Hint

The total is 2π radians. The constant terms in α+β cancel.

Worked solution

(a)

(3k+1)+(k−1)+π/3=2π, so 4k=5π/3 and k=5π/12.

Answer: \(k=5\pi/12\).

(b)

Substitution gives α=5π/4+1 and β=5π/12−1. Since 5π/12≈1.309, β≈0.309 rad is positive. Both are valid angle sizes in the partition.

Answer: \(\alpha=5\pi/4+1\); \(\beta=5\pi/12-1>0\).

(c)

α in degrees is (5π/4+1)(180/π)=225+180/π≈282.296°. To 3 significant figures this is 282°. It lies between 180° and 360°, so it is reflex.

Answer: \(282^\circ\), a reflex angle.

Maths4U indicative marking guidance

Maths4U indicative marking guidance, not an official examination-board mark scheme. Award stated criteria once each, up to the part maximum. Accept valid equivalent methods. Follow-through applies only where explicitly stated; do not award an accuracy criterion for an incorrect final result.

(a) · 3 marks

• 1 mark: Set the sum equal to 2π.
• 1 mark: Simplify to 4k=5π/3.
• 1 mark: Give k=5π/12.

Alternative methods: A correct rearrangement of the angle-sum equation is accepted.

Common error: Setting the sum equal to 360 while retaining radian expressions.

(b) · 2 marks

• 1 mark: Give both exact angle expressions.
• 1 mark: Justify that β>0, for example with a controlled numerical comparison.

Alternative methods: An exact comparison 5π>12 is accepted.

Common error: Accepting a negative angle in the specified non-overlapping partition.

(c) · 2 marks

• 1 mark: Convert to 282° to 3 significant figures.
• 1 mark: Classify it as reflex using the appropriate interval.

Alternative methods: The unrounded degree expression followed by correct rounding is accepted.

Common error: Calling every positive angle below 360° a minor angle.

Total: 7 marks.