AS 91577 · 2023 paper · a remainder, an unknown from a quotient, a surd equation, comparing parts, and a pair of conditions
Question 3(d) Merit
Practises Arithmetic & the conjugate
Five separate parts: a remainder, an unknown found from a quotient, an equation with two square roots, two unknowns found from a reciprocal, and a constant pinned down by two conditions at once.
Find the values of x and y, given that x and y are real and 1/(x + yi) = 1 − 1/(1 + pi), giving your answers in terms of p, where p is a positive real constant.
Work it out
Try it on paper first. Stuck? Take one hint at a time.
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1 Combine the right side into one fraction.
1 − 1/(1 + pi) = (1 + pi − 1)/(1 + pi) = pi/(1 + pi) -
2 Both sides are fractions with 1 on top on the left. Turn both upside down.
x + yi = (1 + pi)/pi -
3 Split the fraction, and simplify 1/i.
(1 + pi)/pi = 1/pi + 1, and 1/i = i/i2 = −i, so 1/pi = −i/p.
x + yi = 1 − (1/p)i -
4 Compare real and imaginary parts.
x = 1 and y = −1/p.
What the marker wanted
- Achieved The right side as one fraction: 1/(x + yi) = pi/(1 + pi).
- Merit Both x and y.
In our words, from NZQA’s 2023 marking schedule.
How did it go?
Got it moves it further out. Review brings it back tomorrow.