Hint
The vertex c is excluded. Use the direction component ratio \(\displaystyle \frac{v}{u}\).
Worked solution
(a)
A positive multiple of u+iv has the stated argument; \(\displaystyle t=0\) and negative t are excluded.
Answer: \(z=8 + 5 i+t(2 + 5 i),\quad t>0\)
(b)
Substitution gives \(\displaystyle t^{2}(u^{2}+v^{2})=261\), hence \(\displaystyle t=\pm 3\). Only \(\displaystyle t=3\)>0 belongs to the ray.
Answer: \(\left( 14, \ 20\right)\)
(c)
The positive ray parameter is one there.
\(10\)
Answer: \(10\)
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: Use a positive parameter in the specified direction and exclude the vertex.
- 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: A full straight line includes an incorrect opposite ray.
(b) · 2 marks
- 1 mark: Substitute the ray parameter in the circle and retain its positive root.
- 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: Do not assign an argument to zero.
(c) · 2 marks
- 1 mark: The positive ray parameter is one there.
- 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: A full straight line includes an incorrect opposite ray.
Total: 6 marks.