Hint
The divisor complex number is nonzero. Use \(\displaystyle i^{2}=-1\) and collect real and imaginary coefficients; conjugate the divisor for division.
Worked solution
(a)
Expand the product with \(\displaystyle i^{2}=-1\). For the quotient, multiply top and bottom by the conjugate of w.
Answer: \(9 + 39 i\); \(\displaystyle \frac{13}{6} - \frac{i}{2}\)
(b)
Use \(\displaystyle i^{2}=-1\) and collect real and imaginary coefficients; conjugate the divisor for division.
\(39\)
Answer: \(39\)
(c)
Use \(\displaystyle i^{2}=-1\) and collect real and imaginary coefficients; conjugate the divisor for division.
\(\displaystyle \frac{13}{6}\)
Answer: \(\displaystyle \frac{13}{6}\)
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: Expand the product and rationalise the quotient using the conjugate denominator.
- 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: The imaginary part does not include i.
(b) · 2 marks
- 1 mark: Use \(\displaystyle i^{2}=-1\) and collect real and imaginary coefficients; conjugate the divisor for division.
- 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: Conjugate every imaginary term in the divisor.
(c) · 2 marks
- 1 mark: Use \(\displaystyle i^{2}=-1\) and collect real and imaginary coefficients; conjugate the divisor for division.
- 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: The imaginary part does not include i.
Total: 6 marks.