle nx=\pi\) for the first positive zero.
- 1 mark: Give the correct requested result, retaining signs, units and any domain restrictions.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Do not forget to divide the angle solutions by n.
(c) · 2 marks
- 1 mark: The upper endpoint is also a root.
- 1 mark: Give the correct requested result, retaining signs, units and any domain restrictions.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: An inverse-trig value gives only one branch.
Total: 6 marks.